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Doctoral Thesis

Mathematical Results on the Foundations of Quantum Mechanics

Author(s): Schubnel, Baptiste

Publication Date: 2014

Permanent Link: https://doi.org/10.3929/ethz-a-010428944

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Diss. ETH No. 22382

Mathematical Results on theFoundations of Quantum Mechanics

A thesis submitted to attain the degree ofDoctor of Sciences of ETH Zurich

(Dr. sc. ETH Zurich)

presented byBaptiste Schubnel

MSc. Physics ETHZborn on 01.04.1986citizen of France

accepted on the recommendation ofHorst Knörrer, examiner

Jürg Fröhlich, co-examinerEugene Trubowitz, co-examiner

2014

À ”mès ¯pà˚rè›n˚ts,

i

Abstract

The purpose of this thesis is to provide new mathematical results related to the foundationsof quantum mechanics. We study the following problems:

(i) the existence of closed quantum systems,

(ii) the preparation of states in quantum mechanics,

(iii) the algebraic characterization of empirical properties of a quantum system,

(iv) the emergence of quantum trajectories.

We also study the analyticity of the resonances and the analyticity of the ground state ofan atom coupled to the quantized electromagnetic field, with respect to the atom momentum~p. This problem is not directly related to the foundations of quantum mechanics. However,the analyticity of the ground state is used in one of our proof concerning the existence ofclosed quantum systems. Furthermore, this model of an atom provides a simple but non-trivial example of a small quantum system interacting with an environment (the quantizedelectromagnetic field).

Our approach is as follows: we first define rigorously physical systems using tools of op-erator theory (C∗-algebras). Then we study the problems (i)-(iv) mentioned above. Wecharacterize them with the algebraic tools introduced, and we examine concrete examples ofquantum systems where these phenomena can be observed and analyzed in detail. All theexamples discussed in this thesis consist of one (or more than one) non-relativistic particle(s),coupled to a quantized field (typically the electromagnetic field, or a phonon field).

Résumé

L’objectif de cette thèse est d’apporter de nouveaux résultats mathématiques tournant autourde questions relatives aux fondements de la mécanique quantique. Nous étudions en détail lesproblèmes suivants:

(i) l’existence de systèmes quantiques fermés,

(ii) la préparation des états en mécanique quantique,

(iii) la caractérisation algébrique des propriétés empiriques d’un système quantique,

(iv) l’émergence de trajectoires quantiques.

Nous nous penchons également sur l’étude de l’analyticité des résonances et de l’étatfondamental d’un atome avec un moment dipolaire éléctrique non nul, couplé au champ éléc-tromagnétique quantifié, en fonction de sa quantité de mouvement ~p. Ce problème n’est pasen lien direct avec les fondements de la mécanique quantique, mais l’analyticité de l’état fon-damental est utilisée afin de prouver l’existence de systèmes quantiques fermés. Cette étudea aussi l’avantage de fournir un exemple simple mais réaliste de système quantique couplé àun environnement (correspondant au champ éléctromagnétique quantifié).

L’ approche suivie dans cette thèse est la suivante: nous cherchons tout d’abord à définirde manière concise un système physique à l’aide d’outils de la théorie des algèbres d’opérateurs

ii

(C∗-algèbres). Nous abordons ensuites les points (i)-(iv) évoqués ci-dessus, en les caractérisantdans un premier temps avec les outils algébriques introduits, puis en étudiant des exemplesconcrets de systèmes quantiques où les phénomènes discutés peuvent être observés et analysésdans le détail. Tous les exemples traités dans cette thèse sont des modèles de systèmes quan-tiques constitués d’une (ou plusieurs) particule(s) non-relativiste(s), couplée(s) à un champquantifié (typiquement le champ électromagnétique, ou un champ de phonons).

iii

Acknowledgments

Mes remerciements les plus grands vont à Jürg, mon directeur de thèse "officieux", pour sonsavoir, sa patience, son sens de la rigueur, et pour toutes les autres choses qu’il m’aura apprisespendant ces années de thèse. Je le remercie en particulier pour sa capacité rare à trouver denouveaux problèmes (qui plus est, la plupart du temps résolubles), pour son enthousiasmecommunicatif pour la recherche et pour sa vision éclairée de la mécanique quantique qui est àla base de cette thèse.

Je remercie aussi bien entendu Horst, pour avoir accepté de me prendre comme doctorantau sein du groupe 5 de mathématiques, et pour son écoute attentive durant ces trois annéeset demie. Merci aussi à Eugene pour accepter d’être le co-référent de cette thèse.

D’autres personnes ont contribué de manière significative aux travaux présentés dans cettethèse. Jérémy, en particulier, avec qui j’ai eu le plaisir de travailler, et que je remercie pourson accueil très chaleureux à Bordeaux (et bientôt, je l’espère, à Metz!). Merci aussi à Miguel,pour nos intéressantes discussions en anglais, entrecoupées de français et d’espagnol, et pournos collaborations en cours et à venir. Je suis également reconnaissant envers Volker pourm’avoir invité à Braunschweig, et Philippe pour ces deux séjours que j’aurais eu le plaisir depasser à Bielefeld.

J’en profite pour remercier mes collègues et amis: merci à mes collègues du groupe 5 ainsiqu’à ceux du groupe de physique mathématique du département de physique et de l’université,avec qui j’ai eu l’occasion de travailler, de discuter, de suivre des cours et/ou d’organiser desexercices. Je pense en particulier aux "autrichiens" du groupe 5, Claudia et Daniel, qui, commeles "français", aiment bien se plaindre de la marche du monde. Je remercie également mesamis du séminaire de statistique, Johannes et Michaël, pour nos intenses parties de badminton,ainsi que Jonas, pour nos fausses notes de musique jouées avec tellement de plaisir.

Enfin je tiens à remercier ma famille. Je pense en particulier à ma soeur et à sa tribu,pour tous les bons moments que nous avons passés en Suisse et en Alsace ensemble, et à mabelle-famille suisse, notamment mes beaux-parents, pour l’aide au "logement" qu’ils m’ontprocurée durant la rédaction de cette thèse... Merci bien entendu à Nadia, pour son soutienconstant et pour le simple fait de rendre la vie plus belle quand nous sommes ensemble. Etsurtout merci à notre petit Lucien, pour ses jolis sourires qui me feraient presque oublier lesproblèmes de la physique mathématique.

Finalement, cette thèse n’aurait jamais pu voir le jour sans le soutien et la confianceinconditionnels de mes parents, Annick et Pierre, qui m’ont toujours encouragé à accomplirce que je voulais réaliser dans la vie, sans jamais chercher à me forcer la main. Je leur dédiecette thèse.

iv

Table of contents

Detailed Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1 Introduction 51.1 Models of physical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Closed systems in quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . 121.3 On the problem of preparing states in quantum mechanics . . . . . . . . . . . . 131.4 The emergence of facts in physical systems . . . . . . . . . . . . . . . . . . . . . 17

2 Analyticity of the resonances 282.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2 Parameters of the problem and Notations . . . . . . . . . . . . . . . . . . . . . 402.3 The first decimation step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.4 The inductive construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.5 Existence and analyticity of the resonances . . . . . . . . . . . . . . . . . . . . 582.6 Imaginary part of the resonances . . . . . . . . . . . . . . . . . . . . . . . . . . 602.7 Appendices for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3 Closed systems in Quantum Mechanics 743.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.2 Models and main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.3 Proof of Theorem 3.2.2.1 and Corollary 3.2.2.1 . . . . . . . . . . . . . . . . . . 833.4 Appendix for Section 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.5 Appendix for Section 3.2.1: Proof of Lemma 3.2.1.1 . . . . . . . . . . . . . . . . 106

4 Preparation of states 1114.1 Model and summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.2 Analysis of Z t,s for t− s ∝ λ−2(s) . . . . . . . . . . . . . . . . . . . . . . . . 1154.3 Rewriting Z t,0 as a sum of terms labelled by graphs . . . . . . . . . . . . . 1244.4 Proof of the Kotecky-Preiss criterion and convergence of the Cluster ex-

pansion when N →∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324.5 Extensions of Theorem 4.1.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1414.6 Appendix for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1444.7 Proof of Proposition 4.3.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1504.8 Proofs of Corollaries 4.5.1.1 and 4.5.2.1 (sketch) . . . . . . . . . . . . . . . . . 151

v

5 Emergence of facts in Quantum Mechanics 1555.1 Decoherence and information loss: two necessary ingredients for the emergence

of facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.2 Illustration of decoherence and information loss: Study of an exactly solvable

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1635.3 When does an observation or measurement of a physical quantity take place? . 1725.4 Non-demolition measurements and quantum trajectories . . . . . . . . . . . . . 1745.5 Appendix to Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

vi

1

Detailed Plan

The work presented in this thesis is largely based on [70, 71, 54, 53, 17]. [70, 71] are resultsof a collaboration with J. Fröhlich; [54, 53] were obtained in collaboration with J. Faupin andJ. Fröhlich; [17] was obtained in collaboration with M. Ballesteros, J. Faupin and J. Fröhlich.Finally, some of the results presented in Chapter 5 are part of an ongoing collaboration withJ. Fröhlich and M. Ballesteros.

Chapter 1: Introduction

We begin by introducing an abstract algebraic framework for the formulation of mathematicalmodels of physical systems, that is general enough to encompass classical and quantum me-chanical models. We then outline the main differences between quantum and classical systems.At the same time, we discuss each of the results presented in depth in later chapters.

Chapter 2: Resonances of a moving atom

We study a simple model of an atom interacting with the quantized electromagnetic field.The atom has a finite mass m, finitely many excited states and an electric dipole moment,~d0 = −λ0

~d, where ‖di‖ = 1, i = 1, 2, 3, and λ0 is proportional to the elementary electriccharge. The interaction of the atom with the radiation field is described with the help ofthe Ritz Hamiltonian, −~d0 · ~E, where ~E is the electric field, cut off at large frequencies. Amathematical study of the Lamb shift, the decay channels and the life times of the excitedstates of the atom is presented. It is rigorously proven that these quantities are analyticfunctions of the momentum ~p of the atom, provided |~p| < mc and |=~p| and |λ0| are sufficientlysmall. The proof relies on a somewhat novel inductive construction involving a sequence of‘smooth Feshbach-Schur maps’ applied to a complex dilatation of the original Hamiltonian,which yields an algorithm for the calculation of resonance energies that converges super-exponentially fast.

Chapter 3: Closed systems

We discuss the notion of “closed systems” in quantum mechanics. For this purpose, we studytwo models of a quantum-mechanical system P spatially far separated from the “rest of theuniverse” Q. Under reasonable assumptions on the interaction between P and Q, we showthat the system P behaves as a closed system if the initial state of S = P ∨ Q belongs to alarge class of states, including ones exhibiting entanglement between P and Q.

Chapter 4: Preparation of states

We address the problem of how to prepare a quantum mechanical system, S, in a specificinitial state of interest - e.g., for the purposes of some experiment. We study a method thathas the attractive feature to prepare S in a preassigned initial state with certainty; i.e., theprobability of success in preparing S in a given state is unity. This method relies on couplingS to an open quantum-mechanical environment, E, in such a way that the dynamics of S ∨Epulls the state of S towards an “attractor", which is the desired initial state of S.

2

Chapter 5: Emergence of facts in Quantum Mechanics

We attempt to elucidate the roles played by entanglement between a system and its envi-ronment and of information loss in understanding "decoherence" and "dephasing", which arekey mechanisms in a quantum theory of measurements and experiments. We illustrate theseprinciples with a fully solvable model of a small quantum mechanical system P coupled toan environment E. We also discuss the problem of "time in quantum mechanics" and sketchan answer to the question when an experiment can be considered to have been completedsuccessfully. Finally, we review some general features of indirect measurements in quantummechanics, and we give some preliminary results concerning the phenomenon of purificationon the spectrum of an observable and the emergence of quantum trajectories.

3

Notations

We give a list of notations used in this thesis. The reader should consult this section again ifhe encounters at some place a notation that looks unfamiliar to him.

Algebras and Hilbert spaces

• A: algebra, generally a C∗-algebra. Elements in A are denoted by a, b, ....

• H, h: Hilbert space. Vectors in H are generally denoted by greek letters ϕ,ψ, ....

• 〈·|·〉; ‖ · ‖: scalar product; norm induced by the scalar product on a given Hilbert space.

• B(H): Banach algebra of bounded operators on the Hilbert space H. Elements in B(H)are generally written with capital letters A,B, .... The operator norm supψ 6=0 ‖Aψ‖/‖ψ‖ isdenoted by ‖A‖. The identity operator in B(H) is denoted by 1H (or 1 if no confusion canarise).

• D(A): domain of a linear operator A on H.

• σ(A): spectrum of the linear operator A : D(A) → H. It is the set of complex number zsuch that A− z1H is not a bijection with bounded inverse. For a s.a. operator A, we use thenotation: σpp(A): pure point spectrum, σac(A): absolutely continuous spectrum, σsing(A):continuous singular spectrum.

• [A,B]: the commutator of A with B: AB −BA (if well-defined).

Functions spaces

• Lp(Rn), p ∈ N: Banach space of equivalence classes of measurable functions f : Rn → C,with

‖f‖Lp :=( ∫

Rndnx |f(x)|p

)1/p<∞.

• R3n := (R3 × 1, 2)n.

• L2(R3) : Hilbert space of equivalence classes of measurable functions f : R3 → C, with

‖f‖L2 :=( ∑λ=1,2

∫R3d3x |f(x, λ)|2

)1/2<∞.

• Ck(Ω): space of k-times continuously differentiable complex valued functions on the openset Ω ⊂ Rn.

• S(Rn): Schwarz space of smooth functions of rapid decrease.

• ∆: Laplace operator.

4

• f , f : Fourier transform, inverse Fourier transform of f .

Fock space

• F(h): Fock space over h. Denoting h⊗n := h⊗ ...⊗ h, h⊗0 := C, one has that

F(h) :=⊕n≥0

h⊗n.

• F+(h),F−(h): Bosonic, Fermionic Fock space over h. Defining the projections P (n)± : h⊗n →

h⊗n by

P(n)± ϕ1 ⊗ ...⊗ ϕn :=

1

n!

∑σ∈Sn

(±1)|σ|ϕσ(1) ⊗ ...⊗ ϕσ(n),

one has thatF±(h) :=

⊕n≥0

P(n)± h⊗n.

• dΓ(A): second quantization of A = A∗, self-adjoint operator on h. Defined on P (n)± h⊗n by

dΓ(A)|C = 0 and by (n ≥ 1)

dΓ(A)|P (n)± h⊗n

:= P(n)±

n∑i=1

1⊗ ...1⊗ A︸︷︷︸i

⊗ 1...⊗ 1.

• Special case: h = L2(R3). Elements of F±(h) are sequences of functions ψ := (ψ(n))n≥0

with ψ(n) ∈ L2(R3n), n ≥ 1, and ψ(0) ∈ C. The functions ψ(n)(k1, ..., kn) are totally symmet-ric/antisymmetric under the exchange of the arguments ki ∈ R3.

• a(k), a∗(k): annihilation and creation "operators" on F±(L2(R3)). Set

DS := ϕ ∈ F±(L2(R3)) | ∃N,ϕ(n) = 0 ∀n ≥ N, and ϕ(n) ∈ S(R3n) ∀n ∈ 1, ..., N − 1.

For all ϕ ∈ DS ,

(a(k)ϕ)(n)(k1, ..., kn) : =√n+ 1 ϕ(n+1)(k, k1, ..., kn),

(a∗(k)ϕ)(n)(k1, ..., kn) : =1√n

n∑i=1

(±1)i−1δ(k − ki) ϕ(n−1)(k1, ..., ki−1, ki+1, ..., kn).

a∗(k) is defined in the sense of the quadratic forms. Its form domain is DS ×DS .

• Finally, we point out that we always work in units where ~ = c = 1, where ~ is the reducedPlanck constant, and c is the velocity of light.

Chapter 1

Introduction

We introduce in Paragraph 1.1.1 algebraic data that are used to characterize physical systems.In Section 1.1.2, we review the specificities of classical and quantummodels of physical systems.In Section 1.1.3-1.4.3, we discuss the results that are obtained in later chapters (Chapters 2-5).We mention that a short review on operator algebras is presented in the appendix for readersnot familiar with this topic. For more details, see e.g. [123].

1.1 Models of physical systems

1.1.1 The algebraic data to characterize a physical system

A model of a physical system S can be specified in terms of a set of algebraic data. Thefirst data is the set of "observables", or “physical quantities", OS , that can be observed ormeasured on the system S.

Observables may be added and multiplied, and it is reasonable to view OS as a self-adjointsubset of an operator algebra, AS , usually taken to be a unital C∗−algebra. AS is the secondalgebraic data that we will use to characterize the system S. It is chosen sufficiently largesuch that time evolution of observables is given in terms of ∗automorphisms of AS : given twotimes, s and t, τt,s is a ∗automorphism of AS that associates with every physical quantity inAS specified at time s an operator in AS representing the same physical quantity at time t.We must require that

τt,s τs,u = τt,u, (1.1.1)

for any triple of times (t, s, u). The smallest C∗−algebra containing OS , 〈OS〉, is generallystrictly contained in AS . 〈OS〉 may actually contain only very few elements.

To make predictions concerning the possible outcomes of measurements and observationscarried out on S, we use the notion of states. A state is a positive linear functional on thedynamical C∗−algebra AS . The expectation value of an observable a = a∗ ∈ OS if the systemis in the state ω is given by ω(a) ∈ R. One usually requires ω to be normalized, in the sensethat ω(1) = 1. The set of all states on AS is denoted by SS . Recapitulating, a model of aphysical system S is specified by the following data.

Definition 1.1.1.1 (Algebraic data specifying a model of a physical system).

(I) A list of physical quantities, or observables, OS = ai = a∗i i∈I , that belong to a certain(unital) C∗−algebra AS , also called the dynamical C∗−algebra of the system.

5

CHAPTER 1. INTRODUCTION 6

(II) The convex set SS of states on the C∗−algebra AS .

(III) A groupoid τt,st,s∈R of ∗automorphisms on AS satisfying (1.1.1) that represents timetranslations in the following sense : For each time t ∈ R, we have copies OS(t) that are∗-isomorphic to OS , and that are contained in AS . If a(s) ∈ OS(s) and a(t) ∈ OS(t)are the operators in AS representing an arbitrary observable a ∈ OS at times s and t,respectively, then

a(t) = τt,s(a(s)), (1.1.2)

with τt,s = τt,u τu,s, for arbitrary times t, u and s in R.

The system S specified by the data (OS ,AS , τt,s,SS) is what we call a closed system.This notion is related to the fact that time evolution on S is given in terms of ∗automorphismsof the dynamical algebra AS . The notion of closed systems is discussed in more details inParagraph 1.2.

We mention a few further rules and notions pertaining to physical systems that will beused throughout this thesis. We say that the system characterized by AS is autonomous if

τt,s = τt−s (1.1.3)

where τtt∈R is a one-parameter group of ∗automorphisms of AS . The composition, S1 ∨ S2,of two systems S1 and S2, characterized by the sets OS1 and OS2 , and the algebras AS1 andAS2 , respectively, can be defined by choosing

OS1∨S2 := OS1 ∪ OS2 . (1.1.4)

The dynamical algebra AS1∨S2 is a C∗-algebra containing the algebraic product 〈OS1〉⊗〈OS2〉.A closed subsystem P of S is characterized by a set of observablesOP ⊆ OS and by a dynamicalalgebra AP ⊆ AS .

1.1.2 Classical and quantum models.

A classical (or realistic) model of a system S is fully characterized by the property that its"dynamical C∗-algebra" AS is abelian. A quantum model of a system differs from a classicalmodel only in that the algebra AS is non-commutative. Apart from this crucial difference, thealgebraic data defining the two possibilities (realistic or quantum) are as specified in points(I)-(III) of Definition 1.1.1.1, Subsect. 1.1.1. We review here the general features of classicaland quantum models of physical systems.

Classical models of physical systems

A well-known theorem, due to I.M. Gel’fand (see e.g. [123]), tells us that every abelianC∗−algebra AS is ∗isomorphic to a C∗-algebra, C0(MS), of continuous functions on MS

vanishing at ∞, where MS is a locally compact space (the spectrum of AS).Every state, ω, on AS is given by a unique (Borel) probability measure, dµω, on MS (and

conversely). Every pure state is given by a Dirac δ−function, δx, onMS , for some x ∈M ; i.e.,the space of pure states can be identified with MS , (which is why MS is called "state space").Thus, the set of pure states of AS cannot be endowed with a linear or affine structure: thereis no superposition principle for pure states. If AS0 ⊂ AS is a subalgebra of AS then any pure


state of AS is also a pure state of AS0 . If S = S1 ∨ S2 is the composition of two subsystems,S1 and S2, these systems are classical, too, and we have that any pure state of S is also a purestate of S1 and of S2: there is no interesting notion of entanglement.

According to point (III) of Definition 1.1.1.1 in Subsect. 1.1.1, time evolution of a systemis given by *automorphisms of its dynamical C∗−algebra AS . If AS is an abelian C∗-algebraand MS denotes its spectrum, then any ∗automorphism, α, of AS , corresponds to a homeo-morphism, φα, of MS : If a is an arbitrary element of AS , thus given by a bounded continuousfunction (also denoted by a) on MS , then

α(a)(ξ) =: a(φ−1α (ξ)), ξ ∈MS . (1.1.5)

Conversely, any homeomorphism, φ, from MS to MS determines a ∗automorphism, αφ, by

αφ(a)(ξ) := a(φ−1(ξ)), ξ ∈MS . (1.1.6)

If αt,st,s∈R is a groupoid of ∗automorphisms of AS , with αt,s αs,u = αt,u, then there existsa groupoid of homeomorphisms, φt,st,s∈R, of MS , with φt,s φs,u = φt,u, such that

αt,s(a)(ξ) = a(φs,t(ξ)), ξ ∈MS , (1.1.7)

where φs,t = φ−1t,s . If we assume that that there is a subalgebra AS ⊂ AS that is norm-dense

in AS such that αt,s(a) is continuously differentiable in t (and in s), for arbitrary a ∈ AS , andif we assume that MS admits a tangent bundle, then φt,s corresponds to the flow of a vectorfield, Xs,

d

dtφt,s(ξ) = −Xt(φt,s(ξ)), ξ ∈MS . (1.1.8)

Hence, at least formally, the homeomorphisms φt,s can be constructed from a family of vectorfields Xss∈R by integrating the ordinary differential equations (1.1.8) if the vector fields Xs

are globally Lipschitz-continuous in s, for all s ∈ R. If Xs ≡ X is independent of s thenφt,s = φt−s is a one-parameter group of homeomorphisms of MS , (and conversely).

Quantum models of physical systems

The only feature distinguishing a quantum-mechanical model of a physical system from aclassical model is that, in a quantum model AS ) OS is a non-commutative algebra. For a longtime, physicists tried to embed quantum theories into classical ones, using in particular hiddenvariables; see e.g. [48]. The study of correlation matrices of families of (non-commuting)possible events in two independent systems rules out this possibility (except in the case of atwo-level system), and shows that the numerical range of possible values of the matrix elementsof such correlation matrices is strictly larger in quantum probability theory than in classicalprobability theory, as discovered by Bell [23, 125]; see also [97] for an alternative approach.The non-commutativity of AS has further profound consequences that will be discussed inmore depth in Section 5.

The GNS theorem (see e.g. [123, 121], and the appendix) tells us that given a state ωon the C∗-algebra AS , AS can be represented as a ∗-subalgebra of the algebra of boundedoperators on a certain Hilbert space Hω; ω is represented by a cyclic vector Ω ∈ Hω, and

ω(a) = 〈Ω|πω(a)Ω〉


for all a ∈ AS . We emphasize that this theorem is valid for abelian as well as for non-abelianalgebras.

Let us focus on the particular case where AS = B(HS). Then the system is a quantumsystem as soon as dim(HS) ≥ 2. If S is autonomous, the one-parameter group τtt∈Rof ∗-automorphisms of B(HS) is represented by a one-parameter group Utt∈R of unitarytransformations on HS ; see [27]. Time evolution of a ∈ B(HS) is given by

a(t) = U∗t aUt (1.1.9)

for all t ∈ R. According to Stone’s theorem (see e.g. [111]), Utt∈R is generated by aself-adjoint operator HS on HS , and Ut = e−itHS for all t ∈ R.

The normalized vectors in HS are the pure states of the system. The normal states ω ofthe systems are represented by trace-class operators on HS : If ω is normal, there is a positiveself-adjoint operator ρω with Tr(ρω) = 1 such that

ω(a) = Tr(ρωa)

for all a ∈ B(HS); see e.g. [27]. The expectation value of time translations of an observablea = a∗ ∈ B(HS) is then given by

ω(a(t)) = Tr(ρωU∗(t)aU(t)).

In this thesis, we treat particular examples of quantum systems to illustrate the maininteresting features of quantum models of physical systems; see Sections 1.1.3-1.3 and Chapters2-5. It is hard to analyze the dynamical properties of complicated quantum systems, and wewill essentially focus on systems S made of a particle P and of an environment E. If we work atzero temperature, the dynamical algebra AS can be taken to be the C∗-algebra B(HS), wherethe Hilbert space HS is the tensor product space HS := HP ⊗HE . HP is the Hilbert space ofpure states of the particle P , and HE the Hilbert space of pure states of the environment E.

We consider models where P is a quantum particle with finitely many internal degrees offreedom, and we will chooseHP = L2(R3,Cn0) (orHP = Cn0 if the particle is infinitely heavy),for a certain n0 ∈ N. The environment E is modeled by a quantized field/ideal quantum gas,HE = F±(h), where h is a separable Hilbert space, and F±(h) is the bosonic/fermionic Fockspace over h. We will have a particular interest in models where a non-relativistic particle iscoupled to the quantized electromagnetic field (called non-relativistic QED models). There,h = L2(R3 × 1, 2) and HE := F+(L2(R3 × 1, 2)); see also 1.1.3. As explained in Section1.4, the presence of the environment E, modeled by a quantum field, is crucial for the existenceof quantum phenomena such as decoherence and information loss.

1.1.3 Example of a quantum system: an atom coupled to the quantizedelectromagnetic field

To give a somewhat non-trivial example of a quantum mechanical system, we propose to de-scribe in more details the model of an atom, P , interacting with the quantized electromagneticfield, E. This model is studied in detail in Chapter 2 and 3, and some of its dynamical prop-erties are investigated. Our model is a variant of the so-called Pauli-Fierz model of quantumelectrodynamics; see e.g. [47]. The atomic degrees of freedom are treated non-relativistically,but photons are massless, and no infrared cutoff is imposed on the interactions between atomsand the quantized radiation field. In order to avoid technical complications, we focus on a


simplistic model of an atom: The mass, m, of an atom is positive and finite (set equal to 1),its kinematics is non-relativistic, but it has only finitely many excited states and cannot beionized. The total electric charge of every atom vanishes and its interaction with the quantizedradiation field arises by coupling its electric dipole moment to the quantized electric field, (i.e.,the interaction Hamiltonian is given by −~d0 · ~E, where ~d0 is the dipole moment operator ofthe atom and ~E is the quantized electric field).

The total Hilbert space of the system S = P ∨E is HS = L2(R3)⊗Cn0⊗F+(L2(R3)). TheHilbert space HE = F+(L2(R3)) is the bosonic Fock space over L2(R3) := L2(R3 × 1, 2),and HP = L2(R3)⊗ Cn0 . The dynamics is generated by the self-adjoint operator

HS := HP ⊗ 1HE + 1HP ⊗HE + λ0HP,E ,

where

HP = −∆

2⊗ 1Cn0 +

n0∑i=1

ei1L2(R3) ⊗Πi.

∆ is the Laplace operator, ei ∈ R for all i = 1, ..., n0, and e1 < .... < en0 . The projections Πi

are one-dimensional orthogonal projections that satisfy ΠiΠj = δijΠi, for all i, j = 1, ..., n0.The electromagnetic field and interaction Hamiltonians are defined in the sense of the quadraticforms by

HE :=∑λ=1,2

∫R3|~k| a∗λ(~k)aλ(~k)d3k,

andHP,E = −λ0

~d · ~E(~x),

where ~x ∈ R3 denotes the position of the (center of mass of the) atom in physical space,λ0 ≥ 0 is the coupling constant, ~d = dx~ex + dy~ey + dz~ez (dx, dy and dz are Hermitian n0 × n0

matrices), and ~E(~x) is the quantized electric field in the Coulomb Gauge. The interactionHamiltonian HP,E can be deduced from a globally neutral system of charges interacting withthe quantized electronagnetic field, under the approximation that the typical wavelength ofthe field is much larger than the typical dimension of the system: this approximation is calledthe “dipole approximation" (see [38] for more details). The exact expression of ~E(~x) is givenin (2.1.5). For the present discussion, it is only important to point out that ~E(~x) is linearin annihilation and creation operators, and that the form factor of the interaction is infraredregular.

The one-parameter group of time translations U(t)t∈R is generated by HS , and thedynamical properties of the model can be inferred from the spectral properties of HS . Ifλ0 = 0, the projections 1L2(R3) ⊗ Πi ⊗ 1HE commute with HS and the subspaces (1L2(R3) ⊗Πi ⊗ 1HE )HS are invariant under the dynamics e−itHSt∈R, for all i = 1, ..., n0: there is notransition between two distinct internal energy levels. It is the interaction Hamiltonian HP,E

that is responsible for transitions between two distinct subspaces (1L2(R3)⊗Πi⊗1HE )HS and(1L2(R3) ⊗Πj ⊗ 1HE )HS , i 6= j. To analyze the spectral properties of HS , it is useful to go to"Fourier space". The model described here is indeed invariant under space translations; seeSection 2.1.1. Applying a generalized Fourier transform, one deduces that HS can be rewritten(up to a unitary transformation) as a direct integral of fiber Hamiltonians, H(~p), where ~p isthe atom+field momentum, and

H(~p) = (~p− ~PE)2/2 +n0∑i=1

eiΠi +HE + λ0HI (1.1.10)


is an operator on Cn0⊗F+(L2(R3)) for all ~p ∈ R3 (we suppressed the tensor products with theidentities 1Cn0 and 1HE to simplify notations). The interaction Hamiltonian HI in (1.1.10)does not depend on the position ~x anymore. It is still linear in annihilation and creationoperators, and infrared regular.

If λ0 = 0, the spectrum of H(~p) is well known: σ(H(~p)) = σpp(H(~p)) ∪ σac(H(~p)), whereσpp(H(~p)) = ~p2/2 + e1, ..., ~p

2/2 + en0 and σac(H(~p)) = [~p2/2 + e1,+∞) \ σpp(H(~p)) arethe pure point spectrum and the absolutely continuous spectrum of H(~p), respectively. Weremark that the eigenvalues ~p2/2 + ei are embedded in the continuous spectrum. If λ0 6= 0,the study of the spectral properties of the Hamiltonian H(~p) is difficult, even for small valuesof |λ0|, because there is no gap between the eigenvalues ~p2/2+ei and the continuous spectrumof the unperturbed Hamiltonian ( with λ0 = 0). Therefore, tools of standard perturbationtheory (see e.g. in [95]) do not apply. The exact spectral properties of H(~p) when λ0 6= 0have been unknown for a long time, until breakthrough techniques were introduced in the lasttwo decades, in particular the Mourre commutator method [94], the Spectral RenormalizationGroup approach [12], and the Pizzo’s method [109].

If λ0 6= 0, the behavior of the system is different regarding whether |~p| ≥ 1 or |~p| < 1.If |~p| < 1, the eigenvalues ei, i = 2, ..., n0, dissolve into the continuous spectrum and excitedstates become resonant. As the interaction is sufficiently regular in the infrared regime, thepoint spectrum of H(~p) is reduced to the minimum of σ(H(~p)), which is shifted from e1 +~p2/2by a small amount: this is the Lamb shift effect. The rest of the spectrum is absolutelycontinuous; see e.g. [11]. If the interaction were not regular in the infrared regime (typicallyfor charged particles like electrons), then the minimum of σ(H(~p)) would not be an eigenvalueof H(~p): In its minimal energy state, the particle would be surrounded by an infrared cloudof infinitely many soft photons that does not correspond to any state in Fock space, see [61].If |~p| ≥ 1, one expects the point spectrum of H(~p) to be empty, as a relic of the Cerenkovradiation. Very little is known in that regime; but see [42] for some results in this direction.

e1 + ~p2/2 e2 + ~p2/2 e3 + ~p2/2 ... eN + ~p2/2

R− axis

R− axise′1

Figure 1.1: The spectrum of H(~p) when λ0 = 0 ( figure on the top ), and when λ0 6= 0 ( figureon the bottom, with |~p| < 1). If λ0 = 0, the eigenvalues ei + ~p2/2, i = 2, ..., n0, are embedded in thecontinuous spectrum, and e1 + ~p2/2 lies at the bottom of the spectrum. If λ0 6= 0, the eigenvaluesei + ~p2/2, i > 1, dissolve into the continuous spectrum. The eigenvalue e1 + ~p2/2 is shifted by a smallamount that depends on λ0. It remains an eigenvalue because the interaction is infrared regular.

The spectral properties of the fiber Hamiltonian H(~p), |~p| < 1, as described above, havebeen already studied extensively for translation invariant models similar to ours. Concerningthe ground state of H(~p) for similar models, one can consult in particular [1, 3, 10, 33, 34, 35,49, 68, 84, 100, 109, 36, 65, 104, 53]. In this thesis, we will be particularly interested in theregularity of inf(σ(H(~p))) and in the regularity of the ground state, with respect to the totalmomentum ~p.


We will use this regularity in our proof on the existence of closed quantum systems inChapter 3. In [53], we were able to show that the bottom of the spectrum of H(~p) is asimple eigenvalue, is real analytic in ~p for |~p| < 1, and that its associated one dimensionalprojection is also real analytic in ~p (these results only hold if the coupling λ0 is sufficientlysmall). The real-analyticity of inf(σ(H(~p))) with respect to the momentum ~p has already beenestablished in [1] for the Nelson model (a toy-model of an electron coupled to the quantizedelectromagnetic field, which is infrared singular) using in particular a cluster expansion anda Feynman-Kac type Formula. In [53], we use the spectral renormalization group techniquedeveloped in [12, 10]. For similar ideas in this direction, one can consult the work in [78] wherethis method is used to show the real analyticity of inf(σ(H(~p)) as a function of the couplingconstant λ0.

The main idea of the renormalization group method (see e.g. [13],[11], [14], [9], [77], [78],[85], [118], [3], [54] for earlier developments) is to decimate iteratively the high energy modesof the field via the use of an "isospectral" transformation, called renormalization map. Ateach step of the procedure, one gets a new operator whose spectral properties near ei + ~p2/2can be used to reconstruct the spectral properties of H(~p) near ei + ~p2/2. The interactionturns out to be irrelevant under the renormalization map and becomes weaker and weaker ateach step, whereas the free field Hamiltonian, HE , and the field momentum operator, ~PE , aremarginal, and do not change much. Therefore, asymptotically, the limiting operator obtainedby iterating arbitrarily many times the renormalization map is a function of HE and ~PE ,whose spectrum is easy to study. If this operator has an eigenvalue (or more), one can thenreconstruct the eigenvalue(s) of the initial operator, H(~p), by using the isospectrality of therenormalization map.

In the first chapter of this thesis, we do not present [53] but we expose instead a general-ization of this result to the resonances of the Hamiltonian H(~p). The study of resonances fornon-relativistic q.e.d models has also a long history; see [13], [11], [12], [14], [79], [118], [7], andreferences given there. However, the atomic nucleus was usually treated as static (infinitelyheavy). In our work, we show that the excited states of the atom turn into resonances whenλ0 is non-zero, and, furthermore, that these resonances are real analytic in ~p for |~p| < 1.

Resonances are the eigenvalues of the dilated Hamiltonians Hθ(~p), where =(θ) > 0, andwhere Hθ(~p) is obtained from H(~p) by "dilating" the field momenta ~k by a factor e−θ; see also[18, 117, 120, 119, 89]. One can show that these eigenvalues do not depend on the value of θ ifθ lies in some domain of the upper-half plane. Resonances correspond to metastable states, i.e.to states with a finite life time. By a contour deformation argument, one can show that theprobability of staying in an excited state of the uncoupled Hamiltonian decays exponentiallyfast in time if Fermi Golden rules are imposed (basically, Fermi Golden rules say that theprobability of transitions between any excited state and the ground state is non-zero); see [11]for an estimate on time scales of order λ−2

0 , and [16] for an estimate at arbitrarily large times.The rate of decay depends on the imaginary part of the resonance, which itself depends onthe coupling constant λ0. In Chapter 2, we prove

Theorem 1.1.3.1. Let 0 < ν < 1. There exists λc(ν) > 0 such that, for all 0 ≤ λ0 < λc(ν)and for all ~p ∈ R3, |~p| < ν, the following properties are satisfied:

a) inf σ(H(~p)) is a non-degenerate eigenvalue of H(~p).

b) For every i0 ∈ 1, · · · , n0 and θ ∈ C with 0 < =θ < π/4 large enough, Hθ(~p) has an


eigenvalue, z(∞)i0

(~p), such that z(∞)i0

(~p)→ ei0, as λ0 → 0. For i0 = 1,

z(∞)1 (~p) = inf σ(H(~p)).

Moreover, for |~p| < ν, |λ0| small enough and 0 < =θ < π/4 large enough, z(∞)i0

(~p), i0 ≥ 1, areanalytic in ~p, λ0 and θ. In particular, they are independent of θ.

The proof of Theorem 1.1.3.1 is based on a variant of the Spectral Renormalization Group.The novelty of our approach resides in the fact that we avoid the use of Banach spaces (onthe contrary to the references cited above). In particular, our construction of the iterativealgorithm is simpler and converges “(super-)super-exponentially” fast, whereas the speed ofconvergence of the renormalized Hamiltonians is exponentially fast in the traditional RGapproach.

1.2 Closed systems in quantum mechanics

The purpose of this section is to show that many reasonable quantum systems can be viewedas closed systems. We start from a closed system S characterized by the datas

(OS ,AS , τt,s,SS).

(AS could contain the observables of the "entire universe" if we want to be sure that S isreally closed). Let OP be a subset of OS and let AP ⊃ OP be a C∗-algebra strictly containedin AS .

Definition 1.2.0.1. If there is a class of states CS ⊆ SS, and a groupoid of ∗-automorphismsτt,s : AP → AP , such that, for any ω ∈ CS,

ω(τt,s(a)) ≈ ω|AP (τt,s(a)), (1.2.1)

for all a ∈ AP and for all t, s ∈ I ⊆ R, then we say that the system P characterized by thedata (OP ,AP , τt,s,SP ) can be viewed as a closed subsystem of S during the time intervalI. In (1.2.1), the state ω|AP : AP → C in (1.2.1) is the reduced state of ω : AS → C to thesubalgebra AP ⊆ AS.

The symbol ≈ means that the two sides of (1.2.1) are so close that the absolute value oftheir difference is below any reasonable experimental resolution thresholds. If P can be viewedas a closed subsystem of S, then, for every state ω in CS , the algebraic data (OP ,AP , τt,s,SP )and (OS ,AS , τt,s,SS) give the same predictions concerning expectation values of observablesa = a∗ ∈ OP , provided one replaces ω by its reduced state ω|AP ∈ SP .

The purpose of the second result presented in this thesis, in Chapter 3, is to exhibit someconcrete models of quantum systems where the no-signaling condition (1.2.1) can be proven.We consider models where AS = B(HS). We assume that S is made of two systems P and Q,that are spatially far separated, and of an environment E that is coupled to P and Q (thatcan be empty). The Hilbert space HS is assumed to be of the form

HS = HP ⊗HQ ⊗HE , (1.2.2)


where HP , HQ and HE are separable Hilbert spaces. We denote by Utt∈R the one-parametergroup of unitary transformations of time translations in HS . Rewriting (1.2.1) in this particu-lar setting, we deduce that P can be viewed as a closed system if there exists a one-parametergroup (UP (t))t∈R of unitary transformations on HP such that

Tr(ρ U(t)∗(O ⊗ 1HQ∨E )U(t)) ≈ TrHP (ρP UP (t)∗OUP (t)), ∀O ∈ B(HP ), (1.2.3)

for a suitably chosen subset of density matrices ρ, and for all times t in some interval containedin R. In (1.2.3), ρP = TrQ∨E(ρ) is the reduced density matrix of ρ on HP . The notation ≈is made mathematically precise in the context of the two models analyzed in Chapter 3. Webriefly list the properties of the two models we consider:

(a) The first model describes a quantum particle, P , with spin 1/2, interacting with a largequantum system Q and moving away from Q. The subsystem Q may consist of anotherquantum particle entangled with P and a “detector”. The two particles are prepared inan initial state chosen such that they move away from each other, with P moving awayfrom the detector.

(b) The second model describes a neutral atom P with a non-vanishing electric dipole momentthat interacts with a large quantum system Q. Both P and Q are coupled to the quantizedelectromagnetic field, E. In presence of a slowly varying external potential, we are ableto construct an effective dynamics for P that does not make any explicit reference to theelectromagnetic field E.

In Chapter 3, we discuss various sufficient conditions implying that, for a large class CSof initial states on AS including ones exhibiting entanglement between P and Q, the time-evolution of expectation values of observables of the subsystem P behaves as if the subsystemsQ and E were absent, in the sense of (1.2.3). The no-signaling property (1.2.3) has beenalready studied by various authors. For instance, one can find related works in [50, 116]. Itis often assumed ( see e.g. [116]) that the dynamics between P and Q ∨ E decouples, in thesense that the total Hamiltonian HS is given by HS = HP ⊗ 1HQ∨E + 1HP ⊗HQ∨E . In thatcase, (1.2.3) holds trivially for all states ρ on HS . This hypothesis is however very simplistic.Our analysis turns out to be more delicate, in particular in the second model, because we addinteractions between P , Q, and E. One of the strength of our result (only shown for the firstmodel), is that our estimates of the error in (1.2.3) are uniform in the number of degrees offreedom of the system Q.

1.3 On the problem of preparing states in quantum mechanics

Knowing the initial state of a system S is crucial to predict probabilities of outcomes ofexperiments and observations carried out on S. Experimental physicists often first preparetheir quantum system S into some specific state, ωS , before involving this system in anexperimental process. It is therefore important to know how to prepare S in ωS , and thisproblem seems to be disregarded by most of the treatises on quantum mechanics. In Chapter4 , we treat a particular method that enables one to prepare a system S in a preassigned initialstate with certainty; i.e., the probability of success in preparing S in a given state is unity.

We propose in this introduction to describe three alternative techniques that can be usedto prepare a quantum-mechanical system S in a specific initial state ωS , the last one being the


one studied in detail in Chapter 4. In concrete applications, the methods of state preparationdiscussed below are often combined with one another.

Preparation of states via adiabatic evolution

We consider a quantum-mechanical system S with AS = B(HS). We assume that we knowhow to prepare the system in an initial state Ω ≡ Ω(0) ∈ HS , where Ω is the ground stateof a Hamiltonian H(0) generating the time evolution of states of S at times t ≤ 0. (Forexample, one may make use of the state preparation procedure outlined in the next paragraphto prepare S in the state Ω at an early time). We would like to find out how, at a later time,one might manage to prepare S in a state, ΩS ≡ Ω(1) 6= Ω(0), of interest for the purposeof some observations or experiments. The idea explored in this subsection is to make useof adiabatic evolution to transform the initial state Ω into the desired state ΩS . By turningon suitable time-dependent external fields one may be able to tune the time evolution of Sto be given by a family of time-dependent Hamiltonians H(s)s∈R with the property thatΩS = Ω(1) is the ground state of the operator H(1).

Given the family of Hamiltonians H(s)s∈R and a time-scale parameter τ > 0, the timeevolution of state vectors inHS from an initial time t0 to time t is given by a unitary propagatorU(t, t0) that solves the equation

d

dtU(t, t0) = −iH(t/τ)U(t, t0), U(t, t) = 1HS , (1.3.1)

for all t0, t ∈ R. The so-called adiabatic limit is the limit where τ tends to ∞.In order to investigate the adiabatic limit mathematically, one has to require some as-

sumptions on the Hamiltonians H(s), s ∈ R, (see, e.g., [6], [124] and [112]): We assumethat all the operators H(s) are self-adjoint on a common dense domain D ⊂ HS , that theresolvents R(s, i) := (H(s)− i)−1 are differentiable in s, with norm-bounded derivatives, andthat the operators H(s) ddsR(s, i) are bounded uniformly in s ∈ R. We assume that all theHamiltonians H(s), s ∈ R, have a non-degenerate ground state energy, e(s), corresponding toa ground state eigenvector Ω(s) ∈ HS , with Ω(0) = Ω. The projections P (s) := |Ω(s)〉〈Ω(s)|are assumed to be twice continuously differentiable in s, with norm-bounded first and secondderivatives. After rescaling the time t by setting s = t/τ , with s0 = t0/τ , Eq. (1.3.1) takesthe form

d

dsUτ (s, s0) = −iτH(s)Uτ (s, s0), (1.3.2)

where Uτ (s, s0) = U(τs, τs0), and one can prove that

sups∈[0,1]

‖Uτ (s, 0)Ω− Ω(s)‖ −→τ→∞

0, (1.3.3)

see [6], [124].Eq. (1.3.3) tells us that it is possible to drive S from an initial state Ω(0) = Ω to the

desired state Ω(1) = ΩS adiabatically. Thus, S can be prepared in the state ΩS . The drawbackof this method is that it presupposes our ability to initially prepare S in the ground state Ωof the Hamiltonian H(0) and that suitable external fields must be turned on that lead to afamily of time-dependent Hamiltonians slowly driving Ω(0) to the desired state ΩS .

There are variants of this method of state preparation that are not based on very slowevolution (i.e., do not involve an adiabatic limit) but require some kind of “optimal control”


used to construct a family of time-dependent Hamiltonians that determine a propagator driv-ing S from its initial state Ω(0) to the desired state ΩS in as short a time as possible; see,e.g., [37].

Preparation of states via duplication of systems and state selection

We next sketch a method for state preparation that is presumably most often used in practice:One attempts to create a large number, n, of independent copies of the system S, which wedenote by Si, i = 1, ..., n. The closed system S1 ∨ S2... ∨ Sn ∨ E is the union of n copies of Sall of which are successively coupled to a measuring device E. The purpose of coupling thesystems S1, ..., Sn to the device E is to perform projective measurements of a physical quantity,represented by a self-adjoint operator a = a∗, common to S1, ..., Sn, one of whose eigenvectorsis the state, ΩS , in which we want to prepare the system S. After a projective measurementof a, the system Si is in an eigenstate, Ωki , of the operator a. Whenever Ωki 6= ΩS , the systemSi is thrown into the waste basket. However, if, in the ith measurement of a, the measured(eigen-)value corresponds to the (eigen-)vector ΩS of a then the system Si is kept and hasbeen successfully prepared in the desired state ΩS .

A typical example of this method for state preparation is a Stern-Gerlach spin measure-ment: In this example, the system S consists of a spin 1/2-particle, (e.g., a silver atom).The experimentalist successively sends a large number, n, of such particles through a veryslightly inhomogeneous magnetic field essentially parallel to the z-axis, which is perpendicularto the initial direction of motion of the particles (parallel to the x-axis). Particles with spinup (S(z) = +1/2) are deflected towards the positive z-direction, whereas particles with spindown (S(z) = −1/2) are deflected towards the negative z-direction. Thus, after traversing themagnetic field, the particle beam is split into two sub-beams that point into slightly distinctdirections. One of these two sub-beams is then targeted towards a screen that destroys it.The remaining beam consists of particles prepared in a fixed eigenstate of S(z) and can beused for further experimentation.

The method of state preparation discussed here demands creating many essentially identi-cal copies of a system S of interest and cannot be applied if the system S cannot be duplicated.For the theorist, this method obviously poses the problem of first understanding what “pro-jective measurements” are; see Chapter 5.

Preparation of states via weak interaction with a dispersive environment

We consider a closed system S characterized by the data (OS ,AS , αt,st,s∈R,SS). We saythat a subsystem P of S can be prepared with certainty in the state ωP if there is a dense setof states CS ⊆ SS , such that, for all ω ∈ CS , for all a ∈ 〈OP 〉 ⊆ AS ,

limt→∞

ω(αt,0(a)) = ωP (a). (1.3.4)

In Chapter 4, we prove (1.3.4) for the so-called generalized spin-boson model. This modelis similar to the model presented in Section 1.1.3, with the additional simplifications that theatom P is infinitely heavy (it does not move: HP = Cn0), and that the photons are replacedby phonons (we neglect the polarization). In that particular case, AS = B(HP ⊗ HE), andthe atom is the system P to be prepared. The algebra 〈OP 〉 is taken to be the C∗-algebra ofelements of the form A⊗ 1HE , where A ∈ B(HP ). Time evolution on S = P ∨E is generated


by the Hamiltonian HS = HP ⊗ 1HE + 1HP ⊗HE +HP,E . We assume that the target stateis the ground state of the atomic Hamiltonian HP , where

HP =n0∑i=1

eiΠi, and e1 < .... < en0 ;

This can be achieved in a laboratory, by manipulating suitable external fields, etc., to tunethe dynamics of P so as to have the property that the state we want P to prepare in is theground state of the given dynamics.

By letting the strength of the interaction between P and E tend to zero sufficiently slowlyin time, we can manage to asymptotically decouple P from E and have P approach its ownground state, which is the desired state ΩP , as time t tends to ∞. To do so, we exploitthe dynamical properties of S = P ∨ E already stated in Section 1.1.3: the excited statesof the atom become resonant if the coupling is non-zero, whereas the ground state energyof the atom stays an eigenvalue (our interaction is infrared regular) of the coupled system,but its energy, e1, is shifted by a small amount (Lamb shift). If we do impose Fermi Goldenrules, it is possible to show that the rate of decay to the ground state is exponentially faston the Van Hove time scale t ∝ λ−2

0 (where λ0 is the coupling constant) using a Contourdeformation argument and the Feshbach-Schur map; see e.g. [14]. It is harder to get decayestimates on larger time scales. This can be done, e.g., by using the Spectral RenormalizationGroup method; see [16]. However, with these techniques involving spectral dilations, one canbasically only control terms of the form

〈Ψ1|e−itHSΨ2〉, (1.3.5)

for Ψ1,Ψ2 ∈ HS = Cn0 ⊗ F+(L2(R3)), and problems appear if one tries to control the limitt→∞ of terms of the type

〈Ψ1|eitHS (O ⊗ 1HE )e−itHSΨ2〉. (1.3.6)

This is unfortunate, because (1.3.6) is exactly the equation we need to study to prove Formula(1.3.4) in our particular setting. The reason why techniques that involve spectral dilations failto get reasonable estimates is the following: To control the large time limit with a deformationcountour argument, the field momenta in HS must be dilated by a factor e−θ for the terme−itHS in (1.3.6), and by a factor eθ for the term eitHS in (1.3.6). This can be done only if O isreplaced by PΩ(O⊗1HE )PΩ, where Ω is the Fock vacuum state (because Ω is invariant underdilations). Note that this problem only appears at zero temperature. If the field is initiallyin the KMS state at temperature 1/β, one can use the spectral properties of the LiouvillianL and the KMS equation to prove thermalization of P as t→∞, and it is sufficient to showthat σ(L) \ 0 is absolutely continuous; see [15]. In that case, P does not reach the groundstate of HP , but the Gibbs equilibrium state at temperature 1/β, e−βHP /Tr(e−βHP ).

In [44], W. De Roeck and A. Kupiainen solved the problem of taking the limit t → ∞in (1.3.6) at zero temperature. Their main tool is a cluster expansion based on the spectralproperties of the map O 7→ PΩe

itHS (O ⊗ 1HE )e−itHSPΩ on the Van Hove time scale. We usetheir technique, and extend their results to settings where the coupling λ0 is not constant, butchanges slowly with time. Indeed, if λ0 stays constant, due to the Lamb shift effect, the atomreaches a state that is closed to the ground state of HP , but it is not the ground state of HP !Switching slowly the coupling λ0 to zero, we prove that preparing P in the ground state ofHP can be done with certainty.


1.4 The emergence of facts in physical systems

1.4.1 Probabilities of events in physical systems

The purpose of the algebraic datas in 1.1.1.1 is to make predictions of the following kind:Suppose we are interested in testing some potential properties (or, put differently, measuresome physical quantities) a1, ..., an ∈ OS during intervals of time ∆1 ≺ ∆2 ≺ ... ≺ ∆n, where

∆ ≺ ∆′ iff, ∀t ∈ ∆, ∀t′ ∈ ∆′ : t ≤ t′. (1.4.1)

We assume that the system is in a state ω. Then a physical model ought to tell us whethera1, ..., an will actually be measurable (i.e., are "empirical" properties) and predict the proba-bility (frequency) that, in a test or measurement of ai at some time ti ∈ ∆i, a given outcomeof the observables ai is measured. These alternative outcomes, or “events", correspond tospectral projections Π

(i)αi := Pa(ti)(I

(i)αi ), αi = 1, ..., ki, where I

(i)αi ∩ I

(i)βi

= ∅, for αi 6= βi, and

∪kiαi=1I(i)αi ⊇ spec a(ti), for all i = 1, ..., n. Then

Π(i)αiΠ

(i)βi

= δαiβiΠ(i)αi ,

ki∑αi=1

Π(i)αi = 1, (1.4.2)

for an arbitrary time ti. A time-ordered sequence Π(1)α1 , ...,Π

(n)αn of possible events Π

(i)αi is

called a "history". Given a model of a physical system, as specified in points (I)-(III) ofDefinition 1.1.1.1, Sect. 1.1.1, the probability of a history Π(1)

α1 , ...,Π(n)αn in a state ω ∈ SS is

predicted to be given by

ProbωΠ(1)α1, ...,Π(n)

αn := ωÄΠ(1)α1...Π(n−1)

αn−1Π(n)αn Π(n−1)

αn−1...Π(1)

α1

ä; (1.4.3)

see also [115, 129, 107]. Formula (1.4.3) fulfills standards requirements for probabilities.Namely,

(1) Probω satisfies0 ≤ ProbωΠ(1)

α1, ...,Π(n)

αn ≤ 1, (1.4.4)

for every state ω ∈ SS and every history Π(1)α1 , ...,Π

(n)αn .

(2) If we introduce Nk1,...,kn := 1, ..., k1 × ...× 1, ..., kn, then∑(α1,...,αn)∈Nk1,...,kn


αn = 1, (1.4.5)

for arbitrary operators a1, ..., an and time intervals ∆1 ≺ ... ≺ ∆n.

Properties (1) and (2) show that Probω is a probability functional. Furthermore, as ob-served in [75, 91, 107], Formula (1.4.3) represents the "only possible" definition of a probabilityfunctional on the lattice of possible events. The problem of (1.4.3) is that it is perfectly mean-ingful if the algebra AS is abelian, but it is most often meaningless if AS is non-commutative.


The abelian case

If AS is abelian, observables are continuous functions on MS , the spectrum of AS . Eventscorrespond to characteristic functions, χ

Ω(i)αi

(ti), of open subsets, Ω

(i)αi (ti), of MS given by

ξ ∈ Ω(i)αi (ti)⇔ ai(ti)(ξ) ∈ I(i)

αi , (1.4.6)

where ai(ti) = τti,t0(ai), I(i)αi is an open subset of R, and t0 is a fiducial time at which the

state of S is specified. We denote by φt,s the homeomorphism of MS corresponding to τt,s.Setting Ω

(i)αi := φt0,ti(Ω

(i)αi (ti)), it is clear that ξ ∈ Ω

(i)αi (ti) iff φt0,ti(ξ) ∈ Ω

(i)αi . Let µ be a

probability measure on MS (i.e. a state of the physical system). The probability of a history,χ

Ω(1)α1

(t1), ..., χ

Ω(n)αn (tn)

, is given by

ProbµχΩ(1)α1

(t1), ..., χ

Ω(n)αn (tn)

:=

∫MS

dµ(ξ)n∏i=1

χΩ

(i)αi

(ti)(ξ) =

∫MS

dµ(ξ)n∏i=1

χΩ

(i)αi

(φt0,ti(ξ)).

Probµ satisfies the sum rule: if two outcomes are complementary (in the sense that Ω(i)αi (ti)∩

Ω(i)βi

(ti) = ∅), then

ProbµχΩ(1)α1

(t1), ...,χ

Ω(i)αi

(ti)∪Ω(i)βi

(ti), ...., χ

Ω(n)αn (tn)

= ProbµχΩ(1)α1

(t1), ..., χ

Ω(i)αi

(ti), ...., χ

Ω(n)αn (tn)

+ ProbµχΩ(1)α1

(t1), ..., χ

Ω(i)βi

(ti), ...., χ

Ω(n)αn (tn)

.

(1.4.7)

If µ is a pure state, i.e., µ = δξ0 , for some ξ0 ∈MS then

ProbµχΩ(1)α1

(t1), ..., χ

Ω(n)αn (tn)

=n∏i=1

χΩ

(i)αi

(ti)(ξ0) =

n∏i=1

χΩ

(i)αi

(φt0,ti(ξ0)), (1.4.8)

i.e., the possible values of Probδξ0 are 0 and 1, for any ξ0 ∈ MS and all histories. If ξt :=φt0,t(ξ0) is the trajectory of states with initial condition ξ0 at time t0 then

Probδξ0χΩ(1)α1

(t1), ..., χ

Ω(n)αn (tn)

= 1⇐⇒ ξti ∈ Ω(i)αi , (1.4.9)

for all i = 1, ..., n; otherwise, Probδξ0 vanishes. If ξ0 /∈ Ω(i)αi , then the event φt0,t(ξ) ∈ Ω

(i)αi is

first observed at time t = ti, where

ti := inf t | ξ0,t = φt0,t(ξ0) ∈ Ω(i)αi , (1.4.10)

and it is last seen at time ti, where

ti := sup t | ξ0,t = φt0,t(ξ0) ∈ Ω(i)αi . (1.4.11)

These features of classical physical systems, in particular the "0-1 laws" in (1.4.9), are char-acteristic of realism and determinism: Given that we know the state, ξ0, of a system S atsome time t0, we know its state, ξt = φt0,t(ξ0), and the value, ai(ξt), of an arbitrary property,ai ∈ PS , of S, at an arbitrary (earlier or later) time t.


What happens if AS is non-commutative?

Let Π(i)1 , ...,Π

(i)ki

be complementary outcomes such that∑kiαi=1 Π

(i)αi = 1. If AS is not abelian,

ki∑αi=1

ProbωΠ(1)α1, ...,Π(i)

αi , ...,Π(n)αn 6= ProbωΠ(1)

α1, ...,Π(i−1)

αi−1,Π(i+1)

αi+1, ...,Π(n)

αn (1.4.12)

in general, and the "sum rule" of probability

ki∑αi=1

ProbωΠ(1)α1, ...,Π(i)

αi , ...,Π(n)αn = ProbωΠ(1)

α1, ...,Π(i−1)

αi−1,Π(i+1)

αi+1, ...,Π(n)

αn (1.4.13)

(which holds for classical models) is violated. If Π(i)αi commutes with the operator Π

(n)αn ... Π

(i+1)αi+1 ,

for all αi, then (1.4.13) holds true. If

[Π(i)αi ,Π

(n)αn ... Π(i+1)

αi+1)] 6= 0, (1.4.14)

then interference terms

<ω(Π(1)α1...Π(i−1)

αi−1Π(i)αiΠ

(i+1)αi+1

...Π(n)αn ...Π

(i+1)αi+1

Π(i)βi

Π(i−1)αi−1

...Π(1)α1

)(1.4.15)

are in general non-zero for αi 6= βi, and (1.4.13) is violated. Therefore, in quantum models ofphysical systems, complementary possible events do not mutually exclude one another, givenfuture events that cause interference. A very famous example of this property of quantumsystems is illustrated by the double slit experiment of R. Feynman; see [58]. In this experiment,the projection Π

(2)∆ represents the possible event that an electron, after having passed a shield

with two slits, reaches a region ∆ of a screen, where it triggers the emission of a flash of light.The projection Π

(1)1 represents the event that the electron has passed through the slit on the

right of the shield, while Π(1)2 stands for the possible event that the electron has passed through

the slit on the left of the shield. Due to non-vanishing interference terms, <ω(Π(1)1 Π

(2)∆ Π

(1)2 ),

the sum rule in (1.4.15) is usually violated:

ω(Π(2)∆ ) 6= ω(Π

(1)1 Π

(2)∆ Π

(1)1 ) + ω(Π

(1)2 Π

(2)∆ Π

(1)2 )

This can be tested, experimentally, because ω(Π(1)1 Π

(2)∆ Π

(1)1 ) can be determined from experi-

ments where the slit on the left of the shield is blocked, while ω(Π(1)2 Π

(2)∆ Π

(1)2 ) can be deter-

mined from experiments where the slit on the right is blocked. ω(Π(2)∆ ) can be determined

from experiments where both slits are left open.

screen

e− ∆


If the sum rule is violated as in (1.4.12), the operator ai(ti) does not represent an empiricalproperty of S, given later observations of physical quantities ai+1, ..., an. Apparently, theoperators a ∈ OS do, in general, not represent properties of S that exist a priori, but onlypotential properties of S whose empirical status depends on the choice of the time evolutionτt,sτ,s∈R of S and of the state ω. We discuss this problem in Paragraph 1.4.2.

One of the consequence of (1.4.12) is the absence of a good notion of conditional probabilityin quantum mechanics. Indeed, given a history of events

hni (α) := Π(1)α1, ...,Π(i−1)

αi−1,Π(i+1)

αi+1, ...,Π(n)

αn ,

one may try to define a conditional probability

ProbµωΠ(i)αi | h

ni (α) :=

µω(α1, ..., αi, ..., αn)∑kiβi=1 µω(α1, ..., βi, ..., αn)

. (1.4.16)

Unfortunately, this definition is meaningless: Recall that Π(i)βi

is a shorthand for the spectral

projection Pai(ti)(I(i)βi

). We fix a subset I(i)αi , but introduce a new decomposition of spec ai into

subsetsI

(i)1 := I(i)

αi , R \ I(i)αi = ∪miβ=2I

(i)β ,

with I(i)β ∩ I

(i)γ = ∅, for β 6= γ, and define

Π(i)βi

:= Pai(ti)(I(i)βi

),

βi = 1, ...,mi. We define

µω(α1, ..., βi, ..., αn) := ProbωΠ(1)α1, ..., Π

(i)βi, ...,Π(n)

αn .

Thenµω(α1, ..., 1, ..., αn) = µω(α1, ..., αi, ..., αn);

but, most often, the putative "conditional probabilities" are different,

ProbµωΠ(i)αi | h

ni (α) 6= ProbµωΠ(i)

αi | hni (α), (1.4.17)

unless all possible interference terms vanish. It may be of interest to note that if the operatorsai have pure-point spectrum with only two distinct eigenvalues then

Π(i)αi αi=1,2 = Π(i)

βiβi=1,2,

and we have equality in (1.4.17). This is related to the Kochen-Specker theorem [97].

Another interesting feature of quantum mechanics is that it is, intrinsically, non deter-ministic. To illustrate this point, we look at a simple simple experimental setup, that werefirst described in [113]. We consider a closed system S = P ∨ Q, where Q is composed of aparticle P ′ of spin 1/2 (electron) and of a spin filter D, and P is another particle of spin 1/2.The particles P and P ′ are scattered into opposite cones, and the spin filter D selects theparticle P ′ according to its spin component along the axis corresponding to a unit vector ~n.A Stern-Gerlach-type experiment is added to the setup to measure a component of the spinof the particle P .


PP

P ′

P ′50%

50%

Spin filter

particle Pparticle P ′

The Hilbert space of the system S = P ∨ Q is assumed to be HP ⊗ HP ′ ⊗ HD, whereHP = HP ′ = L2(R3;C2). We set

| ↑〉 :=

Ç10

å, | ↓〉 :=

Ç01

å.

We assume that the initial state is an entangled state of the form

Ψ0 =1√2n

n∑j=1

(|φj,P , ↓〉 ⊗ |ψj,P ′ , ↑〉 − |φj,P , ↑〉 ⊗ |ψj,P ′ , ↓〉

)⊗ |χj〉, (1.4.18)

where ‖φj,P ‖L2 = ‖ψj,P ′‖L2 = 1, 〈χi|χj〉HD = δij for all i, j = 1, ..., n. We denote by~SP = (σx, σy, σz) the spin operator of the particle P , where σx, σy, σz are the Pauli matrices.Our study of isolated systems (see Chapter 3) shows that, under reasonable assumptionsconcerning the interaction between P , P ′ and Q, the system P can be considered as a closedsystem, and the non-signaling condition (1.2.3) is satisfied. More precisely, if we denote by dthe initial "distance" between P and Q at the fiducial time t = 0, one finds the lemma:

Lemma 1.4.1.1 (No-signaling). Let η > 0. We require assumptions (A1), (A2) and (A3)of Section 3.2.1 for P , P ′ and Q. Then there is a distance d(η, v) > 0 such that, for anyd > d(η, v),

|〈e−itHSΨ0|~SP e−itHSΨ0〉| < η, ∀t ≥ 0. (1.4.19)

For sufficiently large values of d, Corollary 1.4.1.1 shows that the mean value of the spinoperator of the particle P very nearly vanishes for all times t, regardless of the initial statevectors χi of the filter D. In particular, the expectation value of the spin operator of P isindependent of the kind of measurement on P ′ performed by the spin filter D. Here we assumethat a particle P ′ with ~SP ′ · ~n = 1/2 passes the filter, while a particle with ~SP ′ · ~n = −1/2is absorbed by D, with probability very close to 1. A realistic interpretation of quantummechanics, in the sense that the time evolution of pure states in the Schrödinger picturewould completely predict what will happen, necessary fails. It would lead to the predictionthat

〈e−itHSΨ0|(~SP · ~n)e−itHSΨ0〉 ≈ −1

2(1.4.20)

for sufficiently large times t if the particle P ′ has passed the filter D. This contradicts Eq.(1.4.19). It shows that choosing a unitary time evolution and specifying an initial statedoes not predict the results of measurement, but only probabilities for the outcomes of suchmeasurement. Our conclusion remains valid if the particles P and P ′ are indistinguishableparticles; see [113].


1.4.2 The role of entanglement and information loss in the emergence offacts in quantum mechanics

To outline the role played by entanglement and information loss in the emergence of facts inquantum mechanics, we start our discussion with the double slit experiment of R. Feynmanalready mentioned above. If a laser lamp, emitting light of a wave length much smaller thanthe distance between the two slits in the shield, is turned on in the cavity between the shieldand the screen, we expect that the interference pattern, observed on the screen when both slitsin the shield are open and the laser lamp is turned off, gradually disappears when the laserlamp is turned on and its intensity is increased. This is due to scattering processes betweenthe electron and the photons in the laser beam, which serve to track the trajectory of theelectron.

screen

e−

lamp

∆

detecting wall

detecting wall

If the electromagnetic field emitted by the laser is included in the theoretical description ofthe equipment, E, used in this experiment, then the disappearance of the interference patternon the screen can be understood as the result of information loss of the electron into the e.m.field, and decoherence, which makes the interference term ω(Π

(1)1 P

(2)∆ Π

(1)2 ) tend to 0, as the

wave length of the laser decreases and its intensity is cranked up, and, hence, renders thepossible events Π

(1)1 and Π

(1)2 empirical. Decoherence is made possible because the electron

and the field get entangled: the reduced state of the electron becomes mixed even if it wereinitially pure.

In Chapter 5, we give an algebraic characterization of the phenomena of information lossand decoherence, and we illustrate these mechanisms by treating a simple example where afixed particle is coupled to a quantized field. Our main task is to clearly distinguish potentialproperties of S from empirical properties of S. A potential property of S is an elementa ∈ OS . To understand how facts arise, it is necessary to take into account the role oftime. We introduce an algebra, E≥t, of potential properties observable after time t. This isthe C∗-subalgebra of AS generated by arbitrary finite linear combinations of arbitrary finiteproducts

a1(t1)...an(tn), n = 1, 2, 3, ..., (1.4.21)

where ti ≥ t and ai ∈ OS , i = 1, ..., n; in Eq. (1.4.21), a(t) = τt,t0(a) ∈ AS where t0 is afiducial time at which the state of S is specified. One of the fundamental reason why potentialproperties can become empirical relies on the fact that, in general,

E≥t ( E≥t′ . (1.4.22)

for some t ≥ t0 ≥ 0 and for all t′ < t. We call this phenomenon information loss. In local,relativistic quantum theory, the finiteness of the speed of light, i.e., of the speed of propagation


of arbitrary signals, and locality, lead to an intrinsic notion of information loss [62, 31] – atleast in theories with massless particles that satisfy Huyghens’ Principle [29] and are allowedto escape to spatial ∞. This is not so when one considers non-relativistic models of physicalsystems, with signals propagating arbitrarily fast, which is our case here. Then, the systemS must be composed of an environment E and a small system P , and the environment E isused to "retrieve" information from P .

x

t

yt′

t

O

The event at time t′ < t involving photons can never be observed by the observer O

Let t such that E≥t ( E≥t′ ⊆ AS for all t′ < t, and let ω ∈ SS . The restriction of ω to E≥t,ωt, is generally a mixed state, even if ω is pure as a state on AS or E≥t′ , with t′ < t. Thistransformation of a pure state into a mixture is necessary for potential properties to becomeempirical. If a(t) =

∑ki=1 αiΠi(t) (we choose a with finite spectrum to simplify matters)

becomes empirical at time t, conventional wisdom tells us that

ω(b) ≈k∑i=1

ω(Πi(t)bΠi(t)), (1.4.23)

for all b ∈ E≥t. In particular, ωt is generally mixed. The process that leads to (1.4.23) is calleddephasing/decoherence; see e.g. [114]. In Section 5.1, we give an algebraic characterization ofempirical properties based on (1.4.23) and the process of information loss. To state a correctdefinition of empirical properties of S, we introduce the centralizer Cω≥t of ω after time t. It isthe subalgebra of E≥t defined by

Cω≥t := a ∈ E≥t | ω(ab) = ω(ba) ∀b ∈ E≥t. (1.4.24)

Our investigations lead to the following definition of empirical properties of a physical system.

Definition 1.4.2.1. Empirical properties of SAn observable a = a∗ ∈ OS is empirical at time t if ω((a(t) − a)2) ≈ 0 for some elementa = a∗ ∈ Z(Cω≥t), where Z(Cω≥t) is the center of the centralizer Cω≥t.If E≥t is a von Neumann algebra and if ω is a normal state on E≥t, then the restriction ωt of ωto the algebra E≥t is represented by a density "matrix", ρ(t). If E≥t is a type I von Neumannalgebra, a(t) is empirical iff ω((a(t)− f(ρ(t)))2) ≈ 0 for some real valued measurable functionf .

The notation ω((a(t)− a)2) ≈ 0 means that a(t) cannot be distinguished from a becauseω((a(t)− a)2) is below the resolution threshold of any reasonable experiment.

If the outcome, αi, of the empirical property a at time t is measured, then the state of thesystem ω can be modified to make further predictions relative to S, and one should replaceωt by

ω(αi)t :=

1

Tr(ρΠi(t))Tr(ρΠi(t) ·Πi(t)) (1.4.25)


on E≥t to improve our predictions concerning future events. One then says that a projective(or von Neumann) measurement of a ∈ OS has been made at time t, with outcome αi. Thiscan be represented by the map ω 7→ ω

(αi)t (this is reminiscent of the projection of the wave

packet postulate).If αi is not registered, (1.4.23) is the only equation that can be used to make predictions

concerning future events pertaining to S. Formula (1.4.25) allows one to describe experimentalprocesses where repeated projective measurements are carried out on S, see Paragraph 1.4.3and Section 5.3. We emphasize that a second potential property b ∈ OS may be empiricalafter time t′ > t only for a subset of states ω(α)

t with α ∈ K ( 1, ..., k. In that case, onemust use POVM’s (Projection operator valued measures) to describe the observation of a andb as a single measurement; see Remark 5.3.0.1 in Section 5.1.

1.4.3 Quantum trajectories and non-demolition measurements

The last topic addressed in this thesis, in Section 5.4, concerns the emergence of quantumtrajectories, and ultimately, the emergence of particle tracks (produced, for instance, by anα-particle in a cloud chamber).

Motivation

The problem of the appearance of tracks produced by alpha particles in a cloud/bubble cham-ber has been studied for a long time. The question to considering why an α-particle, initiallymodeled by spherical wave packet, produces straight line tracks in a cloud chamber, wasaddressed by Einstein to Born in the 1920’s; see e.g. [59] for an historical review.

This problem has been partly solved by Mott in 1929 in [105], but within a very restrictivesetting. He considered an α-particle interacting with two atoms, S1 and S2. Under thecondition that the wave function of α is spherically symmetric and centered at the origin attime t = 0, Mott shows in [105] that the probability that S1 and S2 get both ionized by α atlatter times is almost zero, unless S1 and S2 are both located on a straight line pointing towardsthe origin. His result is a first step towards understanding the formation of particle tracks,but it is too limited. In particular, we would like to replace S1 and S2 by an environment, E,with infinitely many atoms, or by a quantized field. Another related problem is the formationof tracks when a charged particle move faster than light in a dielectric medium, also calledCerenkov effect. The study of the Cerenkov effect is "in principle" possible by looking at thetranslation invariant Nelson model (see e.g. [42]) at large momentum (|~p| > 1).

The trajectory of a particle in a cloud chamber is not revealed by a direct projectivemeasurement on the particle itself, but rather by a direct detection (and absorption by adetector) of the photons produced by the atoms that become ionized when the particle passesnearby. The particle is therefore only weakly disturbed by the measurement process ( the samehappens with the Cerenkov effect, even if the particle experiences friction and slows down; see[60] for a semi-classical analysis of this phenomenon). The process of measuring indirectly apotential property a = a∗ ∈ AP of a subsystem of a quantum system S = P ∨ E by carryingout a series of projective measurements on a probe E is called an indirect measurement.

Indirect measurements and quantum non-demolition measurement

Let us first clearly distinguish indirect measurements from projective measurements. LetS = P ∨ E be a quantum system, and let AP be the dynamical algebra of the subsystem P .


We assume that OS = OE . Let a = a∗ ∈ AP . The property a cannot be directly measured bya projective measurement as we described in the previous section because it does not belong tothe set OS of observable properties of S. However, if the probe E and P interact and becomeentangled, a sequence of projective measurements of potential properties x1, ..., xn ∈ OE atvarious times t1,...,tn, may provide information on the property a of P , such that a is indirectlymeasured in the limit where n tends to +∞.

To give a more precise definition, let us work in the realm of Hilbert spaces. We usehere capital letters for the observables. We assume that AS = B(HP ⊗ HE), where HPis finite dimensional, and that OS is only made of elements of the form 1HP ⊗ X, whereX = X∗ ∈ B(HE). We would like to measure

A = A∗ =n∑i=1

αiΠi ∈ B(HP ), (1.4.26)

where A⊗1HE /∈ OS . We assume that the state of S at the fiducial time t = 0 is the productstate ρ := ρ ⊗ ρE . Von Neumann measurements of a property X = X∗ ∈ B(HE) are carriedout repeatedly on E at every time t multiple of ∆, for some ∆ > 0. Let πξ, ξ ∈ σ(X), be thefamily of orthogonal projections on HE associated to X, with∑

ξ∈σ(X)

πξ = 1HE . (1.4.27)

We assume that S is closed between two probes measurements, and we denote by HS theHamiltonian on HP ⊗ HE generating time evolution between two projective measurementscarried out on E.

A projective measurement of X carried out at time t = ∆ with outcome ξ may be repre-sented by the map ρ 7→ ρ(1)(ξ), where

ρ(1)(ξ) =TrHE ((1HP ⊗ πξ)e−i∆HS ρei∆HS (1HP ⊗ πξ))Tr((1HP ⊗ πξ)e−i∆HS ρei∆HS (1HP ⊗ πξ))

(1.4.28)

is the reduced state of the system P directly after the measurement of X. The repetition ofprojective measurements of X with outcomes ξ1, ..., ξk at times t1 = ∆,..., tk = k∆, leads toa sequence ρ(k) of reduced states of P , where

ρ(k)(ξ1, ..., ξk) =TrHE ((1HP ⊗ πξk)e−i∆HS (1HP ⊗ πξk−1

)...ρ...(1HP ⊗ πξk−1)ei∆HS (1HP ⊗ πξk))

Tr((1HP ⊗ πξk)e−i∆HS (1HP ⊗ πξk−1)...ρ...(1HP ⊗ πξk−1

)ei∆HS (1HP ⊗ πξk)).

The probability of observing a sequence ξkof k-measurements ξ

k= (ξ1, ..., ξk) is given by

µ(k)(ξk) = Tr((1HP ⊗πξk)e−i∆HS (1HP ⊗πξk−1

)...ρ...(1HP ⊗πξk−1)ei∆HS (1HP ⊗πξk)). (1.4.29)

We introduce the measurable space (Ω,Σ), where Ω is made of all possible infinite sequencesξ = (ξ1, ξ2, ..., ), and where the σ-algebra Σ is generated by the cylinder sets

Λx1,...,xk := ξ | ξ1 = x1, ..., ξk = xk,

for all x1, ..., xk ∈ σ(X) and for all k ∈ N. Using Kolmogorov extension theorem, and the factthat

∑ξ∈σ(X) πξ = 1HE , it is not difficult to realize that µ(k) induces a probability measure,

µ, on the measurable space (Ω,Σ); see e.g. [101].We denote by L1

+(HP ) the set of density matrices on HP .


Definition 1.4.3.1. Indirect measurement of AWe say that the sequence of probe measurements detailed above leads to an indirect measure-ment of the property A on P if the sequence of reduced states ρ(k) converges almost surelyto a random variable Γ : Ω→ B(HP ) that takes values in the set

Πiρ′Πi

Tr(Πiρ′Πi)| i = 1, ..., n, ρ′ ∈ L1

+(HP ).

If the Hamiltonian HS commutes with A, we say that the indirect measurement of A isa quantum non-demolition measurement of A. Quantum non-demolition measurements havebeen used quite successfully in recent years. For instance, the group of S. Haroche has usedQND measurements to measure the number of photons in a pure optical cavity (using Rydbergatoms as probes); see e.g. [82].

Finally, we say that P "purifies" on the spectrum of A if

µ(ξ∣∣∣∃αk(ξ) ∈ 1, ..., n, ∥∥∥ρ(k)(ξ)−

Παk(ξ)ρ(k)(ξ)Παk(ξ)

Tr(Παk(ξ)ρ(k)(ξ)Παk(ξ))

∥∥∥ 1)≈ 1 (1.4.30)

for all sufficiently large values of k ∈ N. The notations "" and "≈" are informal andare made precise in concrete examples. The word "purifies" might be misleading, since theprojections Πi are not necessarily one-dimensional. However, in concrete applications, weare often interested in observables with simple eigenvalues, and there the word "purifies" iswell-appropriated.

A first step towards understanding the formation of particle tracks is to consider a particlehopping on the lattice Z3. We should then show that the reduced state of the particle Ppurifies on the spectrum of the position operator of the particle. This problem is difficult,because the Hilbert space of the particle is an infinite dimensional Hilbert space, and becausethe position operator is unbounded. In this thesis, we are only able to prove purification for asystem P with dim(HP ) <∞. Moreover, we choose an Hamiltonian HS such that the reduceddynamics of P is a Markov chain, with

ρ(k+1)(ξ1, ..., ξk, ξ) =(C

(k)ξ )∗ρ(k)(ξ1, ..., ξk)C

(k)ξ

Tr((C(k)ξ )∗ρ(k)(ξ1, ..., ξk)C

(k)ξ )

. (1.4.31)

The operators C(k)ξ ∈ B(HP ) are independent of the previous outcomes ξ1, ..., ξk. They are

assumed to satisfy the sum rule∑ξ∈σ(X)

C(k)ξ (C

(k)ξ )∗ =

∑ξ∈σ(X)

(C(k)ξ )∗C

(k)ξ = 1HP , (1.4.32)

for all k ∈ N. As we already discussed, (1.4.32) is important, because it implies that thefunctional

µ(ξ1, ..., ξk) := Tr((C

(k)ξk

)∗...(C(1)ξ1

)∗ ρ C(1)ξ1...C

(k)ξk

)generates a probability measure (denoted by the same symbol) on the measurable space (Ω,Σ).We remark that (1.4.31) preserves pure states. Actually, under some non-degeneracy condi-tions, one can even show that pure states are the attractors of the reduced dynamics of P ;see [101]: the reduced state of P almost surely purifies as k tends to infinity. If the operators


C(k)ξ commute with A for all k ∈ N (i.e. in a non-demolition measurement of A), then the

eigenstates of A are invariant under (1.4.31). In that case, under (again) some non-degeneracyconditions, one can show that the eigenstates of A are actually the attractors of the reduceddynamics of P ; see [22] and Section 5.4.1.

If [A,C(k)ξ ] 6= 0 (which is probably the most relevant problem), little seem to be known

concerning the phenomenon of purification on the spectrum of A. We expect this phenomenonto happen if the commutator [A,C

(k)ξ ] is small in norm for all k ∈ N and for all ξ ∈ σ(X), or

if this commutator is not "too often" non-zero. In Section 5.4, we give first results towardsthe understanding of (1.4.30) in these cases. We describe a model where (1.4.30) can beestablished. In particular, we are able to calculate the transition probability between twoeigenprojections Πi and Πj of A. A first (and naive) result in the continuous time limit isfinally given in Section 5.4.3. As for the purification result presented in [22] and [101], weneed to impose some non-degeneracy conditions to prove purification on the spectrum of A.Details are given in Section 5.

Chapter 2

Study of a quantum system: an atomcoupled to the quantized e.m. field.Analyticity of the resonances

2.1 Introduction

Our aim in this chapter is to determine the radiative corrections to the ground-state dispersionlaw of the model of an atom presented in Section 1.1.3, and to calculate atomic resonanceenergies, decay channels and life times of excited states. Among our new results are proofsof real analyticity of these quantities as functions of the total momentum, ~p, of the dressedatom, for |~p| < 1 and of analyticity in the elementary electric charge near the origin.

The main mathematical tools we will employ to prove our main results are based on acombination of dilatation analyticity with a novel method of “spectral renormalization” (inthe guise of an inductive construction based on a sequence of smooth iso-spectral Feshbach-Schur maps). In previous work (see [11], [14], [9], [77], [78], [85], [118], [3], [54]), spectralrenormalization is cast in the form of a renormalization group construction involving iterationof a renormalization map (constructed from a Feshbach-Schur map that lowers an energy scaleby a fixed factor ρ < 1), which maps a suitably chosen Banach space of effective Hamiltonianson Fock space into itself. One then attempts to determine the fixed points and the stable andunstable manifolds of the renormalization map – in accordance with the general philosophyof the renormalization group. While this approach is conceptually transparent and yieldsvery detailed information on the spectral problem under consideration, it leads to certainsomewhat artificial technical complications, and it is numerically quite inefficient. In this work,we present an inductive construction involving a sequence of smooth iso-spectral Feshbach-Schur maps indexed by a sequence of energy scales that converges to 0 in a “(super-)super-exponential” fashion and hence is very efficient numerically. One of our main aims in thiswork is to describe this method and to demonstrate its basic simplicity and efficiency on anexample that is of interest to physicists.

2.1.1 The Model

We already introduced the model considered here in Section 1.1.3. We remind the reader thatthe atom P is treated as a moving n0 level system. The Hilbert space of pure states of the

28

CHAPTER 2. ANALYTICITY OF THE RESONANCES 29

atom is L2(R3)⊗ Cn0 and its Hamiltonian

HP = −∆

2+

n0∑i=1

eiΠi,

with en0 > · · · > e1. The orthogonal projections Πi are one-dimensional and satisfy ΠiΠj =δijΠi for all i, j = 1, ..., n0. The energy scale of transitions between internal states of the atomis measured by the quantity

δ0 := mini 6=j|ei − ej |. (2.1.1)

The domain ofHP isH2(R3)⊗Cn0 , whereH2(R3) denotes the Sobolev space of wave functionswith square-integrable derivatives up to order 2. The Hilbert space of states of photons is givenby

HE := F+(L2(R3)), (2.1.2)and the Hamiltonian of the free electromagnetic field is given by

HE =

∫R3|~k| a∗(k)a(k)dk, (2.1.3)

which is a densely defined, positive operator on HE . Our goal is to study an atom inter-acting with the quantized electromagnetic field. The Hilbert space of states of this system(atom∨photons) is the tensor product space

HS = HP ⊗HE .

We choose the interaction of the atom with the quantized electromagnetic field to be given bythe Ritz Hamiltonian

λ0HP,E := −~d0 · ~E(~x), (2.1.4)

where ~d0 = −λ0~d is the atomic dipole moment, λ0 > 0 is a coupling constant proportional to

the elementary electric charge, and ‖di‖ = 1, i = 1, ..., 3. Furthermore, ~x is the position ofthe (center of mass of the) atom and ~E denotes the quantized electric field, cut off at largephoton frequencies. It is given by the operator

~E(~x) := i

∫R3

Λ(~k)|~k|12~ε(k)

(ei~k·~x ⊗ 1Cn0 ⊗ a(k)− e−i~k·~x ⊗ 1Cn0 ⊗ a∗(k)

)dk, (2.1.5)

acting on HS . In (2.1.5), k 7→ ~ε(k) ∈ R3 represents the polarization vector. It is a measurablefunction with the properties

|~ε(k)| = 1, ~ε(k) · ~k = 0, ~ε(r~k, λ) = ~ε(~k, λ), ∀r > 0, ∀k ∈ R3. (2.1.6)

The function Λ : R3 7→ R is an ultraviolet cut-off. To be concrete, we take it to be theGaussian

Λ(~k) = e−|~k|2/(2σ2

Λ) (2.1.7)for some cut-off constant σΛ ≥ 1. (Obviously, one may consider a more general class of cut-offfunctions.)The total Hamiltonian of the system is the sum of the Hamiltonians of the atom and theelectromagnetic field, plus an interaction term. It is given by

HS := HP +HE + λ0HP,E . (2.1.8)

Using the Kato-Rellich theorem, one shows that the HamiltonianHS is defined and self-adjointon the dense domain D(HP ⊗ 1HE + 1HP ⊗HE), where D(A) represents the domain of thelinear operator A.


The Fibre Hamiltonian

The photon momentum operator is the vector operator defined by

~PE :=

∫R3

~k a∗(k)a(k)dk. (2.1.9)

Let F denote Fourier transformation in the electron position variable ~x ∈ R3. We define theunitary operator

U := Fei~x·~PE (2.1.10)

on HS . We conjugate the Hamiltonian HS in (2.1.8) by the unitary operator U introduced in(2.1.10) and subtract the trivial term ~p2

2 , to obtain the operator

H := UHSU∗ − ~p2

2=

1

2(~p− ~PE)2 − ~p2

2+

n0∑i=1

eiΠi +HE + λ0HI , (2.1.11)

whereHI := i

∫R3

Λ(~k)|~k|12

Ä~ε(k) · ~d⊗ a(k)− ~ε(k) · ~d⊗ a∗(k)

ädk (2.1.12)

and ~p = −i~∇+ ~PE denotes the total momentum operator. The operator H introduced in Eq.(2.1.11) is the main object of study of this work. We remark that

L2(R3)⊗ Cn0 ⊗HE ∼= L2(R3;Cn0 ⊗HE). (2.1.13)

Using (2.1.13) we see that, for an arbitrary φ ∈ L2(R3~p;Cn0 ⊗HE),

(Hφ)(~p) = H(~p)φ(~p), (2.1.14)

where the fibre Hamiltonian, H(~p), is the operator acting on the fibre space

H~p := Cn0 ⊗HE

given by

H(~p) :=1

2~P 2E − ~p · ~PE +HE +

n0∑i=1

eiΠi + λ0HI . (2.1.15)

Using the fact that HI is relatively bounded with respect to H1/2E and applying the Kato-

Rellich theorem, one sees that, for all ~p ∈ R3, H(~p) is a self-adjoint operator on its domain

D(H(~p)) = D(HE) ∩ D(~P 2E). (2.1.16)

Eqs (2.1.14)-(2.1.15) can be reformulated in the formalism of direct integrals:

HS =

∫ ⊕R3H~pd~p, H =

∫ ⊕R3H(~p)d~p, (2.1.17)

This work is devoted to studying properties of the fibre Hamiltonians H(~p), ~p ∈ R3.


Complex Dilatations

For θ ∈ R, we define the (unitary) dilatation operator γ(θ) by setting

γ(θ)(φ)(~k, λ) := e−3θ/2φ(e−θ~k, λ), for φ ∈ L2(R3). (2.1.18)

By Γ(θ) := Γ(γ(θ)) we denote the operator on Fock space HE obtained by “second quantiza-tion” of γ(θ): For an operator ω acting on the one-photon Hilbert space L2(R3), Γ(ω) denotesthe operator defined on HE whose restriction to the n-photon subspace is given by

Γ(ω)|L2(R3)⊗ns := ⊗nω. (2.1.19)

A straightforward computation shows that

Hθ(~p) := Γ(θ)H(~p)Γ(θ)∗ =1

2e−2θ ~P 2

E − e−θ~p · ~PE +n0∑i=1

eiΠi + e−θHE + λ0HI,θ, (2.1.20)

where

HI,θ := ie−2θ∫R3

Λ(e−θ~k)|~k|12

Ä~ε(k) · ~d⊗ a(k)− ~ε(k) · ~d⊗ a∗(k)

ädk. (2.1.21)

The operator Hθ(~p) can be analytically extended to the complex domain

D(0, π/4) := θ ∈ C : |θ| < π/4. (2.1.22)

We will verify in Appendix 2.7.1, below, that, for all ~p ∈ R3, the map θ 7→ Hθ(~p) is an analyticfamily of type (A) on D(0, π/4), in the sense that Hθ(~p) is closed on D(HE) ∩ D(~P 2

E), forall θ ∈ D(0, π/4), and the vector function θ 7→ Hθ(~p)u is analytic in θ on D(0, π/4), for allu ∈ D(HE) ∩ D(~P 2

E). The study of resonances of the operator H(~p) amounts to studyingnon-real eigenvalues of Hθ(~p), for θ belonging to a suitable open subset of D(0, π/4) \ R.

Analyticity in the Total Momentum

We pick a vector ~p∗ in R3 of length smaller than 1 and a complex number θ = iϑ with0 < ϑ < π/4. We set

µ =1− |~p∗|

2(2.1.23)

and define an open set Uθ[~p∗] in complexified momentum space C3 by

Uθ[~p∗] := ~p ∈ C3 | |~p− ~p∗| < µ ∩ ~p ∈ C3 | |=~p| < µ

2tan(ϑ). (2.1.24)

For ~p ∈ Uθ[~p∗], we consider the operator

Hθ(~p) := e−2θ~P 2E

2− e−θ~p · ~PE +

n0∑i=1

eiΠi + λ0HI,θ + e−θHE . (2.1.25)

Our main interest, in this work, will be to analyze the ~p-dependence of the ground-state, theground-state energy and the resonance energies of the Hamiltonian Hθ(~p), for ~p ∈ Uθ[~p∗]. ByHθ,0(~p) we denote the operator given by

Hθ,0(~p) := e−2θ~P 2E

2− e−θ~p · ~PE +

n0∑i=1

eiΠi + e−θHE (2.1.26)


corresponding to a vanishing coupling constant, λ0 = 0. It is easy to verify that, for δ0 > 0,e1, . . . , eN are simple eigenvalues of Hθ,0(~p). Moreover, it is easy to see that, for |~p| < 1 and~p ∈ R3, the spectrum of Hθ,0(~p) is included in a region of the form depicted in Figure 2.1.

e1 e2 e3 ... eNϑ

2ϑ

Figure 2.1: Shape of the spectrum of the unperturbed operator Hθ,0(~p) for ~p ∈ R3, |~p| < 1. e1,...,eNare eigenvalues of Hθ,0(~p), the essential spectrum is located inside the cuspidal grey regions.

2.1.2 Main Results

Theorem 2.1.2.1, below, claims that, for |~p| < 1, a ground-state and resonances exist and thatthe ground-state, the ground-state energy and the resonance energies are analytic in ~p ∈ Uθ[~p∗](and in λ0, for |λ0| small enough). If |~p| > 1 one expects that the operator H(~p) does not havea stable ground-state, due to emission of Cherenkov radiation; see [42]. Assuming that theso-called Fermi-Golden-Rule condition holds, the imaginary parts of the resonance energiesare strictly negative, i.e., the life times of the excited states of an atom are strictly finite dueto radiative decay; see Proposition 2.1.2.1.

Theorem 2.1.2.1. Let 0 < ν < 1. There exists λc(ν) > 0 such that, for all 0 ≤ λ0 < λc(ν)and ~p ∈ R3, |~p| < ν, the following properties are satisfied:

a) E(~p) := inf σ(H(~p)) is a non-degenerate eigenvalue of H(~p).

b) For every i0 ∈ 1, · · · , n0 and θ ∈ C with 0 < =θ < π/4 large enough, Hθ(~p) has aneigenvalue, z(∞)

i0(~p), such that z(∞)

i0(~p)→ ei0, as λ0 → 0. For i0 = 1, z(∞)

1 (~p) = E(~p).

Moreover, for |~p| < ν, |λ0| small enough and 0 < =θ < π/4 large enough, the ground stateenergy, E(~p), and the resonance energies, z(∞)

i0(~p), i0 ≥ 2, are analytic in ~p, λ0 and θ. In

particular, they are independent of θ.

Remark 2.1.2.1.

• Existence of a ground state and analyticity of the map ~p 7→ E(~p) are proven in [54].

• For simplicity of exposition, we only prove, in the present work, the (existence and)analyticity of the resonance energies z(∞)

i0(~p) in ~p, for i0 ≥ 2. In the following, we fix i0

and write z(∞)i0

(~p) =: z(∞)(~p); (dependence on i0 suppressed). Our proof can be adaptedin a straightforward way to establish the statements concerning analyticity in λ0 and θ.For different models similar to the model of non-relativistic QED studied in this work,analyticity in the coupling constant has been proven previously in [78, 86, 85].

• The fact that z(∞)(~p) is independent of θ is a direct consequence of the analyticity ofz(∞)(~p) in θ, together with unitarity of the dilatation operator Γ(θ) for real θ’s and withthe existence of a normalizable and analytic eigenstate of Hθ(~p) associated to z(∞)(~p).


• Theorem 1.1 extends to more realistic models of atoms (with dynamical electrons) asconsidered for instance in [3, 66, 100]. Such models are not treated in our work in ordernot to hide the basic simplicity of our methods.

Proposition 2.1.2.1. Let i0 > 1 and ~p ∈ R3, |~p| < 1. Suppose that

∑j<i0

∫R3dk∣∣∣ 3∑s=1

(ds)n0−j+1,n0−i0+1εs(k)∣∣∣2|~k||Λ(~k)|2δ

Äej − ei0 + |~k| − ~p · ~k +

~k2

2

ä> 0.

(Fermi-Golden-Rule condition) (2.1.27)

Then, under the conditions of Theorem 2.1.2.1 and for λ0 small enough, the imaginary partof z(∞)(~p) is strictly negative.

The proof of Proposition 2.1.2.1 does not rely on the inductive construction used to es-tablish Theorem 2.1.2.1. A single application of a suitably chosen Feshbach-Schur map, i.e., asingle decimation step, is sufficient to prove this proposition, and our argumentation followsclosely the one presented in [14, 12]. To render this work reasonably self-contained, the proofis given in Section 2.6.2.

2.1.3 Strategy of Proof and Sketch of Methods

Ultimately, our aim is to study spectral properties of the operators Hθ(~p) introduced in(2.1.25). This spectral problem is difficult, because, among other things, it involves the studyof eigenvalues imbedded in continuous spectrum and located at thresholds of the continuousspectrum of Hθ(~p). Standard analytic perturbation theory is therefore not applicable. Thekey tool we will use to prove our results is the isospectral Feshbach-Schur map, which wasoriginally developed to cope with problems of this kind in [13]. In this work we will use thesmooth Feshbach-Schur map introduced in [9] and further studied in [77] and [78], which hasmajor technical advantages (and, alas, some conceptual disadvantages), as compared to theoriginal Feshbach-Schur map.

The Feshbach-Schur map is tailor-made for the analysis of small regions in the spectra ofclosed operators on Hilbert space, in particular regions of their spectra near thresholds. Itenables one to construct “effective operators” that, on the part of the spectrum of interest,have the same spectrum (with the same multiplicity) as the original operator, i.e., are iso-spectral to the original operator. By iterating the Feshbach-Schur map one is able to zoominto tiny regions in the spectrum of an operator of interest and extract ever more accurateinformation on such parts of the spectrum. In particular, by constructing an infinite sequenceof Feshbach-Schur maps, we will be able to determine the exact location of the ground-state-and the resonance energies and the corresponding eigenstates of the deformed Hamiltonians,Hθ(~p), =θ > 0, of atoms coupled to the radiation field. The Feshbach-Schur maps will beadapted to the particular resonance that one wishes to analyze. It is a novel aspect of ourconstruction that it yields an algorithm that converges super-exponentially fast.

Mathematical Tools

The Feshbach-Schur Map The fundamental tool used to prove our main results is thesmooth Feshbach-Schur map; see [9, 77]. A key property of this map is its iso-spectrality,which we now describe in more precise terms.


Definition 2.1.3.1 (Feshbach-Schur Pairs). Let P be a positive operator on a separable Hilbertspace H whose norm is bounded by 1, 0 ≤ P ≤ 1. Assume that P and P :=

√1− P 2 are both

non-zero. Let H and T be two closed operators on H with identical domains D(H) and D(T ).Assume that P and P commute with T . We set W := H − T and we define

HP :=T + PWP HP := T + PWP.

The pair (H,T ) is called a Feshbach-Schur pair associated with P iff

(i) HP and T are bounded invertible on P [H]

(ii) H−1PPWP can be extended to a bounded operator on H

For an arbitrary Feshbach-Schur pair (H,T ) associated with P , we define the smooth Feshbach-Schur map by

FP (·, T ) : H 7→ FP (H,T ) := T + PWP − PWPH−1PPWP. (2.1.28)

Theorem 2.1.3.1. Let 0 ≤ P ≤ 1, and let (H,T ) be a Feshbach-Schur pair associated with P(i.e., satisfying properties (i) and (ii) in Definition 2.1.3.1). Let V be a closed subspace withP [H] ⊂ V ⊂ H, and such that

T : D(T ) ∩ V → V, PT−1PV ⊂ V.

Define

QP (H,T ) := P − PH−1PPWP.

Then the following hold true:

(i) H is bounded invertible on H if and only if FP (H,T ) is bounded invertible on V .

(ii) H is not injective if and only if FP (H,T ) is not injective as an operator on V :

Hψ = 0, ψ 6= 0 =⇒ FP (H,T )Pψ = 0, Pψ 6= 0,

FP (H,T )φ = 0, φ 6= 0 =⇒ HQP (H,T )φ = 0, QP (H,T )φ 6= 0.

Remark 2.1.3.1.

• Items (i) and (ii) of Theorem 2.1.3.1 describe what we call iso-spectrality. This notiondoes not mean that the spectra of H and of FP (H,T ) are identical. Rather, iso-spectralityis a local property: One uses the Feshbach-Schur map to explore spectral properties of anoperator within specific, small regions in the complex plane.

• As emphasized in [77], if T is bounded invertible in P [H], if T−1PWP and PWT−1Pare bounded operators with norm strictly less than one, and if T−1PWP is bounded,then items (i) and (ii) of Definition 2.1.3.1 are satisfied. We will often use these criteriato show that a pair (H,T ) is a Feshbach-Schur pair associated with P .


Wick Monomials We now describe the general class of operators to which the methodsdeveloped in this work, based on the smooth Feshbach-Schur map, can be applied.

Setting N0 := N ∪ 0, we denote by

w := wm,nm,n∈N0 (2.1.29)

a sequence of bounded measurable functions,

∀m,n : wm,n : R× R3 × R3m × R3n → C, (2.1.30)

that are continuously differentiable in the variables, r ∈ σ(HE) ⊂ R, ~l ∈ σ(~PE) = R3,respectively, appearing in the first and the second argument, and symmetric in them variablesin R3m and the n variables in R3n. We suppose furthermore that

w0,0(0,~0) = 0. (2.1.31)

With a sequence, w, of functions, as specified above, and a positive number 1 ≥ ρ > 0, weassociate an operator

Wm,n(w) := 1HE≤ρ

∫R3m×R3n

a∗(k1) · · · a∗(km)wm,n(HE , ~PE , k1, · · · , km, k1, · · · , kn) (2.1.32)

a(k1) · · · a(kn)m∏i=1

dki

n∏j=1

dkj1HE≤ρ. (2.1.33)

It is easy to show thatWm,n(w) is actually a bounded operator onHE . The operatorsWm,n(w)defined in (2.1.32) are called (generalized) Wick-monomials (at the energy scale ρ). For everysequence of functions w and every E ∈ C we define

H[w, E ] =∑

m+n≥0

Wm,n(w) + E , W≥1(w) :=∑

m+n≥1

Wm,n(w). (2.1.34)

The complex number E is the vacuum expectation value of H[w, E ]:

〈Ω|H[w, E ]Ω〉 = E . (2.1.35)

The First Decimation Step of Spectral Renormalization

Recall that we wish to analyze the fate of an excited state of an atom after it is coupledto the radiation field. Let us consider the excited state indexed by i0 ∈ 1, · · · , N, withunperturbed internal energy ei0 . We expect that, after coupling the atom to the quantizedradiation field, an excited state (corresponding to an index i0 > 1) is unstable, i.e., is turnedinto a resonance. Our goal is to determine its life time and the real part of the resonanceenergy (Lamb shift). For this purpose, we introduce a sequence of smooth Feshbach-Schur“decimation” maps that will be successively applied to the deformed Hamiltonians Hθ(~p), withthe goal of constructing a sequence of operators, which – when applied to the vacuum Ω –will converge to z(∞)(~p)Ω, where z(∞)(~p) is the ith0 resonance energy; (as announced, we willomit reference to i0 in our notation, since i0 will be fixed). In this subsection, we sketch theconstruction of the first decimation map.


We define a decreasing function χ ∈ C∞(R) satisfying

χ(r) :=

1, if r ≤ 3/4,0 if r > 1,

(2.1.36)

and strictly decreasing in (3/4, 1). Furthermore, we choose a constant ρ0 ∈ (0, 1) and define

χρ0(r) := χ(r/ρ0), χρ0(r) :=

»1− χ2

ρ0(r). (2.1.37)

Let ψi0 denote the normalized eigenvector (unique up to a phase) of the operator∑n0i=1 eiΠi

corresponding to the eigenvalue ei0 . The orthogonal projection onto ψi0 is denoted by

Πi0 := |ψi0〉〈ψi0 |. (2.1.38)

Next, we define an operator χi0 by

χi0 := Πi0 ⊗ χρ0(HE). (2.1.39)

In Section 2.3 we will prove that, for |z − ei0 | ρ0µ sin(ϑ), (Hθ(~p) − z,Hθ,0(~p) − z) is aFeshbach-Schur pair associated to χi0 and that, as a consequence, there is a sequence offunctions w(0)(~p, z) [see (2.1.29)] and a complex number E(0)(~p, z) such that an application ofthe Feshbach-Schur map, Fχi0

(·, Hθ,0(~p)− z), to the operator Hθ(~p)− z yields an operator ofthe form specified in Eq. (2.1.34). More precisely,

Fχi0(Hθ(~p)− z,Hθ,0(~p)− z)|Πi0⊗1HE≤ρ0 [H~p] = Πi0 ⊗H[w(0)(~p, z), E(0)(~p, z)]. (2.1.40)

We simplify our notation by writing [see (2.1.34)]

H(0)(~p, z) := H[w(0)(~p, z), E(0)(~p, z)] = W(0)≥1 (~p, z) + w

(0)0,0(~p, z,HE , ~PE) + E0(~p, z), (2.1.41)

whereW

(0)≥1 (~p, z) :=

∑m+n≥1

Wm,n(w(0)(~p, z)).

One expects that it is easier to analyze the operator H(0)(~p, z), rather than the originaloperator Hθ(~p) − z, because the former acts on a subspace, Πi0 ⊗ 1HE≤ρ0 [H~p] ⊂ H~p (withall internal states corresponding to indices i 6= i0 eliminated), and the operator W (0)

≥1 (~p, z) isbounded in norm by some power of ρ0. If ρ0 could be chosen to be very small, the spectrumof H0(~p, z) near 0, and hence of the operator Hθ(~p) near ei0 (by iso-spectrality), could belocated fairly accurately.

Inductive Construction of Effective Hamiltonians

The accuracy of the information on the spectrum of the operator H(0)(~p, z) near 0, and henceon the spectrum of the operator Hθ(~p) near ei0 , that can be achieved (after one application ofthe Feshbach-Schur map) is limited by the circumstance that ρ0 cannot be taken to be verysmall. Luckily, it turns out that this limitation can be removed by successive applicationsof Feshbach-Schur maps that lower the energy range of the states in the subspaces on whichthe Feshbach-Schur operators act further and further towards 0 and, hence, determine the


location of the spectrum of Hθ(~p) near ei0 ever more accurately. Successive applications ofFeshbach-Schur maps yield Hamiltonians

H(j)(~p, z) = H[w(j)(~p, z), E(j)(~p, z)], j ∈ N0, (2.1.42)

as in Eq. (2.1.34), with the following properties:

Hθ(~p)− z is bounded invertible⇐⇒ H(j)(~p, z) is bounded invertible. (2.1.43)

Hθ(~p)− z is not injective⇐⇒ H(j)(~p, z) is not injective. (2.1.44)

These “iso-spectrality properties” permit us to trade the analysis of the spectrum of Hθ(~p)near the energy ei0 of an excited state of the atom for the analysis of the spectrum of theoperators H(j)(~p, z) near the origin. This turns out to simplify matters considerably: Thestudy of the operators H(j)(~p, z) is much easier than the study of the original Hamiltonian,because H(j)(~p, z) is the sum of a diagonal operator, whose spectrum is known explicitly, and aperturbation term whose operator norm will turn out to decrease to zero super-exponentially,as j → ∞. Below, we describe in somewhat more detail how this idea, which was originallydeveloped in [13], [12], can be implemented, technically; (details will be presented in Section2.4).

Let ~p∗ ∈ R3, with |p∗| < 1, and let ~p ∈ Uθ[~p∗]. We define two sequences of numbers

(ρj)j∈N0 , (rj)j∈N0 by

ρj = ρ(2−ε)j0 , with ε ∈ (0, 1), rj :=

µ sin(ϑ)

32ρj , (2.1.45)

where 0 < ρ0 < 1 is a suitably chosen parameter; (see Section 2.4). The rate of convergenceof the sequence ρj depends on the infrared behavior of the interacting Hamiltonian HI . Ingeneral, if HI behaves like |~k|α−1/2 in the infrared and α > 0, the sequence ρj can be chosento be equal to ρ(1+ε)j

0 for any 0 < ε < α. A filtration of Hilbert spaces (H(j))j∈N0 is given bysetting

H(j) = 1HE≤ρj [HE ]. (2.1.46)

We construct inductively a sequence of complex numbers z(j−1)(~p)j∈N0 , z(−1)(~p) := ei0 ,and, for every z ∈ D(z(j−1)(~p), rj), a sequence of functions w(j)(~p, z) and a complex numberE(j)(~p, z) [see (2.1.29)-(2.1.35)] , with the following properties:

(a) Let

W (j)m,n(~p, z) := Wm,n(w(j)(~p, z)), H(j)(~p, z) := H[w(j)(~p, z), E(j)(~p, z)], (2.1.47)

acting on H(j), (with m,n ∈ N0); see (2.1.32) and (2.1.34). Then we have that

‖W (j)0,0 (~p, z)Ψ‖ = ‖w(j)

0,0(~p, z,HE , ~PE)Ψ‖ ≥ ε‖HEΨ‖, ∀Ψ ∈ H(j) (2.1.48)

for some constant ε > 0 depending on ~p, but independent of j. The pair of operatorsÄH(j)(~p, z),W

(j)0,0 (~p, z)+E(j)(~p, z)

äis a Feshbach-Schur pair associated to χρj (HE). Thus

H(j+1)(~p, z) = Fχρj+1 (HE)[H(j)(~p, z),W

(j)0,0 (~p, z) + E(j)(~p, z)]|H(j+1) (2.1.49)

is well defined.


(b) The complex number z(j)(~p) is defined as the only zero of the function

D(z(j−1)(~p),

2

3rj)3 z −→ E(j)(~p, z) = 〈Ω| H(j)(~p, z)Ω〉, (2.1.50)

and the following inequalities hold:

|z(j)(~p)− z(j−1)(~p)| < rj2,

∣∣∣E(j)(~p, z)∣∣∣ ≤ µ

16ρj+1, for z ∈ D

(z(j)(~p),

2

3rj+1

).

(2.1.51)

Ei0z(0)z(1)

D(Ei0 , r0)

D(z(0), r1)

D(z(1), r2)

Figure 2.2: For fixed ~p ∈ Uθ[~p∗], the sets D(z(j)(~p), rj+1) are shrinking super-exponentially fast withj and, for every j ∈ N0, D(z(j)(~p), rj+1) ⊂ D(z(j−1)(~p), rj).

By Eqs. (2.1.34) and (2.1.47),

H(j)(~p, z) = W(j)0,0 (~p, z) + E(j)(~p, z) +W

(j)≥1 (~p, z). (2.1.52)

As a function of HE and ~PE , the operator W (j)0,0 (~p, z) = w

(j)0,0(~p, z,HE , ~PE) is defined by func-

tional calculus and satisfies (2.1.48). Given w(j)0,0, the spectrum of

W(j)0,0 (~p, z) + E(j)(~p, z) (2.1.53)

can be determined explicitly. This operator is therefore considered to be the unperturbedHamiltonian (the operator T in Definition 2.1.3.1) in the next application of the Feshbach-Schur map. Eq. (2.1.52) shows that the operator H(j)(~p, z) is the sum of the unperturbedHamiltonian, T = W

(j)0,0 (~p, z) + E(j)(~p, z), and a perturbation given by W = W

(j)≥1 (~p, z) whose

norm tends to zero, as j tends to∞, super-exponentially rapidly. We will actually prove that,for every j ∈ N0,

‖W (j)≥1 (~p, z)‖ ≤ Cjρ2

j , (2.1.54)


for some constant C > 1. Recalling formula (2.1.28) for the Feshbach-Schur map, just aboveTheorem 2.1.3.1, we find that the bound (2.1.54), the lower bound in (2.1.48) and (2.1.51) en-able us to construct Fχρj+1 (HE)[H

(j)(~p, z),W(j)0,0 (~p, z)+E(j)(~p, z)] with the help of a convergent

Neumann expansion in powers of the perturbationW (j)≥1 (~p, z). Thanks to (2.1.54), (2.1.48) and

(2.1.51) and using “iso-spectrality”, the sequence H(j)(~p, z) of effective Hamiltonians enablesus to locate the spectrum of the deformed Hamiltonian Hθ(~p), =θ > 0, near the energy ei0with ever higher precision as the resonance energy z(∞)(~p) is approached. (We remark thatz(∞)(~p) is an eigenvalue of the operator Hθ(~p), =θ > 0, as proven in the next subsection.)

It is a characteristic feature of multi-scale renormalization, as well as of KAM theory, thata problem of singular perturbation theory involving an infinite range of scales is decomposedinto a sequence of infinitely many regular perturbation problems, one for every finite rangeof scales, solved iteratively, with the splitting of an effective Hamiltonian into an unperturbedpart and a perturbation chosen anew, in every step, j, of the iterative perturbative analysis.These are key features enabling one to successfully cope with problems of singular perturbationtheory. They will become manifest in the analysis presented in this work.

Construction of Eigenvalues and Analyticity in ~p, θ and λ0

In this section we sketch the main ideas of our construction of the ground-state-(i0 = 1) andresonance-(i0 > 1) energy z

(∞)i0

(~p) of Hθ(~p) (for some fixed i0 ≥ 1) and of the proof thatz(∞)(~p) = z

(∞)i0

(~p) is an eigenvalue of Hθ(~p), for ~p ∈ Uθ[~p∗], =θ < π4 large enough, and λ0 ≥ 0

small enough. Note that, for the ground-state, i.e., for i0 = 1, we can choose θ to vanish,and z(∞)(~p) is shown to be a simple eigenvalue of the self-adjoint operator H(~p), for ~p ∈ R3,with |~p| < 1, and λ0 positive and small enough. As a function of ~p this is the renormalizeddispersion law of the atom.

We start our considerations by observing that the sequence of approximate resonanceenergies (z(j)(~p))j∈N0 is Cauchy, as follows from Eq. (2.1.51). It is not difficult to show (seeSection 2.5) that it actually is a Cauchy sequence of analytic functions of the momentum ~p,for ~p ∈ Uθ[~p∗]. Analyticity in θ, for =θ < π

4 large enough, and in λ0, for |λ0| small enough,can be shown by very similar arguments, which we skip here. We then define

z(∞)(~p) := limj→∞

z(j)(~p) =⋂j∈N0

DÄz(j−1)(~p), rj

ä, (2.1.55)

which is analytic in ~p ∈ Uθ[~p∗]. The complex number z(∞)(~p) is an eigenvalue of Hθ(~p);

it is the resonance energy that we are looking for. It is convenient to extend the operatorH(j)(~p, z(∞)(~p)), for j ∈ N0, to an operator defined on the entire Fock space HE by defining itto vanish on the orthogonal complement of the subspace H(j) = Ran(1HE≤ρj ). We continueto use the same symbol, H(j)(~p, z(∞)(~p)), for this extension. Similarly, we extend the otheroperators in (2.1.52) to operators acting on the entire Fock space.

We then show that

limj→∞

H(j)(~p, z(∞)(~p)) = 0 = H(∞)(~p, z(∞)(~p)). (2.1.56)

In the proof of (2.1.56), we use (2.1.31).With some further effort, using iso-spectrality, one then shows that

z(∞)(~p) is an eigenvalue of Hθ(~p), (2.1.57)


for =θ < π4 large enough. Analyticity of z(∞)(~p) in θ then implies that this quantity is actually

independent of θ (and this is the reason why the index θ for z(∞)(~p) is omitted).Next, we sketch the proofs of (2.1.56) and of (2.1.57). Using (2.1.51), (2.1.52) and (2.1.54),

we see thatlimj→∞

‖H(j)(~p, z(∞)(~p))−W (j)0,0 (~p, z(∞)(~p))‖ = 0. (2.1.58)

As explained in Section 2.1.3, see (2.1.41) and (2.1.46),

W(j)0,0 (~p, z(∞)(~p)) = w

(j)0,0(~p, z(∞)(~p), HE , ~PE)1HE≤ρj . (2.1.59)

The derivatives of w(j)0,0(~p, z(∞)(~p), r,~l) in the variables r and ~l are uniformly bounded, for

r ∈ [0, ρj ], |~l| ≤ r (and |w(j)0,0(~p, z(∞)(~p), r,~l)| ≥ ε · r, for some constant ε > 0 independent of

j), for all j ∈ N0. These properties and the normalization condition (2.1.31) imply that

limj→∞

W(j)0,0 (~p, z(∞)(~p)) = 0, (2.1.60)

which, together with (2.1.58), implies (2.1.56).By (2.1.46), ⋂

j∈N0

H(j) = CΩ.

Eq. (2.1.56) then shows that the vacuum Ω is an eigenvector of H(∞)(~p, z(∞)(~p)) with eigen-value 0. To prove (2.1.57) we apply part (ii) of Theorem 2.1.3.1 iteratively, after each appli-cation of a Feshbach map.We define

Qχρj := Qχρj

(H(j−1)(~p, z(∞)(~p)),W

(j−1)0,0 (~p, z(∞)(~p)) + E(j−1)

0,0 (~p, z(∞)(~p))), (2.1.61)

where the operator Q has been defined in Theorem 2.1.3.1. One can then prove that

ψ := limj→∞

Qχρ1 · · ·Qχρj (Ω) (2.1.62)

exists. Using that H(∞)(~p, z(∞)(~p))Ω = 0, we are able to show that

H(0)(~p, z(∞)(~p))ψ = 0.

Then, using Theorem 2.1.3.1 once more, we conclude that [see (2.1.38)-(2.1.39)]

Qχi0

(Hθ(~p)− z(∞)(~p), Hθ,0(~p)− z(∞)(~p)

)(ψi0 ⊗ ψ) (2.1.63)

is an eigenvector of Hθ(~p) with eigenvalue z(∞)(~p).

2.2 Parameters of the problem and Notations

In this section, we present a list of all the parameters appearing in the analysis of the spec-tral problems solved in this work. In Subsection 2.2.2, we introduce the main symbols andnotations used in subsequent sections.


2.2.1 System- and algorithmic parameters

The quantities λ0 (coupling constant), δ0 (spacing between energies of excited states of theatom), n0 (number of internal energy levels of the atom), σΛ (ultraviolet cu-off imposed onthe quantized electric field), and µ (bound on the momentum of the atom) are parameterscharacteristic of the physical system under investigation. They are henceforth called systemparameters. All our estimates depend on the choice of these parameters, and our main re-sults only hold if suitable restrictions on the values of these parameters are imposed. Otherparameters appearing in our analysis are related to the mathematical methods applied to es-tablish our main results, in particular to the algorithm (inductive construction) used to derivethe main estimates needed in our proofs. We call them algorithmic parameters. Amongthese parameters are the dilatation parameter, ϑ, appearing in the complex deformation ofthe basic Hamiltonian used to locate the resonance energies, and the scale parameter ρ0, aswell as the parameter ε appearing in the definition of the Feshbach maps; see Eq. (2.1.45).These (auxiliary) parameters are chosen so as to ensure (and “optimize”) the convergence ofthe inductive construction outlined above. Constraints on the choice of the parameters ϑ andρ0 are discussed in Sections 2.3 and 2.4, respectively.

2.2.2 Notations relative to creation/annihilation operators and integrals

We introduce the notations

k(m) := (k1, . . . , km) ∈ R3m, k(n) := (k1, . . . , kn) ∈ R3n,

K(m,n) := (k(m), k(n)), dK(m,n) :=m∏i=1

dki

n∏j=1

dkj ,

|K(m,n)| := |k(m)| |k(n)|, |k(m)| :=m∏i=1

|~ki|, |k(n)| :=n∏j=1

|~kj |,

a∗(k(m)) :=m∏i=1

a∗λi(~ki), a(k(n)) :=

n∏j=1

aλj (~kj).

For ρ ∈ C, we set

ρk(m) := (ρ~k1, λ1, . . . , ρ~km, λm), ρK(m,n) := (ρk(m), ρk(n)).

For m := (m1, ...,mL), n := (n1, ..., nL), we set

k(mi)i := (ki1, ..., kimi) ∈ R3mi , k(m) := (k

(m1)1 , ..., k

(mL)L ) ∈ R3[

∑imi],

K(m,n) := (k(m1)1 , ..., k

(mL)L , k

(n1)1 , ..., k

(nL)L ) ∈ R3[

∑i(mi+ni)].


For ρ ∈ R+, and m,n ∈ N, we introduce

Bρ := k ∈ R3 | |~k| ≤ ρ,

B(m)ρ := (k1, ..., km) ∈ R3m |

m∑i=1

|~ki| ≤ ρ,

B(m,n)ρ := B(m)

ρ ×B(n)ρ .

2.2.3 Kernels and their domain of definition

Let ρ > 0. We set

Bρ := (r,~l) ∈ [0, ρ]× R3, |~l| ≤ r. (2.2.1)

Let wm,n be a functionwm,n : Bρ ×B(m,n)

ρ → C.

We introduce

‖wm,n‖ 12

:= sup(r,~l)∈Bρ

ess supK(m,n)∈B(m,n)

ρ

|wm,nÄr,~l,K(m,n)

ä|

|k(m)|1/2|k(n)|1/2. (2.2.2)

The choice of the norm ‖ ·‖ 12is motivated by the infrared behavior of the interaction Hamilto-

nian HI , which behaves like ~k 7→ |~k|1/2 for small values of |~k|. Lemma 2.3.1.1 below establishesthe link between the norm of the operatorWm,n and the norm ‖wm,n‖ 1

2of its associated kernel.

Finally, if w0,0 : Bρ → C is essentially bounded, we set

‖w0,0‖∞ = ess sup(r,~l)∈Bρ

|w0,0

Är,~lä|. (2.2.3)

2.2.4 Notations relative to estimates

Many numerical constants appear in our estimates. Keeping track of all these constants wouldbe very cumbersome and is not necessary for mathematical rigor. Let a, b > 0. We write

a = O(b) (2.2.4)

if there is a numerical constant C > 0 independent of the system and algorithmic parameterssuch that a ≤ Cb.

The shorthand“for all a b, . . . ” (2.2.5)

means that “there exists a (possibly very small, but) positive numerical constant C independentof the system and algorithmic parameters such that, for all a ≤ Cb, . . . ”


2.3 The first decimation step

Recall that

Hθ(~p) = e−2θ~P 2E

2− e−θ~p · ~PE +

n0∑i=1

eiΠi + λ0HI,θ + e−θHE .

In Appendix 2.7.1, we verify that the map (θ, ~p, λ0) 7→ Hθ(~p) defines an analytic familyof type (A) in D(0, π/4) × U × C, where U := ~p ∈ C3 | |<(~p)| + |=(~p)| < 1 ⊃ Uθ[~p

∗]. Inparticular, Hθ(~p) is closed on its domain D(Hθ(~p)) = D(HE) ∩ D(~P 2

E).The purpose of this section is to highlight how to construct an effective Hamiltonian

H(0)(~p, z) on 1HE≤ρ0(HE) from the Hamiltonian Hθ(~p) − z with the help of the Feshbach-Schur map, and to exhibit some remarkable properties of H(0)(~p, z) that are used in ourinductive construction.

In Subsection 2.3.1, we state two standard lemmas that we use repeatedly in our analysis.The proofs are postponed to Appendix 2.7.2 for the reader’s convenience. The first lemmashows that the norm of Wick monomials of the form (2.1.32) are controlled by the norm ofthe integral kernels appearing in their definition.

We prove in Subsection 2.3.1 that the pair (Hθ(~p)−z,Hθ,0(~p)−z) is a Feshbach-Schur pairassociated to the generalized projection χi0 = Πi0 ⊗ χρ0(HE) (defined in (2.1.39)), providedthat the coupling constant λ0 is sufficiently small. This result holds for all (~p, z) in the openset Uρ0 [ei0 ], where

Uρ0 [ei0 ] := Uθ[~p∗]×D(ei0 , r0) and r0 =

ρ0µ sin(ϑ)

32. (2.3.1)

H(0)(~p, z) is the partial trace over the internal degrees of freedom of the restriction of theFeshbach operator Fχi0

(Hθ(~p)− z,Hθ,0(~p)− z) to Ran(χi0).In Subsection 2.3.2, we show that H(0)(~p, z) can be rewritten as a convergent series of

Wick monomials,H(0)(~p, z) =

∑m+n≥0

W (0)m,n(~p, z) + E(0)(~p, z), (2.3.2)

where E(0)(~p, z) = 〈Ω|H(0)(~p, z)Ω〉, and that H(0)(~p, z) is analytic in (~p, z) on the open setUρ0 [ei0 ]. Upper bounds on the norm of the perturbation W

(0)≥1 (~p, z) and on the sequence of

kernels w(0)m,n are stated. Details of the proofs are postponed to Appendix 2.7.3 and 2.7.3. We

finally prove in Lemma 2.3.2.2 that there exists a unique element z(0)(~p) ∈ D(ei0 , r0) for each~p ∈ Uθ[~p], such that E(0)(~p, z(0)(~p)) = 0. The properties of the kernels w(0)

m,n and the functionE(0) established in Lemmas 2.3.2.1 and 2.3.2.2 are the basis of the inductive constructiondescribed in Section 2.4.

2.3.1 Feshbach-Schur Pair

Two Lemmas

We begin with a lemma showing that the norm of the Wick monomials is controlled by thenorm of their associated kernels. The proof is standard and deferred to Appendix 2.7.2.


Lemma 2.3.1.1. Let ρ > 0. Let wm,n be a function wm,n : Bρ×B(m,n)ρ → C with ‖wm,n‖ 1

2<

∞, and let Wm,n be the Wick monomial on 1HE≤ρHE, defined in the sense of quadratic formsby

Wm,n := 1HE≤ρ

Ç∫B

(m,n)ρ

dK(m,n)a∗(k(m))wm,nÄHE , ~PE ,K

(m,n)äa(k

(n))

å1HE≤ρ.

Then‖Wm,n‖ ≤ (8π)

m+n2 ρ2(m+n)‖wm,n‖ 1

2. (2.3.3)

The next lemma will be used in the remainder of this section. Again, its proof is deferredto Appendix 2.7.2. We remind the reader that σΛ is the ultraviolet cut-off parameter thatappears in the interacting Hamiltonian HI and that δ0 denotes the minimal distance betweentwo distinct eigenvalues of

∑n0i=1 eiΠi.

Lemma 2.3.1.2.

• Let 0 < ρ < 1. For all θ = iϑ, 0 < ϑ < π/4, we have that

∥∥∥(HE + ρ)−1/2HI,θ(HE + ρ)−1/2∥∥∥ = O

(σ

3/2Λ

ρ1/2

). (2.3.4)

• Let 0 < ρ0 < min(1, δ0). For all θ = iϑ, 0 < ϑ < π/4, and (~p, z) ∈ Uρ0 [ei0 ], the operator[Hθ,0(~p)− z]|Ran(χi0 ) is bounded invertible and satisfies the estimates

∥∥∥[Hθ,0(~p)− z]−1|Ran(χi0 )

∥∥∥ = OÇ

1

µρ0 sin(ϑ)

å, (2.3.5)

∥∥∥ î(Hθ,0(~p)− z)−1(HE + ρ0)ó|Ran(χi0 )

∥∥∥ = OÇ

1

µ sin(ϑ)

å. (2.3.6)

(Hθ(~p)− z,Hθ,0(~p)− z) is a Feshbach-Schur pair

We now show that the pair (Hθ(~p) − z,Hθ,0(~p) − z) is a Feshbach-Schur pair, provided thatthe coupling constant λ0 is small enough and that the scale parameter ρ0 satisfies ρ0 λ2

0(µ sinϑ)−2.

Lemma 2.3.1.3. There exists λc > 0 such that, for all 0 ≤ λ0 ≤ λc, θ = iϑ satisfying0 < ϑ < π/4 and σ−3/2

Λ µ sinϑ λ0, (~p, z) ∈ Uρ0 [ei0 ], and ρ0 such that λ20σ

3Λ(µ sinϑ)−2

ρ0 < min(1, δ0), the pair (Hθ(~p)− z,Hθ,0(~p)− z) is a Feshbach-Schur pair associated to χi0 .

Proof. Lemma 2.3.1.2 shows that [Hθ,0(~p) − z]|Ran(χi0 ) is bounded invertible for all (~p, z) ∈Uρ0 [ei0 ]. We prove that

Hχi0(~p, z) := Hθ,0(~p)− z + λ0χi0HI,θχi0 (2.3.7)


is bounded invertible on Ran(χi0). The proof is standard and relies on Equation (2.3.4) inLemma 2.3.1.2. By (2.3.4), the Neumann series for [Hχi0

(~p, z)]−1|Ran(χi0 ) is estimated as

∥∥∥[Hχi0(~p, z)]−1

|Ran(χi0 )

∥∥∥ ≤ ρ−10

∞∑n=0

îCσ

3/2Λ λ0ρ

−1/20

ón ×∥∥∥∥ [(Hθ,0(~p)− z)−

12 (HE + ρ0)(Hθ,0(~p)− z)−

12

]|Ran(χi0 )

∥∥∥∥n+1

,(2.3.8)

for some numerical constant C > 0. Using (2.3.6), we see that the Neumann series convergesuniformly in (~p, z) ∈ Uρ0 [ei0 ] provided that λ0 σ

−3/2Λ ρ

1/20 µ sinϑ. Moreover,∥∥∥[Hχi0

(~p, z)]−1|Ran(χi0 )

∥∥∥ = OÇ

1

µρ0 sin(ϑ)

å. (2.3.9)

Since in addition HI,θχi0 and χi0HI,θ extend to bounded operators on H~p, it follows that(Hθ(~p)− z,Hθ,0(~p)− z) is a Feshbach-Schur pair associated to χi0 .

2.3.2 Wick-ordering and analyticity of H(0)(~p, z)

We assume that the parameters ρ0, λ0 and θ satisfy the hypotheses of Lemma 2.3.1.3, sothat the smooth Feshbach-Schur map associated to χi0 can be applied to the pair (Hθ(~p) −z,Hθ,0(~p) − z) for all (~p, z) ∈ Uρ0 [ei0 ]. Let H(0)(~p, z) be the bounded operator on H(0) =1HE≤ρ0HE associated with the bounded quadratic form defined by

〈ψ|H(0)(~p, z)φ〉 := 〈ψi0 ⊗ ψ|Fχi0(Hθ(~p, z), Hθ,0(~p)− z)ψi0 ⊗ φ〉, (2.3.10)

for all ψ, φ ∈ H(0), where ψi0 is a normalized eigenvector associated to the eigenvalue ei0 of∑n0i=1 eiΠi. Here we omit the argument θ to simplify notations. Lemma 2.3.2.1 below shows

that H(0)(~p, z) can be rewritten as a convergent series of Wick monomials on H(0),

H(0)(~p, z) =∑

m+n≥0

W (0)m,n(~p, z) + E(0)(~p, z). (2.3.11)

The convergence is uniform on the open set Uρ0 [ei0 ]. The main tool used in the proof is thepull-though formula

a(k)g(HE , ~PE) = g(HE + |~k|, ~PE + ~k)a(k), (2.3.12)

which holds for any measurable function g : R4 → C, and which enables us to normal orderthe creation and annihilation operators that appear in H(0)(~p, z). Lemma 2.3.2.1 also showsthat H(0)(~p, z) can be made arbitrary close (in norm) to the operatorÄ

e−θHE − e−θ~p · ~PE + ei0 − zä|H(0)

(2.3.13)

by an appropriate tuning of the coupling constant λ0, and that the map (~p, z) 7→ H(0)(~p, z) ∈B(H(0)) is analytic on Uρ0 [ei0 ]. The proof of Lemma 2.3.2.1 is postponed to Appendix 2.7.3.

Lemma 2.3.2.1. Let γ > 0. There exists λc(γ) > 0 such that, for all 0 ≤ λ0 ≤ λc(γ) and θand ρ0 as in Lemma 2.3.1.3, H(0) can be rewritten as a uniformly convergent series of Wickmonomials on Uρ0 [ei0 ],

H(0)(~p, z) = H[w(0)(~p, z), E(0)(~p, z)] =∑

m+n≥0

W (0)m,n(~p, z) + E(0)(~p, z). (2.3.14)


The associated kernelsw(0)m,n : Uρ0 [ei0 ]× Bρ0 ×B(m,n)

ρ0→ C

and the function E(0) : Uρ0 [ei0 ]→ C satisfy the following properties:

• w(0)0,0(~p, z, ·, ·) is C1 on Bρ0 and w(0)

0,0(~p, z, 0,~0) = 0 for all (~p, z) ∈ Uρ0 [ei0 ],

• w(0)m,n(~p, z, ·, ·,K(m,n)), m + n ≥ 1, are C1 on Bρ0 for almost every K(m,n) ∈ B(m,n)

ρ0 andevery (~p, z) ∈ Uρ0 [ei0 ],

• For all (~p, z) ∈ Uρ0 [ei0 ],

‖w(0)m,n(~p, z)‖ 1

2≤ γµ sin(ϑ)ρ

− 12

(m+n)+10 , (2.3.15)

‖∂#w(0)m,n(~p, z)‖ 1

2≤ γρ−

12

(m+n)0 , (2.3.16)

for all m+ n ≥ 1, where ∂# stands for ∂r or ∂lj , and

|(ei0 − z)− E(0)(~p, z)| ≤ γµρ0 sin(ϑ), (2.3.17)

‖∂rw(0)0,0(~p, z)− e−θ‖∞ +

3∑q=1

‖∂lqw(0)0,0(~p, z) + pqe

−θ‖∞ ≤ γ +√

3ρ0. (2.3.18)

Moreover, the bounded operator-valued function (~p, z) 7→ H(0)(~p, z) ∈ B(H(0)) is analytic onUρ0 [ei0 ].

Since (~p, z) 7→ H(0)(~p, z) ∈ B(H(0)) is analytic on the open set Uρ0 [ei0 ], the map (~p, z) 7→E(0)(~p, z) = 〈Ω|H(0)(~p, z)Ω〉 ∈ C is also analytic. Our next lemma establishes, for each~p ∈ Uθ[~p∗], the existence of a unique element z(0)(~p) ∈ D(ei0 , r0), such that E(0)(~p, z(0)(~p)) = 0.Here we recall that r0 = ρ0µ sin(ϑ)/32.

Lemma 2.3.2.2. Let 0 < γ 1 and suppose that λ0, ρ0, θ = iϑ are fixed as in Lemma2.3.1.3. Let ~p ∈ Uθ[~p

∗]. The holomorphic function z 7→ E(0)(~p, z) ∈ C possesses a uniquezero z(0)(~p) ∈ D(ei0 , r0). Furthermore, for any η > 0 such that r0η + γρ0µ sin(ϑ) < r0,D(z(0)(~p), r0η) ⊂ D(ei0 , r0), and

|E(0)(~p, z)| ≤ 2γρ0µ sin(ϑ) + r0η, (2.3.19)

|z(0)(~p)− ei0 | ≤ γρ0µ sin(ϑ), (2.3.20)

for all z ∈ D(z(0)(~p), r0η).

Proof. Since γ 1, we have that r0 > γρ0µ sin(ϑ). We apply Rouché’s theorem to thefunctions z 7→ E(0)(~p, z) and z 7→ ei0 − z on D(ei0 , r) with γρ0µ sin(ϑ) < r < r0. For anyz ∈ ∂D(ei0 , r), we have that |ei0 − z| = r, and hence

|E(0)(~p, z)− (ei0 − z)| ≤ γρ0µ sin(ϑ) < |ei0 − z|,


for any r > γρ0µ sin(ϑ). As r can be chosen arbitrarily close to r0, we deduce that z 7→E(0)(~p, z) possesses a unique zero, z(0)(~p), inD(ei0 , r0). Let η > 0 such that r0η+γρ0µ sin(ϑ) <r0. The triangle inequality implies that

|E(0)(~p, z)| ≤ γρ0µ sin(ϑ) + |z − z(0)(~p)|+ |z(0)(~p)− ei0 | ≤ 2γρ0µ sin(ϑ) + r0η,

for all z ∈ D(z(0)(~p), r0η) ⊂ D(ei0 , r0).

2.4 The inductive construction

As described in Subsection 2.1.3, we propose to inductively construct a sequence of effectiveHamiltonians, H(j)(~p, z), j = 0, 1, ..., with the property that H(j)(~p, z(∞)(~p)) is not injective ifand only if z(∞)(~p) is an eigenvalue of Hθ(~p). We use the notations introduced in Section 2.1,and we now present the details of our inductive construction. In particular, one of our purposesin this section is to prove bounds on the perturbation W (j)

≥1 (~p, z); see (2.1.54). We remind thereader that our inductive construction can be summarized by describing the induction step,from j to j + 1:

(i) In passing from j to j + 1, our starting point is the effective Hamiltonian H(j)(~p, z) con-structed in the previous induction step, which is an operator defined on the space H(j) =1HE≤ρjHE , provided (~p, z) is constrained to belong to a certain open subset Uρj [z(j−1)] ofC3 × C. For each ~p ∈ Uθ[~p∗], the admissible values of z are then taken to lie inside a smalldisk centered at the zero, z(j)(~p), of the function

z 7→ E(j)(~p, z) = 〈Ω|H(j)(~p, z)Ω〉.

This will define an open set Uρj+1 [z(j)] ⊂ Uρj [z(j−1)].

(ii) We apply the Feshbach-Schur map to the Feshbach pair (H(j)(~p, z),W(j)0,0 (~p, z) + E(j)(~p, z))

associated to χρj+1(HE), for all (~p, z) in Uρj+1 [z(j)]. We then re-Wick order the resultingoperator (all creation operators moved to the left of all annihilation operators, using the pull-through formula). This yields a new effective Hamiltonian, H(j+1)(~p, z), at step j+1, whichwill be shown to be well-defined on H(j+1) = 1HE≤ρj+1HE , provided (~p, z) ∈ Uρj+1 [z(j)].

2.4.1 Inductive properties of the kernels – from an energy scale ρ to theenergy scale ρ2−ε

In this subsection, we consider an effective Hamiltonian, given as a sum of Wick monomials,at an energy scale ρ, with 0 < ρ < 1. By an application of the smooth Feshbach-Schur map,we obtain a new effective Hamiltonian at an energy scale ρ, with

ρ := ρ2−ε, 0 < ε < 1, (2.4.1)

which has certain properties allowing us to iterate the construction. For a kernel wm,n(~p, z)

defined on a subset S of Bρ × B(m,n)ρ , ‖wm,n(~p, z)‖1/2 is the norm of wm,n(~p, z) as defined in

(2.2.2) with the supremum taken over the subset S.


For f : C3 → C a continuous function and ρ > 0, we define

Uρ[f ] := (~p, z) ∈ Uθ[~p∗]× C | z ∈ D (f(~p), rρ) , (2.4.2)

where D (f(~p), rρ) is the complex open disk centered at f(~p) and with radius

rρ :=ρµ sin(ϑ)

32.

For (~p, z) ∈ Uρ[f ], we consider the operator

H(~p, z) = H[w(~p, z), E(~p, z)] =∑

m+n≥0

Wm,n(~p, z) + E(~p, z), (2.4.3)

on 1HE≤ρHE , associated with a sequence of kernels

wm,n : Uρ[f ]× Bρ ×B(m,n)ρ → C

and a function E : Uρ[f ]→ C. We assume that there exists a constant D > 1 such that

(a) w0,0(~p, z, ·, ·) is C1 on Bρ, and w0,0(~p, z, 0,~0) = 0, for all (~p, z) ∈ Uρ[f ];

(b) the kernels wm,n(~p, z, ·, ·,K(m,n)), m + n ≥ 1, are C1 on Bρ, for almost every K(m,n) ∈B

(m,n)ρ and every (~p, z) ∈ Uρ[f ]. Moreover, wm,n(~p, z, ·, ·,K(m,n)) is symmetric in k(m) and

k(n)

;

(c) for all (~p, z) ∈ Uρ[f ],

‖wm,n(~p, z)‖ 12≤ Dm+nρ−(m+n)+1µ sin(ϑ), (2.4.4)

‖∂#wm,n(~p, z)‖ 12≤ Dm+nρ−(m+n), (2.4.5)

for all m+ n ≥ 1, where ∂# stands for ∂r or ∂lj ;

(d) the maps E : Uρ[f ]→ C and (~p, z) 7→ H(~p, z) ∈ B(1HE≤ρHE) are analytic on Uρ[f ];

(e) for all ~p ∈ Uθ[~p∗], the holomorphic function z 7→ E(~p, z) ∈ C possesses a unique zero

f(~p) ∈ D(f(~p), 2rρ/3), where f is analytic in ~p ∈ Uθ[~p∗]; with ρ defined by (2.4.1), we

then have thatUρ[f ] ⊂ Uρ[f ],

|f(~p)− f(~p)| < rρ2, (2.4.6)

for all ~p ∈ Uθ[~p∗], and|E(~p, z)| ≤ µρ

16, (2.4.7)

for all (~p, z) ∈ Uρ[f ].


Lemma 2.4.1.1. Let ρ, ε ∈ (0, 1) and let D > 1 be as in (2.4.4) and (2.4.5) and such thatρε D−1 and ρ1−ε 1. Let f : C3 → C be a continuous function, and let H(~p, z) be theoperator given in (2.4.3). For (~p, z) ∈ Uρ[f ], this operator is assumed to satisfy properties(a)–(e), above. In addition, we assume that

‖∂rw0,0(~p, z)− e−θ‖∞ +3∑q=1

‖∂lqw0,0(~p, z) + pqe−θ‖∞ ≤

µ

4, ∀(~p, z) ∈ Uρ[f ], (2.4.8)

and that|∂zE(~p, z) + 1| ≤ 1

4, ∀z ∈ D(f(~p),

2

3rρ), (2.4.9)

where D(f(~p), 23rρ) denotes the closed disk with radius 2rρ/3 centered at f(~p).

Then, for ρ = ρ2−ε, the effective Hamiltonian

H(~p, z) := Fχρ(HE)[H(~p, z),W0,0(~p, z) + E(~p, z)]1HE≤ρ[HE ],

is well-defined for all (~p, z) ∈ Uρ[f ], and there exists a sequence of kernels

wm,n : Uρ[f ]× Bρ ×B(m,n)ρ → C,

and a function E : Uρ[f ]→ C, such that

H(~p, z) = H[w(~p, z), E(~p, z)] =∑

m+n≥0

Wm,n(~p, z) + E(~p, z), (2.4.10)

for all (~p, z) ∈ Uρ[f ]. The maps wm,n and E have properties (a)–(e), above, with ρ replaced byρ, f by f , D replaced by DC, for some constant C independent of the ’problem parameters’and D, and f replaced by ˜

f . Here ˜f(~p) denotes the unique zero of the map z 7→ E(~p, z) ∈ C

in D(f(~p), 2rρ/3).Moreover, we have that

‖w0,0(~p, z) + E(~p, z)− w0,0(~p, z)− E(~p, z)‖∞ ≤ D2Cρ2+εµ sin2(ϑ), (2.4.11)

‖∂#(w0,0(~p, z)− w0,0(~p, z))‖∞ ≤ D2Cρ2ε sin(ϑ), (2.4.12)

for all (~p, z) ∈ Uρ[f ].

Remark 2.4.1.1. In our inductive construction, the constant D in Lemma 2.4.1.1 is replacedat step j by Cj, where C is the numerical constant appearing in (2.4.11) and (2.4.12). More-over, ρ is replaced by ρ(2−ε)j

0 . The hypothesis ρ(2−ε)jε0 C−j is fulfilled at any step j if ρ0 is

sufficiently small. Furthermore, Equations (2.4.11) and (2.4.12) imply that (2.4.8) and (2.4.9)hold true at any step j for sufficiently small values of ρ0; see Paragraph 2.4.2.

The proof of Lemma 2.4.1.1 occupies three subsections. In Subsection 2.4.1 we show that thepair

(H(~p, z),W0,0(~p, z) + E(~p, z))

is a Feshbach-Schur pair associated to χρ(HE), for all (~p, z) ∈ Uρ[f ]. In Subsection 2.4.1, weapply the Feshbach-Schur map to this Feshbach-Schur pair and re-Wick order the resultingoperator H(~p, z), so as to bring it into the “normal form” shown in Eq. (2.4.10). We thenverify that the sequence of kernels w has properties (a), (b), (c) and satisfy the estimates in(2.4.11)–(2.4.12). Finally, in Subsection 2.4.1, we prove that properties (d) and (e) hold, too.


Proof of applicability of the Feshbach-Schur map.

Let (~p, z) ∈ Uρ[f ]. We have that

|w0,0(~p, z, r,~l) + E(~p, z)| ≥ |w0,0(~p, z, r,~l)| − |E(~p, z)|

≥ |r − ~p ·~l| − µ

4r − µρ/16

≥ rÅµ− µ

4

ã− µρ/16

≥ µ

2ρ,

for all (~p, z) ∈ Uρ[f ] and r ≥ 3ρ/4. Therefore, the restriction of W0,0(~p, z) + E(~p, z) toRan(χρ(HE)) is bounded-invertible and

‖[W0,0(~p, z) + E(~p, z)]−1χρ(HE)‖ = O( 1

µρ

).

Next, we prove that the restriction of Hχρ(HE)(~p, z) to Ran(χρ(HE)) is bounded-invertible.It follows from (2.4.4) and (2.3.3) that

‖Wm,n(~p, z)‖ ≤ ρ2(m+n)‖wm,n(~p, z)‖ 12(8π)

m+n2 ≤ ρ2(m+n)(8π)

m+n2 ρ−(m+n)+1Dm+nµ.

(2.4.13)Summing (2.4.13) over m+ n ≥ 1, we find that

‖W≥1(~p, z)‖ = OÄDρ2µ

ä, (2.4.14)

which yields

‖[W0,0(~p, z) + E(~p, z)]−1χρ(HE)W≥1(~p, z)‖ = O (Dρε) .

Since ρε D−1, we deduce that (H(~p, z),W0,0(~p, z) + E(~p, z)) is a Feshbach-Schur pair asso-ciated to χρ(HE), for all (~p, z) ∈ Uρ[z(~p)], and that H(~p, z) is well-defined.

Re-Wick ordering and proof of the inductive properties (a), (b) and (c)

We temporarily omit the argument (~p, z). To shorten our notation, we introduce the operators

Hχρ := W0,0 + E + χρ(HE)W≥1χρ(HE) (2.4.15)

and

R( ~HE , ~PE) :=χ2ρ(HE)

W0,0( ~HE , ~PE) + E. (2.4.16)

We have that

H = (W0,0 + E)1HE≤ρ + χρ(HE)W≥1χρ(HE) + V , (2.4.17)

whereV = −χρ(HE)W≥1χρ(HE)

îHχρ

ó−1

|Ran(χρ(HE))χρ(HE)W≥1χρ(HE).


For any L ∈ N and any M1, ...,ML ∈ N, we define M := (M1, ...,ML). The Neumannexpansion for V reads

V = −∞∑L=2

(−1)L∑

M,N ; Mi+Ni≥1

VM,N , (2.4.18)

where

VM,N := χρ(HE)L−1∏i=1

îWMi,NiR( ~HE , ~PE)

óWML,NLχρ(HE). (2.4.19)

To normal order VM,N , we pick out m1/n1 creation/annihilation operators from the M1/N1

creation/annihilation operators available in WM1,N1 ,..., mL/nL creation/annihilation opera-tors from the ML/NL creation/annihilation operators available in WML,NL , and contract allthe remaining annihilation and creation operators. As the monomials WMi,Ni ’s are symmetricin k1, ..., kMi

, and in k1, ..., kNi , there are

CM,Nm,n :=

L∏i=1

ÇMi

mi

åÇNi

ni

åcontraction schemes giving rise to the same contribution. We then pull through all the remain-ing m1 + ....+mL uncontracted creation operators to the left, and the remaining n1 + ....+nLuncontracted annihilation operators to the right. This causes a shift in the arguments of theoperators wMi,Ni(HE , ~PE ,K

(Mi,Ni)i ) and R( ~HE , ~PE) via the pull-through formula


for any measurable function g : R4 → C. The contracted part is expressed as a vacuumexpectation value. VM,N can be rewritten in the form

VM,N =∑

m,n; mi≤Mi,ni≤NiCM,Nm,n

∫dK(m,n) a∗(k(m))〈Ω|VM,N

m,n (r,~l,K(m,n))Ω〉r=HE ,~l=~PE

a(k(n)

),

where a∗(k(m)) =∏Li=1 a

∗(k(mi)i ), see Paragraph 2.2.2. Furthermore, if mi (or ni) is equal to

zero, a∗(k(mi)i ) (or a(k

(ni)i )) is replaced by 1 in the above formula. A precise expression for

the operator VM,Nm,n (r,~l,K(m,n)) is given in [54]. The reader should notice that this precise

expression is not necessary to pursue our analysis. What matters is that VM,Nm,n (r,~l,K(m,n))

is a product of L− 1 operators R( ~HE + r, ~PE +~l) with shifted arguments, together with thetruncated kernels WMi,Ni

mi,ni (r,~l, k(mi)i , k

(ni)i ) with shifted arguments, where

WMi,Nimi,ni (r,~l,k

(mi)i , k

(ni)i ) = 1HE≤ρ

∫dX

(Mi−mi,Ni−ni)i a∗(x

(Mi−mi)i )

wMi,Ni (HE + r, ~PE +~l,K(mi,ni)i , X

(Mi−mi,Ni−ni)i ) a(x

(Ni−ni)i ) 1HE≤ρ

(2.4.21)

and

VM,Nm,n (r,~l,K(m,n)) = χρ(r + r0)

L−1∏i=1

[WMi,Nimi,ni (r + ri,~l +~li, k

(mi)i , k

(ni)i )

R(HE + r + ri, ~PE +~l +~li)]WML,NLmLnL

(r + rL,~l +~lL, k(mL)L , k

(nL)L )χρ(r + rL).

(2.4.22)


The terms ri’s, ri’s, ~li’s,~li’s are the shifts that come from the pull-through formula, see (2.7.19).

Therefore, we deduce that there exists a sequence of kernels wM,N : Uρ[f ]×Bρ ×B(m,n)ρ → C

and a function E : Uρ[f ]→ C such that

H(~p, z) = H[w(~p, z), E(~p, z)] =∑

M+N≥0

WM,N (~p, z) + E(~p, z), (2.4.23)

where we have introduced the arguments (~p, z) again in (2.4.23). The associated kernels aregiven by

wM,N (~p, z, r,~l, k(M), k(N)

) =∑L=1

(−1)L+1∑

m1 + ... +mL = Mn1 + ... + nL = N

∑p1, ..., pLq1, ..., qL

mi + ni + pi + qi ≥ 1

Cm+p,n+qm,n

〈Ω|V m+p,n+qm,n (~p, z, r,~l,K(M,N))Ω〉sym, (2.4.24)

for M +N ≥ 1, where m+ p = (m1 + p1, ...,mL + pL), and fsym denotes the symmetrization

of f with respect to the variables k(M) and k(N)

. Notice that we have set pi := Mi − mi,qi := Ni − ni, i = 1, ..., L. Also note that we started the sum in (2.4.24) at L = 1, and notL = 2, since we included the contribution of the term χρ(HE)W≥1(~p, z)χρ(HE) in (2.4.17).Similarly,

w0,0(~p, z, r,~l)+E(~p, z) =[w0,0(~p, z, r,~l) + E(~p, z)

+∞∑L=2

(−1)L−1∑

p1, ..., pLq1, ..., qLpi + qi ≥ 1

〈Ω|V p,q

0,0 (r,~l)Ω〉]1r≤ρ. (2.4.25)

We now bound (2.4.24) and (2.4.25). Thanks to (2.3.3),

‖WMi,Nimi,ni (r,~l, k

(mi)i , k

(ni)i )‖ ≤ |k(mi)

i |1/2 |k(ni)i |1/2‖wMi,Ni‖ 1

2(√

8πρ2)Mi+Ni−mi−ni . (2.4.26)

The same bounds are satisfied by the partial derivatives with respect to r and lq with ‖wMi,Ni‖ 12

replaced by ‖∂#wMi,Ni‖ 12. Since

|w0,0(~p, z, r,~l) + E(~p, z)| ≥ µ

2ρ,

there exists a numerical constant C > 0 such that

sup(r,~l),r≥ 3

4ρ,|~l|≤r

|R(~p, z, r,~l)| ≤ C

ρµ, (2.4.27)

sup(r,~l),r≥ 3

4ρ,|~l|≤r

|∂]R(~p, z, r,~l)| ≤ C

ρ2µ2. (2.4.28)

Suppressing the argument (~p, z, r,~l,K(m,n)) on the left side to shorten our formulas, it followsimmediately from (2.4.26) and (2.4.22) that

‖V m+p,n+qm,n ‖ ≤ CL

(µρ)L−1

L∏i=1

ï|k(mi)i |1/2 |k(ni)

i |1/2‖wmi+pi,ni+qi(~p, z)‖ 12(√

8πρ2)pi+qiò,


and,

‖∂rVm+p,n+qm,n ‖ ≤ L CL

(µρ)L−1supj

L∏i=1

[|k(mi)i |1/2 |k(ni)

i |1/2((1− δij)‖wmi+pi,ni+qi(~p, z)‖ 1

2

+ δij‖∂rwmi+pi,ni+qi(~p, z)‖ 12

)(√

8πρ2)pi+qi]

+ (L+ 1)CL

(µρ)L

L∏i=1

ï|k(mi)i |1/2 |k(ni)

i |1/2‖wmi+pi,ni+qi(~p, z)‖ 12(√

8πρ2)pi+qiò

Using the inequality Çmi + pimi

å≤ 2mi+pi ,

we deduce that

‖wM,N (~p, z)‖ 12≤∞∑L=1

4M+NCL

(µρ)L−1

∑m,n,p,q

mi+ni+pi+qi≥1

L∏i=1

(‖wmi+pi,ni+qi(~p, z)‖ 1

2

(2√

8πρ2)pi+qiÅ

1

2

ãmi+ni ).

Inequality (2.4.4) for the norms of the kernels implies that

‖wmi+pi,ni+qi(~p, z)‖ 12≤ ρ−(mi+ni+pi+qi)+1Dmi+ni+pi+qiµ sin(ϑ)

≤Ä 1

32Cρä−(mi+ni+pi+qi)+1

Dmi+ni+pi+qiµ sin(ϑ),

where we have used that mi + ni + pi + qi ≥ 1. It then follows that

‖wM,N (~p, z)‖ 12≤(128CD

ρ

)M+N ∞∑L=1

(µ sin(ϑ)ρ/32)L

(µρ)L−1

Ñ ∑p+q≥0

(64C√

8πρ2D

ρ

)p+q ∑m+n≥0

1

2m+n

éL

≤(128CD

ρ

)M+Nµρ sin(ϑ),

where we have used that ρε < (128C√

8πD)−1, in order to sum over p+ q on the right-handside. A similar bound is satisfied by the partial derivatives with respect to r and lq, q = 1, ..., 3,with µρ sin(ϑ) replaced by 1. This shows that wM,N satisfies property (c), with D replacedby CD provided that C is chosen large enough. It is important to remark that the constantC > 0 can be chosen to be independent of the ’problem parameters’ and of the constant D.This property of C is needed to implement our inductive construction.

Likewise,

‖w0,0(~p, z)+E(~p, z)− w0,0(~p, z)− E(~p, z)‖∞

≤∞∑L=2

(C sin(ϑ)ρµ)L

(ρµ)L−1

Ñ ∑p+q≥1

î2D√

8πρóp+qéL

≤ CD2ρ4ρ−1µ sin2(ϑ) = CD2ρ2+εµ sin2(ϑ),

and a similar bound is satisfied for ∂#(w0,0(~p, z)−w0,0(~p, z)), which proves (2.4.11)–(2.4.12),provided that C is chosen large enough.


Properties (d) and (e)

We first observe that the map (~p, z) 7→ H(~p, z) is analytic on Uρ[f ]. This follows from the factsthat the smooth Feshbach-Schur map preserves analyticity in (~p, z) (because the Neumannseries converges uniformly on Uρ[f ]) and that the maps (~p, z) 7→ W≥1(~p, z) ∈ B(HE) and(~p, z) 7→W0,0(~p, z)+E(~p, z) are analytic on Uρ[f ] (because, by assumption, (~p, z) 7→ H(~p, z) isanalytic on the open set Uρ[f ] ⊃ Uρ[f ]). We refer the reader to [54] for more details. Therefore(~p, z) 7→ E(~p, z) = 〈Ω|H(~p, z)Ω〉 is analytic in (~p, z), too.

We now prove property (e). Let ~p ∈ Uθ[~p∗] and z = f(~p) + βrρ ∈ D(f(~p), rρ), with

0 ≤ |β| ≤ 2/3. The triangle inequality, the inequality (2.4.6), and the hypothesis ρ1−ε 1imply that z ∈ D(f(~p), 2rρ/3). We consider the circular contour C centered at f(~p) and withradius rC = 3rρ/4. We have that ~p × C ⊂ Uρ[f ] ⊂ Uρ[f ], and, by Cauchy’s formula,

|∂zE(~p, z)− ∂zE(~p, z)| ≤ 1

2π

∣∣∣∣ ∫Cdz′E(~p, z′)− E(~p, z′)

(z − z′)2

∣∣∣∣≤ 3

4(3/4− |β|)2

CD2µ sin2(ϑ)ρ2+ε

rρ,

where we have used (2.4.11) and the fact that w0,0(~p, z, 0,~0) = 0 = w0,0(~p, z, 0,~0), see property(a). It follows that

|∂zE(~p, z)− ∂zE(~p, z)| ≤ 96

4(3/4− |β|)2CD2ρ2ε. (2.4.29)

Since

|∂zE(~p, z) + 1| ≤ 1/4,

for any z ∈ D(f(~p), 23rρ), we deduce that

|∂zE(~p, z) + 1| ≤ 1

2, (2.4.30)

for any z ∈ D(f(~p), 23rρ), where we use the hypothesis that ρ

ε D−1. We estimate |E(~p, z)−f(~p) + z| on the circle ∂D(f(~p), 2

3rρ). Let z ∈ D(f(~p), 23rρ). We obtain from (2.4.30) that

|E(~p, z)− f(~p) + z| ≤ |E(~p, z)− E(~p, f(~p)) + z − f(~p)|+ |E(~p, f(~p))|

≤ 1

2|z − f(~p)|+ |E(~p, f(~p))− E(~p, f(~p))|

≤ 1

2|z − f(~p)|+ CD2µ sin2(ϑ)ρ2+ε, (2.4.31)

where we have used the fact that E(~p, f(~p)) = 0, along with (2.4.11). If ρ is sufficiently smallthen

CD2 sin2(ϑ)ρ2+εµ <rρ3,

and hence|E(~p, z)− f(~p) + z| < |z − f(~p)|, (2.4.32)

for any z ∈ ∂D(f(~p), 23rρ). Rouché’s theorem then implies that z 7→ E(~p, z) has a unique zero,

˜f(~p), inside the disk of radius 2rρ/3.


To prove (2.4.6), we observe that (2.4.31), with z =˜f(~p), yields

1

2| ˜f(~p)− f(~p)| ≤ CD2µ sin2(ϑ)ρ2+ε = 32CD2 sin(ϑ)ρ2εrρ.

The hypothesis that ρε D−1 implies that

| ˜f(~p)− f(~p)| ≤ rρ2. (2.4.33)

Next, we verify that E satisfies (2.4.7) with ρ replaced by ˜ρ = ρ2−ε. The triangle inequalitydirectly implies that U ˜ρ[

˜f ] ⊂ Uρ[f ], and we have that

|E(~p, z)| ≤ |E(~p, z) + z − ˜f(~p)|+ |z − ˜

f(~p)|

≤ 3

2|z − ˜

f(~p)| ≤ 2r ˜ρ ≤1

16˜ρµ, (2.4.34)

for any z ∈ D(˜f(~p), r ˜ρ) ⊂ D(f(~p), 2rρ/3), where (2.4.30) has been used in the second inequal-

ity.

To complete our proof, we show that the map ~p 7→ ˜f(~p) is analytic on Uθ[~p

∗] and thatthe set U ˜ρ[

˜f ] is open. Let (~p0,

˜f(~p0)) ∈ U ˜ρ[

˜f ] ⊂ Uρ[f ]. Since |∂zE(~p0, z) + 1| < 1/2, for all

z ∈ D(f(~p0), 2rρ/3), the inverse function theorem implies that the map

(~p, z) 7→ (~p, E(~p, z)) (2.4.35)

is biholomorphic in a small polydisk D0 ⊂ Uρ[f ] around (~p0,˜f(~p0)). We denote its inverse by

h. The image of D0, denoted D1, contains the point (~p0, 0), because E(~p0,˜f(~p0)) = 0. As D1

is open, it contains (~p, 0), for any ~p sufficiently close to ~p0. Therefore, h(~p, 0) coincides with(~p,

˜f(~p)), for ~p near ~p0, and we deduce that ~p 7→ ˜

f(~p) is holomorphic in a neighborhood of~p0. By letting ~p0 vary one sees that this implies that ~p 7→ ˜

f(~p) is holomorphic on Uθ[~p∗]. Inparticular the set

U ˜ρ[˜f ] :=

(~p, z) ∈ Uθ[~p∗]× C | z ∈ D(

˜f(~p), r ˜ρ)

. (2.4.36)

is an open subset of C4.

2.4.2 Inductive construction of the operators H(j)(~p, z)

Let 0 < ε < 1 and let ρ0 be such that 0 < ρ0 min(1, δ0)µ. Let γ < ρ0, where γ is theconstant that appears in the estimates (2.3.15)–(2.3.18). For λ0 and θ = iϑ as in Lemma2.3.2.1, we deduce from Section 2.3 that the sequence of kernels w(0)

m,n and the function E(0)

corresponding to the operator H(0)(~p, z) satisfy properties (a)–(e) and (2.4.8)–(2.4.9), withD = 1, f(~p) = ei0 , f(~p) = z(0)(~p), and ρ = ρ0. We consider the scaling sequence

ρj = ρ(2−ε)j0 , (2.4.37)

and letrj := rρj =

µρj sin(ϑ)

32. (2.4.38)


By repeated application of Lemma 2.4.1.1, we construct a sequence of effective HamiltoniansH(j)(~p, z) at scale ρj by setting


(j)0,0 (~p, z) + E(j)(~p, z)]1HE≤ρj+1

[HE ]. (2.4.39)

In order to be able to apply Lemma 2.4.1.1 in each induction step, we only need to show thatthe properties (2.4.8)–(2.4.9) hold in each step of the induction and that ρεjCj 1, for allj ∈ N0. This is accomplished in the following lemma.

Lemma 2.4.2.1. Suppose that

∞∑k=1

C2k−1ρε(2−ε)k−1

0 +√

3ρ0 µ and ρ1−ε0 1. (2.4.40)

Let γ < ρ0 and choose λ0 and θ = iϑ as in Lemma 2.3.2.1. For any j ∈ N0, there exists afunction z(j−1) : Uθ[~p

∗]→ C, a sequence of kernels w(j)m,n : Uρj [z(j−1)]×Bρj ×B

(m,n)ρj → C and

a function E(j) : Uρj [z(j−1)]→ C such that (2.4.39) holds, and

(a) w(j)0,0(~p, z, ·, ·) is C1 on Bρj , and w

(j)0,0(~p, z, 0,~0) = 0, for all (~p, z) ∈ Uρj [z(j−1)].

(b) w(j)m,n(~p, z, ·, ·,K(m,n)), m + n ≥ 1, are C1 on Bρj for almost every K(m,n) ∈ B(m,n)

ρj and

every (~p, z) ∈ Uρj [z(j−1)]. Moreover, w(j)m,n(~p, z, ·, ·,K(m,n)) is symmetric in k(m) and k

(n).

(c) For all (~p, z) ∈ Uρj [z(j−1)],

‖w(j)m,n(~p, z)‖ 1

2≤ Cj(m+n)ρ

−(m+n)+1j µ sin(ϑ), (2.4.41)

‖∂#w(j)m,n(~p, z)‖ 1

2≤ Cj(m+n)ρ

−(m+n)j , (2.4.42)

for all m+ n ≥ 1, where ∂# stands for ∂r or ∂lp and C is the positive constant appearingin Lemma 2.4.1.1.

(d) The maps E : Uρj [z(j−1)] → C and (~p, z) 7→ H[w(j)(~p, z), E(j)(~p, z)] ∈ B(1HE≤ρjHE) areanalytic on Uρj [z(j−1)].

(e) For all ~p ∈ Uθ[~p∗], the holomorphic function z 7→ E(j)(~p, z) ∈ C has a unique zero z(j)(~p) ∈D(z(j−1)(~p), 2rρj/3). The function z(j) is analytic on Uθ[~p

∗]. Moreover, Uρj+1 [z(j)] ⊂Uρj [z(j−1)], and

|E(j)(~p, z)| ≤ µρj+1

16, (2.4.43)

|z(j)(~p)− z(j−1)(~p)| < rj/2, (2.4.44)

for all (~p, z) ∈ Uρj+1 [z(j)] and ~p ∈ Uθ[~p∗], respectively.


Proof. The hypothesis that∞∑k=1

C2k−1ρε(2−ε)k−1

0 +√

3ρ0 µ

implies that Cjρεj < c, for a positive numerical constant c, where c is small enough such thatLemma 2.4.1.1 holds at any step j. As mentioned at the beginning of this subsection, theproperties (a)–(e) and the inequalities (2.4.8) and (2.4.9) are valid are step j = 0. Proceedingby induction, we assume that the properties (a)–(e) and the inequalities (2.4.8) and (2.4.9)are valid at any step k ≤ j ∈ N0. If we prove that

‖∂rw(j)0,0(~p, z)− e−θ‖∞ +

3∑q=1

‖∂lqw(j)0,0(~p, z) + pqe

−θ‖∞ ≤µ

4, ∀(~p, z) ∈ Uρj [z(j−1)], (2.4.45)

and ∣∣∣∂zE(j)(~p, z) + 1∣∣∣ < 1

4, ∀z ∈ D(z(j−1)(~p),

2

3rj), (2.4.46)

then Lemma 2.4.1.1 shows that


(j)0,0 (~p, z) + E(j)(~p, z)]|H(j+1)

is well-defined on Uρj+1 [z(j)] and satisfies properties (a)-(e) at step j + 1, and hence theinduction step will be completed.

The bound (2.4.45) is a direct consequence of estimate (2.4.11) in Lemma 2.4.1.1. Sinceproperties (a)–(e) are valid for any k ≤ j, we deduce that Uρj [z(j−1)] ⊂ Uρj−1 [z(j−2)] ⊂ ... ⊂Uρ0 [ei0 ]. Moreover, (2.4.11)–(2.4.12) are valid at any step k = 1, ..., j, with ρ replaced by ρk−1

and D2C by C2k−1. Therefore,

‖∂rw(j)0,0(~p, z)− e−θ‖∞ ≤

j∑k=1

‖∂rw(k)0,0(~p, z)− ∂rw(k−1)

0,0 (~p, z)‖∞ + ‖∂rw(0)0,0(~p, z)− e−θ‖∞

≤j∑

k=1

C2k−1ρ2εk−1 sin(ϑ) + γ +

√3ρ0,

for any (~p, z) ∈ Uρj [z(j−1)]. Our assumptions on ρ0 and γ imply that the sum on the rightside is smaller than µ/4.

The bound (2.4.46) is proven by a similar argument: For any ~p ∈ Uθ[~p∗] and any z ∈

D(z(j−1)(~p), 23rj), we have that

∣∣∣∂zE(j)(~p, z) + 1∣∣∣ ≤ j∑

k=1

∣∣∣∂zE(k)(~p, z)− ∂zE(k−1)(~p, z)∣∣∣+ ∣∣∣∂zE(0)(~p, z) + 1

∣∣∣≤ C

Ñj∑

k=1

C2k−1ρ2εk−1 + γ

é,

where C is a positive numerical constant independent of the ’problem parameters’. The lastinequality follows from (2.4.29) in the proof of Lemma 2.4.1.1. Also note that we have used thefact that D(z(j−1)(~p), 2

3rj) ⊂ D(z(j−2)(~p), 23rj−1), which follows from (2.4.44). Thus, the right

side is smaller than 1/4 if ρ0 is sufficiently small. This completes the proof of the lemma.


2.5 Existence and analyticity of the resonances

In Paragraph 2.5.1, we prove that the sequence of functions ~p 7→ z(j)(~p) constructed in theprevious subsection converges uniformly on Uθ[~p∗]. Identifying 1Cn0 ⊗Qχρj with Qχρj for anyj ≥ 1 (see (2.1.61)), we show in Subsection 2.5.2 that the sequence of vector valued functions

Ψ(j)(~p) := Qχi0Qχρ1 ...Qχρj (ψi0 ⊗ Ω) (2.5.1)

converges uniformly on Uθ[~p∗] to a non-vanishing function ~p 7→ Ψ(∞)(~p). We remind the

reader that ψi0 is a unit eigenvector of∑n0i=1 eiΠi associated with the eigenvalue ei0 and

that the operators Q# are defined as in Theorem 2.1.3.1. For all ~p ∈ Uθ(~p∗), Ψ(∞)(~p) is aneigenvector of Hθ(~p) associated with the eigenvalue z(∞)(~p).

2.5.1 Convergence of z(j) and analyticity of the limit

Lemma 2.5.1.1. Suppose that the parameters ε, γ, ρ0, λ0 and θ are fixed as in Section 2.4.The sequence of holomorphic functions (z(j)) converges uniformly to an holomorphic functionz(∞) on Uθ[~p∗].

Proof. The estimate

|z(j+1)(~p)− z(j)(~p)| ≤ rj+1

2=µρj+1 sin(ϑ)

64, ∀~p ∈ Uθ[~p∗], (2.5.2)

obtained in Lemma 2.4.2.1, implies that (z(j)) is uniformly Cauchy. Hence (z(j)) convergesuniformly on Uθ(~p∗). Since (z(j)) is analytic on Uθ(~p∗) for all j, by Lemma 2.4.2.1, the uniformlimit z(∞) is also analytic.

2.5.2 Existence and analyticity of the eigenvector Ψ(∞)(~p)

Lemma 2.5.2.1. Suppose that the parameters ε, γ, ρ0, λ0 and θ are fixed as in Section 2.4.The sequence (Ψ(j)) converges uniformly on Uθ[~p∗]. The limit, Ψ(∞), satisfies Ψ(∞)(~p) 6= 0 forall ~p ∈ Uθ[~p∗]. Furthermore, (Hθ(~p) − z(∞)(~p))Ψ(j)(~p) converges to zero uniformly on Uθ[~p∗]and

dim Ker(Hθ(~p)− z(∞)(~p)) = 1.

Proof. We use the formulaHQχ(H,T ) = χFχ(H,T ) (2.5.3)

for the Feshbach pair (H,T ), see [9]. It implies that

[Hθ(~p)− z(∞)(~p)]Qχi0Qχρ1 ...Qχρj

= χi0χρ1(HE)...χρj (HE)Fχρj (HE)[H(j−1),W

(j−1)0,0 + E(j−1)],

(2.5.4)

where we omitted the argument (~p, z(∞)(~p)) on the right-hand side of (2.5.4). Applying (2.5.4)to ψi0 ⊗ Ω, we deduce that

[Hθ(~p)− z(∞)(~p))]Ψ(j)(~p) = χi0χρ1(HE)...χρj (HE) H(j)(~p, z(∞)(~p))|ψi0 ⊗ Ω〉. (2.5.5)


We have that

H(j)(~p, z(∞)(~p))|ψi0⊗Ω〉 =∑M>0

W(j)M,0(~p, z(∞)(~p))|ψi0⊗Ω〉+E(j)(~p, z(∞)(~p))|ψi0⊗Ω〉. (2.5.6)

The bound (2.4.41), the key estimate (2.3.3), and the inequality (2.4.43) imply that the right-hand side is bounded in norm by some numerical constant multiplied by ρj , i.e. tends to zerosuper-exponentially fast as j tends to infinity. We show that Ψ(j) converges to a non-vanishingvector-valued function. We remark that

Ψ(j+1)(~p)−Ψ(j)(~p) = Qχi0Qχρ1 ...Qχρj (Qχρj+1

− χρj+1(HE))|ψi0 ⊗ Ω〉

= (Qχi0− χi0 + χi0)...(Qχρj − χρj (HE) + χρj (HE))(Qχρj+1

− χρj+1(HE))|ψi0 ⊗ Ω〉.

Using that exp(x) ≥ 1 + x for any x ≥ 0, we deduce that

‖Ψ(j+1)(~p)−Ψ(j)(~p)‖ ≤ eÄ∑j

k=1 ‖Qχρk−χρk (HE)‖+‖Qχi0−χi0‖

ä‖Qχρj+1

− χρj+1(HE)‖. (2.5.7)

Furthermore,

‖Qχρk − χρk(HE)‖ = ‖χρk(HE)[H(k−1)χρk

(HE)]−1|Ran(χρk

(HE))χρk(HE)W(k−1)≥1 χρk(HE)‖

= OÄCk−1ρ2

k−1µä‖[H(k−1)

χρk(HE)]

−1|Ran(χρk

(HE))‖ = OÄCk−1ρεk−1

äfor any k ≥ 1. The right-hand side converges super-exponentially fast to zero and the sumover j of ‖Qχρj − χρj (HE)‖ is smaller than one if ρ0 is small enough. We deduce that‖Ψ(j+1)(~p) − Ψ(j)(~p)‖ converges super-exponentially fast to zero. This implies that (Ψ(j)) isuniformly Cauchy and hence converges uniformly. Since each Ψ(j) is analytic in ~p, the limitΨ(∞) is also analytic on Uθ[~p∗]. Furthermore, a similar argument as above shows that

‖Ψ(j)(~p)− Ω‖ < 1

for all ~p. Therefore, Ψ(∞)(~p) 6= 0 for all ~p, which implies that

dim Ker(Hθ(~p)− z(∞)(~p)) ≥ 1.

Using the “iso-spectrality” of the Feshbach-Schur map together with the fact that any vectorΨ ∈ HE satisfies the lim

j→∞χρj(HE)Ψ = 〈Ω|Ψ〉Ω, we verify that

dim Ker(Hθ(~p)− z(∞)(~p)) ≤ 1.

Our argumentation is similar to [87]. Let n ∈ N. We assume that z(∞)(~p) is degenerate. Wecan find a non-zero vector Ψ ∈ HE such that PΩΨ = 0 and H(n)(~p, z∞(~p))χρn(HE)Ψ = 0. Wetake n large enough such that χρn(HE)χρj (HE) = χρj (HE) for all j > n. We use that

1HE≤ρn ≥ χρn(HE),

1HE≤ρn = PΩ +∞∑

j=n+1

1 34ρj+1<HE≤ 3

4ρj

+ 1 34ρn+1<HE≤ρn ,


to estimate the norm of χρn(HE)Ψ. We get:

‖χρn(HE)Ψ‖2 ≤∞∑

j=n+1

‖1 34ρj+1<HE≤ 3

4ρjχρn(HE)Ψ‖2 + ‖1 3

4ρn+1<HE≤ρnχρn(HE)Ψ‖2

≤ 4

µ2

∞∑j=n

ρ−2j+1

∥∥∥(W (j)0,0 (~p, z∞(~p)) + E(j)(~p, z∞(~p)))1 3

4ρj+1<HE≤ρjχρj (HE)Ψ

∥∥∥2.

To go from the first line to the second line, we have used the estimate

|w(j)0,0(~p, z∞(~p), r,~l) + E(j)(~p, z∞(~p))| ≥ µ

2ρj+1, ∀r ≥ 3

4ρj+1, |~l| ≤ r,

proven at the beginning of Paragraph 2.4.1, and the equality

1 34ρj+1<HE≤ 3

4ρjχρn(HE) = 1 3

4ρj+1<HE≤ 3

4ρjχρj (HE).

Since 13ρj+1/4<HE≤ρj commutes with W (j)0,0 (~p, z∞(~p)) + E(j)(~p, z∞(~p)), and since

H(j)(~p, z∞(~p))χρj (HE)Ψ = 0,

we deduce that

‖χρn(HE)Ψ‖2 ≤ 4

µ2‖χρn(HE)Ψ‖2

∞∑j=n

ρ−2j+1

∥∥∥W (j)≥1 (~p, z∞(~p))

∥∥∥2.

We have shown in (2.4.14) that there exists a numerical constant C independent of j and theproblem parameters, such that

‖W (j)≥1 (~p, z)‖ ≤ CCjρ2

jµ,

for all j ∈ N. Since∑j C

2jρ4j/ρ

2j+1 converges, ‖χρn(HE)Ψ‖ must be zero for large values of

n. This contradicts Theorem 2.1.3.1, and, therefore, z∞(~p) cannot be degenerate.

2.6 Imaginary part of the resonances

In this section, assuming that Fermi’s Golden Rule holds, we prove that the imaginary partof z(∞)(~p) is strictly negative for small enough values of the coupling constant λ0. More pre-cisely, using the isospectrality of the Feshbach-Schur map (see Definition 2.1.3.1 and Theorem2.1.3.1), we verify that the operatorHθ(~p)−z is invertible for any z ∈ C such that =z ≥ −c0λ

20,

where c0 is a positive constant.

2.6.1 Computing the leading part of the Feshbach-Schur Map

We recall from Lemma 2.3.1.3 that if the parameters λ0, ρ0 and θ = iϑ satisfy the conditionsλ2

0σ3Λ(µ sinϑ)−2 ρ0 < min(1, δ0), then (Hθ(~p) − z,Hθ,0(~p) − z) is a Feshbach-Schur pair


associated to χi0 for any (~p, z) ∈ Uρ0 [ei0 ]. The corresponding Feshbach-Schur operator isgiven by

Fχi0(Hθ(~p)− z,Hθ,0(~p)− z) = Hθ,0(~p)− z + λ0χi0HI,θχi0

− λ20χi0HI,θχi0 [Hχi0

(~p, z)]−1|Ran(χi0 )χi0HI,θχi0 . (2.6.1)

We recall from Theorem 2.1.3.1 the iso-spectral property:

Hθ(~p)− z is bounded invertible⇐⇒ Fχi0

(Hθ(~p)− z,Hθ,0(~p)− z)|Πi0⊗1HE≤ρ0 [Cn0⊗HE ] is bounded invertible. (2.6.2)

The estimation of the imaginary part of z(∞)(~p) relies on the analysis of Fχi0(Hθ(~p) −

z,Hθ,0(~p)− z). We set

wθ0,1(k) := ie−2θΛ(e−θ~k)|~k|1/2~ε(k) · ~d, wθ1,0(k) = −wθ0,1(k), (2.6.3)

and

Zod(~p) :=

∫dkΠi0w

θ0,1(k)Πi0

Ä n0∑i=1

eiΠi − ei0 + e−θ|~k|+ e−2θ~k2

2− e−θ~p · ~k

ä−1Πi0w

θ1,0(k)Πi0 ,

(2.6.4)

Zd(~p) :=

∫dkΠi0w

θ0,1(k)Πi0

Äe−θ|~k|+ e−2θ

~k2

2− e−θ~p · ~k

ä−1wθ1,0(k)Πi0 ,

where, recall, Πi0 is the orthogonal projection onto the one-dimensional eigenspace associatedto the eigenvalue ei0 of

∑n0i=1 eiΠi.

In the next lemma we identify the leading order term of

Fχi0(Hθ(~p)− z,Hθ,0(~p)− z)|Πi0⊗1HE≤ρ0 [Cn0⊗HE ]

in terms of powers of λ0.

Lemma 2.6.1.1. Under the conditions of Lemma 2.3.1.3, there is a bounded operator Remsuch that

Fχi0(Hθ(~p)−z,Hθ,0(~p)− z)|Πi0⊗1HE≤ρ0 [Cn0⊗HE ] =

îei0 − z + e−θHE + e−2θ

~P 2E

2− e−θ~p · ~PE

ó(2.6.5)

− λ20Z

d(~p)⊗ χ2ρ0

(HE)− λ20Z

od(~p)⊗ χ2ρ0

(HE) + Rem,

and

‖Rem‖ ≤ Cλ20

( σ9/2Λ

µ2 sin(ϑ)2 min(1, δ20)

( λ0

ρ01/2

+ρ0

2

λ0

)+ ρ0

), (2.6.6)

where C is a positive constant.

Proof. The proof follows the lines of Lemma 3.16 of [14], where similar results are shown fora different model (see also [12], where all details are included).


Remark 2.6.1.1. Lemma 2.6.1.1 gives the leading order contribution of Fχi0(Hθ(~p)−z,Hθ,0(~p)−

z) provided that we choose, for instance,

ρ0 = λ4/50 . (2.6.7)

The condition λ20σ

3Λ(µ sinϑ)−2 ρ0 is then satisfied if we require that λ6/5

0 σ−3Λ (µ sinϑ)2.

Equations (2.6.5) and (2.6.6) give

Fχi0(Hθ(~p)− z,Hθ,0(~p)− z)|Πi0⊗1HE≤ρ0 [Cn0⊗HE ] =

îei0 − z + e−θHE + e−2θ

~P 2E

2− e−θ~p · ~PE

ó− λ2

0Zd(~p)⊗ χ2

ρ0(HE)− λ2

0Zod(~p)⊗ χ2

ρ0(HE) + λ

2+3/50 O

( σ9/2Λ

sin(ϑ)2µ2 min(1, δ20)

).

(2.6.8)

The remainder term is small compared to λ20 provided that

λ3/50 σ

−9/2Λ (µ sinϑ)2 min(1, δ2

0).

2.6.2 The Imaginary Part of z(∞)(~p)

In this section we estimate the imaginary part of z(∞)(~p), assuming here that ~p has real entries.We study the leading order term of Fχi0

(Hθ(~p)−z,Hθ,0(~p)−z)|Πi0⊗1HE≤ρ0 [Cn0⊗HE ], that wedenote by HL(~p)− z (see (2.6.23) below). It is a normal operator whose spectrum is explicitlycomputable: As Zod(~p) and Zd(~p) are rank-one operators, we can write Zod(~p) = zod(~p)Πi0

and Zd(~p) = zd(~p)Πi0 , for some complex numbers zod(~p) and zd(~p). Then we can write HL(~p)as the sum of (ei0 −λ2

0zd(~p)−λ2

0zod(~p))Πi0 plus an operator that is a function of HE and ~PE .

The spectrum of the latter operator lies in the lower half plane, which can be easily shownfrom geometrical considerations, using the spectral theorem. Using analyticity in θ we showthat zd(~p) is real, which implies that the imaginary part of the spectrum of HL(~p) is below−λ2

0=zod(~p). Proving that −=zod(~p) < 0 is, thus, essential to show that =z(∞)(~p) < 0. Thisis where the Fermi Golden Rule is used, see Subsection 2.6.2.

Once we have proven that the imaginary part of the spectrum ofHL(~p) is below−λ20=zod(~p),

which is negative, we conclude by a perturbative argument, using a Neumann series expan-sion, that HL(~p)− z + Rem is invertible if =z is larger than a (strictly) negative number (forsmall λ0). This assertion and the iso-spectrality of the Feshbach-Schur map then imply thatHθ(~p)− z is invertible for such z’s, from which we conclude in Subsection 2.6.2 that =z(∞)(~p)is (strictly) smaller than zero.

Analysis of Zod(~p)

Proposition 2.6.2.1. Let ~p ∈ Uθ[~p∗] ∩ R3. We define (see (2.1.38))

zod(~p) := 〈ψi0 | Zod(~p)ψi0〉. (2.6.9)

The following holds true:

=zod(~p) = π∑j<i0

∫R3dk∣∣∣ ∑s∈1,2,3

(ds)N−j+1,N−i0+1εs(k)∣∣∣2 (2.6.10)

|~k||Λ(~k)|2δÄej − ei0 + |~k| − ~p · ~k +

~k2

2

ä.


Proof. We denote by

k :=1

|k|~k, rk := (r~k, λ), k := (k, λ), ∀k ∈ R3, k 6= (0, λ), ∀r ≥ 0. (2.6.11)

Let

fodk

(z) :=z3e−z2/σ2

Λ (2.6.12)

· 〈ψi0 |~ε(k) · ~dΠi0

Ä n0∑i=1

eiΠi − ei0 + z +z2

2− z~p · k

ä−1Πi0~ε(k) · ~d ψi0〉.

Using spherical coordinates we obtain that (see (2.6.3) and (2.6.4))

zod(~p) =

∫dk

∫ ∞0

dre−θfodk

(e−θr), (2.6.13)

where∫dk denotes the surface integral over the 2 dimensional sphere in R3.

Let γθ : [0,∞)→ C be the path defined by the formula

γθ(r) := e−θr. (2.6.14)

We denote furthermore, for every R > 0, by γθ,R the restriction of γθ to the interval [0, R]and by γθ,R the straight (oriented) line segment with starting point γθ(R) and ending pointγθ(R).

Eq. (2.6.13) implies that we can view the integral with respect to r as a complex integralwith respect to γθ. It follows furthermore that

=zod(~p) =1

2i

∫dk[ ∫

γθ

fodk

(z)dz −∫γθ

fodk

(z)dz]. (2.6.15)

The function fodk

is meromorphic in the region delimited by the curves γθ and γθ containingthe positive part of the real axis. The poles of fod

kin this region are the positive real numbers

r such that ei0 − r− r2

2 + r~p · k belongs σ(∑n0i=1 eiΠi) \ ei0. Since −r− r2

2 + r~p · k is strictlynegative and strictly decreasing as a function of r for r > 0, there are only i0 − 1 poles andthey correspond to the positive real numbers rodj such that

ei0 − rodj −(rodj )2

2+ rodj ~p · k = ej (2.6.16)

for some j < i0. In fact the explicit solutions of Eq. (2.6.16) are given by the formula

ej − ei0 +r2

2+ r(1− ~p · k) =

1

2

[r −

(− (1− ~p · k) +

√2(ei0 − ej) + (1− ~p · k)2

)](2.6.17)

·[r −

(− (1− ~p · k)−

√2(ei0 − ej) + (1− ~p · k)2

)].

Let R > 0 such that the poles are contained in the interior of the (closed) curve γθ,R+γθ,R−γθ,R(the curve γθ,R followed by γθ,R and this last one followed by −γθ,R, which is the curve γθ,Rgoing in the contrary direction). It follows from the exponential decay of fod

kthat∫

γθ

fodk

(z)dz −∫γθ

fodk

(z)dz =

∫γθ,R+γθ,R−γθ,R

fodk

(z). (2.6.18)


From the residue theorem and (2.6.15)-(2.6.18) we conclude that

=zod(~p) = π

∫dk

∑j∈1,··· ,i0−1

Res(fodk, rodj ). (2.6.19)

We obtain finally (2.6.10) from (2.6.19) and from the fact that

Res(fodk, rodj ) = lim

z→rodj(z − rodj )fod

k(z) (2.6.20)

= limz→rodj

(z − rodj )z3e−z2/σ2

Λ

· 〈ψi0 |~ε(k) · ~dΠj

Ä n0∑i=1

eiΠi − ei0 + z +z2

2− z~p · k

ä−1Πj~ε(k) · ~d ψi0〉

=(rodj )3e−(rodj )2/σ2Λ

· 〈ψi0 |~ε(k) · ~dΠj1»

2(ei0 − ej) + (1− ~p · k)2Πj~ε(k) · ~d ψi0〉,

where we used (2.6.16)-(2.6.17).

Estimations of =z(∞)(~p)

Theorem 2.6.2.1. Suppose that the parameters θ, λ0, ρ0 satisfy the conditions of Lemma2.3.1.3 and λ3/5

0 σ−9/2Λ (µ sinϑ)2 min(1, δ2

0). There exists a positive constant C independentof the problem parameters such that, for all ~p ∈ Uθ[~p∗] ∩ R3 and z ∈ C such that |z − ei0 | <ρ0µ sin(ϑ)/32 and

λ20

[C

σ9/2Λ

sin(ϑ)2µ2 min(1, δ20)λ

3/50 −=zod(~p)

]< =z,

the operator Hθ(~p) − z is (bounded) invertible. In particular, if =zod(~p) > 0 (i.e. the FermiGolden Rule is satisfied), the imaginary part of z(∞)(~p) is strictly negative.

Proof. We definezd(~p) := 〈ψi0 | Zd(~p)ψi0〉. (2.6.21)

Applying the procedures of Paragraph 2.6.2, it is easy to prove that zd(~p) does not depend onθ. zd(~p) is therefore real, since for θ = 0, zd(~p) ∈ R. Using (2.6.5) and (2.6.6) we obtain that

Fχi0(Hθ(~p)− z,Hθ,0(~p)− z)|Πi0⊗1HE≤ρ0 [Cn0⊗HE ] =(HL(~p)− z)|Πi0⊗1HE≤ρ0 [Cn0⊗HE ] + Rem,

(2.6.22)

where

HL(~p) :=ei0 −îλ2

0<zod(~p) + λ20zd(~p)óχ2i0 − iλ

20=zod(~p)χ2

i0 (2.6.23)

+îe−θHE + e−2θ

~P 2E

2− e−θ~p · ~PE

óand

‖Rem‖ ≤ Cλ2+3/50

σ9/2Λ

sin(ϑ)2µ2 min(1, δ20). (2.6.24)


As |~p| < 1, it follows that for every r ≥ 0 and every ~l ∈ R3 with |~l| ≤ r ,

r − ~p ·~l ≥ 0, (2.6.25)

which implies that

=[e−θr + e−2θ

~l2

2− e−θ~p ·~l

]≤ 0. (2.6.26)

Eq.(2.6.26) and the Spectral Theorem applied to the normal operator HL(~p) imply thatHL(~p)− z restricted to the range of χi0 is invertible for =z > −λ2

0=zod(~p) and that

‖(HL(~p)− z)−1|Πi0⊗1HE≤ρ0 [Cn0⊗HE ]‖ ≤

1

|=z + λ20=zod(~p)|

. (2.6.27)

A Neumann series expansion together with (2.6.24) and (2.6.27) imply that Fχi0(Hθ(~p) −

z,Hθ,0(~p)− z)|Πi0⊗1HE≤ρ0 [Cn0⊗HE ] is invertible, for =z > −λ20=zod(~p), whenever

Cλ2+3/50

σ9/2Λ

sin(ϑ)2µ2 min(1, δ20)

1

|=z + λ20=zod(~p)|

< 1. (2.6.28)

The conclusions of Theorem 2.6.2.1 follow from this last assertion and the iso-spectrality ofthe Feshbach-Schur map.


2.7 Appendices for Chapter 2

2.7.1 Analyticity of Type (A)

We recall that a map η 7→ Aη from an open connected set V ⊂ Cn0 to the set of (unbounded)operators in a Hilbert space H is called analytic of type (A) if there is a dense domain D ⊂ Hsuch that, for all η ∈ V , Aη is closed on D, and for all ψ ∈ D, the map η 7→ Aηψ is analyticon V .

Lemma 2.7.1.1. Let U := ~p ∈ C3 | |<(~p)| + |=(~p)| < 1. The map (θ, ~p, λ0) 7→ Hθ(~p) isanalytic of type (A) on D(0, π/4)× U × C.

Proof. Let θ ∈ D(0, π/4), ~p ∈ C3 with |<~p|+|=~p| < 1, and λ0 ∈ C. For all ψ ∈ D(HE)∩D(~P 2E),

we have that∥∥∥Äe−2θ~P 2E

2− e−θ~p · ~PE + e−θHE

äψ∥∥∥2

= e−2<θ∥∥∥(HE − ~p · ~PE)ψ

∥∥∥2+

1

4e−4<θ

∥∥∥~P 2Eψ∥∥∥2

+ <Äe−θe−2θ

¨(HE − ~p · ~PE)ψ| ~P 2

Eψ∂ä

≥ e−2<θ(1− |~p|)2‖HEψ‖2 +1

4e−4<θ

∥∥∥~P 2Eψ∥∥∥2

+ e−3<θ<Äei=θ¨(HE − ~p · ~PE)ψ| ~P 2

Eψ∂ä.

Using that | sin=θ| ≤ cos=θ, a direct computation gives

<Äei=θ¨(HE − ~p · ~PE)ψ| ~P 2

Eu〉ä

= cos(=θ)¨(HE −<~p · ~PE)ψ| ~P 2

Eu∂

+ sin(=θ)¨=~p · ~PEψ| ~P 2

Eψ∂

≥ cos(=θ)(1− |<~p| − |=~p|)¨HEψ| ~P 2

Eψ∂≥ 0,

and hence∥∥∥Äe−2θ~P 2E

2− e−θ~p · ~PE + e−θHE

äψ∥∥∥2≥ e−2<θ(1− |~p|)2‖HEψ‖2 +

1

4e−4<θ

∥∥∥~P 2Eψ∥∥∥2.

This implies that e−2θ ~P 2E/2− e−θ~p · ~PE + e−θHE is closed on D(HE) ∩ D(~P 2

E). Since HI,θ isrelatively H1/2

E -bounded, it is infinitesimally small with respect to HE , and therefore, since inaddition

∑n0i=1 eiΠi is bounded, we easily deduce that Hθ(~p) is closed on D(HE) ∩ D(~P 2

E).Verifying that (θ, ~p, λ0) 7→ Hθ(~p) is analytic onD(0, π/4)×U×C for all ψ ∈ D(HE)∩D(~P 2

E)is straightforward (Here we need in particular that the ultraviolet cutoff function Λ is realanalytic).

2.7.2 Proof of Lemmas 2.3.1.1 and 2.3.1.2

Proof of Lemma 2.3.1.1

Let ϕ,ψ ∈ Ran(1HE≤ρHE). We have that

|〈ψ|Wm,nϕ〉| ≤ ‖wm,n‖ 12

∫B

(m,n)ρ

dK(m,n)‖a(k(m))ψ‖‖a(k(n)

)ϕ‖ |k(m)|1/2|k(n)|1/2

≤ ‖wm,n‖ 12V1/2m V1/2

n Dn(ϕ)1/2D1/2m (ψ),


where

Vm :=

∫B

(m)ρ

dk(m),

Dn(ϕ) :=

∫B

(n)ρ

dk(n)|k(n)| ‖a(k(n))ϕ‖2.

A direct computation gives Vm ≤ (m!)−1(8π)mρ3m and an easy argument by induction showsthat Dn(ϕ) ≤ ‖Hn/2

E ϕ‖2 ≤ ρn‖ϕ‖2. Putting all the bounds together, we find that

|〈ψ|Wm,nϕ〉| ≤ ρ2(m+n)‖wm,n‖ 12

(8π)m+n

2

√m!n!

‖ψ‖‖ϕ‖, (2.7.1)

which implies (2.3.3).


Proof of the estimate (2.3.4) . Let j ∈ 1, ..., 3. We introduce

fj(k) = −iΛ(e−θ~k)|~k|1/2εj(k).

It is sufficient to bound a(fj)(HE + ρ)−1/2. We set ω(~k) = |~k|. Thanks to the pull-throughformula, we have that for any ψ ∈ HE ,

‖a(fj)(HE + ρ)−1/2ψ‖ ≤∫dk ‖(HE + ρ+ |~k|)−1/2a(k)fj(k)ψ‖

≤∥∥∥ fjω1/2

∥∥∥L2(R3)

Å∫dk‖(HE + ρ+ ω)−1/2ω1/2a(.)ψ‖2

ã1/2

≤ 2‖Λ(e−iϑ·)‖L2(R3)‖ψ‖

where the last line comes from the equality

||(HE + ρ+ ω(k))−1/2ω1/2a(k)ψ||2 = 〈(HE + ρ)−1/2ψ| ω(k)a∗(k)a(k)(HE + ρ)−1/2ψ〉.

This proves (2.3.4).

Proof of the estimate (2.3.5) . Let (~p, z) ∈ Uρ0 [ei0 ]. We have that

[Hθ,0(~p)− z]|Ran(χi0 ) =n0∑j=1

Πj ⊗ bj(~p, z,HE , ~PE), (2.7.2)

where

bi0(~p, z, r,~l) =

(e−2θ

~l2

2+ e−θr − e−θ~p ·~l + ei0 − z

)1r≥3ρ0/4, (2.7.3)

bj(~p, z, r,~l) = e−2θ~l2

2+ e−θr − e−θ~p ·~l + ej − z (j 6= i0). (2.7.4)

Any vector ϕ ∈ HE can be represented as a sequence (ϕ(n)) of completely symmetric functionsof momenta, ϕ(n) ∈ L2

s(R3n). The operators bj(~p, z,HE , ~PE) are multiplication operators in


this representation. Therefore, we only need to show that |bi0(~p, z, r,~l)| and |bj(~p, z, r,~l)|,j 6= i0, are bounded below by strictly positive constants. This amounts to estimate thedistance between z and the range of bi0(~p, 0, ·, ·) and bj(~p, 0, ·, ·), j 6= i0; see Figure 2.3 below.

... ei0−1 ei0 ei0+1 ...ϑ

2ϑr0

Figure 2.3: The spectral parameter z is located inside the disk D(ei0 , r0) of center ei0 and radius r0.|ei0 − ej | ≥ δ0 for all j 6= i0. The grey shaded regions contain the range of bi0(~p, 0, ·, ·) and bj(~p, 0, ·, ·),j 6= i0.

Let r ≥ 3ρ0

4 and |~l| ≤ r. Estimating |bi0(~p, z, r,~l)| from below by the absolute value of itsreal part, we obtain

|bi0(~p, z, r,~l)| ≥∣∣∣∣ cos(2ϑ)

~l2

2+ cos(ϑ)

Är −<(~p) ·~l

ä−=(~p) ·~l sin(ϑ) + ei0 −<(z)

∣∣∣∣≥ r cos(ϑ) (1− |~p| − |=(~p)| tan(ϑ))− |ei0 − z|,

as θ = iϑ, with 0 < ϑ < π/4. Since |~p| < 1− µ, and |=~p| < µ tan(ϑ)/2 < µ/2, we finally findthat

|bi0(~p, z, r,~l)| ≥ r cos(ϑ)µ

2− ρ0µ

32>ρ0µ

8.

For j < i0, the difference ej − z can be cancelled by e−2θ~l2

2 + e−θr − e−θ~p ·~l if ρ0 is not smallenough, and we have to impose a constraint on ρ0 that depends on the minimal separationbetween the eigenvalues, δ0. Let ρ0 < δ0. We assume that 0 < δ0 < 1. For δ0 ≥ 1, itsuffices to replace δ0 by 1 in the next estimates. We split the interval [0,∞) into two disjointssubintervals I1 and I2, with I1 = [0, δ0/8] and I2 = (δ0/8,∞). We give a lower bound for bjon both intervals. Let r ∈ I1. We have that

|bj(~p, z, r,~l)| ≥ |ei0 − ej | − |z − ei0 | − r cos(ϑ)− rÅ|~p| cos(ϑ) + |=(~p)| sin(ϑ) +

r

2

ã> |ei0 − ej | −

δ0

2≥ δ0

2.

(2.7.5)

Let r ∈ I2. We estimate |bj(~p, z, r,~l)| from below by the absolute value of its imaginary part.An easy calculation shows that

|bj(~p, z, r,~l)| ≥ r sin(ϑ) (1− |~p| − |=(~p)| cot(ϑ))− ρ0µ sin(ϑ)

32.

Since we have chosen |=(~p)| ≤ µ tan(ϑ)/2, it follows that

|bj(~p, z, r,~l)| ≥δ0µ sin(ϑ)

32. (2.7.6)

We deduce that [Hθ,0(~p)−z]|Ran(χi0 ) is bounded invertible and that its inverse satisfies (2.3.5).The proof of (2.3.6) is similar.


2.7.3 Proof of Lemma 2.3.2.1

Proof of the estimates

To prove Lemma 2.3.2.1, we re-Wick order the product of creation and annihilation operatorsthat appear in

Πi0 ⊗H(0)(~p, z) = Fχi0(Hθ(~p)− z,Hθ,0(~p)− z)|Πi0⊗1HE≤ρ0 [Cn0⊗HE ]. (2.7.7)

We recall that we want to find a sequence of kernels (w(0)M,N ), M + N ≥ 0, with w

(0)M,N :

Uρ0 [ei0 ]× Bρ0 ×B(M,N)ρ0 → C, and a map E(0) : Uρ0 [ei0 ]→ C, such that

H(0)(~p, z) =∑

M+N≥0

W(0)M,N (~p, z) + E(0)(~p, z). (2.7.8)

We remind the reader that the operators W (0)M,N (~p, z) are defined in the sense of quadratic

forms by

W(0)M,N (~p, z) = 1HE≤ρ0

Ç∫B

(M,N)ρ0

dK(M,N)a∗(k(M))w(0)M,N

Ä~p, z,HE , ~PE ,K

(M,N)äa(k

(N))

å1HE≤ρ0 ,

for all M +N ≥ 1.For any bounded operator A on Cn0 ⊗HE , we denote by 〈A〉i0 the bounded operator on

HE associated to the bounded quadratic form

(ψ, φ) 7→ 〈ψi0 ⊗ ψ|A(ψi0 ⊗ φ)〉 ∈ C. (2.7.9)

The bounded operator H(0)(~p, z) in (2.7.7) identifies with the following operator (denoted bythe same symbol) on HE :

H(0)(~p, z) =( ÄHE − ~p · ~PE

äe−θ +

~P 2E

2e−2θ + ei0 − z

)1HE≤ρ0 + λ0〈χi0HI,θχi0〉i0

− λ20〈χi0HI,θχi0 [Hχi0

(~p, z)]−1|Ran(χi0 )χi0HI,θχi0〉i0 .

(2.7.10)

The operators on the first line are already Wick-ordered. The first operator contributes toW

(0)0,0 (~p, z) + E(0)(~p, z), and the second to W

(1)1,0 (~p, z) and W

(1)0,1 (~p, z). To normal order the

operator on the second line, we use the Neumann expansion for [Hχi0(~p, z)]−1

|Ran(χi0 ). It is easyto show that

〈χi0HI,θχi0 [Hχi0(~p, z)]−1

|Ran(χi0 )χi0HI,θχi0〉i0 =∞∑L=2

(−λ0)L−2VL(~p, z), (2.7.11)

where

VL(~p, z) := 〈χi0ÅHI,θχ

2i0 [Hθ,0(~p)− z]−1

|Ran(χi0 )

ãL−1

HI,θχi0〉i0 .

We first split HI,θ into the sum of two operators (HI,θ)0,1 and (HI,θ)

1,0, where

(HI,θ)0,1 = i

∫R3dk |~k|1/2e−2θΛ(e−θ~k)~ε(~k) · ~d a(k), (2.7.12)

(HI,θ)1,0 = −i

∫R3dk |~k|1/2e−2θΛ(e−θ~k)~ε(~k) · ~d a∗(k). (2.7.13)


This yields

VL(~p, z) =∑

p1, ..., pLq1, ..., qLpi + qi = 1

〈χi0L−1∏j=1

Å(HI,θ)

pj ,qjχ2i0 [Hθ,0(~p)− z]−1

|Ran(χi0 )

ã(HI,θ)

pL,qLχi0〉i0 . (2.7.14)

Let j ∈ 1, ..., L. The operator-valued distribution a(kj)/a∗(kj) that appears in (HI,θ)pj ,qj

can either be contracted with another creation/annihilation operator appearing in (HI,θ)pj′ ,qj′ ,

j′ 6= j, or left uncontracted. In the latter case, we pull it to the right of (2.7.14) if it is anannihilation operator, or to the left of (2.7.14) if it is a creation operator. This modifies theoperators χ2

i0[Hθ,0(~p)− z]−1

|Ran(χi0 ) via the pull-though formula


which is valid for any measurable function g : R4 → C. We introduce the notations

(HI,θ)0001 = i|~k|1/2e−2θΛ(e−θ~k)~ε(~k) · ~d,

(HI,θ)0010 = −i|~k|1/2e−2θΛ(e−θ~k)~ε(~k) · ~d,

(HI,θ)1000 = (HI,θ)

1,0, (HI,θ)0100 = (HI,θ)

0,1.

The contracted part is expressed as a vacuum expectation value, and

VL(~p, z) =∑

m,n,p,qmi+ni+pi+qi=1

∫a∗(k(m))〈ψi0⊗Ω|V m+p,n+q

m,n (~p, z,HE , ~PE ,K(m,n))(ψi0⊗Ω)〉a(k

(n))dK(m,n)

(2.7.16)where m = (m1, . . . ,mL),

Vm+p,n+qm,n (~p, z, r,~l,K(m,n)) = χρ0(r + r0)

L−1∏j=1

[(HI,θ)

pjqjmjnj

R(~p, z,HE + r + rj , ~PE +~l +~lj)](HI,θ)

pLqLmLnL

χρ0(r + rL),

(2.7.17)

and

R(~p, z,HE + r, ~PE +~l) :=∑j 6=i0

Pj ⊗ [bj(~p, z,HE + r, ~PE +~l)]−1

+ Pi0 ⊗ χ2ρ0

(HE + r)[bi0(~p, z,HE + r, ~PE +~l)]−1;

(2.7.18)

(see Eqs. (2.7.3)-(2.7.4)). The shifts ri’s, ri’s, ~li’s,~li’s come from the Pull-through formula

and are given by

ri :=i−1∑j=1

Σ[k(nj)j ] +

L∑j=i+1

Σ[k(mj)j ], ~li :=

i−1∑j=1

~Σ[k(nj)j ] +

L∑j=i+1

~Σ[k(mj)j ],

ri :=i∑

j=1

Σ[k(nj)j ] +

L∑j=i+1

Σ[k(mj)j ],

~li :=

i∑j=1

~Σ[k(nj)j ] +

L∑j=i+1

~Σ[k(mj)j ]. (2.7.19)


where Σ[k(m)] :=∑mi=1 |~ki|, Σ[k(n)] :=

∑nj=1 |

~kj |, ~Σ[k(m)] :=

∑mi=1

~ki, ~Σ[k(n)] :=∑nj=1

~kj . We

rewrite


(~p, z)]−1|Ran(χi0 )χi0HI,θχi0〉i0 =

∑M+N≥0

W(0)M,N (~p, z), (2.7.20)

where the kernels w(0)M,N are given by

w(0)M,N (~p, z, r,~l, k(M), k

(N)) =

−∑L=2

(−λ0)L∑

m,n, p, q

m1 + ... +mL = Mn1 + ... + nL = N

mi + ni + pi + qi = 1

〈ψi0 ⊗ Ω|V m+p,n+qm,n (~p, z, r,~l,K(M,N))ψi0 ⊗ Ω〉sym,

(2.7.21)

for M +N ≥ 1, and

w(0)0,0(~p, z, r,~l) = −

∑L=2

(−λ0)L∑p, q

pi + qi = 1

〈ψi0 ⊗ Ω|V p,q

0,0 (~p, z, r,~l)ψi0 ⊗ Ω〉(2.7.22)

for M +N = 0. fsym denotes the symmetrization of f with respect to the variables k(M) andk

(N). Using the estimate (2.3.4),

‖(HE + ρ0)−1/2HI,θ(HE + ρ0)−1/2‖ = O(ρ−1/20 σ

3/2Λ ), (2.7.23)

we obtain the bound

‖V m+p,n+qm,n (~p, z, r,~l,K(m,n))‖ ≤ ρ0

L∏j=1

∥∥∥∥(HE + ρ0)−1/2(HI,θ)pjqjmjnj (HE + ρ0)−1/2

∥∥∥∥supj

∥∥∥∥(HE + ρ0)R(HE + r + rj , ~PE +~l +~lj)

∥∥∥∥L−1

.

(2.7.24)

Distinguishing the cases pj + qj = 1 and pj + qj = 0 and using Estimate (2.3.6) of Lemma2.3.1.2, we deduce that there exists a positive constant C such that

‖V m+p,n+qm,n (~p, z, r,~l,K(m,n))‖ ≤ ρ0

CL

(µ sin(ϑ))L−1

L∏j=1

(σ3/2Λ ρ

−1/20 )pj+qjρ

−(mj+nj)0 |k(mj)

j |1/2|k(nj)j |1/2.

(2.7.25)

Since mj +nj + pj + qj = 1, it follows that (up to a multiplication of C by a numerical factor)

‖w(0)M,N (~p, z)‖ 1

2≤∞∑L=2

(Cσ3/2Λ λ0ρ

−1/20 )L

(µ sin(ϑ))L−1ρ− 1

2(M+N)

0 ρ0. (2.7.26)

Similarly,

‖w(0)0,0(~p, z)‖∞ ≤

∞∑L=2

(Cσ3/2Λ λ0ρ

−1/20 )L

(µ sin(ϑ))L−1ρ0, (2.7.27)


and proceeding in the same way for the derivatives ∂#w(0)M,N , where ∂# stands for ∂r or ∂lj ,

we conclude that

‖w(0)M,N (~p, z)‖ 1

2= O

Ñλ2

0σ3Λ

µ sin(ϑ)ρ12

(M+N)0

é, (2.7.28)

‖∂#w(0)M,N (~p, z)‖ 1

2= O

Ñλ2

0σ3Λ

µ2 sin2(ϑ)ρ12

(M+N)+10

é, (2.7.29)

uniformly with respect to M +N ≥ 1, and that

‖w(0)0,0(~p, z)‖∞ = O

Çλ2

0σ3Λ

µ sin(ϑ)

å, (2.7.30)

‖∂#w(0)0,0(~p, z)‖∞ = O

Çλ2

0σ3Λ

µ2 sin2(ϑ)ρ0

å. (2.7.31)

Since λ0 ≥ 0 is chosen such that λ0 µ sin(ϑ)ρ1/20 σ

3/2Λ , this concludes the proof of the

estimates of the lemma.

Proof of the analyticity of the map (~p, z) 7→ H(0)(~p, z) ∈ B(H(0))

We start from the expression of H(0)(~p, z) given in (2.7.10),

H(0)(~p, z) =( ÄHE − ~p · ~PE

äe−θ +

~P 2E

2e−2θ + ei0 − z

)1HE≤ρ0 + λ0〈χi0HI,θχi0〉i0


(~p, z)]−1|Ran(χi0 )χi0HI,θχi0〉i0 .

(2.7.32)

The first term on the right-hand side of (2.7.32) is analytic on Uρ0 [ei0 ]. We have seen inthe proof of Lemma 2.3.1.3 that the Neumann series for [Hχi0

(~p, z)]−1|Ran(χi0 ) is uniformly

convergent on Uρ0 [ei0 ]. It is therefore sufficient to check that the map

(~p, z) 7→ñHE + ρ0

Hθ,0(~p, z)

ô|Ran(χi0 )

∈ B(Ran(χi0))

is analytic on Uρ0 [ei0 ]. Since weak and strong analyticity are equivalent, we only need to showthat the maps

(~p, z) 7→¨ψj ⊗ ϕ

∣∣∣ ñ HE + ρ0

Hθ,0(~p, z)

ô|Ran(χi0 )

ψj ⊗ ϕ∂∈ C (2.7.33)

are analytic for any ψj ⊗ ϕ ∈ Ran(χi0), where ψj , j = 1, ..., N , are unit eigenvectors of∑n0i=1 eiΠi associated to the eigenvalues ej . Using, for any ϕ in Fock space, the representation

ϕ = (ϕ(n)) with ϕ(n) ∈ L2s(R3n), we find that

〈ψj ⊗ ϕ∣∣∣ ñ HE + ρ0

Hθ,0(~p, z)

ô|Ran(χi0 )

ψj ⊗ ϕ∂

= (1− δi0j)ρ0

ej − z|ϕ(0)|2

+∑n≥1

∫R3n

dk(n) |k1|+ · · ·+ |kn|+ ρ0

bj(~p, z,Σ[k(n)], ~Σ[k(n)])|ϕ(n)(k1, ..., kn)|2,


where bj is defined in (2.7.3)–(2.7.4), and we have set Σ[k(n)] = |k1|+ · · ·+ |kn| and ~Σ[k(n)] =~k1 + · · ·+~kn. The functions (~p, z) 7→ (r+ρ0)/bj(~p, z, r,~l) are analytic on Uρ0 [ei0 ] for any fixed(r,~l) ∈ R+ × R3 with |~l| ≤ r, and |(r + ρ0)/bj(~p, z, r,~l)| is uniformly bounded as follows fromthe proof of Lemma 2.3.1.3. Using Morera’s theorem for several complex variables (see [127]),we deduce that

(~p, z) 7→¨ψj ⊗ ϕ

∣∣∣ ñ HE + ρ0

Hθ,0(~p, z)

ô|Ran(χi0 )

ψj ⊗ ϕ∂∈ C (2.7.34)

is analytic, and hence, that H(0)(~p, z) is analytic in (~p, z) ∈ Uρ0 [ei0 ].

Chapter 3

Closed systems in Quantum Mechanics

3.1 Introduction

We discuss two models exhibiting closed subsystems. These models are introduced in Sections3.2.1 and 3.2.2. We remind the reader that the first model describes a quantum particle, P ,with spin 1/2 interacting with a large quantum system Q and moving away from Q. (Thesubsystem Q may consist of another quantum particle entangled with P and a “detector”.The two particles are prepared in an initial state chosen such that they move away from eachother, with P moving away from the detector.) The second model describes a neutral atom Pwith a non-vanishing electric dipole moment that interacts with a large quantum system Q.Both P and Q are coupled to the quantized electromagnetic field, E. The goal is to identifyan effective dynamics for P that does not make any explicit reference to the electromagneticfield E. Our results are stated in subsections 3.2.1 and 3.2.2. Proofs are presented in section3.3. Many of the techniques used in our proofs are inspired by scattering theory; see, e.g.,[113, 46, 63, 64, 55, 56]. Some technical lemmas are proven in two appendices.

3.2 Models and main results

3.2.1 Model 1: A quantum particle P interacting with a large quantumsystem Q

Description of the model

We consider a quantum particle, P , of mass m = 1 and spin 1/2. The particle interacts witha large quantum system, Q, which we keep as general as possible. The pure states of thecomposed system, P ∨Q, correspond to unit rays in the Hilbert space H = HP ⊗HQ, whereHP := L2(R3)⊗C2 and HQ is a separable Hilbert space. The dynamics of P ∨Q is specifiedby a selfadjoint Hamiltonian

H = HP ⊗ 1HQ + 1HP ⊗HQ +HP,Q (3.2.1)

defined on a dense domain D(H) ⊂ H. In (3.2.1),

HP := −∆

2⊗ 1C2 .

74

CHAPTER 3. CLOSED SYSTEMS IN QUANTUM MECHANICS 75

The operator HP and HQ, defined on their respective domains, are self-adjoint.

Remark. It is not important to exclude the presence of external fields or potentials acting onthe particle. All that matters is that the propagation of the particle approaches the one of afree particle as time tends to ∞. To keep our analysis simple we assume that if the interactionbetween P and Q is turned off then P propagates freely.

To identify P as a closed subsystem of P ∨Q, one assumes that

1. the strength of the interaction between P and Q (described by the operator HP,Q) de-cays to zero rapidly as the “distance” between P and Q tends to ∞; and

2. the initial state of the system is chosen such that the particle P propagates away fromQ, the distance between P and Q growing ever larger. (We will actually choose theinitial state such that, with very high probability, the particle P is scattered into a conefar separated from the subsystem Q.)

A graphical illustration of Assumptions (1) and (2) is given below.

OxO2θ0

Q

Ω

d

Figure 3.1: The system Q is localized inside the domain Ω. The particle P scatters inside the grey coloredset with a probability very close to 1.

Next, we reformulate Assumptions (1) and (2) in mathematically precise terms. It is conve-nient to identify L2(R3) ⊗ C2 ⊗ HQ with L2(R3;C2 ⊗ HQ). We denote by (~ex, ~ey, ~ez) threeorthonormal vectors in R3.

(A1) (Location of Q and properties of the interaction Hamiltonian) There is an open subsetΩ ⊂ R3 (possibly unbounded), the “spatial location” of the subsystem Q, separated fromthe cone

C2θ0 := ~k ∈ R3 | ~k · ~ex ≥ |~k| cos(2θ0) (3.2.2)

with π/4 > θ0 > 0, by a distance d > 0, and a covering

Ω =⋃n∈I

Ωn, I ⊆ N


of Ω by open cubes Ωn of uniformly bounded diameter such that

(i) the interaction Hamiltonian HP,Q can be written as a strongly convergent sum ofoperators,

HP,Q =∑n∈I

HP,Qn , (3.2.3)

on the dense domain D(HP,Q) ⊇ D(HP )⊗D(HQ).The operator HP,Qn encodes the interaction between the particle P and the sub-system of Q located in the cube Ωn. The distance between Ωn and the cone C2θ0 isdenoted by dn and is supposed to tends to +∞, as n tends to ∞;

(ii) there is a constant α > 1 and a sequence Nnn∈I of operators on HQ with theproperties that

‖(HP,QnΨ)(~x)‖C2⊗HQ ≤‖(1HP ⊗Nn)Ψ(~x)‖C2⊗HQ

[dist(Ωn, ~x)]α, ~x ∈ Ωc, (3.2.4)

for all n ∈ I and for all Ψ ∈ D(HP )⊗D(HQ), and∑n∈I

d1−α

2n ≤ Cd−β, for some β > 0, C <∞. (3.2.5)

Furthermore, [HP,Qn , ~x] = 0 for all n ∈ I.

(A2) (Choice of initial state) The initial state Ψ0 ∈ S(R3;C2 ⊗HQ), ‖Ψ0‖ = 1, is a smoothfunction of ~x of rapid decay with values in C2 ⊗HQ. Its Fourier transform,“Ψ0(~k) :=

1

(2π)3/2

∫R3e−i

~k·~xΨ0(~x) d3x, (3.2.6)

has support in the conical region Cθ0;v defined by

Cθ0;v := ~k ∈ R3 | ~k · ~ex ≥ |~k| cos(θ0), |~k| > v (3.2.7)

for some v > 0.

(A3) (Bound on the number of particles in Ωn) For s ∈ 1, 2,

‖(1HP ⊗Nn)e−itHQΨ0‖Ls(R3;C2⊗HQ) < C, ∀t ≥ 0,∀n ∈ I. (3.2.8)

Remark 3.2.1.1. Assumption (A2) guarantees that the distance between P and Q growsin time with very high probability. The hypotheses (A1) and (A3) are mathematical refor-mulations of Assumption (1). The operator Nn can be thought of as counting the numberof “particles” of the system Q contained in the subset Ωn, for all n ∈ I. If the system Qis composed of identical particles, HQ is the bosonic/fermionic Fock space over L2(R3;Cp),(p = 1, 2, ...) and the operator Nn is the second quantization of the multiplication operator bythe characteristic function 1Ωn.

The decomposition of Ω into cubes is used to get bounds that are uniform in the number ofdegrees of freedom of the system Q.

We observe that the decay in (3.2.4) is faster than the one of the Coulomb potential. Tojustify (3.2.4) one would have to invoke screening.


Result

The reduced density matrix, ρP , of the particle P corresponding to the state Ψ0 ∈ L2(R3;C2⊗HQ) of the entire system P is defined by

〈ϕ1|ρPϕ2〉L2(R3;C2) :=∑j∈J〈ϕ1 ⊗ ej |Ψ0〉〈Ψ0|ϕ2 ⊗ ej〉,

where ϕ1, ϕ2 are arbitrary vectors in L2(R3;C2) and ejj∈J is an orthonormal basis in HQ.

Lemma 3.2.1.1. We require assumptions (A1), (A2) and (A3). Then, for all η > 0, thereexists a length d(η, v) > 0 such that, for any d > d(η, v),∣∣∣〈e−itHΨ0|(OP ⊗ 1HQ)e−itHΨ0〉 − TrHP (ρP e

itHPOP e−itHP )

∣∣∣ ≤ η‖OP ‖, (3.2.9)

for all OP ∈ B(L2(R3;C2)) and for all t ≥ 0.

Lemma 3.2.1.1 justifies considering P as a closed subsystem: any observable of the subsys-tem P evolves as if Q were absent, up to an error term that can be made arbitrarily small byincreasing the separation between P and Q. A similar result was already discussed in [113],but with a finite range interaction between P and Q. The proof of Lemma 3.2.1.1 is given inAppendix 3.5.

3.2.2 Model 2: A neutral atom coupled to a quantum system Q and to thequantized electromagnetic field

The model

We consider a neutral atom P that interacts with a quantum system Q and the quantizedelectromagnetic field. The atom either moves freely or moves in a slowly varying externalpotential. We assume that, initially, it is localized (with a probability close to one) far awayfrom the system Q, and we allow the system Q to create and annihilate photons. Our aimis to prove a result of the form of (2.1.44). Our estimates for this model are however notuniform in the number of degrees of freedom of the subsystem Q. This problem could besolved by decomposing Q into small subsystems. This complication is avoided to keep ourexposition as simple as possible. We do not specify the nature of Q, but we emphasize thatit could represent another atom or a molecule. The internal degrees of freedom of the atomP are described by a two-level system. The total Hilbert space of the system P is the tensorproduct space

H := HP ⊗HQ ⊗HE ,

whereHP := L2(R3)⊗ C2 and HE := F+(L2(R3))

are the Hilbert spaces associated to the atom and to the electromagnetic field, respectively.Here F+(L2(R3)) is the (symmetric) Fock space over L2(R3). We use the notation

R3 := R3 × 1, 2 =¶k := (~k, λ) ∈ R3 × 1, 2

©, dk =

∑λ=1,2

d3k,


where ~k is the photon momentum and λ denotes the polarization of the photon. Any elementΦ ∈ F+(L2(R3)) can be represented as a sequence (Φ(n)) of totally symmetric n-photonsfunctions. We remind the reader that the scalar product on F+(L2(R3)) is defined by

〈Φ|Ψ〉 =∑n≥0

∫R3n

Φ(n)

(k1, ..., kn)Ψ(n)(k1, ..., kn)dk1...dkn

for all Φ,Ψ ∈ F+(L2(R3)).The Hamiltonian of the total system is written as

H :=HP ⊗ 1HQ ⊗ 1HE + 1HP ⊗HQ ⊗ 1HE + 1HP ⊗ 1HQ ⊗HE

+HP,E +HP,Q +HQ,E ,

where

HP := −∆

2+

Çω0 00 0

åis the free atomic Hamiltonian, with ω0 the energy of the excited internal state of the atom,HQ the Hamiltonian for the system Q, and

HE := dΓ(|~k|) ≡∫R3|~k|a∗(k)a(k)dk

is the second quantized Hamiltonian of the free electromagnetic field. The operator-valueddistributions a(k) := aλ(~k) and a∗(k) := a∗λ(~k) are the photon annihilation and creationoperators. We suppose that HQ is a semi-bounded self-adjoint operator on HQ. In whatfollows, we write HP for HP ⊗1HQ⊗1HE , and likewise for HQ and HE , unless confusion mayarise.

The interaction Hamiltonians, HP,E , HP,Q and HQ,E describe the interactions betweenthe atom, the system Q, and the quantized field. The atom-field interaction is of the formHP,E = −~d · ~E, where ~d = −λ0~σ is the dipole moment of the atom, ~σ is the vector of Paulimatrices, and ~E is the quantized electric field, i.e.,

HP,E := iλ0

∫R3χ(~k)|~k|

12~ε(k) · ~σ

(ei~k·~xa(k)− e−i~k·~xa∗(k)

)dk,

where χ ∈ C∞0 (R3; [0, 1]) is an ultraviolet-cutoff function such that χ ≡ 1 on ~k ∈ R3, |~k| ≤1/2 and χ ≡ 0 on ~k ∈ R3, |~k| ≥ 1, and ~ε(k) := ~ελ(~k) are polarization vectors of theelectromagnetic field in the Coulomb gauge. With the usual notations, HP,E can be rewrittenin the form

HP,E = Φ(hx) ≡ a∗(hx) + a(hx), (3.2.10)

withhx(k) := −iλ0χ(~k)|~k|

12~ε(k) · ~σe−i~k·~x. (3.2.11)

By standard estimates (see Lemma 3.4.4.1), HP,E isHP +HE-bounded with relative bound0. We suppose that HP,Q and HQ,E are symmetric operators relatively bounded with respectto HP + HQ and HQ + HE , respectively, and that H is a self-adjoint operator with domainD(H) = D(HP +HQ+HE) ⊃ H2(R3)⊗C2⊗D(HQ)⊗D(HE). Further technical assumptionson HP,Q and HQ,E needed to state our main theorem will be described below.


Assumptions

We assume that:

1. The support of the initial atomic wave function is contained inside a ball of radius Rcentered at the origin.

2. The interaction Hamiltonian HP,Q1|~x|≤d between the atom and Q, restricted to the ballof radius d R centered at the origin, is bounded by Cd−β , for some constants C andβ > 0.

3. With very high probability, the subsystem Q does not emit nor absorb photons insidethe ball of radius 3d centered at the origin.

Q

Q

3d

R

Assumptions (2) and (3) imply that the system Q does not penetrate into the ball ofradius 3d centered at the origin. This hypothesis can be weakened for concrete choices of thesubsystem Q.

We recall the definition of the scattering identification operator (see [90], [46] or [64] formore details) and a few other standard tools from scattering theory to rewrite Assumptions (1)through (3) in mathematically precise terms. Let Ffin denote the set of all vectors Φ = (Φ(n)) ∈F+(L2(R3)) such that Φ(n) = 0 for all but finitely many n’s. The map I : Ffin ⊗ Ffin → Ffin

is defined as the extension by linearity of the map

I : a∗(g1) · · · a∗(gm)Ω⊗ a∗(h1) · · · a∗(hn)Ω 7→ a∗(h1) · · · a∗(hn)a∗(g1) · · · a∗(gm)Ω, (3.2.12)

for all g1, . . . , gm, h1, . . . , hn ∈ L2(R3). The closure of I on HE ⊗HE is denoted by the samesymbol and is called the scattering identification operator. Observe that I is unbounded. Let

H0 := HP ⊗HE , H∞ := HQ ⊗HE .

The Hilbert space H0 corresponds to the atom together with photons located near the origin,whereas H∞ corresponds to the system Q together with photons located far from the origin.We extend the operator I to the space H0 ⊗H∞ by setting

I : H0 ⊗H∞ → H.


We use I to “amalgamate” H0 with H∞.We recall that the Hamiltonian HP∨E on H0 = HP ⊗ HE associated with the atom and

the quantized radiation field,

HP∨E := HP +HE +HP,E ,

is translation-invariant, in the sense that HP∨E commutes with each component of the totalmomentum operator

~PP∨E := ~PP + ~PE = −i~∇x +

∫R3

~ka∗(k)a(k)dk.

This implies (see e.g. [8] or [53] for more details) that there exists a unitary map

U : HP ⊗HE →∫ ⊕R3

C2 ⊗HE d3p,

such thatUHP∨EU

−1 =

∫ ⊕R3H(~p)d3p.

For any fixed total momentum ~p ∈ R3, the Hamiltonian H(~p) is a self-adjoint, semi-boundedoperator on C2 ⊗HE . Its expression is given in Appendix 3.4.1. It turns out that, for |~p| < 1and for small coupling λ0, H(~p) has a ground state with associated eigenvalue E(~p), and thatthis ground state, ψ(~p), is real analytic in ~p, for |~p| < 1; see [53], Theorem 3.4.1.1 in theappendix, and Chapter 2. Given 0 < ν < 1, we assume in the rest of this chapter that |λ0| <λc(ν), where λc(ν) > 0 is the critical coupling constant such that H(~p) has a ground state forall ~p ∈ R3 with |~p| < ν. We introduce a dressing transformation J : L2(R3) → HP ⊗ HE ,defined, for all u ∈ L2(R3) and for a.e. ~x ∈ R3, by the expression

J (u)(~x) :=1

(2π)32

∫R3u(~p)ei~x·(~p−

~PE)χBν/2(~p)ψ(~p) d3p, (3.2.13)

where χBν/2 ∈ C∞0 (R3; [0, 1]) is such that χBν/2 ≡ 1 on Bν/4 = ~p ∈ R3, |~p| < ν/4, andχBν/2 ≡ 0, outside Bν/2 := ~p ∈ R3, |~p| ≤ ν/2. The state J (u) describes a dressed single-atom state. We recall that, for any operator a on L2(R3), the second quantization of a, Γ(a),is the operator defined on HE by its restriction to the n-photons Hilbert space, which is givenby

Γ(a)|L2(R3)⊗ns := ⊗na, n = 0, 1, 2, ... (3.2.14)

and ⊗0a = 1. We denote byN :=

∫R3a∗(k)a(k)dk

the photon number operator on Fock Space. We are ready to state our main assumptions.

(B1) (Initial state of atom) Let v ∈ L2(R3) be such that supp(v) ⊂ ~x ∈ R3, |~x| ≤ 1. Theinitial orbital wave function of the atom is supposed to be of the form

u(~x) = R−3/2v(R−1~x),


for some R ≥ 1. In particular,

supp(u) ⊂ ~x ∈ R3, |~x| ≤ R,

and ‖u‖L2 = ‖v‖L2 is independent of R.

(B2) (Initial state of photons far from the atom) The state ϕ ∈ H∞ = HQ ⊗HE satisfiesÄ1HQ ⊗ Γ(1|~y|≥3d)

äϕ = ϕ,

for some d > 0, where ~y := i~∇k denotes the “photon position variable”, and

ϕ ∈ D(HQ∨E) ∩ D(1HQ ⊗ eδN ),

for some δ > 0.

(B3) (Interaction P −Q) The interaction Hamiltonian between the atom and the subsystemQ, HP,Q, is a symmetric operator on HP ⊗HQ, relatively bounded with respect to H,and satisfying

‖HP,Q1|~x|≤d‖ ≤ Cd−β (3.2.15)

for some constants C and β > 0.

(B4) (Interaction Q − E) The interaction Hamiltonian between the subsystem Q and theradiation field, HQ,E , is a symmetric operator on HQ ⊗ HE such that HQ∨E = HQ +HE +HQ,E is self-adjoint on

D(HQ∨E) = D(HQ +HE).

Moreover, in the sense of quadratic forms, HQ,E satisfiesîHQ,E , a

](1|~y|≤3dh)ó

= 0,îHQ,E ,Γ(j(~y))

ó= 0, (3.2.16)

for all h ∈ L2(R3) and for all Fourier multiplication operators j(~y) on L2(R3) such thatj(~y)1|~y|≥3d = 1|~y|≥3d, where a] stands for a or a∗.

(B5) (Number of photons emitted by Q) The initial state ϕ satisfies e−itHQ∨Eϕ ∈ D(N) for alltimes t ≥ 0, and ∥∥∥dΓ(1|~y|≥ct)e

−itHQ∨Eϕ∥∥∥ ≤ C〈t〉, (3.2.17)

for some c > 1, where C is a positive constant depending on ϕ and 〈t〉 := (1 + t2)1/2.


Remark 3.2.2.1. (B1), (B3) and (B4) are direct mathematical reformulations of the hy-potheses (1), (2) and (3) above. In (B2), we assume that, initially, photons in contact withQ are “localized” outside the ball of radius 3d centered at the origin.

The constant C in (3.2.15) depends a priori on the number of degrees of freedom of thesubsystem Q. This problem could be circumvented by decomposing Q into subsystems locatedever further away from P , as we did for the first model.

Assumption (3.2.16) is very strong and can be relaxed in concrete examples for the subsys-tem Q. For instance, if HQ,E is linear in annihilation and creation operatorP,Eq. (3.2.16) isnot relevant and the estimate of the norm of the commutator of HQ,E with other operators onFock space can be carried out directly. The calculations are the same as for the operator HP,E.

Assumption (B5) implies that the number of photons created by Q does not grow fasterthan linearly in time. Indeed, using Hardy’s inequality and the fact that D(HQ∨E) ⊂ D(HE),we have that∥∥∥dΓ(1|~y|≤ct)e

−itHQ∨Eϕ∥∥∥ ≤ ct∥∥∥dΓ

Ä|~y|−1

äe−itHQ∨Eϕ

∥∥∥ ≤ ct‖HEe−itHQ∨Eϕ

∥∥∥ ≤ Ct.(3.2.17) says that the number of photons emitted by Q and traveling faster than light growsat most linearly in time. Eq. (3.2.17) could be weakened by a polynomial growth. This wouldlead to worse estimates in Theorem 3.2.2.1 below. Assumption (B5) is not fully satisfactory,since the upper bound may depend on the number of degrees of freedom of Q. The main reasonwhy we impose (3.2.17) is that photons are massless. The operator N is not HE-bounded, andsome of our estimates cannot be proven if we do not control the time evolution of the totalnumber of photons.

For massive particles, the dispersion law ω(~k) = |~k| is replaced by ω(~k) =»~k2 +m2, where

m > 0 is the mass of the particles of the field. Since N is HE-bounded, and since, under ourassumptions, D(HQ∨E) ⊂ D(HE), we have that∥∥∥Ne−itHQ∨Eϕ∥∥∥ ≤ CÄ‖HQ∨Eϕ‖+ ‖ϕ‖

ä.

Hypothesis (B5) is therefore obviously satisfied for massive particles. To simplify our presen-tation, we only state and prove our main result for photons.

Main Result

Our aim is to show that, under Assumptions (B1)-(B5), P behaves as a closed system overa finite interval of times. For a, b > 0, we write a = O(b) if there is a constant C > 0independent of t, d and R, such that a ≤ Cb.

Theorem 3.2.2.1. Consider an initial state ψ ∈ H of the form

ψ =1

‖∑li=1 I(J (ui)⊗ ϕi)‖

l∑j=1

I(J (uj)⊗ ϕj),

where ui and ϕi satisfy Assumptions (B1), (B2) and (B5) with d > R2 ≥ 1, for i = 1, . . . , l,and 〈ϕi|ϕj〉 = δij for all i, j = 1, . . . , l. We introduce the density matrix

ρP :=1∑l

i=1 ‖J (ui)‖2l∑

j=1

|J (uj)〉〈J (uj)| ∈ B(H0).


Suppose, moreover, that Asumptions (B3) and (B4) are satisfied. Thenë−itHψ|(OP ⊗ 1HQ ⊗ 1HE )e−itHψ

∂= TrH0(ρP e

itHP∨E (OP ⊗ 1HE )e−itHP∨E )

+ ‖OP ‖(OÄ(d/R)

−1+γ2

ä+OÄ〈t〉(d/R2)−

12

ä+O(t2d−

12 ) +O(td−β)

)for all 0 < γ ≤ 1, all t ≥ 0, and all OP ∈ B(HP ).

A corollary: the dressed atom in a slowly varying external potential

We now assume that the atom is placed in a slowly varying external potential, Vε(~x) ≡ V (ε~x),with Vε ∈ L∞(R3;R). We set

HεP := HP + Vε(~x), Hε

P∨E := HP∨E + Vε(~x), Hε := H + Vε(~x).

We define the effective Hamiltonian HεP,eff on L2(R3) as

HεP,eff := E(−i~∇x) + Vε(~x),

where E(~p) is the ground state energy of the fiber Hamiltonian H(~p). Since Vε is bounded,HεP , H

εP∨E , and H

ε are self-adjoint on D(HP ), D(HP∨E) and D(H), respectively.

Corollary 3.2.2.1. Suppose that V ∈ L∞(R3;R) satisfies supp(V ) ⊂ B1 = ~x ∈ R3, |~x| < 1.Set ui(t) := e−itH

εP,effui with ui ∈ H2(R3), and

ρε(t) :=1∑l

i=1 ‖J (ui(t))‖2l∑

j=1

|J (uj(t))〉〈J (uj(t))|

for all t ≥ 0. Under the assumptions of Theorem 3.2.2.1, we have thatë−itH

εψ|(OP ⊗ 1HQ ⊗ 1HE )e−itH

εψ∂

= TrH0(ρε(t)(OP ⊗ 1HE )) + ‖OP ‖(O(tε)

+OÄ(d/R)

−1+γ2


12

ä+O(t2d−

12 ) +O(td−β)

),

for all 0 < γ ≤ 1, all t ≥ 0, and all OP ∈ B(HP ).

This result is similar to the one proven in [8].

3.3 Proof of Theorem 3.2.2.1 and Corollary 3.2.2.1

3.3.1 Plan of the proof

The estimates used in the proof of Theorem 3.2.2.1 are insensitive to the presence of thepotential Vε (see Corollary 3.2.2.1). The bounds derived in the next sections are valid forboth Hε and H. To keep consistent notations, we prove Theorem 3.2.2.1 with H replaced byHε and HP∨E by Hε

P∨E . In Section 3.3.3, we prove that, in the dressed atom state J (u),with u as in Hypothesis (B1), most photons are localized in the ball of radius d R centeredat the origin. Using the fact that the propagation velocity of photons is finite, we show, inaddition, that after time t, for the dynamics generated by the atom-field Hamiltonian Hε

P∨E ,


most photons in the state e−itHεP∨EJ (u) remain localized in the ball of radius d centered at

the origin.In Section 3.3.4, we introduce a partition of unity in Fock space (see [46]) separating pho-

tons localized near the origin from photons localized near infinity. We rewrite the HamiltonianHε in the factorization of the Fock space determined by this partition of unity.

In Section 3.3.5, we prove Theorem 3.2.2.1, using Cook’s method, the partition of unity ofSection 3.3.4 and the localization lemmas of Section 3.3.3.

Proofs of some technical lemmas are postponed to the appendix.

3.3.2 Notations and conventions

We remind the reader that for a, b > 0, we write

a = O(b)

if there is a constant C > 0 independent of t, d and R such that a ≤ Cb. For two vectorsΨ1,Ψ2 ∈ H and a constant b > 0, we write Ψ1 = Ψ2 +O(b) if ‖Ψ1 −Ψ2‖ = O(b).

Given two self-adjoint operators A and B, the commutator [A,B] is defined in the senseof quadratic forms on D(A) ∩ D(B) by

〈u|[A,B]v〉 = 〈Au|Bv〉 − 〈Bu|Av〉.

In our proof, we will encounter such a commutator that extends continuously to some suitabledomain. The corresponding extension will be denoted by the same symbol, unless confusionmay arise. In the same spirit, we will often make use of “Cook’s method” to compare twodifferent dynamics. Suppose, for instance, that B is A-bounded. Then we will write∥∥∥e−itBu− e−itAu∥∥∥ ≤ ∫ t

0

∥∥∥(A−B)e−isAu∥∥∥ds,

for u ∈ D(A). A proper justification of the previous inequality would be∥∥∥e−itBu− e−itAu∥∥∥ = supv∈D(B),‖v‖=1

∣∣∣¨v|u− eitBe−itAu∂∣∣∣= sup

v∈D(B),‖v‖=1

∣∣∣ ∫ t

0〈v|eisB(A−B)e−isAu

∂ds∣∣∣

=

∥∥∥∥ ∫ t

0eisB(A−B)e−isAu ds

∥∥∥∥,the last equality being a consequence of the fact that A−B extends to an A-bounded operator.We will proceed similarly to estimate quantities like ‖Be−itAu‖ = ‖eitABe−itAu‖ assumingfor instance that B is bounded and that the commutator [A,B] extends to an A-boundedoperator. Since such arguments are standard, we will not repeat them in the rest of the paper.

3.3.3 Localization of photons

In this section, we begin by verifying that in the dressed atom state

J (u)(~x) =1

(2π)32

∫R3u(~p)ei~x·(~p−

~PE)χBν/2(~p)ψ(~p) d3p, (3.3.1)


(with u ∈ L2(R3) as in Hypothesis (B1)), most photons are localized near the origin. Hereψ(~p) is a non-degenerate ground state of H(~p), and ~p 7→ ψ(~p) is real analytic on ~p ∈ R3, |~p| <ν, for any 0 < ν < 1; see Theorem 3.4.1.1 of Appendix 3.4.1 for more details.

Next, we consider the evolution of the state J (u) under the dynamics generated by HεP∨E ;

(we recall that HεP∨E is the Hamiltonian for the atom in the external potential Vε and in-

teracting with the photon field). Using that the propagation velocity of photons is finite, weare able to prove that, for times not too large, most photons remain localized in the ball ofradius d 1 centered at the origin. This property will be important in the proof of our maintheorem (see Section 3.3.5), since it will allow us to show that the interaction between photonsclose to the atom and the system Q remains small for times not too large.

We begin with three lemmas whose proofs are postponed to Appendix 3.4.1. The first oneestablishes polynomial decay in |~x| in the state J (u), assuming that u is compactly supported.It is a simple consequence of standard properties of the Fourier transform combined with theanalyticity of ~p 7→ ψ(~p) (where, recall, ψ(~p) is a ground state of the fiber Hamiltonian H(~p)).In what follows, we use the identification HP ⊗HE ' L2(R3;C2 ⊗HE).

Lemma 3.3.3.1. Let u ∈ L2(R3) be as in Hypothesis (B1). Then∥∥∥(1 + |~x|)µJ (u)∥∥∥HP⊗HE

≤ CµRµ,

for all µ ≥ 0, where Cµ is a positive constant independent of R > 1.

In the next lemma, we control the number of photons in the fibered ground state ψ(~p).Based on the pull-through formula, the proof of Lemma 3.3.3.2 follows the one of Lemma 1.5in [61].

Lemma 3.3.3.2. Let 0 < ν < 1. For all ~p ∈ R3 such that |~p| < ν and δ ∈ R, we have that

ψ(~p) ∈ D(eδN ).

Moreover,sup

~p∈Bν/2‖eδNψ(~p)‖ ≤ Cδ.

Next, using Lemmas 3.3.3.1 and 3.3.3.2, we prove that, in the dressed atom state J (u)(with u as in Hypothesis (B1)), most photons are localized in the ball ~y ∈ R3, |~y| ≤ d.

Lemma 3.3.3.3. Let u ∈ L2(R3) be as in Hypothesis (B1). Then, for all d > R ≥ 1 and0 < γ ≤ 1, ∥∥∥Γ(1|~y|≤d)J (u)− J (u)

∥∥∥ = OÄ(d/R)−1+γ

ä.

The next lemma is another, related consequence of Lemmas 3.3.3.1 and 3.3.3.2. Consid-ering the dynamics generated by Hε

P∨E and the initial state J (u), with u as in Hypothesis(B1), Lemma 3.3.3.4 shows that, after times t such that 0 ≤ t d/R, most photons remainlocalized in the ball ~y ∈ R3, |~y| ≤ d. This is a consequence of the fact that the propagationvelocity of photons is finite.

Lemma 3.3.3.4. Let u ∈ L2(R3) be as in Hypothesis (B1). Then, for all d > R ≥ 1 andt ≥ 0, ∥∥∥Ä1− Γ(1|~y|≤d)

äe−itH

εP∨EJ (u)

∥∥∥ = OÄt

54d−

12

ä+OÄ〈t〉

34 (d/R)−

12

ä, (3.3.2)

where we recall the notation 〈t〉 := (1 + t2)1/2.


Proof. We begin with establishing two preliminary estimates.

Step 1. We have that ∥∥∥Ne−itHεP∨EJ (u)

∥∥∥ = O(〈t〉). (3.3.3)

By Lemma 3.3.3.2, we know that J (u) ∈ D(N). To see that e−itHεP∨EJ (u) ∈ D(N) for all

t ∈ R, we observe that D(HP ) ⊗ Ffin(D(|~k|)) is a common core for HεP∨E and N , and that

(N + 1)−1 preserves D(HP )⊗Ffin(D(|~k|)). Here we use the notation

Ffin(V ) = Φ = Φ(n) ∈ HE , Φ(n) ∈ ⊗nsV for all n, Φ(n) = 0 for all but finitely many n.

Since the commutator îN,Hε

P∨Eó

=îN,HP,E

ó= −iΦ(ihx), (3.3.4)

is bothHεP∨E- andN -relatively bounded, we deduce from [73, Lemma 2] that e−itHε

P∨ED(N) ⊂D(N) and hence in particular that e−itHε

P∨EJ (u) ∈ D(N). Here, as in (3.2.10)–(3.2.11),Φ(ihx) := a∗(ihx) + a(ihx).

To prove (3.3.3), we use that∥∥∥Ne−itHεP∨EJ (u)

∥∥∥ =∥∥∥eitHε

P∨ENe−itHεP∨EJ (u)

∥∥∥≤∥∥∥NJ (u)

∥∥∥+

∫ t

0

∥∥∥îN,HεP∨Eóe−isH

εP∨EJ (u)

∥∥∥ds.It follows from Lemma 3.3.3.2 that ‖NJ (u)‖ = O(1). Since D(Hε

P∨E) ⊂ D(HE) ⊂ D(H1/2E )

and since Φ(ihx) is H1/2E -relatively bounded, we deduce from (5.2.19) that there are positive

constants a and b such that∥∥∥îN,HεP∨Eóe−isH

εP∨EJ (u)

∥∥∥ ≤ a∥∥∥HεP∨EJ (u)

∥∥∥+ b‖J (u)‖.

Since J (u) ∈ D(HP∨E) = D(HεP∨E), this concludes the proof of Step 1.

Step 2. We have that ∥∥∥|~x|e−itHεP∨EJ (u)

∥∥∥ = O(t) +O(R).

The proof of this estimate is similar to Step 1. The only differences are that ‖|~x|J (u)‖ = O(R)by Lemma 3.3.3.1, and that the commutator [|~x|, Hε

P∨E ] = [|~x|,−∆~x]/2 is relatively boundedwith respect to (−∆~x + 1)1/2. The latter property follows from the computation

[|~x|,−∆~x] = ~∇x ·~x

|~x|+

~x

|~x|· ~∇x = 2

~x

|~x|· ~∇x +

2

|~x|,

and the fact that |~x|−1 is relatively bounded with respect to (−∆~x+1)1/2 by Hardy’s inequalityin R3.

Step 3. Now we proceed to the proof of Lemma 3.3.3.4. We introduce a smooth functionχ·≤d ∈ C∞0 ([0,∞); [0, 1]) satisfying χr≤d1r≤d/2 = 1r≤d/2 and χr≤d1r≥d = 0. We observe that,to prove (3.3.2), it is sufficient to establish that∥∥∥Ä1− Γ(χ|~y|≤d)

äe−itH

εP∨EJ (u)

∥∥∥ = OÄt

54d−

12

ä+OÄ〈t〉

34 (d/R)−

12

ä. (3.3.5)


The proof of (3.3.5) relies on the following argument:∥∥∥Ä1− Γ(χ|~y|≤d)äe−itH

εP∨EJ (u)

∥∥∥2

≤¨J (u)|eitHε

P∨EÄ1− Γ(χ|~y|≤d)

äe−itH

εP∨EJ (u)

∂=∥∥∥Ä1− Γ(χ|~y|≤d)

äJ (u)

∥∥∥2+ i

∫ t

0

¨J (u)|eisHε

P∨EîΓ(χ|~y|≤d), H

εP∨Eóe−isH

εP∨EJ (u)

∂ds.

The first term on the right side of this inequality is of order O((d/R)−2+2γ) for all 0 < γ ≤ 1,by Lemma 3.3.3.3. Here we choose γ = 1/2. To estimate the second term, we compute thecommutator î

Γ(χ|~y|≤d), HεP∨Eó

=îΓ(χ|~y|≤d), HE

ó+îΓ(χ|~y|≤d), HP,E

ó,

and estimate each term separately. In Appendix 3.4.4 (see Lemma 3.4.4.7), we verify that∥∥∥îΓ(χ|~y|≤d), HE

ó(N + 1)−1

∥∥∥ = O(d−1).

Together with Step 1, this shows that∫ t

0

¨J (u)|eisHε

P∨EîΓ(χ|~y|≤d), HE

óe−isH

εP∨EJ (u)

∂ds = O

Ät2d−1

ä. (3.3.6)

Using again Appendix 3.4.4 (see Lemma 3.4.4.5), we have that∥∥∥〈~x〉−1îΓ(χ|~y|≤d), HP,E

ó(N + 1)−

12

∥∥∥ = O(d−1).

Combined with Step 1 and Step 2, this implies that∫ t

0

¨J (u)|eisHε

P∨EîΓ(χ|~y|≤d), HP,E

óe−isH

εP∨EJ (u)

∂ds = O

Ät

52d−1

ä+OÄt

32 (d/R)−1

ä.

This estimate and (3.3.6) imply (3.3.5), which concludes the proof of the lemma.

We conclude this subsection with another localization lemma that will be useful in theproof of Theorem 3.2.2.1. In spirit, Lemma 3.3.3.5 is similar to Lemma 3.3.3.4 and followsfrom the fact that the propagation velocity of photons is finite. For the dynamics generatedby HQ∨E , it shows that, if in the initial state ϕ all photons are localized in the region ~y ∈R3, |~y| ≥ 3d, then in the evolved state e−itHQ∨Eϕ, with t d, most photons are localized in~y ∈ R3, |~y| ≥ 2d.

Lemma 3.3.3.5. Let ϕ ∈ H∞ = HQ ⊗ HE be as in Hypothesis (B2) and suppose thatHypotheses (B4) and (B5) hold. For all d > 0 and t ≥ 0,∥∥∥Ä1− Γ(1|~y|≥2d)

äe−itHQ∨Eϕ

∥∥∥ = OÄt2d−1

ä. (3.3.7)


Proof. Let χr≥2d ∈ C∞0 ([0,∞); [0, 1]) be a smooth function satisfying χr≥2d1r≥3d = 1r≥3d andχr≥2d1r≤2d = 0. With this definition, we see that, in order to prove (3.3.7), it is sufficient toshow that ∥∥∥Ä1− Γ(χ|~y|≥2d)

äe−itHQ∨Eϕ

∥∥∥ = OÄt2d−1

ä. (3.3.8)

As in Lemma 3.3.3.4, we use that∥∥∥Ä1− Γ(χ|~y|≥2d)äe−itHQ∨Eϕ

∥∥∥ ≤ ∫ t

0

∥∥∥îΓ(χ|~y|≥2d), HQ∨Eóe−isHQ∨Eϕ

∥∥∥ds.We have that [Γ(χ|~y|≥2d), HQ] = 0, and it follows from Hypothesis (B4) that [Γ(χ|~y|≥2d), HQ,E ] =0. Therefore∥∥∥[Γ(χ|~y|≥2d), HQ∨E ](N + 1)−1

∥∥∥ =∥∥∥[Γ(χ|~y|≥2d), HE ](N + 1)−1

∥∥∥ = O(d−1), (3.3.9)

the last estimate being proven in Appendix 3.4.4 (see Lemma 3.4.4.7). By Hypothesis (B5)together with the assumption that ϕ ∈ D(N), we have that∥∥∥(N + 1)e−isHQ∨Eϕ

∥∥∥ = O(〈s〉). (3.3.10)

Equations (3.3.9) and (3.3.10) imply (3.3.8), which concludes the proof of the lemma.

3.3.4 Factorization of Fock space

We introduce a factorization of Fock space (see [46] or [64] for more details), which will beused to factorize e−itHε into a tensor product of the form e−itH

εP∨E ⊗ e−itHQ∨E plus an error

term. This factorization is carried out in Section 3.3.5 and is one of the main ingredients ofour proof. Let j0 ∈ C∞0 ([0,∞); [0, 1]) be such that j0 ≡ 1 on [0, 1] and j0 ≡ 0 on [2,∞),and let j∞ be defined by the relation j2

0 + j2∞ ≡ 1. Recall that ~y := i~∇k denote the “photon

position variable”. Given d > 0, we introduce the bounded operators j0 := j0(|~y|/d) andj∞ := j∞(|~y|/d) on L2(R3). We set

j : L2(R3)→ L2(R3)⊕ L2(R3)

u 7→ (j0u, j∞u).

Next we lift the operator j to the Fock space HE = F+(L2(R3)) defining a map

Γ(j) : HE → F+(L2(R3)⊕ L2(R3)),

with Γ(j) defined as in (3.2.14). Let

U : F+(L2(R3)⊕ L2(R3))→ HE ⊗HE ,

be the unitary operator defined by

UΩ := Ω⊗ Ω (3.3.11)Ua∗(u1 ⊕ u2) = (a∗(u1)⊗ 1 + 1⊗ a∗(u2))U . (3.3.12)

The factorization of Fock space that we consider is defined by

Γ(j) : HE → HE ⊗HE , Γ(j) = UΓ(j).


Using the relation j20 + j2

∞ ≡ 1, one can verify that Γ(j) is a partial isometry. The adjoint ofΓ(j) can be represented as

Γ(j)∗ = I(Γ(j0)⊗ Γ(j∞)), (3.3.13)

where I denotes the identification operator defined in (3.2.12).On the total Hilbert spaceH = HP⊗HQ⊗HE , we denote by the same symbol the operator

Γ(j) : H → H0 ⊗H∞,

where, recall, H0 = HP ⊗HE and H∞ = HQ ⊗HE . We introduce the bounded operator

χ|~y|≤d := j0(2|~y|/d), (3.3.14)

on L2(R3). As in Section 3.3.3, it corresponds to a smooth version of the projection 1|~y|≤dsatisfying χ|~y|≤d ∈ C∞0 ([0,∞); [0, 1]), χ|~y|≤d1|~y|≤d/2 = 1|~y|≤d/2 and χ|~y|≤d1|~y|≥d = 0.

We begin with a localization lemma for the initial state IJ (u) ⊗ ϕ, which will be usefulin the sequel. For the convenience of the reader, the proof of Lemma 3.3.4.1 is deferred toAppendix 3.4.1.

Lemma 3.3.4.1. Let u ∈ L2(R3) be as in Hypothesis (B1) and ϕ ∈ HQ ⊗ HE be as inHypothesis (B2), with d > R ≥ 1. Then J (u) ⊗ ϕ ∈ D(I) and, for all 0 < γ ≤ 1, we havethat

IJ (u)⊗ ϕ = IÄΓ(χ|~y|≤d)J (u)

ä⊗ÄΓ(1|~y|≥3d)ϕ

ä+O((d/R)

−1+γ2 ).

A few remarks concerning the statement of Lemma 3.3.4.1 are in order. In more preciseterms, the lemma means that∥∥∥IJ (u)⊗ ϕ− I

ÄΓ(χ|~y|≤d)J (u)


ä∥∥∥ ≤ C(d/R)−1+γ

2 ,

where C is a positive constant depending on γ, v (v being the function of Hypothesis (B1))and ϕ, but not on R and d such that d > R. We mention that the exponent (−1 + γ)/2 ispresumably not sharp. We do not make any attempt to optimize it. We also observe thatthe fact that J (u) ⊗ ϕ ∈ D(I) follows from Lemma 3.3.3.2 and Hypothesis (B2). The factthat



äalso belongs to D(I) is a consequence of (3.3.13). More

precisely, since j0 ≡ 1 on [0, 1] and j∞ ≡ 1 on [3,∞), (3.3.13) implies that

IÄΓ(χ|~y|≤d)J (u)


ä= Γ(j)∗



ä,

and therefore the boundedness of Γ(j)∗ yields∥∥∥IÄΓ(χ|~y|≤d)J (u)ä⊗ÄΓ(1|~y|≥3d)ϕ

ä∥∥∥ ≤ ∥∥∥J (u)∥∥∥‖ϕ‖.

In what follows, we denote by Hε the total Hamiltonian onH where the interaction betweenthe atom P and the system Q has been removed, that is

Hε := Hε −HP,Q. (3.3.15)


Moreover, to shorten notations, we introduce the definition

ψloc := IÄΓ(χ|~y|≤d)J (u)


ä= Γ(j)∗



ä. (3.3.16)

We prove in Lemma 3.3.4.2 that, if the system is initially in the (non-normalized) state ψloc,the contribution of the interaction between the atom and the subsystem Q, HP,Q, to thedynamics, remains small for times not too large.

Lemma 3.3.4.2. Let u ∈ L2(R3) be as in Hypothesis (B1) and ϕ ∈ HQ ⊗ HE be as inHypothesis (B2), with d > R ≥ 1. Assume Hypothesis (B3). For all times t ≥ 0, we havethat

e−itHεψloc = e−itH

εψloc +O(td−1) +O((d/R)−∞) +O(td−β).

Proof. We estimate the norm of

e−itHεψloc − e−itH

εψloc = (e−itH

εχ|~x|≤d − χ|~x|≤de−itH

ε)ψloc (3.3.17)

+ (χ|~x|≤d − 1)e−itHεψloc + e−itH

ε(1− χ|~x|≤d)ψloc.

Using unitarity of e−itHε , we compute∥∥∥(e−itHεχ|~x|≤d − χ|~x|≤de−itH

ε)ψloc

∥∥∥ (3.3.18)

=

∥∥∥∥ ∫ t

0e−isH

ε(−Hεχ|~x|≤d + χ|~x|≤dH

ε)e−i(t−s)Hεψloc ds

∥∥∥∥≤∫ t

0

∥∥∥HP,Qχ|~x|≤de−i(t−s)Hε

ψloc

∥∥∥ ds+

∫ t

0

∥∥∥[Hε, χ|~x|≤d]e−i(t−s)Hε

ψloc

∥∥∥ ds. (3.3.19)

By Hypothesis (B3), we have that∥∥∥HP,Qχ|~x|≤de−isHε

ψloc

∥∥∥ ≤ Cd−β. (3.3.20)

To estimate the second term on the right side of (3.3.19), we compute∥∥∥îHε, χ|~x|≥dó(−∆~x + 1)−

12

∥∥∥ =1

2

∥∥∥î−∆~x, χ|~x|≥dó(−∆~x + 1)−

12

∥∥∥ = O(d−1). (3.3.21)

Since D(Hε) = D(Hε) ⊂ D(−∆~x) ⊂ D((−∆~x + 1)12 ), there exist positive constant a, b such

that‖(−∆~x + 1)

12u‖ ≤ a‖Hεu‖+ b‖u‖,

for all u ∈ D(Hε), which, combined with (3.3.21), yields∥∥∥îHε, χ|~x|≥dóe−i(t−s)H

εψloc

∥∥∥ = O(d−1). (3.3.22)

Here we used that ψloc belongs to D(Hε). Indeed, by Hypothesis (B3), HP,Q is Hε-relativelybounded, and hence Hε is also Hε-relatively bounded. Moreover D(Hε) = D(HP +HQ+HE)by assumption, and it is not difficult to verify that ψloc ∈ D(HP +HQ +HE).


We now introduce χ|~x|≥d := 1 − χ|~x|≤d. We estimate the second term in the right side of(3.3.18) by using

∥∥∥χ|~x|≥de−itHεψloc

∥∥∥ ≤ ∥∥∥χ|~x|≥dψloc

∥∥∥+

∥∥∥∥ ∫ t

0eisH

εîHε, χ|~x|≥d

óe−isH

εψloc ds

∥∥∥∥≤∥∥∥χ|~x|≥dψloc

∥∥∥+

∫ t

0

∥∥∥îHε, χ|~x|≥dóe−isH

εψloc

∥∥∥ ds. (3.3.23)

The first term on the right side of (3.3.23) is estimated as follows. The definition (3.3.16) ofψloc gives ∥∥∥χ|~x|≥dψloc

∥∥∥ =∥∥∥χ|~x|≥dΓ(j)∗



ä∥∥∥≤∥∥∥χ|~x|≥dJ (u)

∥∥∥‖ϕ‖.Applying Lemma 3.3.3.1 we then find that∥∥∥χ|~x|≥dψloc

∥∥∥ = O((d/R)−∞). (3.3.24)

The second term in the right side of (3.3.23) has been already estimated in (3.3.22) aboveand the first term in the right side of (3.3.18) is estimated by (3.3.24). Equations (3.3.19),(3.3.20), (3.3.22), (3.3.23) and (3.3.24) prove the statement of the lemma.

On the Hilbert space H0 ⊗H∞, we abbreviate

N0 := N ⊗ 1H∞ , N∞ := 1H0 ⊗N,

where, recall, N is the photon-number operator on Fock space. In the next lemma, we rewritethe Hamiltonian Hε defined in (3.3.15) (total Hamiltonian without the interaction betweenthe atom and the system Q) in the representation corresponding to the factorization of Fockspace given by Γ(j). Combined with Lemma 3.3.4.2, Lemma 3.3.4.3 will allow us to comparethe dynamics e−itHε on H with the tensor product e−itHε

P∨E ⊗ e−itHQ∨E on H0 ⊗ H∞. Theproof of Lemma 3.3.4.3 is somewhat technical. It will be given in Appendix 3.4.2.

Lemma 3.3.4.3. Assume Hypothesis (B4). On D(HP∨E)⊗D(HQ∨E), the following relationholds:

HεΓ(j)∗ = Γ(j)∗ÄHεP∨E ⊗ 1H∞ + 1H0 ⊗HQ∨E

ä+ Rem1 + Rem2,

with ∥∥∥Rem1(N0 +N∞ + 1)−1∥∥∥ = O(d−1),

and ∥∥∥Rem2

ÄN0 +N∞ + 〈~x〉4−2δ

ä−1∥∥∥ = O(d−2+δ),

for all 0 < δ ≤ 2, where we used the usual notation 〈~x〉 :=√

1 + ~x2.


3.3.5 Proof of Theorem 3.2.2.1

In this section, we prove our main result, Theorem 3.2.2.1.

Proof of Theorem 3.2.2.1. To simplify the exposition, we assume that the initial (non-normalized)state ψ is given by ψ = IJ (u)⊗ϕ. The more general initial condition presented in the state-ment of Theorem 3.2.2.1 can be directly deduced from this special case. We begin by applyingLemma 3.3.4.1. Using unitarity of e−itHε , this gives

e−itHεψ = e−itH

εψloc +O

Ä(d/R)

−1+γ2

ä,

for all 0 < γ ≤ 1, with ψloc defined in (3.3.16). Lemma 3.3.4.2 then implies that

e−itHεψ = e−itH

εψloc +O

Ä(d/R)

−1+γ2

ä+O(td−1) +O(td−β), (3.3.25)

where Hε is defined in (3.3.15).Next, we show that

Γ(j)e−itHεψloc =

Äe−itH

εP∨EJ (u)

ä⊗Äe−itHQ∨Eϕ


12

ä+O(t2d−

12 ). (3.3.26)

We begin by using the localization lemmas established above in order to rewrite the tensorproduct (e−itH

εP∨EJ (u)) ⊗ (e−itHQ∨Eϕ). Applying Lemma 3.3.3.4 and Lemma 3.3.3.5, we

obtain thatÄe−itH

εP∨EJ (u)


ä=ÄΓ(1|~y|≤d)e

−itHεP∨EJ (u)

ä⊗ÄΓ(1|~y|≥2d)e

−itHQ∨Eϕä

+OÄt

54d−

12

ä+OÄ〈t〉

34 (d/R)−

12

ä+O(t2d−1). (3.3.27)

We observe that, since Γ(j) is a partial isometry, we have the relation

Γ(j)Γ(j)∗ = 1Ran(Γ(j)), (3.3.28)

where 1Ran(Γ(j)) stands for the projection onto the (closed) subspace Ran(Γ(j)) of H0 ⊗H∞.Moreover it is not difficult to verify that Ran(Γ(1|~y|≤d) ⊗ Γ(1|~y|≥2d)) ⊂ Ran(Γ(j)). Thus wededuce from (3.3.27) and (3.3.28) thatÄ

e−itHεP∨EJ (u)


ä(3.3.29)

= Γ(j)Γ(j)∗ÄΓ(1|~y|≤d)e

−itHεP∨EJ (u)

ä⊗ÄΓ(1|~y|≥2d)e

−itHQ∨Eϕä

+OÄt

54d−

12

ä+OÄ〈t〉

34 (d/R)−

12

ä+O(t2d−1). (3.3.30)

Next we rewriteÄe−itH

εP∨EJ (u)


ä= e−it(H

εP∨E⊗1H∞+1H0

⊗HQ∨E)ÄJ (u)⊗ ϕ

ä= e−it(H

εP∨E⊗1H∞+1H0

⊗HQ∨E)ÄÄ

Γ(χ|~y|≤d)J (u)ä⊗ÄΓ(1|~y|≥2d)ϕ

ää+O((d/R)−1+γ), (3.3.31)

for all 0 < γ ≤ 1, the last equality being a consequence of Lemma 3.3.3.3 and Hypothesis(B2). To shorten notations, we set

ψloc :=ÄΓ(χ|~y|≤d)J (u)


ä,


so that ψloc = Γ(j)∗ψloc according to (3.3.16). Remark that ψloc ∈ D(HP∨E) ⊗ D(HQ∨E)because χ|·|≤d is a smooth function with compact support. Combining (3.3.30) and (3.3.31)(with 0 < γ ≤ 1/2), we obtain thatÄ

e−itHεP∨EJ (u)


ä= Γ(j)Γ(j)∗e−it(H

εP∨E⊗1H∞+1H0

⊗HQ∨E)ψloc

+OÄt

54d−

12

ä+OÄ〈t〉

34 (d/R)−

12

ä+O(t2d−1). (3.3.32)

Now we prove (3.3.26). It follows from (3.3.32) and ψloc = Γ(j)∗ψloc that∥∥∥Γ(j)e−itHεψloc −

Äe−itH

εP∨EJ (u)


ä∥∥∥=∥∥∥Γ(j)e−itH

εΓ(j)∗ψloc − Γ(j)Γ(j)∗e−it(H

εP∨E⊗1H∞+1H0

⊗HQ∨E)ψloc

∥∥∥+OÄt

54d−

12

ä+OÄ〈t〉

34 (d/R)−

12

ä+O(t2d−1)

=∥∥∥e−itΓ(j)HεΓ(j)∗ψloc − Γ(j)Γ(j)∗e−it(H

εP∨E⊗1H∞+1H0

⊗HQ∨E)ψloc

∥∥∥+OÄt

54d−

12

ä+OÄ〈t〉

34 (d/R)−

12

ä+O(t2d−1). (3.3.33)

We compute ∥∥∥e−itΓ(j)HεΓ(j)∗ψloc − Γ(j)Γ(j)∗e−it(HεP∨E⊗1H∞+1H0

⊗HQ∨E)ψloc

∥∥∥≤∫ t

0

∥∥∥Γ(j)ÄHεΓ(j)∗ − Γ(j)∗(Hε

P∨E ⊗ 1H∞ + 1H0 ⊗HQ∨E)ä

e−is(HεP∨E⊗1H∞+1H0

⊗HQ∨E)ψloc

∥∥∥ds, (3.3.34)

where we used that Γ(j)Γ(j)∗ψloc = ψloc and that Γ(j)∗Γ(j) = 1H. Applying Lemma 3.3.4.3with δ = 3/2, we obtain that∥∥∥ÄHεΓ(j)∗ − Γ(j)∗

ÄHεP∨E ⊗ 1H∞ + 1H0 ⊗HQ∨E

ääÄN0 +N∞ + 〈~x〉

ä−1∥∥∥ = O(d−

12 ). (3.3.35)

Adapting in a straightforward way the proof of Lemma 3.3.3.4 and using that Γ(χ|~y|≤d)J (u) ∈D(N) ∩ D(HP∨E), we deduce that∥∥∥(N + 〈x〉)e−isHε

P∨EΓ(χ|~y|≤d)J (u)∥∥∥ = O(s) +O(R). (3.3.36)

By Hypothesis (B5), we also have that∥∥∥Ne−isHQ∨Eϕ∥∥∥ = O(〈s〉). (3.3.37)

Equations (3.3.35), (3.3.36) and (3.3.37) yield that∥∥∥ÄHεΓ(j)∗ − Γ(j)∗(HεP∨E ⊗ 1H∞ + 1H0 ⊗HQ∨E)

äe−is(H

εP∨E⊗1H∞+1H0

⊗HQ∨E)ψloc

∥∥∥= O(sd−

12 ) +O(Rd−

12 ).

Integrating this estimate and using again that Γ(j) is isometric, we deduce from (3.3.34) that∥∥∥e−itΓ(j)HεΓ(j)∗ψloc − Γ(j)Γ(j)∗e−it(HεP∨E⊗1H∞+1H0

⊗HQ∨E)ψloc

∥∥∥= O(t2d−

12 ) +O(tRd−

12 ). (3.3.38)


Putting together (3.3.33) and (3.3.38), we obtain (3.3.26).To conclude the proof, it suffices to combine (3.3.25) and (3.3.26), which gives

Γ(j)e−itHεψloc =

Äe−itH

εP∨EJ (u)


ä+OÄ(d/R)

−1+γ2


12

ä+O(t2d−

12 ) +O(td−β), (3.3.39)

for all 0 < γ ≤ 1. Since Γ(j) is an isometry commuting with any bounded operator OP onHP , the last equation directly implies the statement of the theorem.

3.4 Appendix for Section 3.3

In this Appendix, we gather the proofs of several technical lemmas that were used in the proofof our main results. In Section 3.4.1, we prove Lemmas 3.3.3.1, 3.3.3.2, 3.3.3.3 and 3.3.4.1.In Section 3.4.2 we prove Lemma 3.3.4.3. Section 3.4.3 is devoted to the proof of Corollary3.2.2.1. Finally, in Section 3.4.4, we recall a few well-known relative bounds for operators onFock space, and we estimate some commutators.

3.4.1 Proofs of the localization lemmas

We begin with recalling the expression of the fibre Hamiltonian H(~p). The unitary map

U : HP ⊗HE →∫ ⊕R3

C2 ⊗HE d3p,

such thatUHP∨EU

−1 =

∫ ⊕R3H(~p)d3p,

is the “generalized Fourier transform”, defined by

(Uϕ)(~p) =1

(2π)3/2

∫R3e−i(~p−

~PE)·~xϕ(~x)d3x (3.4.1)

for all ϕ ∈ HP ⊗ HE such that each ϕ(n) decays sufficiently rapidly at infinity. Introducingthe notations

b(k) := Uei~k·~xa(k)U−1, b∗(k) := Ue−i

~k·~xa∗(k)U−1, (3.4.2)

one verifies that

H(~p) =1

2

Ä~p− ~PE

ä2+

Çω0 00 0

å+ iλ0

∫R3χ(~k)|~k|

12~ε(k) · ~σ (b(k)− b∗(k)) dk +HE , (3.4.3)

where HE =∫R3 |~k|b∗(k)b(k)dk. It follows from the Kato-Rellich theorem that the fiber

Hamiltonians H(~p) are self-adjoint operators on D(HE + ~P 2E).

We recall the main result of [53], which is used in our proofs. This result is proven similarlyas in Chapter 2.


Theorem 3.4.1.1 (Real analyticity of ~p 7→ E(~p) [53]). Let 0 < ν < 1. There exists a constantλc(ν) > 0 such that, for any coupling constant λ0 ∈ R satisfying |λ0| < λc(ν), the ground stateenergy E(~p) of H(~p) is a non-degenerate eigenvalue of H(~p), and the map ~p 7→ E(~p) and itsassociated eigenprojection ~p 7→ π(~p) are real analytic on Bν := ~p ∈ R3, |~p| < ν.

We mention that, in [53], for simplicity, we have used a sharp ultraviolet cutoff 1|·|≤1(~k)

instead of the smooth ultraviolet cutoff χ(~k) used in the present paper. This modification,however, does not affect the proof given in [53].

We also observe that the uncoupled Hamiltonian

H0(~p) :=1

2

Ä~p− ~PE

ä2+

Çω0 00 0

å+HE

has a unique ground state (up to a phase) associated with the eigenvalue ~p 2/2, given by

ψ0 :=

Ç01

å⊗ Ω,

where Ω denotes the vacuum Fock state. It is not difficult to verify that the ground state ofH(~p) overlaps with the ground state of H0(~p), in the sense that∥∥∥π(~p)ψ0

∥∥∥ = 1−O(|λ0|),

for small enough |λ0|. Therefore,ψ(~p) := π(~p)ψ0

is a (non-normalized) ground state of H(~p), and the map ~p 7→ ψ(~p) is real analytic on Bν . Inwhat follows, we keep the notation ψ(~p) for π(~p)ψ0.

We now prove Lemma 3.3.3.1.


Proof. By an interpolation argument, we see that it suffices to establish the statement of thelemma for µ = N ∈ N ∪ 0. Recall from Hypothesis (B1) that u(~x) = R−3/2v(~x/R) wherev is a function independent of R such that Supp(v) ⊂ ~x ∈ R3, |~x| ≤ 1. Using (3.3.1), wecompute ∥∥∥(1 + |~x|)NJ (u)(~x)

∥∥∥HP⊗HE

=1

(2π)32

∥∥∥∥(1 + |~x|)N∫R3u(~p)ei~x·~pχBν/2(~p)ψ(~p) d3p

∥∥∥∥HP⊗HE

=R

32

(2π)32

∥∥∥∥(1 + |~x|)N∫R3v(R~p)ei~x·~pχBν/2(~p)ψ(~p) d3p

∥∥∥∥HP⊗HE

.

Since v is compactly supported, v ∈ C∞(R3), and hence, since in addition χBν/2 ∈ C∞0 (R3)

and since ~p 7→ ψ(~p) is smooth on the support of χBν/2 by Theorem 3.4.1.1, we deduce that

~p 7→ v(R~p)χBν/2(~p)ψ(~p) ∈ C∞0 (R3;C2 ⊗HE).

The result then follows from standard properties of the Fourier transform, using in particularthat |∂α~p v(R~p)| ≤ CαR

|α| for all multi-index α.


To prove Lemma 3.3.3.2, we need to establish a preliminary lemma. Let 0 < ν < 1. It isshown in [53], Section 5.1, that there is a critical coupling constant λc(ν) > 0 such that themap ~k 7→ E(~k) is analytic on the open set

U [~p] :=¶~k ∈ C3 | |~p− ~k| < 1− ν

2

©for all |~p| < ν and all λ0 ∈ R with |λ0| < λc(ν). Moreover, in the proof of Lemma 4.4. in [53],we show that the choice made for λc(ν) implies that∣∣∣∣E(~k)− |

~k|2

2

∣∣∣∣ < Å1− ν6

ã2

(3.4.4)

for all |~p| < ν and for all ~k ∈ U [~p], if |λ0| < λc(ν).

Lemma 3.4.1.1. Let 0 < ν < 1. We assume that |λ0| < λc(ν). Then

E(~p− ~k)− E(~p) + |~k| ≥ 1− ν2

∣∣∣~k∣∣∣ (3.4.5)

for all ~p ∈ Bν and for all ~k ∈ B(1−ν)/6.

Proof. Let ~p ∈ Bν = ~p ∈ R3 | |~p| < ν. We set

E(~k) := E(~k)− |~k|2

2.

Since E is analytic on U [~p], we have that

|E(~p− ~k)− E(~p)| ≤ sup~l∈U [~p]

|~∇E(~l)| |~k|,

for all complex vectors ~k with |~k| < (1 − ν)/2. Let now ~ξ = (ξ1, ξ2, ξ3) ∈ C3 with |~p − ~ξ| <(1− ν)/6. Using Cauchy formula for holomorphic functions of several complex variables andEq.(3.4.4), we get that∣∣∣∣(∂z1E)(~ξ)

∣∣∣∣ ≤ 1

2π

∣∣∣∣ ∫∂D 1−ν

6(ξ1)

E(z, ξ2, ξ3)

(z − ξ1)2dz

∣∣∣∣ ≤ 1− ν6

, (3.4.6)

where D(1−ν)/6(ξ1) is the complex open disk of radius (1− ν)/6 centered at ξ1 ∈ C and ∂z1Edenotes the partial derivative of E with respect to the first component z1. Similar boundshold for the partial derivatives with respect to z2 and z3, which implies that

|E(~p− ~k)− E(~p)| ≤ 1− ν2|~k| (3.4.7)

for all ~k ∈ C3 with |~k| < (1−ν)/6. The right side of (3.4.7) is independent of ~p for all |~p| < ν.Therefore,

E(~p− ~k)− E(~p) + |~k| = |~k|2

2− ~k · ~p+ E(~p− ~k)− E(~p) + |~k| ≥ 1− ν

2|~k| (3.4.8)

for all ~p ∈ Bν and for all ~k ∈ B(1−ν)/6.


We now proceed to the proof of Lemma 3.3.3.2. The proof follows [61, Lemma 1.5]. Itrelies on Lemma 3.4.1.1 and the pull-through formula

b(k)g(HE , ~PE) = g(HE + |~k|, ~PE + ~k)b(k), (3.4.9)

for any measurable function g : R4 → R. We do not present all the details.


Proof. Let ~p ∈ R3, |~p| < ν. Using Lemma 3.3.3.2, (3.4.9), and adapting the proof of [61,Lemma 1.5] in a straightforward way, we deduce that there exists a constant D(~p) > 0 suchthat, for any n ∈ N,∥∥∥∥ n∏

i=1

b(ki)ψ(~p)

∥∥∥∥ ≤ D(~p)n|λ0|n( n∏i=1

χ(~ki)|~ki|−12

)‖ψ(~p)‖.

The projection of ψ(~p) onto the n-photons sector in Fock space is given by

ψ(n)(k1, . . . , kn)(~p) =1√n!〈Ω|

n∏i=1

b(ki)ψ(~p)〉,

from which we obtain that

eδn∥∥∥ψ(n)(k1, . . . , kn)(~p)

∥∥∥ ≤ eδnÄD(~p)|λ0|

än√n!

χ(k1) · · ·χ(kn)

|k1|12 · · · |kn|

12

‖ψ(~p)‖.

Taking the square and integrating over R3n, a direct computation then gives

e2δn∫R3n

∥∥∥ψ(n)(k1, . . . , kn)(~p)∥∥∥2dk1 · · · dkn ≤

e2δn(4π)n(D(~p)|λ0|)2n

n!‖ψ(~p)‖2,

and therefore ∥∥∥eδNψ(~p)∥∥∥ ≤ e2πe2δ(D(~p)|λ0|)2‖ψ(~p)‖.

This shows that ψ(~p) ∈ D(eδN ). Moreover, one can verify that the constant D(~p) can bechosen to be uniformly bounded on Bν/2 := ~p ∈ R3, |~p| ≤ ν/2, and hence

sup~p∈Bν/2

‖eδNψ(~p)‖ ≤ Cδ.

This concludes the assertion of the proof of the lemma.

Next we prove Lemma 3.3.3.3.



Proof. Since Γ(1|~y|≤d) is a projection, we can write∥∥∥Ä1− Γ(1|~y|≤d)äJ (u)

∥∥∥2=¨J (u)|

Ä1− Γ(1|~y|≤d)

äJ (u)

∂≤¨J (u)|dΓ(1|~y|≥d)J (u)

∂≤ d−2+2γ

¨J (u)|dΓ(|~y|2−2γ)J (u)

∂. (3.4.10)

It remains to show that J (u) ∈ D(dΓ(|~y|2−2γ)1/2) and that¨J (u)|dΓ(|~y|2−2γ)J (u)

∂≤ CγR

2−2γ .

Using (3.3.1) and the fact that

ei~x·~PEdΓ(|~y|2−2γ)e−i~x·

~PE = dΓ(|~y + ~x|2−2γ),

we obtain¨J (u)|dΓ(|~y|2−2γ)J (u)

∂=

1

(2π)3

∥∥∥∥dΓ(|~y + ~x|2−2γ)12

∫R3u(~p)ei~x·~pχBν/2(~p)ψ(~p)d3p

∥∥∥∥2

.

The inequality |~y + ~x|2−2γ ≤ cγ(|~y|2−2γ + |~x|2−2γ) then gives¨J (u)|dΓ(|~y|2−2γ)J (u)

∂≤ Cγ

( ∫R3|u(~p)|χBν/2(~p)

∥∥∥dΓ(|~y|2−2γ)12ψ(~p)

∥∥∥d3p)2

+ Cγ

∥∥∥∥dΓ(|~x|2−2γ)12

∫R3u(~p)ei~x·~pχBν/2(~p)ψ(~p) d3p

∥∥∥∥2

. (3.4.11)

The two terms appearing on the right side of the previous inequality are estimated separately.For the second one, we use that dΓ(|~x|2−2γ) = |~x|2−2γN and estimate, thanks to the Cauchy-Schwarz inequality,∥∥∥∥dΓ(|~x|2−2γ)

12


∥∥∥∥2

≤∫R3|u(~p)|χBν/2(~p)

∥∥∥Nψ(~p)∥∥∥d3p×

∥∥∥∥|~x|2−2γ∫R3u(~p)ei~x·~pχBν/2(~p)ψ(~p) d3p

∥∥∥∥=

∫R3|u(~p)|χBν/2(~p)

∥∥∥Nψ(~p)∥∥∥d3p×

∥∥∥|~x|2−2γJ (u)(x)∥∥∥.

By Lemma 3.3.3.2, sup~p∈Bν/2 ‖Nψ(~p)‖ < ∞ and, by Lemma 3.3.3.1, ‖|~x|2−2γJ (u)(x)‖ ≤CγR

2−2γ . This proves that∥∥∥∥dΓ(|~x|2−2γ)12


∥∥∥∥ ≤ CγR2−2γ . (3.4.12)

It remains to estimate the first term on the right side of (3.4.11). Using the pull-throughformula (3.4.9) and setting f(k) := −iλ0χ(~k)|~k|

12~ε(k) · ~σ, we obtain that

b(k)(H(~p) + ξ) = (H(~p− ~k) + |~k|+ ξ)b(k) + f(k). (3.4.13)


A direct application of (3.4.13) then yields

b(k)ψ(~p) = −ÄH(~p− ~k)− E(~p) + |~k|

ä−1f(k)ψ(~p), (3.4.14)

and by Lemma 3.4.1.1, this implies∥∥∥b(k)ψ(~p)∥∥∥ ≤ C1|·|≤1(~k)|~k|−

12 ‖ψ(~p)‖, (3.4.15)

where C is a positive constant.Differentiating (3.4.14) with respect to ~k, we obtain that∥∥∥|i~∇k|b(k)ψ(~p)

∥∥∥ =∥∥∥~∇kb(k)ψ(~p)

∥∥∥ ≤ C1|·|≤1(~k)|~k|−32 ‖ψ(~p)‖. (3.4.16)

Equations (3.4.15) and (3.4.16), together with an interpolation argument, yield∥∥∥|i~∇k|1−γb(k)ψ(~p)∥∥∥ ≤ C1|·|≤1(~k)|~k|−

32

+γ‖ψ(~p)‖, (3.4.17)

for all 0 ≤ γ ≤ 1. This shows that k 7→ |i~∇k|1−γb(k)ψ(~p) ∈ L2(R3;C2 ⊗ HE) for 0 < γ ≤ 1and, more precisely, that

sup~p∈Bν/2

∥∥∥dΓ(|~y|2−2γ)12ψ(~p)

∥∥∥ <∞.Therefore we have proven that∫

R3|u(~p)|χBν/2(~p)

∥∥∥dΓ(|~y|2−2γ)12ψ(~p)

∥∥∥d3p <∞.

Together with (3.4.10), (3.4.11) and (3.4.12), this concludes the proof.

We conclude this paragraph with the proof of Lemma 3.3.4.1.


Proof. We begin with justifying that J (u) ⊗ ϕ ∈ D(I). It is not difficult to verify that, forany 0 ≤ a, b ≤ 1 such that a2 + b2 ≤ 1, the operator IΓ(a1) ⊗ Γ(b1) extends to a boundedoperator satisfying ∥∥∥IΓ(a1)⊗ Γ(b1)

∥∥∥ ≤ 1.

Since ϕ satisfies Hypothesis (B2), there is δ > 0 such that ϕ ∈ D(eδN ). Choosing δ′ > 0 suchthat e−2δ′ + e−2δ ≤ 1, we deduce that∥∥∥IJ (u)⊗ ϕ

∥∥∥ =∥∥∥IÄΓ(e−δ

′1)⊗ Γ(e−δ1)

ä(eδ′NJ (u))⊗ (eδNϕ)

∥∥∥≤∥∥∥eδ′NJ (u)

∥∥∥∥∥∥eδNϕ∥∥∥ <∞,the fact that

∥∥∥eδ′NJ (u)∥∥∥ <∞ being a consequence of Lemma 3.3.3.2. Hence J (u)⊗ϕ ∈ D(I).

Now we prove that



ä+O((d/R)

−1+γ2 ),


for 0 < γ ≤ 1. Since Γ(1|~y|≥3d)ϕ = ϕ by Hypothesis (B2), we have that



ä+ IÄ(1− Γ(χ|~y|≤d))J (u)

ä⊗ ϕ.

We observe that all the terms of the previous equations are well-defined, as follows from thefact that J (u) ⊗ ϕ ∈ D(I) and the remark after the statement of Lemma 3.3.4.1. Thereforewe have to prove that

IÄ(1− Γ(χ|~y|≤d))J (u)

ä⊗ ϕ = O((d/R)

−1+γ2 ). (3.4.18)

Proceeding as above, we estimate∥∥∥IÄ(1− Γ(χ|~y|≤d))J (u)ä⊗ ϕ

∥∥∥=∥∥∥IÄΓ(e−δ

′1)⊗ Γ(e−δ1)

ä(eδ′N (1− Γ(χ|~y|≤d))J (u))⊗ (eδNϕ)

∥∥∥≤∥∥∥eδ′N (1− Γ(χ|~y|≤d))J (u)

∥∥∥∥∥∥eδNϕ∥∥∥.Next, since eδ′N commutes with Γ(χ|~y|≤d), we deduce that∥∥∥eδ′N (1− Γ(χ|~y|≤d))J (u)

∥∥∥2≤∥∥∥e2δ′NJ (u)

∥∥∥∥∥∥(1− Γ(χ|~y|≤d))2J (u)

∥∥∥≤∥∥∥e2δ′NJ (u)

∥∥∥∥∥∥(1− Γ(χ|~y|≤d))J (u)∥∥∥.

It follows from Lemma 3.3.3.2 that∥∥∥e2δ′NJ (u)

∥∥∥ <∞, and by Lemma 3.3.3.3, we have that∥∥∥(1− Γ(χ|~y|≤d))J (u)∥∥∥ = O((d/R)−1+γ), (3.4.19)

for all 0 < γ ≤ 1. The last three estimates prove (3.4.18), which concludes the proof of thelemma.


Proof of Lemma 3.3.4.3. Since Γ(j) only acts on the photon Fock space, we obviously havethat Ä

HP +HQ

äΓ(j)∗ = Γ(j)∗

ÄHP ⊗ 1H∞ + 1H0 ⊗HQ

ä. (3.4.20)

Moreover, it follows from Hypothesis (B4) that

HQ,EΓ(j)∗ = Γ(j)∗Ä1H0 ⊗HQ,E

ä. (3.4.21)

It remains to consider HEΓ(j)∗ and HP,EΓ(j)∗. A direct computation (see e.g. [46, Lemma2.16]) gives

HEΓ(j)∗ = Γ(j)∗ÄHE ⊗ 1H∞ + 1H0 ⊗HE

ä− dΓ(j∗, ad(|k|, j∗))U∗, (3.4.22)


where U is the unitary operator defined in (3.3.11)–(3.3.12) and, given a, b : L2(R3)⊕L2(R3)→L2(R3), the operator dΓ(a, b) : F+(L2(R3) ⊕ L2(R3)) → HE is defined by its restriction to⊗ns (L2(R3)⊕ L2(R3)) as

dΓ(a, b)|C := 0, (3.4.23)

dΓ(a, b)|⊗nsL2(R3)⊕L2(R3) :=n∑j=1

a⊗ · · · ⊗ a︸︷︷︸j−1

⊗b⊗ a⊗ · · · ⊗ a︸︷︷︸n−j

. (3.4.24)

The operators j∗ and ad(|k|, j∗) : L2(R3)⊕ L2(R3)→ L2(R3) in (3.4.22) are defined by

j∗(h0, h∞) = j0h0 + j∞h∞,

ad(|k|, j∗)(h0, h∞) := [|k|, j0]h0 + [|k|, j∞]h∞,

for all (h0, h∞) ∈ L2(R3) ⊕ L2(R3). By Lemma 3.4.4.6 of Appendix 3.4.4, we have that‖[|k|, j0]‖ = O(d−1) and ‖[|k|, j∞]‖ = O(d−1). This yields∥∥∥dΓ(j∗, ad(|k|, j∗))U∗(N0 +N∞ + 1)−1

∥∥∥ = O(d−1). (3.4.25)

Equations (3.4.22) and (3.4.25) yield

HEΓ(j)∗ = Γ(j)∗ÄHE ⊗ 1H∞ + 1H0 ⊗HE

ä+ Rem1, (3.4.26)

with ∥∥∥Rem1(N0 +N∞ + 1)−1∥∥∥ = O(d−1). (3.4.27)

Now we treat the interaction Hamiltonian HP,E . Using the notations (3.2.10)–(3.2.11), wehave that (see e.g. [46, Lemma 2.15]),

HP,EΓ(j)∗ = Φ(hx)Γ(j)∗ = Γ(j)∗ÄΦ(j0hx)⊗ 1H∞ + 1H0 ⊗ Φ(j∞hx)

ä. (3.4.28)

Here it should be understood that the operators j0, j∞ are applied to the L2(R3;HP ) functionsk 7→ hx(k) defined in (3.2.11). By Lemma 3.4.4.4 of Appendix 3.4.4, we have that∥∥∥Φ(j∞hx)〈~x〉−2+δ(N + 1)−

12

∥∥∥ = O(d−2+δ),

for all 0 < δ ≤ 2 , and likewise that∥∥∥ÄΦ(hx)− Φ(j0hx)ä〈~x〉−2+δ(N + 1)−

12

∥∥∥ = O(d−2+δ).

Therefore we can conclude that

HP,EΓ(j)∗ = Γ(j)∗ÄHP,E ⊗ 1H∞

ä+ Rem2, (3.4.29)

with ∥∥∥Rem2

ÄN0 +N∞ + 〈~x〉4−2δ

ä−1∥∥∥ = O(d−2+δ). (3.4.30)

Equations (3.4.20), (3.4.21), (3.4.26), (3.4.27), (3.4.29) and (3.4.30) prove the statement ofthe lemma.


3.4.3 Proof of Corollary 3.2.2.1

Corollary 3.2.2.1 is a direct consequence of Theorem 3.2.2.1 and the following lemma. Theproof of Lemma 3.4.3.1 uses Theorem 3.4.1.1 and follows the lines of [8].

Lemma 3.4.3.1. Let u ∈ H2(R3) with ‖u‖L2(R3) = 1. Suppose that V ∈ L∞(R3;R) satisfiessupp(V ) ⊂ B1 = ~x ∈ R3, |~x| < 1. Then there exists a constant C > 0 such that∥∥∥e−itHε

P∨EJ (u)− J (e−itHεP,effu)

∥∥∥ ≤ Ctεfor all 0 < ε < 1 and for all t ≥ 0.

Proof. We defineAtu := e−itH

εP∨EJ (u)− J (e−itH

εP,effu) (3.4.31)

for all u ∈ H2(R3). Since u ∈ D(HεP,eff) and J (u) ∈ D(Hε

P∨E), e−i(t−s)HεP∨EJ (e−isH

εP,effu) is

differentiable with respect to s, and we find that

Atu = −ie−itHεP∨E

∫ t

0eisH

εP∨EÄHεP∨EJ (us)− J (Hε

P,effus)äds (3.4.32)

for all u ∈ H2(R3), where us := e−isHεP,effu. We remind the reader that ψ(~p) is the ground

state of the Hamiltonian H(~p) = UHP∨EU−1 with corresponding eigenvalue E(~p), where

U : HP ⊗HE →∫⊕R3 C2 ⊗HE d3p is the generalized Fourier transform defined by

(Uϕ)(~p) =1

(2π)3/2

∫R3e−i(~p−

~PE)·~yϕ(~y) d3y, (3.4.33)

for all ϕ ∈ L1(R3;C2 ⊗HE). It follows that

J (us)(x) = U−1(χBν/2ψus

)(x),

and we have that

HP∨E(J (u)) = HP∨EU−1(χBν/2ψus

)= U−1(EψχBν/2 us), (3.4.34)

where E(~p) is the self-energy of the atom. Furthermore,

J ((HεP,eff − Vε)us) =

1

(2π)3/2

∫R3E(~p)us(~p)e

i~x·(~p−~PE)χBν/2(~p)ψ(~p) d3p. (3.4.35)

We observe that (3.4.34) and (3.4.35) imply that (3.4.32) is equal to

Atu = −ie−itHεP∨E

∫ t

0eisH

εP∨E (VεJ (us)− J (Vεus)) ds. (3.4.36)

Since e−itHεP∨E is an isometry, it is sufficient to bound the norm of

φs := VεJ (us)− J (Vεus). (3.4.37)

We have that

(Uφs)(~p) = U[VεU

−1(usχBν/2ψ)− U−1(’VεusχBν/2ψ)]

(~p)

=[Vε ∗ (usχBν/2ψ)−’VεusχBν/2ψ] (~p)


which can be rewritten as

(Uφs)(~p) =1

(2π)3/2

∫R3Vε(~p− ~q)us(~q)

(χBν/2(~q)ψ(~q)− χBν/2(~p)ψ(~p)

)d3q. (3.4.38)

Since U is an isometry,

‖Atu‖ ≤∫ t

0ds‖Uφs‖. (3.4.39)

We setΨ(~p) := χBν/2(~p)ψ(~p)

for all ~p ∈ R3. As χ is smooth and ψ is real analytic, Ψ is smooth with compact support andis consequently Lipschitz continuous in R3: There exists a constant Mν > 0 such that

‖Ψ(~q)−Ψ(~p)‖C2⊗HE ≤Mν |~p− ~q| (3.4.40)

for all ~p, ~q ∈ R3. Introducing the function

gε(~p) := ε+ |~p|, (3.4.41)

we get that

‖Atus‖ ≤∫ t

0(2π)−3/2

∥∥∥∥ ∫R3Vε(~p− ~q)us(~q) (Ψ(~q)−Ψ(~p)) d3q

∥∥∥∥ds≤∫ t

0(2π)−3/2

∥∥∥∥ ∫R3Vε(~p− ~q)gε(~p− ~q)us(~q)

1

gε(~p− ~q)(Ψ(~q)−Ψ(~p)) d3q

∥∥∥∥ds≤Mν

∫ t

0(2π)−3/2‖Vεgε ∗ us‖L2(R3)ds ≤ (2π)−3/2Mν

∫ t

0‖Vεgε‖L1(R3)‖us‖L2(R3)ds,

where we have used Young’s inequality in the last line. Since Supp(V ) ⊂ B1(0),

‖Vεgε‖L1(R3) ≤ ε‖Vε‖L1(R3) + ε‖V ‖L1(R3). (3.4.42)

Together with ‖us‖L2(R3) = 1, Vε(~p) = 1ε3VÄ~pε

ä, and ‖Vε‖L1(R3) = ‖V ‖L1(R3), this finally

implies that‖Atus‖ = O(tε). (3.4.43)

3.4.4 Relative bounds in Fock space and commutator estimates

We begin this appendix with some useful estimates concerning creation and annihilation op-erators on Fock space and second quantized operators.

We introduce the notation

h0 :=h ∈ L2(R3), ‖h‖h0 :=

∫R3

(1 + |k|−1)|h(k)|2dk <∞. (3.4.44)

We recall the following standard result (see e.g. [63, Lemma 17]).


Lemma 3.4.4.1. Let fi ∈ L2(R3) for i = 1, . . . , n. Then∥∥∥a#(f1) · · · a#(fn)(N + 1)−n2

∥∥∥ ≤ Cn‖f1‖L2(R3) . . . ‖fn‖L2(R3),

where a# stands for a or a∗. If in addition fi ∈ h0 for i = 1, . . . , n (where h0 is defined in(3.4.44)), then ∥∥∥a#(f1) · · · a#(fn)(HE + 1)−

n2

∥∥∥ ≤ Cn‖f1‖h0 . . . ‖fn‖h0 .

The following lemma was used (sometimes implicitly) several times in the main text. Itsproof can be found in [74, Section 3].

Lemma 3.4.4.2. Let ω, ω′ be two self-adjoint operators on L2(R3) with ω′ ≥ 0, D(ω′) ⊂D(ω) and ‖ωϕ‖L2(R3) ≤ ‖ω′ϕ‖L2(R3) for all ϕ ∈ D(ω′). Then D(dΓ(ω′)) ⊂ D(dΓ(ω)) and‖dΓ(ω)Φ‖ ≤ ‖dΓ(ω′)Φ‖ for all Φ ∈ D(dΓ(ω′)).

Now we turn to a few localization estimates that were used in the main text. The nextlemma is a particular case of [26, Lemma 3.1]. We do not present the proof.

Lemma 3.4.4.3. Let F ∈ C∞(R+; [0, 1]) be a smooth function such that Supp(F ) ⊂ [1,∞).Let a ∈ [0, 3/2), b ∈ R, χ ∈ C∞0 (R3) and hbx(~k) be such that, for all α ∈ N3, |∂α~k h

bx(~k)| .

|~k|b−|α|〈~x〉|α|. Assume that b > a− 3/2. Then, for all c ∈ [0, b− a+ 3/2) and d > 0,

∀~x ∈ R3,∥∥∥|~k|−aF (|i~∇k|/d)χ(~k)hbx(~k)

∥∥∥L2(R3

k)≤ Cd−c〈~x〉a+c.

Combining Lemmas 3.4.4.1 and 3.4.4.3, we obtain the following estimates that have beenused in the proof of Lemma 3.3.4.3. Recall that the operators j0, j∞ are defined at thebeginning of Section 3.3.4 and that the coupling function hx was defined in (3.2.11).

Lemma 3.4.4.4. For all 0 < δ ≤ 2, we have that∥∥∥Φ(j∞hx)〈~x〉−2+δ(N + 1)−12

∥∥∥ = O(d−2+δ),∥∥∥Φ((1− j0)hx)〈~x〉−2+δ(N + 1)−12

∥∥∥ = O(d−2+δ).

Proof. The proofs of the two stated estimates being the same, we only consider the first one.Applying Lemma 3.4.4.1, we obtain that, for all ϕ ∈ HP ⊗HE ' L2(R3;C2 ⊗HE),∥∥∥∥Φ(j∞hx)〈~x〉−2+δ(N + 1)−

12ϕ

∥∥∥∥2

≤ C∫R3〈~x〉−4+2δ

∥∥∥j∞hx(k)∥∥∥2

L2(R3)‖ϕ(~x)‖2C2⊗HE d

3x. (3.4.45)

Lemma 3.4.4.3 (applied with a = 0 and b = 1/2) then yields that∥∥∥∥Φ(j∞hx)〈~x〉−2+δ(N + 1)−12ϕ

∥∥∥∥2

≤ Cd−4+2δ‖ϕ‖2,

for all 0 < δ ≤ 2.


Another related consequence of Lemmas 3.4.4.1 and 3.4.4.3 is given by the following com-mutator estimate used in the proof of Lemma 3.3.3.4. Recall that χ|~y|≤d = j0(2|~y|/d) and thatHP,E = Φ(hx) = a∗(hx) + a(hx).

Lemma 3.4.4.5. For all 0 < δ ≤ 2, we have that∥∥∥〈~x〉−2+δîΓ(χ|~y|≤d), HP,E

ó(N + 1)−

12

∥∥∥ = O(d−2+δ).

Proof. A direct computation shows thatîΓ(χ|~y|≤d), a(hx)

ó= Γ(χ|~y|≤d)a((1− χ|~y|≤d)hx).

Applying Lemma 3.4.4.3 (with a = 0 and b = 1/2), we conclude as in Lemma 3.4.4.4 that∥∥∥〈~x〉−2+δîΓ(χ|~y|≤d), a(hx)

ó(N + 1)−

12

∥∥∥ = O(d−2+δ).

Since the estimate for a∗(hx) instead of a(hx) follows in the same way, the lemma is proven.

The next lemma is similar to [26, Lemma 5.2] and relies on Helffer-Sjöstrand functionalcalculus. We refer the reader to [26] for the proof.

Lemma 3.4.4.6. Let f ∈ C∞0 ([0,+∞);R) be a smooth function satisfying the estimates|∂ms f(s)| ≤ Cm〈s〉−m for all m ≥ 0. For all d > 0, we have thatî

|~k|, f(~y 2/d2)ó

= O(d−1). (3.4.46)

As a consequence of Lemma 3.4.4.6, we prove the following.

Lemma 3.4.4.7. ∥∥∥îHE ,Γ(χ|~y|≤d))ó(N + 1)−1

∥∥∥ = O(d−1)∥∥∥îHE ,Γ(χ|~y|≥2d))ó(N + 1)−1

∥∥∥ = O(d−1).

Proof. The two estimates are proven in the same way, we only establish the first one. On then-photons sector, a direct computation givesî

HE ,Γ(χ|~y|≤d))ó

= dΓ(χ|~y|≤d, [|k|, χ|~y|≤d]),

where dΓ(a, b) is defined by (3.4.23)–(3.4.24). Applying Lemma 3.4.4.6, we immediately de-duce that ∥∥∥îHE ,Γ(χ|~y|≤d))

ó|H(n)

E

∥∥∥ ≤ Cnd−1,

where H(n)E denotes the n-photons subspace. Since the constant C in the previous estimate is

uniform in n ∈ N, the lemma follows.


3.5 Appendix for Section 3.2.1: Proof of Lemma 3.2.1.1

In this section, we use the symbol a > b if a ≤ Cb for a positive constant C independent ofthe problem parameters v and d. The proof relies on a “Cook argument”. The sets

Dt := ~kt | ~k ∈ C2θ0;v (3.5.1)

satisfy ⋃t≥0

Dt = C2θ0 . (3.5.2)

Let Ω′ ⊂ R3 be defined by

Ω′ :=¶~x+ ~k | ~x ∈ Bd/4,~k ∈ C2θ0

©. (3.5.3)

Remark that dist(C2θ0 , ∂Ω′) = d/4. Let χΩ′ be a smooth function with support in Ωc, suchthat χΩ′(~x) = 1 for all ~x ∈ Ω′. We introduce

H0 := HP ⊗ 1HQ + 1HP ⊗HQ.

We have that

(e−itH−e−itH0)Ψ0 = (e−itHχΩ′−χΩ′e−itH0)Ψ0+(χΩ′−1)e−itH0Ψ0+e−itH(1−χΩ′)Ψ0. (3.5.4)

We estimate successively the 2 first terms on the right side of (3.5.4).

3.5.1 Free evolution

We control the free evolution with a stationary phase argument in Lemma 3.5.1.1.

Lemma 3.5.1.1. Let p ∈ N, p ≥ 4, and Ψ0 ∈ H be as in (A2). Then

‖(e−itH0Ψ0)(~y)‖C2⊗HQ ≤K

(|~y|+ vt)p(3.5.5)

for all ~y /∈ C2θ0, where

K := C1 max

Å1,

1

vp

ã1

[sin(θ0)]2p

∑|β|≤p+1

‖∂βΨ0‖L1(R3;C2⊗HQ). (3.5.6)

C1 is a positive constant that does not depend on v, d and θ0, but does depend on p.

Proof. Let t > 0 and let ~y /∈ Dt. We introduce the linear differential operator d~y,t :S(Cθ0;v;C2 ⊗HQ)→ S(Cθ0;v;C2 ⊗HQ), defined by

(d~y,tΨ)(~k) :=3∑j=1

∂kj

Çyj − kjt|~y − ~kt|2

Ψ(~k)

å(3.5.7)

for all Ψ ∈ S(Cθ0;v,C2 ⊗HQ) and for all ~k ∈ Cθ0;v. An easy calculation shows that

(d~y,tΨ)(~k) =3∑j=1

yj − kjt|~y − ~kt|2

∂kjΨ(~k)− t

|~y − ~kt|2Ψ(~k). (3.5.8)


Iterating (3.5.8), we get that

‖(dp~y,tΨ)(~k)‖C2⊗HQ ≤ f(p)(~k, ~y, t)

( ∑|β|≤p

‖∂βΨ(~k)‖C2⊗HQ

), (3.5.9)

where β is a multi index, and f (p)(~k, ~y, t) is a positive function of the form

f (p)(~k, ~y, t) =p∑l=0

a(p)l tl

1

|~y − ~kt|l+p. (3.5.10)

The coefficients a(p)l are positive constants independent of t and ~y. Integrating by parts, we

find that

(2π)3/2(e−itH0Ψ0)(~y) =

∫Cθ0;v

e−itk2/2+i~k·~y e−itHQΨ0(~k) d3k

= −i∫Cθ0;v

3∑j=1

yj − kjt|~y − ~kt|2

∂kj [e−itk2/2+i~k·~y] e−itHQΨ0(~k) d3k

= i

∫Cθ0;v

e−itk2/2+i~k·~y e−itHQ(d~y,tΨ0)(~k) d3k

= ip∫Cθ0;v

e−itk2/2+i~k·~y e−itHQ(dp~y,tΨ0)(~k) d3k

for all ~y /∈ Dt. We deduce that

(2π)3/2‖(e−itH0Ψ0)(~y)‖C2⊗HQ ≤∫Cθ0;v

|f (p)(~k, ~y, t)|( ∑|β|≤p

‖∂βΨ0(~k)‖C2⊗HQ

)d3k

≤∑|β|≤p

‖∂βΨ0‖L1(R3;C2⊗HQ) sup~k∈Cθ0;v

|f (p)(~k, ~y, t)|

for all ~y /∈ Dt. For a fixed set of variables (~y, t) with ~y /∈ C2θ0 ,

|~y − ~kt|2 =Ä|~y| − |~k|t cos(θ

~y,~k)ä2

+Ä|~k|t sin(θ

~y,~k)ä2

=Ä|~y| cos(θ

~y,~k)− |~k|t

ä2+Ä|~y| sin(θ

~y,~k)ä2

where θ~y,~k

is the angle between ~y and ~k. We deduce that

|~y − ~kt| ≥ |~y|+ vt

2sin(θ0),

for all ~k ∈ Cθ0;v. This implies that

sup~k∈Cθ0;v

tl1

|~y − ~kt|l+p≤ 4p

vl1

[sin(θ0)]2p1

(|~y|+ vt)p

for all ~y /∈ C2θ0 and for all 0 ≤ l ≤ p. Consequently,

‖(e−itH0Ψ0)(~y)‖C2⊗HQ ≤K

(|~y|+ vt)p(3.5.11)

for all ~y /∈ C2θ0 , where

K := C1 max

Å1,

1

vp

ã1

[sin(θ0)]2p

∑|β|≤p+1

‖∂βΨ0‖L1(R3;C2⊗HQ). (3.5.12)


3.5.2 Bound for the norm of (e−itHχΩ′ − χΩ′e−itH0)Ψ0

Lemma 3.5.2.1. We require (A1)-(A3). Then

‖(e−itHχΩ′ − χΩ′e−itH0)Ψ0‖ >

K

vdp−3+

1

dβ−12

+α2

+1

dβ∗

for all t ≥ 0, where β∗ = β if α < 2 and β∗ = β + 1/2 if α ≥ 2.

Proof. We define At := e−itHχΩ′ − χΩ′e−itH0 . Since D(HP )⊗D(HQ) ⊂ D(H),

s 7→ e−isHχΩ′e−i(t−s)H0

is strongly differentiable on D(HP )⊗D(HQ) and

AtΨ0 = i

∫ t

0e−isH(−HχΩ′ + χΩ′H0)e−i(t−s)H0Ψ0 ds

= i

∫ t

0e−isH

[∆(χΩ′)

2+ ~∇(χΩ′) · ~∇−HP,QχΩ′

]e−i(t−s)H0Ψ0 ds.

∆χΩ′ and ~∇χΩ′ vanish on Ω′ ∪ Ω. We split the formula above into two terms,

AtΨ0 = (AtΨ0)1 + (AtΨ0)2,

where(AtΨ0)1 := −i

∫ t

0e−isHHP,QχΩ′e

−i(t−s)H0Ψ0 ds, (3.5.13)

and(AtΨ0)2 := i

∫ t

0e−isH

[∆(χΩ′)

2+ ~∇(χΩ′) · ~∇

]e−i(t−s)H0Ψ0 ds. (3.5.14)

We first estimate the norm of (AtΨ0)1. We have that

‖HP,QχΩ′e−i(t−s)H0Ψ0‖ ≤

∑n∈I‖HP,QnχΩ′e

−i(t−s)HP (e−i(t−s)HQΨ0)‖.

Using Assumption (A1), we find that

‖(HP,QnχΩ′e−i(t−s)HP (e−i(t−s)HQΨ0))(~x)‖2 ≤ ‖(χΩ′e

−i(t−s)HP (Nne−i(t−s)HQΨ0))(~x)‖2

dist(~x,Qn)2α

≤ χ2Ω′(~x)

(1t−s≤d

‖Nn(e−i(t−s)H0Ψ0)(~x)‖2

d2αn

+ 1t−s>d‖(e−i(t−s)HPNne

−i(t−s)HQΨ0)(~x)‖2

dist(~x,Qn)2α

)for all ~x ∈ R3. Next we use Hölder’s inequality. We set

f(~x) : = ‖(e−i(t−s)HPNne−i(t−s)HQΨ0)(~x)‖2, (3.5.15)

g(~x) : = χ2Ω′(~x)

1

dist(~x,Qn)2α. (3.5.16)


Using the integral representation

e−i(t−s)∆/2ϕ(~x) =1

(2iπ(t− s))3/2

∫R3ei|~x−~y|22(t−s) ϕ(~y) d3y, for a.e. ~x ∈ R3, (3.5.17)

for all ϕ ∈ L1(R3) ∩ L2(R3), and Eq. (3.2.8), we remark that

‖f‖∞ ≤C2

(t− s)3, ‖f‖1 ≤ C. (3.5.18)

Therefore, f ∈ Lp for all p ≥ 1, and we have the estimate

‖f‖p =

Å∫R3|f |p−1|f |

ã1/p

≤ ‖f‖1/p1 ‖f‖p−1p∞ . (3.5.19)

Using Hölder’s inequality, we deduce that∫R3fg ≤ ‖f‖p‖g‖q (3.5.20)

for all p, q ≥ 1 (such that g ∈ Lq) with p−1 + q−1 = 1. Let ε ∈ (0, α − 1). We only treatthe case where α < 2, the other possibility being trivial. We choose p = 3/(1 − ε). Thenq = 3/(2 + ε), and we deduce that∫

R3fg >

1

(t− s)2+ε‖g‖3/(2+ε). (3.5.21)

Moreover,

‖g‖3/(2+ε) =

(∫χ

62+ε

Ω′ (~x)1

dist(~x,Qn)6α

2+ε

d3x

) 2+ε3

>1

d2α−2−εn

. (3.5.22)

Therefore, we have that

‖HP,QnχΩ′e−i(t−s)HP (e−i(t−s)HQΨ0)‖2 >

(1t−s≤d

1

d2αn

+ 1t−s>d1

(t− s)2+εdα−1n

).

Summing over n, we get that

‖HP,QχΩ′e−i(t−s)H0Ψ0‖ >

(1t−s≤d

1

dβ+ 12

+α2

+ 1t−s>d1

(t− s)1+ε/2dβ

). (3.5.23)

Integrating over s, we obtain∫ t

0‖HP,QχΩ′e

−i(t−s)H0Ψ0‖ds >1

dβ−12

+α2

+1

dβ. (3.5.24)

We now estimate ‖(AtΨ0)2‖. Eq. (3.5.5) implies that∥∥∥∆(χΩ′)

2e−itH0Ψ0

∥∥∥2>∫

Ωc\Ω′d3y

K2

(|~y|+ vt)2p>

K2

(d/4 + vt)2(p−2),

where we have used that |~y| ≥ d/4 if ~y ∈ Ωc \Ω′. A similar inequality is satisfied by the termwith

~∇(χΩ′ )·~∇2 in (3.5.14), as ~∇ and HP commute. We conclude that

‖(AtΨ0)2‖ >∫ t

0ds

K

(d/4 + v(t− s))p−2>

K

vdp−3. (3.5.25)


3.5.3 Final step of the proof

The estimation of the norm of(χΩ′ − 1)e−itH0Ψ0

is similar as what we did above and it is easy to show that

‖(χΩ′ − 1)e−itH0Ψ0‖ >K

vdp−3. (3.5.26)

Collecting the estimates (3.5.24) and (3.5.26), we find that∣∣∣〈e−itHΨ0|(OP ⊗ 1HQ)e−itHΨ0〉 − 〈e−itH0Ψ0|OP e−itHΨ0〉∣∣∣

=∣∣∣〈AtΨ0 + (χΩ′ − 1)e−itH0Ψ0 + e−itH(1− χΩ′)Ψ0|(OP ⊗ 1HQ)e−itHΨ0〉

∣∣∣>Ç‖(1− χΩ′)Ψ0‖+

K

vdp−3+

1

dβ−12

+α2

+1

dβ

å‖OP ‖.

Chapter 4

Preparation of states by weak couplingwith a dispersive environment

4.1 Model and summary of results

The goal of this Chapter is to make a mathematical contribution to our understanding of thelast method of state preparation discussed in Section 1.3. For this purpose, we study a specific(idealized) model of a very heavy atom interacting with the quantized e.m. field, describedin Section 4.1.1, below. Our main results and the underlying hypotheses are described andexplained in Subsections 4.1.2 and 4.1.1, respectively. The proof of our main result is outlinedin Subsection 1.4. All technical matters are treated in the rest of the text.

4.1.1 The model

As already explained in the introduction, the model underlying our analysis is the so-calledgeneralized spin-boson model, which describes an idealized very heavy atom coupled to a quan-tized free scalar field. (It is straightforward to replace the scalar field considered in this paperby the quantized electromagnetic field. We prefer to consider a scalar field purely for reasonsof notational simplicity. But we will call the field quanta “photons”.) The atom represents aquantum (sub)system henceforth denoted by P , while the field represents a system denotedby E. By HP we denote the n0-dimensional Hilbert space describing the internal states ofthe atom before it is coupled to the field; (the center-of-mass motion of the atom is neglected,because it is assumed to be very heavy). By F+(L2(R3)) we denote the symmetric Fock spaceover the one-particle Hilbert space L2(R3). As usual, physical quantities or “observables” ofP ∨ E are represented by certain bounded self-adjoint operators acting on the Hilbert spaceH := HP ⊗ F+(L2(R3)). Bosonic annihilation- and creation operators on F+(L2(R3)) areoperator-valued distributions, a(k) and a∗(k), where k ∈ R3 denotes a wave vector, satisfyingthe usual canonical commutation relations[

a(k), a∗(k′)]

= δ(k − k′),îa](k), a](k′)

ó= 0, k, k′ ∈ R3.

Total Hamiltonian

Let e1 < ... < en0 be real numbers and let (ϕi)n0i=1 be an orthonormal basis of HP . We set

Πi := |ϕi〉〈ϕi|, i = 1, ..., n0. We suppose that ωP (·) = 〈ϕ1, (·) ϕ1〉 is the state in which we

111

CHAPTER 4. PREPARATION OF STATES 112

want to prepare the system P . The time-dependent Hamiltonian of the system P ∨E is givenby

H(t) := H0 + λ(t)HI , (4.1.1)

where

H0 =n0∑i=1

eiΠi ⊗ 1HE + 1HP ⊗∫R3d3k ω(k)a∗(k)a(k) := HP ⊗ 1HE + 1HP ⊗HE , (4.1.2)

with ω(k) = |k|, for all k ∈ R3. In (4.1.1), λ(t) is a positive, monotone-decreasing functionof time t, and HI is the interaction Hamiltonian coupling P to E, which we define next.We assume that the coupling between the “atom” and the field is linear in creation- andannihilation operators. More precisely, the interaction Hamiltonian is given by

HI := G⊗ (a(φ) + a∗(φ)) , (4.1.3)

where the “form factor” φ belongs to L2(R3), and

a∗(φ) =

∫d3k φ(k)a∗(k), a(φ) =

∫d3k φ(k)a(k), (4.1.4)

for arbitrary φ ∈ L2(R3). The physics of exchange of quanta of energy between the atomand the field is characterized by the property that it satisfies a Fermi-Golden-Rule conditionintroduced in Section 4.1.1, below.

Initial states and “observables”

We assume that the system is initially in a state of the form

Ψ = ϕ⊗ Ω, (4.1.5)

where ϕ ∈ HP and Ω is the vacuum Fock state. This choice of a simple initial state is mademerely to avoid cumbersome notations and lengthy formulae. Our results still hold true if theinitial state of the atom, ϕ, is replaced by a density matrix on HP , and if the field is in a statewhere finitely many field modes are excited or in a coherent state; see Section 4.5 and [44].Moreover, the initial state may entangle the atom P with the field E.

The situation where the field is initially prepared in an equilibrium state at positive tem-perature is discussed in Section 4.5.

Basic assumptions

Assumption 4.1.1.1. (Decay of correlations) We assume that the form factor φ in (4.1.3) ischosen such that φ and φ/

√ω belong to L2(R3). We define

f(t) :=

∫d3k |φ(k)|2e−itω(k), (4.1.6)

t ≥ 0. We assume that there exists a constant α > 2 such that

|f(t)| ∝ 1

(1 + t)α. (4.1.7)


Assumption 4.1.1.2. (Fermi-Golden-Rule Condition) For all i ∈ 2, ..., n0,

i−1∑j=1

∫d3k |Gij |2|φ(k)|2δ(ej − ei + ω(k)) > 0. (4.1.8)

Assumption 4.1.1.3. (Evolution of λ(t)) There exists a constant γ, with

−1/2 < γ < 0,

such thatλ(t) = (λ(0)1/γ + t)γ . (4.1.9)

The first part of Assumption 4.1.1.1 ensures that HI is H1/2E -bounded. Therefore, by

Kato’s theorem (see [95]), HE + λ(t)HI is self-adjoint on the domain of HE , for all values ofλ(t). The second part of Assumption 4.1.1.1 specifies the minimal decay rate of the “correlationfunction” f(t) in time t needed to carry out our analysis. The behavior of f(t), for large t, isdetermined by the infrared behavior of the form factor φ. If φ is smooth in k, except at k = 0,with φ(k) ' |k|µ, as |k| → 0, and if φ is invariant under rotation and has compact support,the theory of asymptotic expansions for Fourier integrals shows that

f(t) ∝ t−3−2µ,

see [51]. Therefore, µ must be strictly bigger than −1/2 for (4.1.7) to be satisfied. Eq. (4.1.8)implies that the exited states ϕi (i 6= 1) of the atom decay, i.e., correspond to resonances,when the coupling between the atom and the field is turned on. Assumption 4.1.1.3 is an“adiabatic” condition: The coupling λ(t) must decrease sufficiently slowly in t for a state,ρinv(s), invariant under the reduced dynamics of P to exist on a time scale of t− s ∝ λ−2(s),(Van Hove limit).

4.1.2 Main result

Theorem 4.1.2.1. Suppose that Assumptions 4.1.1.1, 4.1.1.2 and 4.1.1.3 are satisfied. Then,there exists a constant λc > 0, such that, for any 0 < λ(0) < λc,

〈Ψ(t)|(O ⊗ 1HE )Ψ(t)〉 −→t→∞

〈ϕ1|Oϕ1〉 ≡ ωP (O), (4.1.10)

for all initial states Ψ of the form given in (4.1.5) and for all observables O ∈ B(HP ). Hereϕ1 is the ground state (unique up to a phase) of HP corresponding to the eigenvalue e1, andΨ(t) is the state Ψ evolves into, after time t, under the dynamics generated by the family ofHamiltonians H(t).

Remark 4.1.2.1. In Section 4.5, we will generalize Theorem 4.1.2.1 to a larger class ofinitial states (including ones with a non-zero, but finite number of occupied field modes andones exhibiting entanglement). We also present a variant of Theorem 4.1.2.1 where the fieldmodes are at some non-zero temperature T > 0. We will show that P thermalizes at the sametemperature T , as t→∞.


4.1.3 Outline of the proof

We will use expansion methods developed in [44, 43] to prove convergence of the expectationvalues 〈Ψ(t)|(O ⊗ 1HE )Ψ(t)〉 to 〈ϕ1|Oϕ1〉, as t→∞. In [44, 43], the authors consider a cou-pling constant λ independent of time t. We adapt the methods developed in these referencesto apply to the models considered in this paper, with λ depending on time t and decreasingto zero, as t→∞. We attempt to present a somewhat streamlined version of the arguments(in particular of the “polymer expansion”) in [44, 43]. We employ the Heisenberg picture, andwe only investigate the time evolution of observables, O, of the atom P .

Step 1. Analysis of the reduced dynamics on the Van Hove time scale

We introduce a linear operator Zt,s : B(HP ) → B(HP ) describing the effective dynamics ofobservables O ∈ B(HP ), for times = O(λ−2) (van Hove time scale). Let U(t, s) be the unitarypropagator generated by the family of time-dependent Hamiltonians H(t)t∈R+ . We defineZt,s(O) ∈ B(HP ) by

〈ϕ|Zt,s(O)ψ〉 := 〈ϕ⊗ Ω|U∗(t, s)(O ⊗ 1HE )U(t, s)(ψ ⊗ Ω)〉, (4.1.11)

for all ϕ,ψ ∈ HP . In defining Zt,s we take an average in the vacuum vector, Ω, of the fieldvariables, because “photons” emitted by the atom escape towards infinity. On the Van Hovetime scale, O(λ(s)−2), the atom decays to its ground state with a probability very close to 1,and the photons have escaped from the vicinity of the atom and will never return to it. Thus,Zt,s(·) can be expected to describe the Heisenberg time evolution of atomic observables fairlyaccurately if t− s = O(λ(s)−2). If the function λ(t) decays slowly in time, the effective timeevolution Zt,s is well approximated by a semi-group of completely positive maps on B(HP )generated by a Lindblad operator, for time differences, t− s, of order O(λ(s)−2); see (4.2.35)-(4.2.37). (The error actually tends to zero in norm as s → ∞.) We use this result to showthat

Zs+τλ−2(s),s = P (s) +R(s), (4.1.12)

where P (s) : B(HP ) → B(HP ) is a one-dimensional projection, and R(s) : B(HP ) → B(HP )is a perturbation that can be made arbitrarily small by choosing the parameter τ > 0 largeenough. We show in Paragraph 4.2.4 that P (s) converges in norm to the one-dimensionalprojection |1HP 〉〈Π1|, as s → ∞, where, recall, Π1 = |ϕ1〉〈ϕ1|. Here elements of B(HP ) arewritten as vectors, |·〉, (more precisely, as vectors in the Hilbert space of matrices).

Eq. (4.1.12) captures the dissipative behavior of the effective dynamics of the atomicsystem on the Van Hove time scale.

Step 2. Reduced dynamics at arbitrarily large times: the cluster expansion

Eq. (4.1.12) is only valid on the Van Hove time scale t− s = O(λ−2(s)). However, we intendto prove that

Zt,0 −→t→∞

|1HP 〉〈Π1|. (4.1.13)

The polymer expansion introduced in [44, 43] offers a way to pass from the Van Hove timescale to arbitrarily large times, t→∞. It is based on the intuition that the dynamics of atomicobservables is close to one given by a quantum Markov dynamics whose only invariant state


is given by |1HP 〉〈Π1|, with errors that can be controlled with the help of a cluster expansionfor a one-dimensional system of “extended particles”, called “polymers”, of ever smaller densityof O(λ(s)2), as s→∞.

In somewhat more precise terms, our expansion is set up as follows: We start by labelingall terms in the Dyson expansion of Zt,s(·) by Feynman diagrams. For each time intervalIi = [ti, ti+1), with ti+1 = ti + τλ(ti)

−2, we sum all contributions labeled by diagrams withthe property that any “photon” emitted at a time in the intervall Ii is re-absorbed by theatom at another time in the same interval Ii. This yields the contribution correspondingto the operator Zti+τλ−2(ti),ti . It is at this point where the decomposition (4.1.12) comesinto play: We use that P (s) is a one-dimensional projection to rewrite the expectation value〈Ψ(tN )|(O⊗1HE )Ψ(tN )〉 in the form of a cluster expansion for a system of “extended particles”/“polymers” in one dimension. We show in Section 4.3 that

〈Ψ(tN )|(O ⊗ 1HE )Ψ(tN )〉 =N∑q=1

1

q!

∑X1,...,Xq

dist(Xi,Xj)≥2, diam(Xj)≤N

p (X1) ... p (Xq) , (4.1.14)

where p (X ) ∈ C are the statistical weights of certain polymers, X ; see Paragraph 4.3.3.

Step 3. The limit t→∞

In Section 4.4 we prove that our cluster expansion, see Eq. (4.1.14), converges uniformly inN , and, as a corollary, that 〈Ψ(t)|(O ⊗ 1HE )Ψ(t)〉 tends to 〈ϕ1|Oϕ1〉, as t → +∞. In ourproof of convergence of the cluster expansion we have to require that the correlation functionf(t) defined in (4.1.6) decay sufficiently fast in t; see Assumption 4.1.1.1. In Paragraph 4.4.1,we use the decay properties of f(t) to derive an upper bound for the statistical weights p(X )of polymers X appearing in (4.1.14) that implies the so-called Kotecky-Preiss criterion,∑

G′, dist(X ′,X )≤1

|p(X ′)| ea(X ′) ≤ a(X ), (4.1.15)

for a suitably chosen positive function a. Using standard results in the theory of clusterexpansions (see Appendix 4.6.1 for a short recap), it is straightforward to show that (4.1.15)implies that (4.1.14) converges to 〈ϕ1|Oϕ1〉, as N → ∞. A crucial point in this analysis isthat the right side of (4.1.14) can be written as an exponential of a convergent sum. Dividingthis expression by a corresponding expression for 1 = 〈Ψ(tN )|Ψ(tN )〉, one observes that thenumber of terms contributing in the limit N →∞ is quite small; see Section 4.4.2.

Generalization of Theorem 4.1.2.1 to initial states with finitely many “photons”and to thermal equilibrium states

Such generalizations of Theorem 4.1.2.1 are formulated in Section 4.5. Sketches of the proofsare given in Appendix 4.8.

4.2 Analysis of Z t,s for t− s ∝ λ−2(s)

We begin this section with a list of notations and conventions that are used throughout thepaper. In Subsections 4.2.2 and 4.2.3, we compare the Dyson expansion for Zt,s(O) with the


Dyson expansion for Zt,s0 (O), where Zt,s0 (O) ∈ B(HP ) is defined by

〈ϕ|Zt,s0 (O)ψ〉 := 〈ϕ⊗ Ω|ei(t−s)H(s)(O ⊗ 1HE )e−i(t−s)H(s)(ψ ⊗ Ω)〉, (4.2.1)

for arbitrary φ, ψ ∈ HP . We then compare Zt,s0 (·) with the semigroup generated by a Lind-bladian, using results in [44]. The calculation of the Lindbladian is explicitly carried out inAppendix B. Using estimates from perturbation theory, we will prove that, for large values ofthe parameter τ and small values of λ(0),

Zs+τλ−2(s),s = P (s) +R(s), (4.2.2)

for all s ≥ 0, where P (s) : B(HP ) → B(HP ) is a one-dimensional projection, and R(s) :B(HP ) → B(HP ) is a small perturbation. Various straightforward but lengthy calculationsare deferred to Appendix B.

4.2.1 Notations

Inner products and norms

The norm on HP determined by the scalar product 〈·|·〉P is denoted by || · ||P . On the tensorproduct space H = HP ⊗F+(L2(R3)), the scalar product is given by

〈φ⊗ ψ|φ′ ⊗ ψ′〉 := 〈φ|φ′〉P 〈ψ|ψ′〉F+ ,

where 〈·|·〉F+ is the scalar product on F+(L2(R3)) and is defined by

〈ψ|χ〉F+ = ψ(0)χ(0) +∑n≥1

∫d3k1 ... d

3kn ψ(n)(k1, ..., kn)χ(n)(k1, ..., kn), (4.2.3)

for all ψ = (ψ(n))n≥0 and all χ = (χ(n))n≥0.The algebra of bounded operators on HP is denoted by B(HP ). Since dimHP =: n0 <∞,

B(HP ) is a Hilbert space equipped with the scalar product

〈X|Y 〉B(HP ) := Tr(X∗Y ), ∀X,Y ∈ B(HP ), (4.2.4)

and the Hilbert-Schmidt norm. The operator norm on B(HP ) is given by

||X|| = supψ 6=0, ψ∈HP

||Xψ||P||ψ||P

(4.2.5)

for all X ∈ B(HP ). The algebra of bounded linear operators on B(HP ) is denoted byB(B(HP )). It is isomorphic to the finite dimensional Hilbert space of n2

0×n20 complex matrices.

We will use two equivalent norms on B(B(HP )), which are defined by

||B||∞ := supX 6=0, X∈B(HP )

||BX||||X||

and ||B||22 := supX 6=0, X∈B(HP )

〈BX|BX〉B(HP )

〈X|X〉B(HP ), (4.2.6)

for all B ∈ B(B(HP )).


Dyson expansion

The starting point of the methods developed in [44, 43] and used in the present paper is theDyson expansion. We need to introduce some shorthand notations, in order to avoid cumber-some complications in the representation of the Dyson expansion. We define the quantities

Φ(φ) := a(φ) + a∗(φ), (4.2.7)

G(t) := eitHPGe−itHP , (4.2.8)

HI(t) := eitH0HIe−itH0 , (4.2.9)

φ(t)(k) := eitω(k) φ(k), (4.2.10)

for all t ∈ R, where H0 and HP are defined in Eq. (4.1.2). Time-integrations will usuallyextend over an n-dimensional simplex

∆n [t, s] := (t1, ..., tn) | s ≤ t1 < ... < tn ≤ t, 0 < s < t <∞. (4.2.11)

For any n-tuple t := (t1, ..., tn), ordered as in (4.2.11), and for any time-dependent operator-valued function O(t), we set

λ(t) := λ(t1) ... λ(tn), (4.2.12)O(t) := O(t1) ... O(tn). (4.2.13)

It turns out to be useful to introduce operators R(A) (right multiplication by A∗) and L(A)(left multiplication by A), A ∈ B(HP ), by setting

R(A)(O) := OA∗, (4.2.14)L(A)(O) := AO, (4.2.15)

for all A,O ∈ B(HP ).

4.2.2 Dyson expansions for Z t,s and Z t,s0

We first recall the Dyson expansion for the propagator U(t, s) of the system.

Lemma 4.2.2.1. Let t, s ∈ R. The Dyson series

U(t, s) = e−i(t−s)H0 +∞∑k=1

(−i)k∫ t

sduk...

∫ u2

sdu1λ(u1)...λ(uk) e

−itH0HI(uk)...HI(u1)eisH0 ,

(4.2.16)and

e−i(t−s)H(s) = e−i(t−s)H0 +∞∑k=1

(−i)k∫ t

sduk...

∫ u2

sdu1λ

k(s) e−itH0HI(uk)...HI(u1)eisH0

(4.2.17)converge strongly on HP ⊗ F (L2(R3)), where F (L2(R3)) ⊂ F+(L2(R3)) is the dense set ofvectors in Fock space describing configurations of finitely many “photons”.


Lemma 4.2.2.1 is standard, (but see Appendix 4.6.2). It can be used to deriving the Dysonexpansion of the effective atom propagator Zt,s. We remind the reader that Zt,s is defined by

〈ϕ|Zt,s(O)ψ〉 := 〈ϕ⊗ Ω|U∗(t, s)(O ⊗ 1HE )U(t, s)(ψ ⊗ Ω)〉, ∀ϕ,ψ ∈ HP , (4.2.18)

for arbitrary O ∈ B(HP ). We compute the vacuum expectation values of products of inter-action Hamiltonians HI(u) in the Dyson expansion of the propagators on the right side of(4.2.18) using Wick’s theorem.

Theorem 4.2.2.1 (Wick’s Theorem). If Ω denotes the Fock vacuum then

〈Ω|Φ(φ(v1))...Φ(φ(v2k))Ω〉 =∑

pairings π

∏(i,j)∈π

〈Ω|Φ(φ(vi))Φ(φ(vj))Ω〉, (4.2.19)

〈Ω|Φ(φ(v1))...Φ(φ(v2k+1))Ω〉 = 0, (4.2.20)

for arbitrary times v1, ..., v2k, v2k+1 ∈ R, where pairings, π, are sets of pairs (i, j), with i < j,whose union is the set 1, ..., 2k.

The “two-point functions” 〈Ω|Φ(φ(vi))Φ(φ(vj))Ω〉 can all be expressed in terms of thecorrelation function f(t), see (4.1.6), which is given by

f(t) = 〈Ω|Φ(φ(t))Φ(φ)Ω〉. (4.2.21)

for all t ∈ R. We note that f(−t) = f(t), for all t ∈ R.Next, we introduce an index r ∈ 0, 1 attached to each time u labeling an interaction

Hamiltonian that appears in the Dyson expansion of the right side of (4.2.18): r takes thevalue 0 if HI(u) in (4.2.16) appears on the left of O, and r = 1 if HI(u) appears on the rightof O. We then write (u, r), instead of u. We also define

A(u, r) =

L(A(u)) if r = 0,

R(A(u)) if r = 1,(4.2.22)

for all (u, r) and for any time dependent family of operators A(u) on H (or on HP ). Remarkthat A(u, r) is an operator on B(H) for all (u, r) if A(u) is an operator on H.

We introduce some convenient short-hand notations for two-point functions as follows.

Definition 4.2.2.1. (The functions F and F)

F(u, r; v, r′) := 〈Ω|ÄΦ(φ(u, r))Φ(φ(v, r′))

ä(1HE )Ω〉, (4.2.23)

andF(u, r; v, r′) := F(u, r; v, r′)(iG)(u, r)(iG)(v, r′). (4.2.24)

We also introduce a time-ordering operator acting on products of operators in B(B(HP )).We will use it to order products of operators (iG)(u, r).

Definition 4.2.2.2 (Time-ordering). Let 0 < u1 < ... < un be an ordered n-tuple of times,and let A(u1), ..., A(un) ∈ B(B(HP )) be a family of operators. We define an operator Tor :B(B(HP ))→ B(B(HP )) by

Tor(A(π(u1)) ... A(π(un))) := A(u1) ... A(un) (4.2.25)

for all permutations π of 1, ..., n.


We denote by w := (u, r; v, r′) a pair of times, u < v, decorated by indices r and r′, andby w = (w1, ..., wk) a k-tuple of such pairs. A similar underlined notations is used for k-tuplesof times u, denoted by u = (u1, ..., uk), and k-tuples of indices r, denoted by r = (r1, ..., rk).

Next, we introduce a measure on k-tuples of pairs, k = 1, 2, 3, ...

dµk(w) :=∑

r,r′∈0,1kχ(u1 < ... < uk)

k∏i=1

χ(ui < vi) du1 ... duk dv1 ... dvk . (4.2.26)

Our next lemma describes the Dyson expansions of Zt,s(O) and Zt,s0 (O). We make use of thenotations introduced above.

Lemma 4.2.2.2.

eisHPZt,s(O)e−isHP =∞∑k=0

∫[s,t]2k

dµk(w)λ(w)Tor

[k∏i=1

F(ui, ri; vi, r′i)

][O(t)] , (4.2.27)

where λ(w) := λ(u)λ(v); and

eisHPZt,s0 (O)e−isHP =∞∑k=0

∫[s,t]2k

dµk(w)λ2k(s)Tor

[k∏i=1


][O(t)] (4.2.28)

for all s, t ∈ R+ with t ≥ s.The series in (4.2.27) and (4.2.28) converge in norm, for all O ∈ B(HP ), and are bounded

bye4|t−s| ||f ||L1 ||G||2λ2(s)||O||. (4.2.29)

A proof of Lemma 4.2.2.2 is given in Appendix 4.6.2.

4.2.3 Comparison of the effective propagator Z t,s with the semigroup gen-erated by a Lindbladian

In Lemma 4.2.3.1, below, we present an estimate on the norm of the difference

Zt,s −Zt,s0 ,

for t−s = τλ(s)−2. We will then use a result from [44, 43] to compare Zt,s0 with the semigroupgenerated by a Linbladian. These findings will enable us to represent the effective propagatorZt,s, with t = s + τλ(s)−2, as the sum of a one-dimensional projection P (s) and a “smallperturbation” R(s). We will show in subsection 4.2.4 that the parameter τ can be chosen insuch a way that, for any given ε0 > 0, ‖R(s)‖∞ < ε0.

Lemma 4.2.3.1. Let t > s ≥ 0. Then

||Zt,s −Zt,s0 ||∞ ≤ e4(t−s)||f ||L1 ||G||2λ2(s) − e4(t−s)||f ||L1 ||G||2λ2(t). (4.2.30)

If Assumption 4.1.1.3, see Eq. (4.1.9), is satisfied then

||Zs+τλ−2(s),s −Zs+τλ−2(s),s

0 ||∞ −→s→∞

0. (4.2.31)


Furthermore, given any ε > 0, there exists λε > 0 such that, for any 0 < λ(0) < λε,


0 ||∞ ≤ ε, (4.2.32)

for all s ≥ 0.

Proof. Let O ∈ B(HP ). Eq. (4.2.27) implies that

eisHPÄZt,s(O)−Zt,s0 (O)

äe−isHP

=∞∑k=0

∫[s,t]2k

dµk(w) (λ(w)− λ2k(s))Tor

[k∏i=1


][O(t)] .

Using that λ(t) decreases in time t and that ‖G(u)G(v)‖ ≤ ‖G‖2 (see Eq. (2.8) for thedefinition of G(u)), we find that

|λ(w)− λ2k(s)|||Tor

[k∏i=1


][O(t)] ||

≤ ||O||Äλ2k(s)− λ2k(t)

ä||G||2k

k∏i=1

|f(vi − ui)|.

We thus conclude that

||Zt,s(O)−Zt,s0 (O)|| ≤ ||O||∞∑k=1

∫[s,t]2k

dµk(w)Äλ2k(s)− λ2k(t)

ä||G||2k

k∏i=1

|f(vi − ui)|

≤ ||O||∞∑k=1

4k(t− s)k

k!

Äλ2k(s)− λ2k(t)

ä||G||2k ||f ||kL1

by integrating first over all the v- variables and subsequently over all the u- variables. Thefactor (t−s)k

k! comes from integrating over the k-dimensional simplex ∆k[t, s]. Hence


0 ||∞ ≤ e4τ ||f ||L1 ||G||2 − e4τλ−2(s)||f ||L1 ||G||2λ2(t),

with t = s+ τλ−2(s). By (4.1.9), the ratio λ(t)/λ(s) is given by

λ(t)

λ(s)=

Çλ(0)1/γ + t

λ(0)1/γ + s

åγ=Ä1 + τ(λ(0)1/γ + s)−2γ−1

äγ. (4.2.33)

ThusÄ1 + τ(λ(0)1/γ + s)−2γ−1

äγ → 1, as s → ∞, because −2γ − 1 < 0. Furthermore, themaximum in (4.2.33) is reached at s = 0; it is equal to

(1 + τλ(0)(−2γ−1)/γ)γ .

Given any τ , we can choose the coupling λ(0) in such a way that this term is as close to 1 aswe wish, because −1/2 < γ < 0.


We define the Liouvillian L ∈ B(B(HP )) by

L := L(HP )−R(HP ). (4.2.34)

The eigenvalues of L are energy differences εij := ei − ej , i, j ∈ 1, ..., n0. The associatedeigenvectors are the n0 × n0 matrices Πij = |ϕi〉〈ϕj | ∈ B(HP ), where ϕi is the eigenvectorof HP corresponding to the eigenvalue ei, i = 1, ..., n0. The eigenvalues ei are assumed tobe non-degenerate, and we may assume that the eigenvalues εij are non-degenerate, too, fori 6= j. The eigenvalue 0 is n0-fold degenerate, and the corresponding eigenvectors are given byΠ1 ≡ Π11, ...,Πn0 ≡ Πn0n0 . We denote by Pε ∈ B(B(HP )) the orthogonal projection onto theeigenspace of L corresponding to the eigenvalue ε(= εij , for some i and j). This projection isone-dimensional if ε 6= 0 and n0-dimensional if ε = 0. The spectrum of L is denoted by σ(L).Following [44, 43, 40], we define the LindbladianM∈ B(B(HP )).

Definition 4.2.3.1. (Lindbladian) A Lindblad generatorM∈ B(B(HP )) is defined by

M :=∑

ε∈σ(L)

∫ ∞0

e−isεPεKsPε ds, (4.2.35)

whereKu2−u1 := e−iu1L

∑(r1,r2)∈0,12

F(u1, r1;u2, r2) eiu2L. (4.2.36)

An easy calculation shows that the right side of (4.2.36) only depends on u2−u1 and thatthe operatorM is well-defined.

Lemma 4.2.3.2. There exists a constant C > 0 independent of λ(·) such that

||Zt,s0 − ei(t−s)L+(t−s)λ2(s)M||∞ ≤ Cλ2(s)eCλ

2(s)(t−s)| ln(λ(s))|, (4.2.37)

for all t, s ≥ 0.

The proof of Lemma 4.2.3.2 is similar to Proposition 3.3 in [44].

4.2.4 Properties of Zs+τλ−2(s),s

We calculate the Lindblad operator M quite explicitly in Appendix B.3. Using those cal-culations, it is easy to deduce that Π1 is a left-eigenvector of M with eigenvalue 0. Indeed,〈Π1|MX〉 = Tr(Π1MX) = 0, for all X ∈ B(HP ). Using formulae (4.6.23) and (4.6.2), oneverifies thatM|1HP 〉 = 0, where

1HP =n0∑i=1

Πi

is the identity matrix in B(HP ). The one-dimensional projection

P := |1HP 〉〈Π1| ∈ B(B(HP )) (4.2.38)

satisfiesP |X〉 = |1HP 〉〈Π1|X〉B(HP ) = |1HP 〉Tr(Π1X)

for all X ∈ B(HP ). Moreover, P commutes withM:

PM = |1HP 〉〈Π1|M = 0 =M|1HP 〉〈Π1| =MP. (4.2.39)


Lemma 4.2.4.1. If the Fermi-Golden-Rule conditions (4.1.8) are satisfied then 0 is a non-degenerate eigenvalue of the Lindbladian M. The other eigenvalues of M have a strictlynegative real part. Furthermore, the projection P = |1HP 〉〈Π1| satisfies PM =MP = 0.

That the non-zero eigenvalues ofM have a strictly negative real part is a consequence ofthe Fermi-Golden-Rule conditions (4.1.8). Since L and M commute, Lemma 4.2.4.1 yieldsthe following corollary.

Corollary 4.2.4.1. The operator ei(t−s)L+(t−s)λ2(s)M has a non-degenerate eigenvalue 1 cor-responding to the eigenvector 1HP . The projection P given in (4.2.38) commutes with theoperator ei(t−s)L+(t−s)λ2(s)M and Pei(t−s)L+(t−s)λ2(s)M = P .

Spectrum of Zs+τλ−2(s),s

Lemma 4.2.4.2. Suppose that Assumption 4.1.1.1, Eq.(4.1.7), Assumption 4.1.1.2 and As-sumption 4.1.1.3, Eq. (4.1.9), are satisfied. Let 0 < ε0 < 1. There are positive constants τε0and λε0,τ > 0 such that, for any τ > τε0 and for any λ(0) < λε0,τ ,

Zs+τλ−2(s),s = P (s) +R(s), (4.2.40)

for all s ≥ 0. The operators P (s) and R(s) have the following properties:

• R(s) is a small perturbation, with ||R(s)||∞ < ε0;

• P (s) has the form |1HP 〉〈Π(s)|, where Π(s) is a rank-1 projection, with Π(s) ' Π1.More precisely, Π(s) converges to Π1 in norm, as s→∞.

The operator P (s) projects onto the (subspace spanned by the) eigenvector

1HP of Zs+τλ−2(s),s.

• P (s) commutes with Zs+τλ−2(s),s and

P (s)R(s) = R(s)P (s) = 0. (4.2.41)

Proof. We first remark that 1HP is an eigenvector of the operator Zs+τλ−2(s),s with associatedeigenvalue 1. Indeed,

Zs+τλ−2(s),s(1HP )PΩ = PΩU∗(s+ τλ−2(s), s)U(s+ τλ−2(s), s)PΩ = PΩ.

Let 0 < ε0 < 1 and s ≥ 0. We consider the disk Dr of radius 1 > r > 0 centered atthe eigenvalue 1 of eiτλ−2(s)L+τM in the complex plane, and we choose r such that Dr ∩σ(eiτλ

−2(s)L+τM) = 1. We introduce

mσ := max<(z) \ 0|z ∈ σ(M).

The finite set σ(eiτλ−2(s)L+τM) \ 1 lies to the left of the vertical line given by the equation

<(z) = eτmσ . Since mσ < 0, the radius r of the disk Dr can be set to 1/2, for sufficientlylarge τ .


Corollary 4.2.4.1 shows that P = |1HP 〉〈Π1| commutes with eiτλ−2(s)L+τM, and that

Peiτλ(s)−2L+τM = eiτλ(s)−2L+τMP = P.

The projection Π1 is the only operator of trace = 1 such that 〈Π1|eiτλ(s)−2L+τM = 〈Π1|, andP coincides with the Riesz projection

P = − 1

2iπ

∫∂D1/2

1

eiτλ(s)−2L+τM − zdz.

The sequence ||(eM − P )n||1/n∞ tends to emσ , as n tends to infinity, (a consequence of thespectral radius formula). Therefore there exists τε0 > 0 such that, for any τ > τε0 ,

||eiτλ(s)−2L+τM − P ||∞ ≤ ||(eM − P )τ ||∞ < eτmσ/2 < ε0/2, (4.2.42)

for all s ≥ 0.We now choose τ > τε0 and compare the spectra of Zs+τλ−2(s),s and eiτλ(s)−2L+τM. The

second resolvent formula yields the formal Neumann series

1

Zs+τλ−2(s),s − z=

1

eiτλ(s)−2L+τM − z

·∞∑k=0

ï(eiτλ(s)−2L+τM −Zs+τλ−2(s),s)

1

eiτλ(s)−2L+τM − z

òk.

(4.2.43)

The resolvent (eiτλ(s)−2L+τM − z)−1 is bounded in norm on the circle ∂D1/2 by a constantC(τ) > 1 that depends on τ , but not on λ(·) or γ, because σ(L) ⊂ R.

Let 0 < ε 1. Lemmas 4.2.3.1 and 4.2.3.2 show that there exists a constant λε dependingon γ and τ such that, for any λ(0) < λε,

||eiτλ(s)−2L+τM −Zs+τλ−2(s),s||∞ <ε

C2(τ), (4.2.44)

for all s ≥ 0. If (4.2.44) holds then the Neumann series in (4.2.43) converges in norm || · ||∞ ,uniformly in s ≥ 0, and there exists a bounded operator A(s, z), such that

1

Zs+τλ−2(s),s − z=

1

eiτλ(s)−2L+τM − z+A(s, z),

for all s ≥ 0 and all z ∈ ∂D1/2. Furthermore, ||A(s, z)||∞ < Cε, for all s ≥ 0 and all z ∈ ∂D1/2.Here C is a positive constant independent of λ and τ . The Riesz projection P (s), defined by

P (s) := − 1

2iπ

∫∂D1/2

1

Zs+τλ−2(s),s − zdz,

is one-dimensional, and ‖P − P (s)‖∞ = O(ε), for all s ≥ 0. That P (s) is rank-one followsfrom the property that two projections P and Q with dim(RanP ) 6= dim(RanQ) must satisfy||P −Q||2 ≥ 1, where the norm || · ||2 has been defined in (4.2.6). For small ε > 0, P (s) mustbe rank-one, because the norms || · ||2 and || · ||∞ are equivalent. The identity matrix, 1HP , isan eigenvector of Zs+τλ−2(s),s corresponding to the eigenvalue 1, for all s ≥ 0, and the Rieszprojection P (s) must project onto the subspace spanned by 1HP . Therefore

P (s) = |1HP 〉〈Π(s)|,


where Π(s) ∈ B(HP ) is an n× n matrix with trace one.Furthermore, ‖Π(s)−Π1‖ = O(ε), because ‖(P −P (s))(|Π1〉−|Π(s)〉)‖ = 〈Π1−Π(s)|Π1−

Π(s)〉B(HP ). To complete our proof we note that

||R(s)||∞ = ||Zs+τλ−2(s),s − P (s)||∞= ||Zs+τλ−2(s),s − eiτλ(s)−2L+τM + eiτλ(s)−2L+τM − P + P − P (s)||∞≤ ||Zs+τλ−2(s),s − eiτλ(s)−2L+τM||∞ + ||eiτλ(s)−2L+τM − P ||∞ + ||P − P (s)||∞,

and ||eiτλ(s)−2L+τM − P ||∞ is bounded by ε0/2, thanks to our choice of τ ; see (4.2.42). Theother terms are bounded by a constant of order ε. Choosing ε small enough, we can make surethat ||R(s)||∞ < ε0. Furthermore, ε can be made arbitrary small by an appropriate choice ofλ(0).

4.3 Rewriting Z t,0 as a sum of terms labelled by graphs

We rewrite the Dyson expansion for Zt,0 using graphs to label terms arising from partial re-summations of the Dyson series. Our analysis involves three steps.

First step. We discretize time on the Van Hove time scale and consider intervals Ii := [ti, ti+1)with ti+1 = ti + τλ−2(ti), i = 0, ..., N − 1, and t0 = 0. We introduce four Feynman rules thatcorrespond to the four possible contraction schemes in the Dyson expansion (4.2.27) and weassociate a Feynman diagram to each pairing appearing under the integrals in (4.2.27); seeParagraph 4.3.1.

Second step. We re-sum the contributions to the Dyson series corresponding to all diagramsthat have the property that any correlation line starting in an interval Ii is ending in thesame interval Ii, for any i = 0, ..., N − 1. We observe that this re-summation just yields thecontribution of the operator Zti+1,ti to the Dyson series for ZtN ,0; see subsection 4.3.2.

Third step. We use the decomposition (4.2.40) in Lemma 4.2.4.2, in order to express 〈Ψ(t)|OΨ(t)〉in the form of a convergent cluster expansion. The fact that the range of the projectionsP (ti) is one-dimensional plays an important role. Indeed, a product of operators A1, ..., An ∈B(B(HP )), when sandwiched between two projections P (s1) = |1HP 〉〈Π(s1)| and P (s2) =|1HP 〉〈Π(s2)|, is equal to

P (s1)A1...AnP (s2) = P (s2) 〈Π(s2)|A1...An|1HP 〉.

Using this identity, we are able to assign a scalar weight to each element, X , of a set, denotedby PN , of “polymers”. The set PN is constructed from the possible pairings (correlation lines)appearing in Wick’s theorem; see subsection 4.3.3.

4.3.1 Discretization of time and Feynman rules

We introduce a sequence (ti)i≥0 of times with the help of a recursion formula

ti+1 = ti + τλ−2(ti), t0 = 0. (4.3.1)


We set

Ii := [ti, ti+1) . (4.3.2)

t1 t2 t3τλ(0)−2 τλ(t1)−2 τλ(t2)−2

λ(t)

t

We introduce four Feynman rules corresponding to the four possible contractions displayedin (4.2.24):

(r = 0)

(r = 1)

ui vi

ui

vi ui

vi

ui vi

(a) (b) (c) (d)

corresponding to the “operator-valued amplitudes”

(a) := λ(ui)λ(vi)f(ui − vi)L(iG(ui))L(iG(vi)), (4.3.3)(b) := λ(ui)λ(vi)f(ui − vi)L(iG(ui))R(iG(vi)), (4.3.4)(c) := λ(ui)λ(vi)f(vi − ui)R(iG(ui))L(iG(vi)), (4.3.5)(d) := λ(ui)λ(vi)f(vi − ui)R(iG(ui))R(iG(vi)). (4.3.6)

The expressions (4.3.3)-(4.3.6) can be read off directly from (4.2.24). To any operator

λ(u)λ(v)Tor

[k∏i=1


]in (4.2.27) there corresponds a unique Feynman diagram constructed according to the rules(4.3.3)-(4.3.6), above. (Time ordering has to be carried out to determine the correspondingcontribution to (4.2.27)). We are led to considering diagrams of a kind indicated in Figure4.1.

O

(r = 0)

(r = 1)

0 t1 t2 t3 t4 t5...

tN

0 t1 t2 t3 t4 t5...

tN

Figure 4.1: A Feynman diagram.


4.3.2 Resummation

Re-summing the Dyson expansion inside isolated double intervals

We fix a natural number N ∈ N. To each interval Ii, as defined in (4.3.2), we associate anindex r that takes the values 0 or 1. We set Ii := (Ii, 0; Ii, 1) and call Ii a double interval. Itcorresponds to the picture

ti ti+1

ti ti+1

The curved part from (tN , r = 0) to (tN , r = 1) in Figure 4.1, which contains a point rep-resenting the observable O, is also considered to be a double interval and is denoted by IN .There are two types of double intervals in Figure 4.1: either Ii is connected to some Ij (j 6= i)by a wavy (correlation) line; or all wavy (correlation) lines starting in Ii end in Ii. In thislatter case, we say that Ii is isolated. Let Ii0 , with i0 6= N , be an isolated interval. We re-sumthe Dyson expansion inside this interval . Let A ⊆ 0, ..., N − 1. A function 1A is defined onthe set of all possible pairings (Wick contraction schemes) as follows:

1A(w) :=

®1 if ∃i ∈ A such that Ii is isolated,0 otherwise. (4.3.7)

In the example where A = i0 we rewrite

ZtN ,0(O) = ZtN ,0i0 (O) + (ZtN ,0(O)−ZtN ,0i0 (O)) (4.3.8)

where

ZtN ,0i0 (O) =∞∑k=0

∫[0,tN ]2k

dµk(w)λ(w) 1i0(w)Tor [F(w)] [OtN ] . (4.3.9)

To shorten our formulae, we have introduced the notation

F(w) :=k∏i=1

F(ui, ri; vi, r′i), (4.3.10)

and

F (w) :=k∏i=1

F (ui, ri; vi, r′i), (4.3.11)

where F(u, r; v, r′) has been defined in (4.2.24) and F (ui, ri; vi, r′i) in (4.2.23). We now explain

how to split the integrations in the formula for ZtN ,0i0 (O). Interchanging summations andintegrations will be carried out without further mention, because the Dyson series convergesin norm; see (4.2.29). For every pairing w consisting of k pairs, it is convenient to write

1i0(w) =k∑

m=0

1i0,m(w), (4.3.12)


where 1i0,m(w) = 1 if Ii0 is isolated and contains exactly m pairs, and is equal to zerootherwise. We plug (4.3.12) into (4.3.9). This yields

ZtN ,0i0 (O) =∞∑k=0

k∑m=0

∫[0,tN ]2k

dµk(w)λ(w) 1i0,m(w)Tor

îF(w)

ó[OtN ] . (4.3.13)

Given some pairing w, with 1i0,m(w) = 1, for some m ∈ N, there are two unique pairings wand w, with w = w ∪ w, such that all times in w lie in Ii0 , whereas no time in w lies in Ii0 .After a change of variables we can factorize the integral in (4.3.13) into two distinct integrals.Exchanging summation over k with summation over m, we get that

ZtN ,0i0 (O) =∞∑k=0

∫[0,tN ]2k

dµk(w)λ(w) χ0,...N−1\i0(w)

Tor

(F(w)

∞∑m=0

∫Ii0

dµm(w)λ(w)Tor [F(w)])

[OtN ] .

We recognize the expansion of the effective propagator eiti0LZti0+1,ti0e−iti0+1L inside the paren-thesis on the right side of this equation; see (4.2.27). Note that χ0,...N−1\i0 only selectspairings with no pairs inside Ii0 or linked to Ii0 . Repeating this procedure for each interval,we obtain the following lemma.

Lemma 4.3.2.1.

ZtN ,0(O) =∑

A⊆0,...,N−1

∞∑k=0

∫[0,tN ]2k

dµk(w)λ(w)Ä1− 10,...,N−1\A(w)

äχ0,...,N−1\A(w)

Tor

(F(w)

∏j∈A

ÄeitjLZtj+1,tje−itj+1L

ä)[OtN ] ,

(4.3.14)

for all O ∈ B(HP ).

4.3.3 The cluster expansion

We plan to use the Dyson series for Zt,s and Lemma 4.2.4.2, namely the identity

Zti+1,ti = P (ti) +R(ti), (4.3.15)

to rewrite 〈Ψ(tN )|(O⊗1HE )Ψ(tN )〉 in the form of a cluster- or polymer expansion. Observingthat

eiti0LZti0+1,ti0e−iti0+1Leiti0+1LZti0+2,ti0+1e−iti0+2L = eiti0LZti0+1,ti0Zti0+2,ti0+1e−iti0+2L

and thateitN−1LZtN ,tN−1e−itNLOtN = eitN−1LZtN ,tN−1O,

we see that the operators eitjL cancel each other in the product inside the parentheses in(4.3.14) unless they are located at the endpoints of a union of adjacent isolated intervals. We


extend the definition of the time-ordering operator Tor in such a way that Tor places eitlL onthe right of e−itlL and e−itl+1L on the right of Ztl+1,tl . We may then write

ZtN ,0(O) =∑

A⊆0,...,N−1

∞∑k=0

∫[0,tN ]2k

dµk(w)λ(w)Ä1− 10,...,N−1\A(w)

äχ0,...,N−1\A(w)

Tor

(Ä∏l∈A

e−itl+1LäF(w)

Ä∏l∈A

eitlLäÄ∏

j∈AZtj+1,tj

ä)[OtN ].

(4.3.16)

Next, we insert (4.3.15) into (4.3.16) and expand the resulting expression as a sum of productsof P ′s and R′s. This yields

ZtN ,0(O) =∑

A⊆0,...,N−1

∑C⊆A

∞∑k=0

∫[0,tN ]2k

dµk(w)λ(w)Ä1− 10,...,N−1\A(w)

äχ0,...N−1\A(w)

Tor

(Ä∏l∈A

e−itl+1LäF(w)

Ä∏l∈A

eitlLäÄ∏

j∈CR(tj)

äÄ ∏m∈A\C

P (tm)ä)

[OtN ].

(4.3.17)

Construction of the set PN

We now elucidate the structure of the polymer set PN that we use to re-organize the sumsand integrals in (4.3.17).

• We introduce “decorated" vertices by associating capital letters R (for red), P (for pur-ple), or B (for blue) to every integer i ∈ 0, ..., N − 1. An integer i labeled with anR, (i, R), corresponds to the perturbation R(ti). An integer i labelled with a P , (i, P ),corresponds to the projection P (ti). An integer i labelled with a B, (i, B), correspondsto an interval Ii that is not isolated. In formula (4.3.17) above, integers in the sets Care labelled with an R, integers in the sets A \ C are labeled with a P , and integers inthe sets 0, ..., N − 1 \ A are labelled with a B. The integer N (corresponding to theobservable O) is labelled with an R and is considered to be an R-vertex.

• We introduce decorated graphs on the the set 0, ..., N. A decorated graph G is a pair(V(G), E(G)). The vertex set V(G) consists of B and/or R-vertices. The edge set E(G)consists of edges ((i, B); (j, B)) joining two distinct B-vertices. The distance betweentwo decorated graphs is defined by

dist (G1,G2) := mini∈G1, j∈G2

|i− j|. (4.3.18)

• A connected graph is an R-vertex or a decorated graph G = (V(G), E(G)) such that V(G)only contains B-vertices and such that the graph G is connected (in the usual sense ofgraph theory).


• A polymer X ∈ PN is a union of disjoint connected graphs G1, ....,Gn (for some n ∈ N)such that dist (∪j∈JGj ,X \ (∪j∈JGj)) = 1, for all J ( 1, ..., n. The vertices of X aredenoted by V(X ), and the edges by E(X ).

The cluster expansion

We use the polymer set PN to rewrite the Dyson series in (4.3.17) in a more convenient form.Starting from (4.3.17), with contributions corresponding to P - and R-vertices re-summed,we remark that the intervals Ii corresponding to B-vertices (where i ∈ 0, ..., N − 1 \ A)may be connected by pairings, w, with 10,...,N−1\A(w) = 0, in many different ways. Weassociate a decorated graph G to every subset A ⊆ 0, ..., N − 1 and to every pairing w with10,...,N−1\A(w) = 0 by labeling elements of A by an index R or P and by drawing an edgebetween (i, B) and (j, B) if there is a correlation line starting in Ii and ending in Ij . Thevertex (N,R), corresponding to the observable O, is added to G. The decorated graph G canbe rewritten as a disjoint union of connected components. We fuse the adjacent connectedcomponents of G and obtain a collection of non-adjacent polymers X1, ...,Xn ∈ PN , for somen ≤ N .

O

0 t1 t2 t3 t4 t5 t6

0 t1 t2 t3 t4 t5 t6

(0, B) (1, P ) (2, R) (3, P ) (4, B) (5, B) (6, R)

Figure 4.2: A pairing w is represented by a Feynman diagram. An example is given in the upper drawingof this figure. We generate graphs on 0, ...., N by drawing edges between intervals that are paired at least byone correlation line and by assigning an index P or R to isolated intervals. The pairing we chose here generates8 possible decorated graphs on the vertex set 0, ..., 6 because there are 8 possible choices for the operatorsassociated to the isolated vertices (1, ·), (2, ·) and (3, ·). One of them is drawn on the lower picture. Thisdecorated graph has three connected components: (2, R), (6, R), and the connected graph G with vertexset V(G) = (0, B), (4, B), (5, B) and edge set E(G) = ((0, B); (4, B)), ((4, B); (5, B)). Fusing the adjacentconnected components, this generates two-non adjacent polymer: X1 = (2, R) and X2 = G ∪ (6, R).

Summing terms in the Dyson series labelled by subsets A ⊆ 0, ..., N − 1, after integratingover all pairings w such that 10,...,N−1\A(w) = 0 in (4.3.17), amounts to the same as sum-ming terms labelled by arbitrary collections of non-adjacent polymers, after integrating overall pairings compatible with these polymers (pairings whose correlation lines follow the edgesof the polymers). At their boundaries, polymers are surrounded by P -vertices (correspondingto one-dimensional projections), and the contributions corresponding to non-adjacent poly-mers factorize. This observation implies that the expectation value 〈Ψ(tN )|OΨ(tN )〉 can be


represented in the form of a cluster expansion for a one-dimensional gas of polymers, withpolymers corresponding to the elements of the set PN . In what follows, the cardinality of aset X is denoted by |X|.

Proposition 4.3.3.1.

1. There is a complex-valued function p : PN → C such that

〈Ψ(tN )|(O ⊗ 1HE )Ψ(tN )〉 =N∑n=1

1

n!

∑X1,...,Xn∈PN

dist(Xi,Xj)≥2, (N,R)∈∪ni=1V(Xi)

p(X1) ... p(Xn). (4.3.19)

The weight p(X ) depends on the observable O only if (N,R) is a vertex of the polymerX .

2. If λ(0) is sufficiently small, then∑X ′, dist(X ,X ′)≤1

|p(X ′)| e|V(X ′)| ≤ |V(X )|, ∀χ ∈ PN , (4.3.20)

where V(X ) is the vertex set of the polymer X . The critical value of λ(0) such that(4.3.20) is satisfied can be chosen uniformly in N .

Remark 4.3.3.1. The factors 1/n! on the right side of (4.3.19) account for over-countingthat originates in summing over all permutations of the polymers X1, ...,Xn in PN . Clearly,the maximal number, n, of polymers appearing in the sums on the right side of Eq. (4.3.19) isfinite, with n < N .

Remark 4.3.3.2. Let X ∈ PN and let β ∈ (0, α − 2); (we remind the reader that α is thedecay rate of the two point correlation function f with time: |f(s)| ∝ (1 + s)−α). If λ(·) wereconstant, we would have to replace the positive function a(X ′) = |V(X ′)| in the Kotecky-Preisscriterion (4.3.20) by

a(X ′) = |V(X ′)|+ β ln(d(X ′))in order to control the limit N → +∞, where

d(X ′) := 1 + max(i | (i, ·) ∈ V(X ′))−min(i | (i, ·) ∈ V(X ′))

is the diameter of the polymer X ′. We refer the reader to [44] for more details, and to remarks4.4.1.1 and 4.4.1.2 in Section 4.4. We can omit the term β ln(d(X ′)) here, because the couplingλ(t) tends to 0 as t tends to +∞.

Remark 4.3.3.3. Part (1) of Proposition 4.3.3.1 is proven in Appendix 4.7. The proof of theKotecky-Preiss criterion (4.3.20) needs some amount of work and is given in Section 4.4.1.The cluster expansion (4.3.19) can also be applied when O happens to be the identity, 1HP .Then

〈Ψ(tN )|(1HP ⊗ 1HE )Ψ(tN )〉 = 1 +N∑n=1

1

n!

∑X1,...,Xn∈PN

dist(Xi,Xj)≥2, (N,R)/∈∪ni=1V(Xi)

p(X1) ... p(Xn),

(4.3.21)


and the weights p(X ) are then all independent of an “observable”. The right side of (4.3.21)is then equal to 1, because the initial state Ψ is assumed to be normalized.

Remark 4.3.3.4. The cluster expansion converges in the limit where N → ∞, because thecoupling function λ(t) is small, for all times t, and because the two-point correlation functionf(t) is “twice integrable”. We investigate this limit in Section 4.4.2.

We conclude this paragraph by introducing some useful notions and notations that enableus to write the exact expressions for the weights p(·). Let X be a polymer. We say that a setU = (i, .), (i + 1, .), ..., (i + k, .) ⊆ V(X ) is a maximal block of neighboring vertices if thereis no set V ⊆ V(X ) of neighboring vertices such that U ( V . We denote by U(X ) the setof maximal blocks of neighboring vertices of X . Maximal blocks of neighboring vertices aresurrounded at their boundaries (extreme points) by vertices corresponding to double intervalswhere a one-dimensional projection P is chosen. The notion of “maximal blocks of neighboringvertices” is therefore helpful in the formulation of the cluster expansion (4.3.19) and the weightsp(X ). We prove in Appendix 4.7 that

p(X ) :=∞∑k=0

∫[0,tN ]2k

dµk(w)λ(w) χX (w) F(w)∏

U∈U(X )

hU (w), (4.3.22)

with

hU (w) := 〈Π(tm(U))|Tor

( ∏(l,B)∈U

[e−itlL

Ä ∏(t,r)∈w∩Il

(iG)(t, r)äeitl+1L

] ∏(j,R)∈U

R(tj))1HP 〉,

(4.3.23)and where we have set

Π(t−1) := |ϕ〉〈ϕ|, R(tN ) := L(O), and m(U) := mini | (i, ·) ∈ U − 1. (4.3.24)

The function χX (w) is equal to 1 only for pairings w that are compatible with the edges of X(in the sense that the correlation lines of the pairing w “follow" the edges of X , and that thereis at least one correlation line in w for each edge of X ). It takes the value zero otherwise.

The exponentiated form of the cluster expansion

In Appendix 4.6.1, some important results concerning convergence criteria for cluster expan-sions are summarized; (see [126, 57] for more details). We now use Proposition 4.6.1.1 (seeAppendix A) and Proposition 4.3.3.1 to rewrite the cluster expansions given in (4.3.19) and(4.3.21) as exponentials of convergent series. In Section 4.4.1 below, we use the exponentiatedform of the cluster expansion to prove our main result, Theorem 4.1.2.1. We introduce afunction ξ : PN × PN → −1, 0 by setting

ξ(X ,X ′) :=

−1 if dist(X ,X ′) ≤ 1,

0 otherwise,(4.3.25)

for all X ,X ′ ∈ PN . The “Ursell functions”, ϕT , are then defined by

ϕT (X1, ...,Xk) :=

1 k = 1,∑

g∈C(Nk)

∏(i,j)∈E (g)

ξ(Xi,Xj) k ≥ 2,(4.3.26)


using the same notations as in Appendix A, where C(Nk) is the set of all connected graphswith vertex set Nk := 1, ..., k. To exponentiate the cluster expansion for the expectationvalue of the observable O, we start from (4.3.19) and single out the polymer X that containsthe vertex (N,R) in the sum on the right side of (4.3.19). This yields

〈Ψ(tN )|(O⊗1HE )Ψ(tN )〉 =∑X ∈ PN

(N,R) ∈ V(X )

p(X )(1+

N∑n=1

1

n!

∑X1,...,Xn∈PN

dist(Xi,Xj)≥2, dist(Xi,X )≥2

p(X1) ... p(Xn)).

The polymers X1, ...,Xn in the sum above are separated from the polymer X by a distancegreater or equal to 2. The weights p(·) satisfy the Kotecky-Preiss criterion (4.3.20), andProposition 4.6.1.1 shows that we can exponentiate the term inside the parenthesis. We getthat

〈Ψ(tN )|(O ⊗ 1HE )Ψ(tN )〉 =∑

X∈PN , (N,R)∈V(X )

p(X ) z(X ), (4.3.27)

where

z(X ) := exp(∑k≥1

1

k!

∑X1, ...,Xk ∈ PN

dist(X1 ∪ ... ∪ Xk,X ) ≥ 2

p(X1)...p(Xk) ϕT (X1, ...,Xk)). (4.3.28)

The weights p(Xi) on the right side of (4.3.28) do not depend on the observable O. Next,we exponentiate the cluster expansion for the expectation value of the identity operator. IfO = 1HP , we start from (4.3.21). We exponentiate the right side of (4.3.21) and obtain that

1 = exp(∑k≥1

1

k!

∑X1, ...,Xk ∈ PN

(N,R) /∈ Xi

p(X1)...p(Xk) ϕT (X1, ...,Xk))

(4.3.29)

We now divide the right side of (4.3.27) by (4.3.29). Many terms in the exponent cancel, andwe are left with

〈Ψ(tN )|(O ⊗ 1HE )Ψ(tN )〉 =∑

X∈PN , (N,R)∈V(X )

p(X ) z(X ), (4.3.30)

where

z(X ) := exp(−∑k≥1

1

k!

∑X1, ...,Xk ∈ PN , (N,R) /∈ Xi

dist(X1 ∪ ... ∪ Xk,X ) ≤ 1

p(X1)...p(Xk) ϕT (X1, ...,Xk)), (4.3.31)

for all N ∈ N. We show in Section 4.4.1 that the main contribution to the right side of (4.3.30)comes from the polymer X = (N,R) when N is large.

4.4 Proof of the Kotecky-Preiss criterion and convergence of the Clusterexpansion when N →∞

To simplify our exposition, we assume that ‖O‖ ≤ ε0 (see Lemma 4.2.4.2). This amounts torescaling the weights p(X ) by a factor ε0/‖O‖ for all polymers X such that (N,R) ∈ V(X ).We propose to verify the Kotecky-Preiss convergence criterion (2) in Proposition 4.3.3.1 for


the weights p. We establish first an upper bound on |p(X )|; (see subsection 4.4.1). We showthat an edge E = ((i, B); (j, B)) ∈ X contributes to this upper bound by a factor

C2∫ ti+1

ti

du

∫ tj+1

tj

dv|f(v − u)|λ(u)λ(v),

where C is some positive constant larger than 1. This last term has the important propertyto be summable in j, for every fixed i, and satisfies

∑j 6=i

∫ ti+1

ti

du

∫ tj+1

tj

dv|f(v − u)|λ(u)λ(v) = O(λ(0)2), (4.4.1)

uniformly in i and in the parameter N . This is a direct consequence of the hypothesis thatf(t) ∝ (1 + t)−α, with α > 2; see (4.1.7). Since every polymer X ′ is a collection of R-verticesand edges, we can estimate ∑

X ′,dist(X ,X ′)≤1

|p(X ′)|e|V(X ′)| (4.4.2)

using (4.4.1) to sum over all possible polymers X ′ with dist(X ′,X ) ≤ 1. The perturbationsR(·) are norm-bounded by ε0 ∈ (0, 1), and it is not difficult to control (4.4.2) by a term oforder |V (X )|(ε0 + λ(0)2), using (4.4.1); see Section 4.4.1 for a detailed proof.

In Section 4.4.2, we show that the expansion in Eq. (4.3.31) converges, as N →∞, usingthe Kotecky-Preiss criterion for the polymer weights p(X ). For large values of N , we showthat the main contribution to the cluster expansion on the right side of (4.3.30) comes fromthe polymer X = (N,R). Polymers of larger size and containing (N,R) make a negligiblecontribution, for large N , because the coupling λ(t) tends to zero, as t tends to infinity. Arigorous proof is given in Section 4.4.2.

4.4.1 The Kotecky-Preiss criterion for p(X )

Upper bound on |p(X )| and summability of weights

Let X ∈ PN . For any edge E = ((i, B); (j, B)) ∈ E(X ), we define

η(E ) := 4‖G‖2∫ ti+1

ti

du

∫ tj+1

tj

dv|f(v − u)|λ(u)λ(v), (4.4.3)

where G is the form factor introduced in (4.1.3). We remind the reader that the cardinalityof a set X is denoted by |X|.

Lemma 4.4.1.1. Let X belong to PN . Then

|p(X )| ≤ 2|U(X )|e4τ ||f ||L1 ‖G‖2|B(X )|( ∏

E∈E(X )

η(E ))ε|R(X )|0 , (4.4.4)

where B(X ) and R(X ) are respectively the sets of B- and R-vertices of X , and U(X ) is definedto be the set of maximal blocks of neighboring vertices of X , as described at the end of subsection3.3.2.


Before proving Lemma 4.4.1.1, we state an important (though easy) lemma which claimsthat, for an arbitrary but fixed i, the weights η((i, B); (j, B)) are summable in j, uniformly ini. The proof of this Lemma follows by inspection; it is a direct consequence of the definition ofthe weights η((i, B); (j, B)) given in (4.4.3), using that the decay rate α of the function f(t),(|f(t)| ∝ 1

(1+t)α , as t→∞), satisfies α > 2.

Lemma 4.4.1.2. If α > 2, as assumed, there is a constant C(α) <∞ independent of N suchthat ∑

j 6=iλ(ti)

−1λ(tj)−1 η((i, B); (j, B)) ≤ C(α), (4.4.5)

for all i ∈ N.

Remark 4.4.1.1. A better estimate is actually satisfied by the weights η(·). It is fairly easyto show that, given any β ∈ (0, α−2), there is a constant C(α, β) <∞ independent of N suchthat ∑

j 6=i(1 + |i− j|)βλ(ti)

−1λ(tj)−1 η((i, B); (j, B)) ≤ C(α, β), (4.4.6)

for all i ∈ N. Inequality (4.4.6) is useful if the coupling λ(·) stays constant, since it impliesthat polymers X with d(X ) 1 have a negligible weight p(X ); see also Remark 4.3.3.2 inSection 4.3. Eq. (4.4.6) shows that the weight associated to an edge E = ((i, B); (j, B)) tendsto zero, as |j − i| tends to infinity. Eq. (4.4.5) is sufficient for our purpose since the couplingλ(t) tends to zero as t tends to infinity, and the factor λ(ti)

−1λ(tj)−1 in (4.4.5) plays the same

role as (1 + |i− j|)β in (4.4.6).


In Lemma 4.2.4.2 it is shown that the operators P (s) converge in norm to the projectionP = |1HP 〉〈Π1|, as the parameter λ(0) (see Lemma 4.2.4.2) tends to 0. Since ‖P‖∞ = 1,it follows that ‖P (s)‖∞ ≤ 2, for all s ≥ 0, provided that λ(0) is small enough. Let w be apairing. We rewrite the amplitude hU (w) (see Eq. (4.3.23)) as

|1HP 〉hU (w) = P (tm(U))Tor

( ∏(l,B)∈U

[e−itlL


(iG)(t, r)äeitl+1L

] ∏(j,R)∈U

R(tj))|1HP 〉,

(4.4.7)for all U ∈ U(X ) with m(U) > 0. It follows that

|hU (w)| ≤ 2 ε|R(X )∩U |0

∏(l,B)∈U

‖G‖|w∩Il|

If m(U) = −1, 1HP has to be replaced by Pϕ = |ϕ〉〈ϕ| (see (4.1.5)) on the left side of (4.4.7),but the upper bound remains unchanged, because ‖Pϕ‖ = 1. Thus,∏

U∈U(X )

|hU (w)| ≤ 2|U(X )| ε|R(X )|0 ‖G‖2k, (4.4.8)


for all pairings w with precisely k pairs. These bounds are used to estimate the weights p(X );(see right side of (4.3.22)). The characteristic function χX in (4.3.22) selects the pairingscompatible with the polymer X ; (the correlation lines of the pairings w must “follow" theedges of X , and that there must be at least one correlation line in w for each edge of X ). Foreach edge E = ((i, B); (j, B)) ∈ E(X ), there exists at least one pair (u, r; v, r′) ∈ w with u ∈ Iiand v ∈ Ij . We choose such a pair and then estimate the integrals over u and v, factoring outthe rest. Carrying out this procedure for each line E ∈ E(X ), we conclude that∫

[0,tN ]2k

dµk(w)λ(w) χX (w) |F (w)| ≤ ‖G‖−2|E(X )|( ∏

E∈E(X )

η(E ))

·∫

[0,tN ]2(k−|E(X )|)dµk−|E(X )|(w)λ(w) |F (w)|

k−|E(X )|∏i=1

χ(ui, vi ∈ B(X )),

(4.4.9)

where F (w) is defined in (4.3.11), and w contains m := k−|E(X )| pairs. If m = 0 the integralis replaced by 1. To estimate the integral on the right side of (4.4.9), we first integrate overthe vi’s and use that λ(ui) < λ(vi). This yields∫

[0,tN ]2mdµm(w)λ(w) |F (w)|

m∏i=1

χ(ui, vi ∈ B(X )) ≤ 4m∫

[0,tN ]mdu χ(u1 < ... < um) ‖f‖mL1

m∏i=1

χ(ui ∈ B(X ))λ2(ui),

(4.4.10)

for all m ∈ N. We now set q = |B(X )| and assume that B(X ) = (i1, B), ..., (iq, B); (the casewhere B(X ) = ∅ yields a factor 1). For each tuple u = (u1, ..., um) under the integral on theright side of (4.4.10), there are q integers, n1, ..., nq, given by nj = |u1, ..., um ∩ Iij |, withj = 1, ..., q. The times ui are ordered, u1 < ... < um, and∫

tij<u1<...<unj<tij+1

[λ(u1)....λ(unj )]2du1...dunj ≤

τnj

nj !,

with τ as in (4.1.12). Hence∫[0,tN ]m

du χ(u1 < ... < um)m∏i=1

χ(ui ∈ B(X ))λ2(ui) ≤ τm∑

n1+...+nq=m

1

n1!...

1

nq!,

and it follows that

|p(X )| ≤ 2|U(X )| ε|R(X )|0

( ∏E∈E(X )

η(E )) ∞∑m=0

4mτm||f‖mL1‖G‖2m∑

n1+...+nq=m

1

n1!...

1

nq!

≤ 2|U(X )| ε|R(X )|0

( ∏E∈E(X )

η(E ))

expÄ4τ ||f‖L1 ‖G‖2q

ä.

Next, we prove the Kotecky-Preiss criterion for the weight p and the function a(X ) =|V(X )| introduced above.


Proof of the Kotecky-Preiss criterion

We first decompose the set R(X ′) of vertices of X ′ decorated with a perturbation R into adisjoint union of maximal blocks of neighboring vertices. For each polymer X ′ in PN , thereexists an m > 0 such that R(X ′) = A1 ∪ ... ∪ Am, where the sets Ai are maximal blocks ofneighboring R-vertices. The set of maximal blocks of neighboring R-vertices in X ′ is denotedby Rmax(X ′). Every polymer X ′ ∈ PN is entirely characterized by the collection of maximalblocks of neighboring R-vertices in X ′ and by its set of edges E(X ′). The polymer drawnbelow, for instance, has two maximal blocks of neighboring R-vertices, (0, R), (1, R) and(5, R), and three edges, E1 = ((2, B); (4, B)), E2 = ((4, B); (9, B)) and E3 = ((6, B); (8, B)).

(1, R)(0, R) (5, R)(2, B) (4, B) (9, B)(8, B)(6, B)

A polymer is a union of maximal blocks of neighboring R-vertices and edges.

We use the decomposition of polymers into blocks of neighboring R-vertices and edges to provethe Kotecky-Preiss criterion.

We start from the bound (4.4.4). Multiplying both sides by e|V(X )|, we get that

|p(X )| e|V(X )| ≤ 2|U(X )|(eε0)|R(X )|( ∏

E∈E(X )

η(E ))e(4τ ||f ||L1 ‖G‖2+1)|B(X )|,

where the function η has been defined in (4.4.3). The integer |U(X )| is bounded by |B(X )| if|B(X )| 6= 0, because X is a union of adjacent connected graphs. If |B(X )| = 0 then |U(X )| = 1.Defining K := 2e(4τ ||f ||L1 ‖G‖2+1), we introduce

ηK(E ) := K2η(E ) (4.4.11)

for every edge E andηK(A) := (eε0)|A| (4.4.12)

for every block A of neighboring R-vertices. Denoting by Ad(X ) the set of vertices adjacentto the polymer X , we obtain that∑

X ′ ∈ PNdist(X ,X ′) ≤ 1

|p(X ′)| e|V(X ′)| ≤ 6|V(X )| sup(j,·)∈Ad(X )∨V(X )

∑X ′∈PN

(j,·)∈V(X′)

∏A∈Rmax(X ′)∨E(X ′)

ηK(A).

In order to avoid too much over-counting, we must carefully estimate the right side of thisinequality. We use that every polymer X ′ ∈ PN is entirely characterized by its set of edges,E(X ′), and its set of maximal blocks of neighboring R-vertices, Rmax(X ′). We denote byAN the set of all possibles edges and blocks of neighboring R-vertices. It is sometimes usefulto explicitly distinguish edges and block of R-vertices, and we denote the set of all possibleedges by EN and the set of all possible blocks of neighboring R-vertices by RN . We canestimate the right side of the last inequality by summing over collections of elements in theset AN = EN ∨ RN– but not over all of them! To carry out this sum without intolerable


over-counting, we introduce graphs: We denote by g(A1, ..., An) the graph on Nn := 1, ..., nthat has an edge (i, j) between i and j if and only if one of the following properties is satisfied.Namely, Ai and/or Aj belong to the set EN and are adjacent next to each other, or Ai, and Ajboth belong to EN and share a common vertex. If X is a polymer, and if A1, ..., An consist ofits set of edges and maximal blocks of neighboring R-vertices, then g(A1, ..., An) is connected.Singling out (one of ) the decorated edge(s) - or the block of neighboring R-vertices - thatcontains the vertex (j, ·) and belongs to X ′, we get the estimate

∑X ′∈PN

(j,·)∈V(X′)


ηK(A) ≤∑A∈AN(j,·)∈A

ηK(A)(1 +

∞∑n=1

1

n!

∑A1,...,An∈AN

g(A,A1,...,An)∈C(Nn+1)

n∏i=1

ηK(Ai)),

where C(Nn+1) is the set of connected graphs with vertex set Nn+1 = 1, ..., n+ 1. We nowfollow ideas from [32] and sum over spanning trees to bound the right side of the last inequality.We denote by T(Nn+1) the set of labelled trees with vertex set Nn+1 = 1, ..., n+ 1, and wewrite t ⊂ g if V(t) = V(g) = Nn+1 and E(t) ⊂ E(g). Then

∑A1,...,An∈AN

g(A,A1,...,An)∈C(Nn+1)

n∏i=1

ηK(Ai) =∑

g∈C(Nn+1)

∑A1,...,An∈ANg(A,A1,...,An)=g

n∏i=1

ηK(Ai)

≤∑

t∈T(Nn+1)

∑A1,...,An∈AN

g(A,A1,...,An)⊃t, g∈C(Nn+1)

n∏i=1

ηK(Ai),

for all A ∈ AN . Every tree t ∈ T(Nn+1) has n edges and each vertex of t is linked to at leastone other vertex by an edge. Using (4.4.11), we find that

∑A1,...,An∈ANg(A,A1,...,An)⊃t

n∏i=1

ηK(Ai) ≤∏

(i,j)∈E(t)

(sup

Ai∈AN

∑Aj∈ANAj∼Ai

ηK(Aj)),

(4.4.13)

where Aj ∼ Ai if Ai and/or Aj belong to EN and they are adjacent to one another, or if Aiand Aj belong both to EN and share a common vertex. If ε0 < e−1, Lemma 4.4.1.2 and aneasy calculation imply that∑

Aj∈ANAj∼Ai

ηK(Aj) ≤ 4eε0

1− eε0+ C(τ)λ(tm(Ai))λ(0), (4.4.14)

for all Ai ∈ AN , uniformly in N . Here C(τ) is a positive constant that depends on τ throughthe constant K appearing in (4.4.11). The number of labelled trees in T(Nn+1) is equal to(n+ 1)n−1, and we deduce that

1

n!

∑t∈T(Nn+1)

∑A1,...,An∈ANg(A,A1,...,An)⊃t

n∏i=1

ηK(Ai) ≤(n+ 1)n−1

n!

ÅC(τ)λ(0)2 + 4

eε0

1− eε0

ãnby plugging (4.4.14) into the right side of (4.4.13). Using Stirling formula, it is easy to seethat we can sum the right side over n if ε0 and λ(0) are sufficiently small. This yields the


upper bound ∑X ′∈PN

(j,·)∈V(X′)


ηK(A) ≤ C1(τ, λ(0))∑A∈AN(j,·)∈A

ηK(A), (4.4.15)

for some constant C1(τ, λ(0)) of order 1. Using (4.4.11) and (4.4.14), it is easy to see thatthere exist constants τc > 0 and λτ > 0 such that the Kotecky-Preiss criterion (4.3.20) issatisfied for all τ > τc and all 0 < λ(0) < λτ .

Remark 4.4.1.2. Following the lines of the proof we just carried out above, we can use (4.4.6)instead of (4.4.5) to show that∑

X ′ ∈ PNdist(X ,X ′) ≤ 1

|p(X ′)| e|V(X ′)|+β ln(d(X ′)) ≤ |V(X )| (4.4.16)

for all polymers X ∈ PN , where β ∈ (0, α − 2). Indeed, if X ′ is a polymer, and if A1, ..., Anconsists of its set of edges and maximal block of R−vertices, then

d(X ′) ≤ d(A1) + ...+ d(An) ≤ (d(A1) + 1)...(d(An) + 1), (4.4.17)

and we can replace ηK(A) in the equations above by (d(A) + 1)βηK(A). The first inequalityin (4.4.17) holds true because a polymer is a fusion of adjacent connected graphs. The secondinequality follows from the positivity of the diameter. As we already mentioned, (4.4.16) isuseful to investigate the limit N →∞ if the coupling λ(·) is constant.

4.4.2 Convergence of the cluster expansion as N →∞

We use the exponentiated form of the cluster expansion derived in Section 4.3.3; see (4.3.30)and (4.3.31). We show that the main contribution to the right side of (4.3.30) comes fromthe polymer X = (N,R) if N is large. We remark that if X ∈ PN , X 6= (N,R) and(N,R) ∈ V(X ), then necessarily (N −1, ·) ∈ V(X ); see Section 4.3.3. We use this remark, andLemma 4.4.2.1 below, to prove Theorem 4.1.2.1.

Convergence to the ground state

Lemma 4.4.2.1. We introduce

ZN (O) :=∑

X∈PN , (N−1,·),(N,R)∈V(X )

p(X ) z(X ). (4.4.18)

ThenZN (O) −→

N→∞0. (4.4.19)

We postpone the proof of Lemma 4.4.2.1 to the next paragraph and turn to the proof ofour main Result, namely Theorem 4.1.2.1.


Proof. (Theorem 4.1.2.1) If X = (N,R), then

p((N,R)) z((N,R)) = 〈Π(tN−1)|O〉 z((N,R)). (4.4.20)

Lemma 4.2.4.2 shows that 〈Π(tN−1)|O〉 converges to 〈ϕ1|Oϕ1〉. We note that z((N,R))does not depend on the choice of the observable O. It is therefore sufficient to exhibit anobservable O for which we can show that z((N,R)) → 1, as N → ∞, in Eq. (4.3.30):Choosing O = 1HP , we find that

〈Ψ(tN )|Ψ(tN )〉 = ZN (1HP ) + 〈Π(tN−1)|1HP 〉z((N,R)) = ZN (1HP ) + z((N,R)) = 1.(4.4.21)

As ZN (1HP )→ 0, as N →∞ (see (4.4.19)), we deduce that z((N,R)) converges to 1. Thiscompletes the proof of Theorem 4.1.2.1.


We first establish an upper bound on the weights z(X ) introduced in (4.3.31). Clearly

|k∏i=1

(1 + ξ(Xi,X ))− 1| ≤k∑i=1

|ξ(Xi,X )|. (4.4.22)

Moreover, the argument of the exponential in (4.3.31) is bounded by

∑k≥1

1

k!

∑X1, ...,Xk ∈ PN

(N,R) /∈ Xi

|p(X1)...p(Xk)|k∑i=1

|ξ(Xi,X )| |ϕT (X1, ...,Xk)| ≤ |V(X )|. (4.4.23)

Inequality (4.4.23) follows from the inequality

1 +∑k≥1

1

k!

∑X1, ...,Xk ∈ PN

(N,R) /∈ Xi

|p(X1)...p(Xk)| |ϕT (X ′,X1, ...,Xk)| ≤ e|V(X ′)|, (4.4.24)

by multiplying both sides of (4.4.24) by |ξ(X ,X ′)| |p(X ′)| and by summing over X ′ ∈ PN ; seealso Appendix A and [126] for more details. We deduce that

|ZN (O)| ≤∑

X∈PN , (N−1,·), (N,R)∈V(X )

|p(X )|e|V(X )|. (4.4.25)

The upper bound for |p(X )| given in (4.4.4) is not sharp enough to show that the right sideof (4.4.25) tends to zero as N tends to infinity. We derive a slightly refined upper bound byexploiting the particular structure of the polymer set PN . We start again from the definitionof p(X ) given in (4.3.22) and use that

P (ti)R(ti) = R(ti)P (ti) = 0, (4.4.26)

for all i = 0, ..., N . Eq. (4.4.26) follows from the relations [P (ti),Zti+1,ti ] = 0, R(ti) =Zti+1,ti − P (ti), and Zti+1,ti(1HP ) = 1HP . The product P (ti)R(ti) does not appear inour expansion. However, terms of the form P (ti−1)R(ti) do arise. We also remark thatR(ti)P (ti+1) = 0, but we will not use this fact. We define

un := ‖P (tn−1)R(tn)‖∞, n ∈ N. (4.4.27)


The sequence (un)∞n=1 tends to zero, as n→ +∞, because

‖P (tn−1)R(tn)‖∞ = ‖(P (tn−1)− P (tn))R(tn)‖∞ ≤ ε0‖P (tn−1)− P (tn)‖∞.

The main idea of our proof is to use the sequence (un)∞n=1 and the decay of the couplingλ(·) towards zero to prove that the right side of (4.4.25) tends to zero as N tends to +∞.To do so, it is useful to distinguish two classes of polymers X in the sum on the right side of(4.4.25).

Class 1: |B(X )| = 0.Every vertex in V(X ) carries a perturbation R(·). There are only N polymers X ∈ PN with(N − 1, R), (N,R) ∈ V(X ) and |B(X )| = 0. Using Formula (4.3.22), we deduce that

p(X ) = 〈Π(tN−|V(X )|)|R(tN−|V(X )|+1) ... R(tN−1)O〉. (4.4.28)

Consequently,

|p(X )|e|V(X )| ≤

e2‖O‖(eε0)|V(X )|−2uN−|V(X )|+1 2 ≤ |V(X )| ≤ N,

‖O‖(eε0)Ne |V(X )| = N + 1,(4.4.29)

and

∑X∈PN , (N−1,·), (N,R)∈V(X )

|B(X )|=0

|p(X )|e|V(X )| ≤ ‖O‖Äe2

N−1∑k=1

(eε0)k−1 uN−k + e(eε0)Nä. (4.4.30)

Class 2: |B(X )| 6= 0.The polymer X in (4.4.25) must contain the vertex (N − 1, ·) and an edge. The color of(N −1, ·) is either red (R) or blue (B), and we treat differently these two possibilities. We usethe bound (4.4.4) on p(X ) and continue our argument as in the proof of Proposition 4.3.3.1,Property (2); see Section 4.4.1. We use the same notations as in Section 4.4.1. We remind thereader that K := 2e(4τ ||f ||L1 ‖G‖2+1) and that we have defined ηK(E ) := K2η(E ) for a singleedge, and ηK(A) = (eε0)|A| for a union, A, of neighboring R-vertices.

Case 2.a: (N − 1, B) ∈ V(X ). We single out one edge E of X such that (N − 1, B) belongsto E . Following the same arguments as in Section 4.4.1, we then get that∑

X∈PN , (N−1,B), (N,R)∈V(X )

|p(X )|e|V(X )|

≤ ‖O‖e∑

E∈EN(N−1,B)∈E

ηK(E )(1 +

N∑n=1

1

n!

∑A1,...,An∈AN

g(E ,A1,...,An)∈C(Nn+1)

n∏i=1

ηK(Ai))

≤ C(τ, λ(0))‖O‖λ(tN−1)λ(0).

where the constant C(τ, λ(0)) > 0 depends on the parameters τ and λ(0), but not on N andλ(t), t > 0. Furthermore, C(τ, λ(0)) decreases when λ(0) decreases.


Case 2.b: (N − 1, R) ∈ V(X ). We denote by A the maximal block of neighboring R-verticesof X that contains (N − 1, R). The block A must be adjacent to an edge on its left side,because X is a fusion of adjacent connected graphs, and because X contains at least one edge.This is also the reason why (0, R) and (1, R) cannot belong to A. We use these remarks toextract a factor λ(tm(A))

µ, µ ∈ (0, 1), from the edge attached to the left side of A; (m(A) hasbeen defined in (4.3.24)). We proceed as follows. We define ηµ,K(E ) := (λ(ti)λ(tj))

−µηK(E)for every edge E = ((i, B); (j, B)) and ηµ,K(A′) := ηK(A′) for every block A′ of neighboringR-vertices. It is easy to check that the estimates carried out in Section 4.4.1 remain almostthe same if we replace the weights ηK by the new weights ηµ,K . Singling out the maximalblock A that contains (N − 1, R) and extracting a factor λ(tm(A))

µ from the edge attached toit, we get that∑X∈PN , (N−1,R), (N,R)∈V(X )

|p(X )|e|V(X )|

≤ ‖O‖e∑A∈RN

(N−1,R)∈A, (0,·),(1,·),(N,R)/∈A

(eε0)|A|λµ(tm(A))N∑n=1

1

n!

∑A1,...,An∈AN

g(A,A1,...,An)∈C(Nn+1)

n∏i=1

ηµ,K(Ai).

Using similar calculation as in Section 4.4.1, we can bound the last line of the previous equationby

C(τ, λ(0))‖O‖N−2∑k=1

(eε0)kλµ(tN−k−1), (4.4.31)

where C(τ, λ(0)) > 0 is independent of N and λ(t), t > 0, and decreases when λ(0) decreases.The right side of Inequality (4.4.30) and the bound (4.4.31) are of the form

ΣN :=N∑k=1

εkvN−k, (4.4.32)

where(vn)∞n=1

is a sequence of positive numbers converging to zero, and 0 < ε < 1. All vn’s are boundedby some positive constant C , and ΣN is bounded by C ε

1−ε , for all N . We therefore concludethat ΣN → 0, as N tends to ∞. Applying this result to (4.4.30) and (4.4.31), we finally findthat ZN (O)→ 0, as N →∞.

4.5 Extensions of Theorem 4.1.2.1

4.5.1 Extension to initial field states with a finite number of photons

We generalize Theorem 4.1.2.1 to initial field states with a finite number of photons. Weassume that the system P∨E is initially in the state Ψ = ϕ⊗ϕE , where ϕE = Φ(f1)....Φ(fn1)Ωfor a fixed number n1 ∈ N. We assume that the functions fi, i = 1, ..., n1, satisfy

|〈fi|φt〉L2 | ∝1

(1 + t)α(4.5.1)


for all i = 1, ..., n1, where α > 2 is the same number as in Assumption 4.1.1.1. We also assumethat ϕE is normalized.

Corollary 4.5.1.1. We choose n1 functions fi ∈ L2(R3), i = 1, ..., n1. Suppose that assump-tions (4.5.1), 4.1.1.1-4.1.1.3 are satisfied. Then there is a constant λc > 0 such that, for any0 < λ(0) < λc,

〈Ψ(t)|(O ⊗ 1HE )Ψ(t)〉 −→t→∞

〈ϕ1|Oϕ1〉, (4.5.2)

for all O ∈ B(HP ) and for all initial states Ψ = ϕ⊗ϕE. ϕ1 is the ground state of HP (uniqueup to a phase) corresponding to the eigenvalue e1.

4.5.2 Thermalization at positive temperature

The method we used to prove Theorem 4.1.2.1 works in a similar way at positive temperature.We explain below how to show that the system P thermalizes in the limit t→ +∞ if the fieldis initially in thermal equilibrium at temperature T > 0.

We work directly in the thermodynamic limit. We consider the Hilbert space

h := L2(R3, d3k) ∩ L2(R3, |k|−1d3k). (4.5.3)

Im〈f |g〉L2 is a symmetric non-degenerate symplectic bilinear form on h and the C∗-algebraU(h) generated by the Weyl operators

W (−f) = W (f)∗, f ∈ h,

W (f)W (g) = e−iIm〈f |g〉L2/2W (f + g), f, g ∈ h,

is unique up to a ∗-isomorphism; see e.g. [27]. U(h) is the algebra of field observables. Time-evolution on U(h) is given by the one-parameter group of ∗-automorphism, αEt t∈R, definedby

αEt (W (f)) := W (eiωtf) (4.5.4)

for all f ∈ h and for all t ∈ R, where ω(k) = |k|. It is well-known that αEt is not normcontinuous (‖W (f)−1‖ = 2 if f 6= 0), and the dynamical properties of the interacting systemP ∨E can only be understood in a representation dependent way. We consider the KMS stateat temperature 1/β > 0 defined on U(h) by

ρβ(W (f)) = exp(− 1

4

∫R3d3k

1 + e−β|k|

1− e−β|k||f(k)|2

), f ∈ h. (4.5.5)

The function t 7→ ρβ(W (tf)) is real analytic and it is possible to make sense of the infinitesimalgenerators Φρβ (f) of the one-parameter group of unitary transformations t 7→ πρβ (W (tf)) inthe GNS representation (Hρβ , πρβ ,Ωρβ ) of (U(h), ρβ); see [27]. The two-point correlations aregiven by

ρβ(Φρβ (f)Φρβ (g)) = 〈f |(1− e−βω)−1g〉L2 + 〈g|e−βω(1− e−βω)−1f〉L2 (4.5.6)

for all f, g ∈ h, and easy calculations show that the state ρβ is quasi-free. The one parametergroup αEt t∈R is represented on πρβ by

πρβ (αE(t)(OE)) = U∗(t)πρβ (OE)U(t), (4.5.7)


where U(t)t∈R is the one-parameter group of unitary transformations defined by

U(t)πρβ (OE)Ωρβ := πρβ (αE(−t)(OE))Ωρβ , U(t)Ωρβ = Ωρβ ,

for all OE ∈ U(h). Time translation of the operators Φρβ (f) is given by Φρβ (f)(t) =Φρβ (eitωf), for all t ∈ R.

We compose the field E with the atomic system P and we consider the C∗-algebra A =B(HP )⊗U(h) equipped with the projective C∗ cross-norm; see [123]. The free dynamics on Ais generated by the one-parameter group of ∗-automorphisms α0

t t∈R, where α0t is determined

byα0t (O ⊗OE) = eitHPOe−itHP ⊗ αEt (OE) (4.5.8)

for all O ∈ B(HP ) and all OE ∈ U(h). We now turn on the interaction between the atomand the field. The dynamics of the interacting system is defined through a Dyson series. Onehas to be careful here because αEt is not norm continuous, and, hence, the Dyson series onlymakes sense in a representation dependent way. A rigorous construction of the interactingdynamics on A as the limit of a regularized and norm-continuous dynamics on a regularizedalgebra can be found in [67]. We avoid these complications here since we are only interested inthe time evolution of observables of the form O⊗1HE . We work directly in the representation(HP ⊗Hρβ ,1⊗πρβ ). We consider the self-adjoint and densely defined operator on HP ⊗Hρβ ,

HI := G⊗ Φρβ (φ),

where φ ∈ h is the form factor of the interaction Hamiltonian of the last sections. Theinteraction Hamiltonian translated at time t is given by

HI(t) = eitHPGe−itHP ⊗ Φρβ (eitωφ).

Let O ∈ B(HP ). The quadratic form

qt,sO (Ψ,Ψ′) :=∞∑n=0

in∫ t

sdt1...

∫ tn−1

sdtn λ(t1)...λ(tn)〈Ψ|(e−isHP ⊗ U(s))

[HI(tn), ..., [HI(t1), eitHPOe−itHP ]...](eisHP ⊗ U∗(s))Ψ′〉

is well-defined for all Ψ,Ψ′ ∈ HP ⊗F (Hρβ ), where F (Hρβ ) := Φ(f1)...Φ(fn)Ωρβ | n ∈ N, fi ∈h. The quadratic form qt,sO induces a unique operator Zt,sβ (O) ∈ B(HP ), defined by

〈ϕ|Zt,sβ (O)ψ〉 := qt,sO (ϕ⊗ Ωρβ , ψ ⊗ Ωρβ ), ∀ϕ,ψ ∈ HP . (4.5.9)

The expression of Zt,sβ (O) is similar to (4.2.27). The only change consists in the replacementof the correlation function f in (4.2.27) by the correlation function

fβ(t) := ρβ(Φ(φt)Φ(φ)) = 〈φt|(1− e−βω)−1φ〉+ 〈φ|e−βω(1− e−βω)−1φt〉 (4.5.10)

at temperature 1/β.

Corollary 4.5.2.1. Let T > 0. Suppose that Assumptions 4.1.1.1 ( with f replaced by fβ),4.1.1.2 and 4.1.1.3 are satisfied. Then there exists a constant λc > 0, such that, for any0 < λ(0) < λc,

limt→∞

〈ϕ|Zt,0β (O)ϕ〉 = TrHP (ρP,βO), (4.5.11)

for all ϕ ∈ HP with ‖ϕ‖ = 1. The state ρP,β := e−βHP /TrHP (e−βHP ) is the Gibbs equilibriumstate of P at temperature T = 1/β.


4.6 Appendix for Chapter 4

4.6.1 Cluster expansions

We review some standard features of cluster expansions. The reader is referred to [110], [28],[126], [57] for more details. We mainly follow the exposition in [126] and [96]. A set of polymersis a measurable set (X,Σ, µ) where µ is a complex measure with finite total variation |µ|(X).An element x ∈ X is called a “polymer". Let ξ : X×X→ R be a symmetric function with theproperty that

|1 + ξ(x, y)| ≤ 1, ∀x, y ∈ X. (4.6.1)

ξ encodes an adjacency relation ∼, i.e. a symmetric and irreflexive binary relation. Forhardcore polymer models, ξ(x, y) = −1 if x ∼ y, and 0 otherwise. For the polymer set PNintroduced in Section 4.3, X ∼ X ′ if dist(X ,X ′) ≤ 1. We consider the partition function

Z := 1 +∑n≥1

1

n!

∫dµ(x1)...dµ(xn)

∏1≤i<j≤n

(1 + ξ(xi, xj)) . (4.6.2)

Formula (4.6.2) is a cluster expansion. Under certain circumstances, the right side of (4.6.2)can be rewritten as the argument of an exponential. To do so, we define

Nn := 1, ..., n. (4.6.3)

For every A ⊂ N, we denote by G(A) the set of graphs with vertex set A and with edges pairs(i, j) with i 6= j and i, j ∈ A. Among those graphs, the connected ones are denoted by C(A),and the trees are denoted by T(A). To make the distinction with the set PN (see Section 4.3),we denote the graphs in G(A) with small letters, i.e. g, f, .... The set of edges of the graph gis denoted by E(g). One has that∏

1≤i<j≤n(1 + ξ(xi, xj)) =

∑g∈G(Nn)

∏(i,j)∈E(g)

ξ(xi, xj). (4.6.4)

The connected part of∑

g∈G(Nn)

∏(i,j)∈E(g)

ξ(xi, xj) is given by

∑g∈C(Nn)

∏(i,j)∈E(g)

ξ(xi, xj).

We introduce the “Ursell functions"

ϕT (x1, ..., xn) :=

1 if n = 1,∑

g∈C(Nn)

∏(i,j)∈E(g)

ξ(xi, xj) if n 6= 1. (4.6.5)

If sums and integrals can be exchanged, we get that

Z = 1 +∑n≥1

1

n!


∑g∈G(Nn)

∏(i,j)∈E(g)

ξ(xi, xj)

= 1 +∑n≥1

1

n!


n∑k=1

1

k!

∑A1∪...∪Ak=Nn

k∏l=1

Ñ ∑g∈C(Al)

∏(i,j)∈E(g)

ξ(xi, xj)

é= 1 +

∑n≥1

1

n!

n∑k=1

1

k!

∑A1∪...∪Ak=Nn


k∏l=1

Ñ ∑g∈C(Al)

∏(i,j)∈E(g)

ξ(xi, xj)

é.


To go from the first to the second line, we have decomposed every graph g into its connectedcomponents. Furthermore, A1 ∪ ... ∪ Ak is a partition of Nn such that Al 6= ∅ for all l. Wewrite

dµ(xAl) =∏x∈Al

dµ(x).

Then, ∫dµ(xAl)

Ñ ∑g∈C(Al)

∏(i,j)∈E(g)

ξ(xi, xj)

édepends only on the number of elements in Al. There are n!

m1!...mk! partitions of Nn in k subsetAl with ml elements, and we deduce that

Z = 1 +∑n≥1

1

n!

n∑k=1

1

k!

∑A1∪...∪Ak=Nn

k∏l=1

∫dµ(xAl)

Ñ ∑g∈C(Al)

∏(i,j)∈E(g)

ξ(xi, xj)

é= 1 +

∑n≥1

1

n!

n∑k=1

1

k!

∑m1+...+mk=n

n!

m1!...mk!

k∏l=1

∫dµ(xNml )

Ñ ∑g∈C(Nml )

∏(i,j)∈E(g)

ξ(xi, xj)

é= 1 +

∞∑k=1

1

k!

Ñ∞∑

m1=1

1

m1!

∫dµ(xNm1

)

Ñ ∑g∈C(Nm1 )

∏(i,j)∈E(g)

ξ(xi, xj)

éék

= exp

Ñ∑n≥1

1

n!

∫dµ(x1)...dµ(xn)ϕT (x1, ..., xn)

é.

Our calculations are formal and the exchange of sums and integrals must be justified. Thisexchange can be done if the function ξ and the measure µ satisfy specific criteria such that theseries above are absolutely convergent; see e.g. [57]. In this paper, we use the Kotecky-Preisscriterion stated below.

Proposition 4.6.1.1. (KP criterion, see e.g.[98], [126]) Let us assume that there is a non-negative function a : X→ R+ such that∫

d|µ|(x′)|ξ(x, x′)|ea(x′) ≤ a(x) ∀x ∈ X, (4.6.6)

and∫d|µ|(x)ea(x) <∞. Then

Z = exp

Ñ∑n≥1

1

n!

∫dµ(x1)...dµ(xn)ϕT (x1, ..., xn)

é, (4.6.7)

and combined sums and integrals converge absolutely. Furthermore, for all x1 ∈ X,

1 +∑n≥2

1

(n− 1)!

∫d|µ|(x2)...d|µ|(xn)|ϕT (x1, ..., xn)| ≤ ea(x1). (4.6.8)

We work here with a finite polymer set, PN , and the integral over X has to be replaced bya finite sum: ∫

dµ(x)↔∑X∈PN

p(X ),

where p(X ) is the weight of the polymer X .


4.6.2 Proofs of the Lemmas stated in Section 4.2


We introduce the operator

U(t, s) := 1 +∞∑k=1

(−i)k∫ t

sduk...

∫ u2

sdu1λ(uk)HI(uk)...λ(u1)HI(u1) (4.6.9)

for all t, s ∈ R. HI(t) = eitH0HIe−itH0 for all t ∈ R. We denote by F (L2(R3)) ⊂ F+(L2(R3))

the subspace of finite particle vectors. We show that F (L2(R3)) ⊂ D(U(t, s)) and that (4.6.9)converges strongly on F (L2(R3)), for all t, s ∈ R. Let n ∈ N and let ϕ(n) := ϕP ⊗ ψ(n), withϕP ∈ HP and ψ(n) ∈ F (≤n)

+ (L2(R3)). An easy calculation shows that

HI(t) = G(t)⊗ Φ(φ(t)) (4.6.10)

where Φ, G(t) and φ(t) have been defined in (4.2.7), (4.2.8) and (4.2.10), respectively. There-fore,

‖HI(t)ϕ(n)‖ ≤ 2‖G‖(n+ 1)1/2‖φ‖L2 ‖ϕ(n)‖. (4.6.11)

Inserting (4.6.11) into (4.6.9), we get that

‖U(t, s)ϕ(n)‖ ≤ ‖ϕ(n)‖+∞∑k=1

1

k!(2λ(s)‖G‖ ‖φ‖L2 |t− s|)k(n+ 1)1/2...(n+ k)1/2‖ϕ(n)‖

≤ ‖ϕ(n)‖(

1 +∞∑k=1

1√k!

(4λ(s)‖G‖ ‖φ‖L2 |t− s|)k)

2−n/2n∏p=1

Ån

p+ 1

ã1/2

,

which clearly converges for all t, s ∈ R. To go from the first to the second line, we have usedthat

(n+ 1)1/2

11/2...

(n+ k)1/2

k1/2= (n+ 1)1/2...(n/k + 1)1/2

≤n∏p=1

Ån

p+ 1

ã1/2

2−(n−k)/2

for all k ≤ n, and that

(n+ 1)1/2

11/2...

(n+ k)1/2

k1/2≤

n∏p=1

Ån

p+ 1

ã1/2

2(k−n)/2

for all k > n. This shows that the series defining U(t, s) converges strongly on F (L2(R3)).


We prove (4.2.27). We rewrite (4.2.16) with the notations introduced in (4.2.11) and (4.2.12).

U(t, s) = e−i(t−s)H0 +∞∑k=1

(−i)k∫

∆k[s,t]du λ(u) e−itH0(HI(u))∗eisH0 (4.6.12)


for all t, s ∈ R+. We plug (4.6.12) and its adjoint into (4.1.11). We get that

Zt,s(O)PΩ =∞∑

k1,k2=0

(−1)k2ik1+k2

∫∆k1 [s,t]×∆k2 [s,t]

dudu′ λ(u)λ(u′) PΩe−isHPHI(u)O(t)(HI(u

′))∗eisHPPΩ,

where O(t) = eitHPOe−itHP ; see (4.2.8). Formula (4.6.10) implies that

HI(u)O(t)(HI(u′))∗ = G(u)O(t)[G(u′)]∗ ⊗ Φ(φ(u1))...Φ(φ(uk1))Φ(φ(u′k2

))...Φ(φ(u′1)).

We glue the time coordinates u and u′ together and introduce the new coordinate

x := (x1, ..., xk1+k2) := (u1, ..., uk1 , u′k2, ..., u′1).

Wick’s theorem (see (4.2.19)) implies that

PΩHI(u)O(t)(HI(u′))∗PΩ = PΩ

∑pairings π

i<j

G(u)O(t)[G(u′)]∗∏

(i,j)∈πf(xi − xj) (4.6.13)

if k1 + k2 is even. We assign a number ri ∈ 0, 1 to every time xi ∈ x and set ri = 0 if i ≤ k1

and ri = 1 if i > k1. We write (xi, ri) and we use this new index to take into account the factthat the operator G(xi) multiplies O from the left if ri = 0 and from the right if ri = 1. Using(4.2.23) and (4.2.24), we get that

(−1)k2ik1+k2PΩHI(u)O(t)(HI(u′))∗PΩ = PΩ

∑pairings π

i<j

Tor

Ä ∏(i,j)∈π

F(xi, ri;xj , rj)ä

[O(t)] .

(4.6.14)Let k1 + k2 = 2k, and let ((xi1 , xj1), ..., (xik , xjk)) be a tuple of k pairs (il, jl ∈ 1, ..., 2k).We classify the pairs in ((xi1 , xj1), ..., (xik , xjk)) in increasing order. There exists a uniquepermutation σ of 1, ..., 2k, such that the k-tuple ((xi1 , xj1), ..., (xik , xjk)) can be rewrittenas Ä

(xσ(1), xσ(2)), ..., (xσ(k1+k2−1), xσ(k1+k2))ä, xσ(1) < xσ(3) < ... < xσ(k1+k2−1), (4.6.15)

and xσ(2i−1) < xσ(2i), for all i = 1, ..., k. Every tuple of k pairs arises 4k times by summingover the indices ri ∈ 0, 1. Using a change of variables for each permutation σ and summingover all possible permutations, we get that

eisHPZt,s(O)e−isHP

=∞∑k=0

∫s<x1<x3<...<x2k−1<t

dx λ(x)∑

r∈0,12kTor

[ k∏i=1

Äχ(x2i−1 < x2i) F(x2i−1, r2i−1;x2i, r2i)

ä][O(t)] .

Introducing (ui, ri) := (x2i−1, r2i−1), (vi, r′i) := (x2i, r2i), wi = (ui, ri; vi, r

′i), and using the

measure (4.2.26), we finally get that

eisHPZt,s(O)e−isHP =∞∑k=0

∫[s,t]2k

dµk(w) λ(w)Tor

[ k∏i=1

F(ui, ri; vi, r′i)]

[O(t)] .


We now show that the series converges strongly. If ui, vi ∈ [t, s], ui < vi, we remind thereader that λ(vi) < λ(ui) ≤ λ(s). One has that

λ(u)λ(v)∥∥∥Tor

[ k∏i=1


[O(t)]∥∥∥ ≤ λ2k(s) ‖O‖ ‖G‖2k

k∏i=1

|f(vi − ui)|. (4.6.16)

We plug this bound into (4.2.27), and we get that

∫[s,t]2k

dµk(w)∥∥∥Tor

[ k∏i=1


[O(t)]∥∥∥

≤ λ2k(s)

∫[s,t]2k

dµk(w) ‖O‖ ‖G‖2kk∏i=1

|f(vi − ui)|.

Then we integrate over the v′s, which leads us to

∫[s,t]2k

dµk(w)

∥∥∥∥Tor

[ k∏i=1


[O(t)]

∥∥∥∥≤ λ2k(s) 4k

∫s<u1<u2<...<uk<t

du ‖O‖‖G‖2k||f ||kL1 .

Finally, we can integrate over the k-dimensional simplex and sum over k to obtain that

∞∑k=0

∫[s,t]2k

dµk(w)

∥∥∥∥Tor

[ k∏i=1


[O(t)]

∥∥∥∥≤∞∑k=0

1

k!λ2k(s) 4k|t− s|k‖G‖2k||f ||kL1‖O‖.

(4.6.17)


We compute Ks|Πij〉, where Πij = |ϕi〉〈ϕj | ∈ B(HP ), and ϕ1, ..., ϕn0 are the normalizedeigenvectors of HP (unique up to a phase). We get that

Ks|Πij〉 = f(s)R(G)eisLL(G)|Πij〉+ f(−s)L(G)eisLR(G)|Πij〉− f(−s)L(G)eisLL(G)|Πij〉 − f(s)R(G)eisLR(G)|Πij〉.

Using the equality

ΠijG =∑k,l

GklΠijΠkl =∑k,l

Gkl|ϕi〉〈ϕj |ϕk〉〈ϕl| =∑l

GjlΠil,

GΠij =∑k,l

GklΠklΠij =∑k,l

Gkl|ϕk〉〈ϕl|ϕi〉〈ϕj | =∑k

GkiΠkj ,


we obtain that

Ks|Πij〉 = f(s)∑k,m

eisεkjGkiGjm|Πkm〉+ f(−s)∑l,m

eisεilGjlGmi|Πml〉

− f(−s)∑k,m

eisεkjGmkGki|Πmj〉 − f(s)∑l,m

eisεilGjlGlm|Πim〉.

If i 6= j, then

PεijKs|Πij〉 =Äf(s)eisεijGiiGjj + f(−s)eisεijGjjGii

ä|Πij〉

−(f(−s)

∑k

eisεkjGikGki + f(s)∑l

eisεilGjlGlj)|Πij〉

and we deduce that

M|Πij〉 = 2

∫ ∞0

ds <(f(s)) GiiGjj |Πij〉

−(∑

k

∫ ∞0

ds f(−s)eisεki |Gik|2 +∑l

∫ ∞0

ds f(s)eisεjl |Gjl|2)|Πij〉.

If i = j, we get that

M|Πii〉 =

∫ ∞0

ds f(s)∑k

eisεkiGkiGik|Πkk〉+

∫ ∞0

ds f(−s)∑l

eisεilGilGli|Πll〉

−∫ ∞

0ds f(−s)

∑k

eisεkiGikGki|Πii〉 −∫ ∞

0ds f(s)

∑l

eisεilGilGli|Πii〉

= 2∑k

∫ ∞0

ds <(f(s)eisεki)|Gki|2|Πkk〉 − 2∑k

∫ ∞0

ds <(f(−s)eisεki)|Gki|2|Πii〉.

Since

f(t) =

∫R3d3k |φ(k)|2e−itω(k),

we deduce that

<Å∫ ∞

0ds f(s)eisεji |Gij |2

ã= π

∫R3d3k |φ(k)|2δ(εji − ω(k))|Gij |2 (4.6.18)

<Å∫ ∞

0ds f(−s)eisεji |Gij |2

ã= π

∫R3d3k |φ(k)|2δ(εji + ω(k))|Gij |2. (4.6.19)

The Fermi golden rules in (4.1.8) imply that

M|Πij〉 = mij |Πij〉

for all i 6= j, with

<(mij) = −∫ ∞

0ds <(f(s))(Gii −Gjj)2

−<(∑k<i

∫ ∞0

ds f(−s)eisεki |Gki|2 +∑l<j

∫ ∞0

ds f(s)eisεjl |Gjl|2).

(4.6.20)


Plugging (4.6.18) and (4.6.19) into (4.6.20), we deduce that

<(mij) ≤ −π∫R3d3k

(∑k<i

|φ(k)|2δ(εki+ω(k))|Gik|2+∑l<j

|φ(k)|2δ(εjl−ω(k))|Glj |2). (4.6.21)

Equations (4.6.21) and (4.1.8) show that <(mij) < 0 for all i 6= j. The eigenvalue mij ofMassociated to the normalized eigenvector |Πij〉 has therefore a strictly negative real part, forall i 6= j.

If i = j,

M|Πii〉 = 2∑i<k

∫ ∞0

ds <(f(s)eisεki)|Gki|2|Πkk〉 − 2∑k<i

∫ ∞0

ds <(f(−s)eisεki)|Gki|2|Πii〉,

which we can rewrite using (4.6.18) and (4.6.19) as

M|Πii〉 = 2π

∫R3d3k

(∑i<k

|φ(k)|2δ(εki−ω(k))|Gik|2|Πkk〉−∑k<i

|φ(k)|2δ(εki+ω(k))|Gki|2|Πii〉).

(4.6.22)Using (4.6.21) and (4.6.22), we representM as a n2

0 × n20 bloc matrix in the basis (Πij).

It takes the form

M =

ÇMD 00 MT

å, (4.6.23)

whereMD is the (n20 − n0)× (n2

0 − n0) diagonal matrix

MD =

ám12 0 ... 0

0 m13 0 00 ... .. 00 0 ... mn0(n0−1)

ë, (4.6.24)

andMT is the lower triangular n0 × n0 matrix given by

MT =

â0 0 ... ... 0

(MT )21 −(MT )21 0 ... 0(MT )31 (MT )32 −(MT )31 − (MT )32 0 ...... ... ... ... 0

(MT )n01 (MT )n02 ... (MT )n0(n0−1) −∑n0−1i=1 (MT )n0i

ì.

The coefficients (MT )ij are positive; see (4.6.22). They satisfy∑i−1j=1(MT )ij > 0 because of

the rules (4.1.8).

4.7 Proof of Proposition 4.3.3.1

We rewrite (4.3.17) as〈Ψ(tN )|OΨ(tN )〉 =

∑G, (N,R)∈G

p(G)


where the sum is carried out over all decorated graphs G on 0, ..., N with (N,R) ∈ V(G)(see Section 4.3.3) and,

p(G) =∞∑k=0

∫[0,tN ]2k

dµk(w)λ(w) F(w)χG(w)〈Π(t−1)|

Tor

( ∏(l,B)∈V(G)

[e−itlL


(iG)(t, r)äeitl+1L

] ∏(j,R)∈V(G)

R(tj)∏

(m,·)/∈V(G)

P (tm))|1HP 〉.

(4.7.1)

The function χG is used to select pairings w whose correlation lines only join intervals that arelinked by an edge in G, and that are such, that, for each edge ((i, B); (j, B)) ∈ E(G), there is(u, r; v, r′) ∈ w with u ∈ Ii and v ∈ Ij . We then use that P (ti) is a one-dimensional projectionand that

P (ti)P (tj) = |1HP 〉〈Π(tj)|. (4.7.2)

Equation (4.7.2) implies that P (ti)P (ti+1)...P (tj) = |1HP 〉〈Π(tj)|. As mentioned in Section4.3.3, we can decompose the set V(G) into maximal blocks of neighboring vertices. AnyU ∈ U(G) is surrounded by a projection P (tm(U)) on its left, and by a projection P (tmax(U)+1)on its right. Together with (4.7.2), this implies that

p(G) =∞∑k=0

∫[0,tN ]2k

dµk(w)λ(w) χG(w) F(w)∏

U∈U(X )

hU (w), (4.7.3)

where hU has been defined in (4.3.23). By construction of the polymer set PN , G = X1∨...∨Xnfor some n, where Xi ∈ PN and dist(Xi,Xj) ≥ 2 for all i 6= j. Since the polymers Xi aresurrounded by one dimensional projections P (·), we see using (4.7.2) that we can factorize

∏U∈U(X )

hU (w) =n∏i=1

Ñ ∏U∈U(Xi)

hU (w ∩ V(Xi))

éfor every pairing w. The integral over pairings can be splitted into the product of n integrals,and we deduce that

p(G) = p(X1) ... p(Xn). (4.7.4)

4.8 Proofs of Corollaries 4.5.1.1 and 4.5.2.1 (sketch)

Some modifications have to be done to adapt the proof of Theorem 4.1.2.1 to Corollar-ies 4.5.1.1 and 4.5.2.1. The construction of connected graphs has to be modified to proveCorollary 4.5.1.1. The projection P = |1HP 〉〈Π1| in Lemma 4.2.4.1 has to be replaced byPβ := |1HP 〉〈ρP,β| to prove Corollary 4.5.2.1, where ρP,β is the Gibbs equilibrium state of Pat temperature T = 1/β.


Corollary 4.5.2.1

The operator Zt,s of Section 4.2 has to be replaced by the new operator Zt,sβ ∈ B(B(HP ))defined in (4.5.9). The calculations carried out in Sections 1,2,3,4 remain valid with f(t)replaced by fβ(t). The only difference occurs in Paragraph 4.2.4, as the LindbladianM nowdepends on the inverse temperature β. Mβ is a block matrix of the form

Mβ =

ÇMβD 00 Mβ

å, (4.8.1)

whereMβD is a (n20 − n0)× (n2

0 − n0) diagonal matrix,

MβD =

ámβ12 0 ... 0

0 mβ13 0 00 ... .. 00 0 ... mβn0(n0−1)

ë, (4.8.2)

and Mβ is a n0 × n0 matrix,

Mβ =

á−∑i 6=1 ai1 a21 ... an01

eβε21a21 −a21eβε21 −∑i>2 ai2 ... an02

... ... ... ...

eβεn01an01 eβεn02an02 ... −∑n0−1i=1 eβεn0ian0i

ë.

The off-diagonal entries of Mβ are positive, and∑j 6=i Mβ;ij > 0, for all i = 1, ..., n0; see

(4.1.8). It is clear that Mβ|1HP 〉 = 0. The reader can check that the Gibbs equilibrium stateat temperature T = 1/β, ρP,β := 1

tr(e−βHP )

∑n0i=1 e

−βεiΠi, satisfies

〈ρP,β|Mβ = 0.

We use a Perron-Frobenius argument to show that any z ∈ σ(Mβ) \ 0 satisfies <(z) < 0,and that 0 is a non-degenerate eigenvalue of Mβ . We introduce the matrix

M′ := Mβ + xM1n0×n0 ,

where xM := maxi≥1(−Mβ;ii). M′ is irreducible non-negative. This follows from (4.1.8) andfrom the characterization of irreducible matrices with strongly connected directed graphs; see[103]. The theorem of Perron-Frobenius for non-negative irreducible matrices implies that themaximal eigenvalue of M′ is unique and that it is equal to xM (because

∑n0j=1 Mβ;ij = 0,

for all i = 1, ..., n0.). Furthermore, the left- and right eigenspaces of M′ associated to xMare one-dimensional. We deduce that 0 is a non-degenerate eigenvalue of Mβ , and that ρP,βis the only left-eigenvector of Mβ with associated eigenvalue 0 and trace one. The rest ofthe spectrum of Mβ lies on the left side of the imaginary axis in the complex plane. Theprojection P in Lemma 4.2.4.1 must be replaced by Pβ := |1HP 〉〈ρP,β|, and the analysis isthen completely similar to what has been done in Sections 2,3, and 4.


Corollary 4.5.1.1

We only sketch the modifications that need to be done to adapt the proof presented in Sections2-4.

Modifications in Section 4.3. Let ϕf = Φ(f1)....Φ(fn1)Ω be the initial state of the field. Weadd a discrete set of points I−1 := t1, ..., tn1 to the time axis to represent the contributionof ϕf to the Dyson expansion.

O

(r = 0)

(r = 1)

0 t1 t2 t3 t4 t5...

tN

0 t1 t2 t3 t4 t5...

tN

t1 t2...

tn1

t1 t2...

tn1

We introduce 7 Feynman rules corresponding to contractions involving the field operatorsΦ(fi).

(r = 0)

(r = 1)

ti tj

ti

tj

ti tj

(a’) (b’) (c’)

(r = 0)

(r = 1)

tivi

ti

vi ti

vi

ti vi

(e’) (f’) (g’) (h’)

(a’) := 〈Φ(fi)|Φ(fj)〉, (4.8.3)(b’) := 〈Φ(fi)|Φ(fj)〉, (4.8.4)(c’) := 〈Φ(fj)|Φ(fi)〉, (4.8.5)(e’) := λ(vi)〈Φ(fi)|Φ(φ(vi))〉L(iG(vi)), (4.8.6)(f’) := λ(vi)〈Φ(fi)|Φ(φ(vi))〉R(iG(vi)), (4.8.7)(g’) := λ(vi)〈Φ(φ(vi))|Φ(fi)〉L(iG(vi)) (4.8.8)(h’) := λ(vi)〈Φ(φ(vi))|Φ(fi)〉R(iG(vi)). (4.8.9)


The function F in (4.2.24) has to be modified to take the rules (4.8.3)-(4.8.9) into account.The set of polymers PN is constructed from the pairings as in Paragraph 4.3.2. The set ofvertices of a polymer can now contains the vertex (−1, B), corresponding to contractions withthe initial field operators Φ(fi). When s = 0, Formula (4.2.27) is replaced by

Zt,0(O) =∞∑k=0

∫(I−1∪[0,t])2k

dµk(w)λ(w)Tor

[k∏i=1


][O(t)] , (4.8.10)

The integral over I−1 is an abuse of notations. It is actually a discrete sum and the measureµk(·) is modified such that pairs (ti, ri; tj , rj) in (4.8.10) are classified in lexicographic order:(ti, ri) < (tj , rj) iff (i, ri) < (j, rj) in the lexicographic sense. The sum over isolated intervalssketched in Section 4.3.2 remains the same, up to some change of notations similar to (4.3.9)→ (4.8.10). The contribution of the isolated vertex I−1 to the Dyson series corresponds tothe left multiplication by the identity operator, because ϕf is normalized. The vertex (−1, ·)can carry two decorations: it is decorated with a B if a correlation line starts in I−1 and endsin another time interval; or it is decorated with a P if I−1 is isolated. In the latter case, weset P (t−1) := 1B(B(HP )). The integral over pairings in (4.3.22) must be modified as (4.3.9) →(4.8.10) if (−1, B) ∈ V(X ).

Modifications in Section 4.4 The function η(E ) has to be modified to take edges that startfrom (−1, B) into account. We set

η(E ) :=

4‖G‖2∫ ti+1ti

du∫ tj+1

tj dv|f(v − u)|λ(u)λ(v) if E = ((i, B); (j, B)), i, j 6= −1,

4‖G‖∫ tj+1

tj dv|〈fj |φ(v)〉|λ(v) if E = ((−1, B); (j, B)).

(4.8.11)The bound (4.4.4) remains true with e4τ‖f‖L1‖G‖2|B(X )| replaced by M |B(X )|

τ,n1 , where

Mτ,n1 := max(e4τ‖f‖L1‖G‖2 , C(n1))

and C(n1) > 0 is a constant that depends on the absolute values of the scalar products(fi, fj)L2 and n1. The rest of Section 4.4 is mainly unchanged (even if we loose a factor λ(·)for correlations involving the vertex (−1, B)) and the convergence of the cluster expansion asN → ∞ can be carried out by inspection, following the proofs given in Sections 4.4.1 and4.4.2.

Chapter 5

Emergence of facts in QuantumMechanics

In this chapter, we attempt to elucidate the roles played by entanglement between a systemand its environment and of information loss in understanding "decoherence" and "dephasing",which are key mechanisms in a quantum theory of measurements and experiments; see also[88, 23, 72, 93]. We provide a very simple example of a quantum system where the phenomenaof information loss and decoherence can be explicitly analyzed. We also discuss the problem of"time in quantum mechanics", and we sketch an answer to the question when an experimentcan be considered to have been completed successfully; ( i.e. "when does a detector click?").In Section 5.4, we discuss the emergence of quantum trajectories. After a short review ofwell-known results, we state and prove new lemmas related to the notion of purification onthe spectrum of an observable.

5.1 Decoherence and information loss: two necessary ingredi-ents for the emergence of facts

5.1.1 Preliminary definitions

Let S be a physical system modeled by the datas

(OS ,AS , τt,st,s∈R,SS) (5.1.1)

with properties as specified in points (I) through (III) of Definition 1.1.1.1, Subsect. 1.1.1,and such that OS ( AS . We have already mentioned in Section 1.4 that the probability of ahistory Π(1)

α1 , ...,Π(n)αn in a state ω ∈ SS is predicted to be given by


αn := ωÄΠ(1)α1...Π(n−1)

αn−1Π(n)αn Π(n−1)

αn−1...Π(1)

α1

ä. (5.1.2)

We pointed out in Section 1.4 that (5.1.2) is problematic for quantum models of physicalsystems, because the presence of interferences implies that potential properties a ∈ OS arenot necessarily empirical. In this section, we introduce some new definitions that are usedlater on to elucidate this problem.

Let t0 be a fiducial time at which the state of S is specified. An observation of a potentialproperty, a, of S at time t, will be described in terms of the operator a(t) = τt,t0(a) ∈ AS .

155

CHAPTER 5. EMERGENCE OF FACTS IN QUANTUM MECHANICS 156

To arrive at a mathematically precise concept of information loss (as time goes by), it isconvenient to introduce the following algebras.

Definition 5.1.1.1. The algebra, E≥t, of potential properties observable after time t is the C∗-subalgebra of AS generated by arbitrary finite linear combinations of arbitrary finite products

a1(t1)...an(tn), n = 1, 2, 3, ...,

where ti ≥ t and ai ∈ OS, i = 1, ..., n, (with a(s) the operator in AS representing the operatora ∈ OS at time s).

It follows from this definition thatE≥t ⊆ E≥t′ (5.1.3)

whenever t > t′, with E≥t ⊆ AS , for all t ∈ R. We also introduce an algebra ES defined by

ES :=∨t∈RE≥t‖·‖. (5.1.4)

The algebra ES carries an action of the group, R, of time translations by ∗automorphisms,τ tt∈R, defined as follows: For a1(t1)...an(tn) ∈ ∨

t∈RE≥t, with ti ∈ R, ai ∈ OS , i = 1, ..., n,

τ t(a1(t1)...an(tn)) := a1(t1 + t)...an(tn + t). (5.1.5)

The definition of τ t extends to all of ES by linearity and continuity. One then has that

τ t : E≥t′ −→ E≥t′+t ⊆ E≥t′ , (5.1.6)

for arbitrary t ≥ 0.

Definition 5.1.1.2. Information loss. We speak of loss of information if

E≥t ( E≥t′ , (5.1.7)

for some times t and t′, with t > t′.

Let ω ∈ SS be a state of the system. Let (Hω, πω,Ω) denote the Hilbert space, therepresentation of AS on Hω, and the cyclic vector in Hω, respectively, associated to the pair(AS , ω) by the GNS construction. By AωS we denote the von Neumann algebra correspondingto the weak closure of πω(AS) in the algebra, B(Hω), of all bounded operators on Hω. Towork in the realm of Hilbert spaces, it is useful to introduce the following algebra

Definition 5.1.1.3. The algebra, Eω≥t, of all possible events observables at times ≥ t, is the vonNeumann algebra corresponding to the weak closure of πω(E≥t) in B(Hω). The von Neumannalgebra EωS is defined similarly.

The algebra Eω≥t contains the spectral projections describing possible events at times s ≥ t.It is therefore justified to call Eω≥t the "algebra of possible events observable at times ≥ t".


5.1.2 Consistent histories : a first but incomplete step towards a notion ofempirical properties

Consistent histories

Let Π(i)αi ni=1 be a sequence of events (also called history) as in (1.4.2). We introduce the

matrixPωα,α′ := ω(Π(1)

α1...Π(n)

αn Π(n)α′n...Π

(1)α′1

), (5.1.8)

with αn = α′n, and α = (α1, ..., αn). In order to select histories of events that exhibit almostno interferences, we introduce the notion of δ-consistent histories; see [80, 114, 69]. We denoteby K a subset of 1, ..., k1 × ... × 1, ..., kn. We remind the reader that αi ∈ 1, ..., ki forall i = 1, ..., n.

Definition 5.1.2.1. Let (Π(1)α1 , ...,Π

(n)αn )α∈K be a family of histories and let δ ∈ [0, 1]. We

say that this family is δ − consistent if

‖[Π(i)αi ,Π

(i+1)αi+1

...Π(n)αn ...Π

(i+1)αi+1

]‖ ≤ 1− δ. (5.1.9)

for all i and for all α ∈ K.

A 1−consistent family of histories is consistent, in the sense that the sum rules in (1.4.13)are valid for all α and all i = 1, ..., n. We define a diagonal matrix ∆ω by

∆α,α′ :=

®Pωα,α if α = α′

0 else

Clearly inequality (5.1.9) implies that

‖Pω −∆ω‖ ≤ const.(1− δ). (5.1.10)

Dynamical mechanisms that imply that ‖Pω−∆ω‖ becomes small, in suitable limiting regimesare known under the names of dephasing and decoherence; see [88, 72, 93, 128]. These mecha-nisms are discussed in Sections 5.1.3 and 5.2. Apparently, if δ is very close to 1, then everythingmight appear to be fine, and one could be tempted to interpret the events Π

(i)αi of consistent

histories as empirical events. However, this is not what we should do, and we will explainwhy further below.

We remark that if δ is very close to 1, then the observation of the events Π(1)α1 , ...,Π

(n)αn can

be reinterpreted as the observations of events Π(1)α1 , ..., Π

(n)αn that differ only very slightly from

the spectral projections Π(1)α1 , ...,Π

(n)αn , and that do not exhibit any interference. This is the

content of Lemma 5.1.2.1 below. Its proof is given in Section 5.5.2.

Lemma 5.1.2.1. Let Π(1)α1 , ...,Π

(n)αn be orthogonal projections. Suppose that

‖[Π(i)αi ,Π

(i+1)αi+1

...Π(n)αn ...Π

(i+1)αi+1

]‖ < ε, (5.1.11)

for all i = 1, ..., n− 1 and all α = (α1, ..., αn), with ε sufficiently small (depending on the totalnumber,

∑ni=1 ki, of n−tuples α, with αi = 1, ..., ki). Then there exist orthogonal projections

Π(i)αi , αi = 1, ..., ki, i = 1, ..., n, with

Π(i)αi Π

(i)βi

= δαiβiΠ(i)αi ,

ki∑αi=1

Π(i)αi = 1, (5.1.12)


such that‖Π(i)

αi −Π(i)αi ‖ ≤ Cε, (5.1.13)

and[Π(i)

αi , Π(i+1)αi+1

...Π(n)αn ...Π

(i+1)αi+1

] = 0, (5.1.14)

for all α and all i ≤ n − 1. The constant C in (5.1.13) depends on∑ni=1 ki, and ε must be

chosen so small that Cε < 1; (in which case Π(i)αi and Π

(i)αi are unitarily equivalent).

Let us now insist on the following important remark, which constitutes a first hint towardsa correct algebraic characterization of empirical properties of physical systems. If there is lossof information and if the relative commutants

(Eω≥ti+1)′ ∩ Eω≥ti , ti−1 < ti ≤ ti, (5.1.15)

are non-trivial, for suitable choices of sequences of times t1 < t2 < ... < tn, t1 < t2 < ... < tn,and if the operator

ai(ti) ∈ (Eω≥ti+1)′ ∩ Eω≥ti , (5.1.16)

and hence Π(i)αi belongs to (Eω≥ti+1

)′ ∩ Eω≥ti , for all αi = 1, ..., ki, with ti−1 < ti ≤ ti, then

[Π(i)αi ,Π

(i+1)αi+1

...Π(n)αn ...Π

(i+1)αi+1

] = 0, (5.1.17)

for all αi and all α. If (5.1.16) and hence (5.1.17) hold, for all i ≤ n, then there is perfectdecoherence, and the histories (Π(1)

α1 , ...,Π(n)αn )α∈K form a consistent family.

Why is the concept of consistent histories not good enough to make sense ofempirical properties ?

First reason: the problem with the future. Given a measurement of a potential propertyai ∈ OS of S at some time ti, the success of this measurement, as expressed in the decoherenceof (absence of interference between) the events Π

(i)1 , ...,Π

(i)ki, apparently not only depends on

the past but seems to depend on the future, namely on subsequent measurements of potentialproperties ai+1, ..., an at times > ti. This is how conditions such as (5.1.9) and (5.1.11) mustbe interpreted. The consistency of a family of stretches of histories can apparently only beassured if one also knows the family of stretches of histories in the future of this history.

Second reason: the necessity of a many worlds interpretation. Let S be characterizedby the algebraic datas (OS ,AS , τt,st,s∈R, ω ∈ SS). We may consider two (or more) familiesof potential properties of S,

a1, ..., an and b1, ..., bm, (5.1.18)

measured at times t1 < ... < tn and t′1 < ... < t′m, respectively, with ai ∈ OS and bj ∈ OS , forall i and j. Both families may give rise to consistent families of histories (e.g., if conditions(5.1.16) hold for the ai’s and the bj ’s). Yet, there may not exist any family

c1, ..., cN, N ≥ n+m,

of potential properties of S (cj ∈ OS , for all j) measured at times T1 < ... < TN , with

T1, ..., TN ⊇ t1, ..., tn ∪ t′1, ..., t′m,


encompassing the two families in (5.1.18) and giving rise to a consistent family of histories.Since the algebraic data characterizing S are fixed, the confusing question arises which one ofthe two or more incompatible families of potential properties a1, ..., an, b1, ..., bm, ... willactually be observed in the course of time, i.e., become real, (or, put differently, correspondto empirical properties). Some people suggest, following Everett [52], that there is a world forevery family of potential properties of S giving rise to a consistent family of histories to be ob-served. This is the "many-worlds interpretation of quantum mechanics". Such interpretationsare unacceptable.

Third reason: times are precisely fixed. It has tacitly been assumed, so far, that thetimes at which quantum-mechanical measurements of potential properties of a system S arecarried out (we are talking of the times ti at which potential properties ai of S are observed) canbe fixed precisely (by an "observer"?). – Obviously, this assumption is nonsense in quantummechanics, (as opposed to classical physics).

5.1.3 An algebraic characterization of "empirical properties" of quantummechanical systems

The search

Let a = a∗ ∈ ES be an operator representing a potential property (or physical quantity) of S,and let ω denote the state of S. As before, we assume that a has a finite spectrum,

a =k∑i=1

αiΠi, k <∞, (5.1.19)

where α1, ..., αk are the eigenvalues of a (now viewed as a self-adjoint operator in the vonNeumann algebra EωS ), and Πi ≡ Π

(i)αi ∈ EωS is the spectral projection of a corresponding to αi,

i = 1, ..., k, with Πi = Π∗i , ΠiΠl = δilΠl. The conventional wisdom is that if a is an empiricalproperty of S at some time t′ earlier than t, i.e., that a is measured (or observed) before timet, then

ω(b) ≈k∑i=1

ω(ΠibΠi), (5.1.20)

for all b ∈ Eω≥t; i.e., ω|Eω≥t is close to an incoherent superposition (mixture) of eigenstates,p−1i ω(Πi(·)Πi) (pi 6= 0), of a, where pi = ω(Πi), (and pi > 0, for at least one choice of i). We

remark that a sufficient condition for (5.1.20) to hold is that

a ∈ (Eω≥t)′ ∩ EωS . (5.1.21)

If there existed a sequence of times, t1 < t2 < ... < tn, and self-adjoint operators a1, ..., an,with finite point spectra, as above, and

al ∈ (Eω≥tl+1)′ ∩ Eω≥tl ,

l = 1, ..., n−1, an ∈ Eω≥tn , then the family of histories Π(1)j1, ...,Π

(n)jn, where Π

(l)jl

is the spectral

projection of al corresponding to the eigenvalue α(l)jl

of al, l = 1, ..., n, is consistent ; see (5.1.16)– (5.1.17). For this observation to be interesting, the relative commutants (Eω≥tl+1)′ ∩ Eω≥tlwould have to be non-trivial, and if we wish to escape from the critiques discussed at the end


of Section 5.1.2, the algebras (Eω≥tl+1)′ ∩ Eω≥tl would have to be abelian, for all l, otherwise thepotential properties al cannot be qualified as empirical. This is certainly not true in general,and the good notion of empirical properties can be actually obtained with a little more work,once we have introduced the centralizer of the state ω.

Definition 5.1.3.1. (i) Given von Neumann algebras M ⊆ N , a state ω on N and anoperator a ∈ N , we define a, ω]M to be the bounded linear functional onM defined by

a, ω]M(b) := ω([a, b]), b ∈M. (5.1.22)

(ii) The centralizer (or stabilizer), CωM, of ω is the subalgebra ofM defined by

CωM := a ∈M | a, ω]M = 0. (5.1.23)

It is easy to see that ω defines a trace on CωM. This means that CωM is a direct sum (orintegral) of finite-dimensional matrix algebras, type-II1 factors, and abelian algebras.

Remark 5.1.3.1. Centralizers of states or weights on von Neumann algebras play an inter-esting role in the classification of von Neumann algebras, (in particular in the study of type-IIIfactors); see [83],[39]. In the appendix 5.5.1, we recall a few relevant results on centralizers.

The following result can be proven by inspection.

Lemma 5.1.3.1. The following assertions are equivalent:(i) |a, ω]Eω≥t(b)| ≤ ε‖b‖, ∀b ∈ E

ω≥t

(ii) |ω(b)−∑ki=1 ω(ΠibΠi)| ≤ const.ε‖b‖, ∀b ∈ Eω≥t.

In view of Lemma 5.1.3.1, one might be tempted, again, to identify elements of the centralizer

Cω≥t := CωEω≥t (5.1.24)

with empirical properties of S observable at times ≥ t. Yet, this is not quite appropriate,again, because of the following reasons:

First reason: In general, the centralizers Cω≥t are non-abelian algebras. If the centralizers Cω≥tare non-commutative algebras then identifying empirical properties of S observable at times≥ t with elements of Cω≥t is subject to the second critic formulated for consistent histories.

Second reason A family of operators, a1, ..., an, with ai ∈ Cω≥ti , i = 1, ..., n, t1 < t2 <

... < tn, does not necessarily give rise to a family of consistent histories. Indeed, let Π(i−1)l ,

l = 1, ..., ki−1, be the spectral projections of ai−1 ∈ Cω≥ti−1. Let ωl denote the state ωl(b) =

p−1l ω(Π

(i−1)l bΠ

(i−1)l ), where pl = ω(Π

(i−1)l ) > 0. Let us assume that pl > 0 for at least two

distinct values of l. The problem is that, in general, the assumption that ai ∈ Cω≥ti doesnot imply that ai ∈ Cωl≥ti , for all l = 1, ..., ki−1 for which pl > 0; this is the phenomenon of"spontaneous symmetry breaking". This means that the "sum rule" (1.4.13), Sect.1.4, may beviolated at the ith slot, for some 1 < i < n. Hence the family a1, ..., an may not give rise to afamily of consistent histories.


Empirical properties

We denote the center of Cω≥t by Zω≥t. Zω≥t is abelian, and it is the right algebra we were lookingfor. More precisely,

Definition 5.1.3.2. Let a = a∗ ∈ Eω≥t. We say that a is an empirical property of S at time twithin an uncertainty (of size) δ ≥ 0 if there is a = a∗ ∈ Zω≥t such that

ω((a− a)2) ≤ δ2. (5.1.25)

Lemma 5.1.3.2. If Cω≥t is decomposable into a direct sum of type In factors,

Cω≥t = ⊕λ∈ΛCω≥t,λ, (5.1.26)

then every element of Zω≥t is a function of the density matrix ρω≥t, where ω|Eω≥t =: Tr(ρω≥t ·).The density matrix ρω≥t is given by

ρω≥t =∑λ∈Λ

pλ(t)Πλ(t), (5.1.27)

where Πλ(t) is the projection on the finite factor Cω≥t,λ. Furthermore, if a = a∗ ∈ Eω≥t, then

a = aω :=∑λ∈Λ

aλ Πλ(t), with aλ = Tr(aΠλ(t))/dim(Πλ(t)), (5.1.28)

is the unique selfadjoint element in Zω≥t minimizing ω((a− a)2).

Before proving Lemma 5.1.3.2, we argue why Definition (5.1.3.2) is the right one. Lemma5.1.3.2 shows that Zω≥t is non-trivial in most cases. Furthermore, our preceding discussionsshow that elements of Zω≥t satisfy all requirements that should be fulfilled by empirical prop-erties. In particular, since Zω≥t is abelian, the critics raised against consistent histories do notapply to Zω≥t. Moreover, (5.1.25) implies that

|a, ω]Eωt (b)| = |ω([a, b])| = |ω([a− a, b])|

≤ 2»ω((a− a)2)ω(b∗b) ≤ 2δ ‖b‖,

(5.1.29)

for arbitrary b ∈ Eω≥t. Thus, if δ is small then ‖a, ω]Eω≥t‖ is small, too. Lemma 5.1.3.1 tells usthat, in the type I case, ω|Eω≥t is close to an incoherent superposition of eigenstates of a.

Proof. (Lemma 5.1.3.2).We have seen, after definition (5.1.23), that ω|Cω≥t is a trace on Cω≥t. This implies that

Cω≥t =

∫ ⊕ΛCω≥t,λ , (5.1.30)

where every algebra Cω≥t,λ, λ ∈ Λ ≡ Λω, is either a finite-dimensional matrix algebra, ≈Mnλ(C), of nλ × nλ matrices, with 1 ≤ nλ < ∞, or a type-II1 factor; (see [123], Theorem


8.21 in Chapter 4, and Theorem 2.4 in Chapter 5). If Cω≥t,λ is isomorphic to Mnλ(C) thenω|Cω≥t,λ ∝ TrCnλ (·). By assumption, Λ is discrete, and

Cω≥t = ⊕λ∈ΛCω≥t,λ, (5.1.31)

with Cω≥t,λ 'Mnλ(C), nλ <∞, for all λ ∈ Λ. Therefore,

ω|Eω≥t =: Tr(ρω≥t ·), (5.1.32)

where ρω≥t is a density matrix, and


pλ(t)Πλ(t). (5.1.33)

The operators Πλ(t) ≡ Πωλ(t) are the eigenprojections of ρω≥t, with dim(Πλ(t)) = nλ < ∞.

The weights pλ(t) ≡ pωλ(t) ≥ 0 are the eigenvalues of ρω≥t, and

Tr(ρω≥t) =∑λ∈Λ

pλ(t)dim(Πλ(t)) = 1.

The algebra Cω≥t,λ ' Mnλ(C) is the algebra of all bounded operators from the eigenspaceRan Πλ(t) to itself, and

ω|Cω≥t,λ = pλ(t)Tr(Πλ(t)(·)).

Any operator a ∈ Eω≥t commuting with all the projections Πλ(t), λ ∈ Λ, belongs to Cω≥t, and anyoperator in the center Zω≥t of Cω≥t is a function of the projections Πλ(t), λ ∈ Λω. In particularΠλ(t) ∈ Zω≥t ⊂ Cω≥t, for all λ, (and hence the eigenprojections of ρω≥t might qualify as empiricalproperties of S). Furthermore, for any a = a∗ ∈ Eω≥t, for any a = a∗ =

∑λ αλΠλ(t) ∈ Zω≥t,

ω((a− a)2) =∑λ∈Λ

pλ(t) Tr(Πλ(t)(a− αλ)2). (5.1.34)

Eq. (5.1.34) is minimized if Tr(Πλ(t)(a− αλ)2) is minimized for all λ ∈ Λ, and the minimizerof Tr(Πλ(t)(a− αλ)2) is αλ = Tr(Πλ(t)a)/Tr(Πλ(t)).

We assume that the hypotheses of Lemma 5.1.3.2 are fulfilled. We adopt the same notationsas in the proof of Lemma 5.1.3.2. We write

a = a∗ :=∑λ∈Λ

αλΠλ(t) ∈ Zω≥t ⊂ Cω≥t. (5.1.35)

Let dµλ(α) denote the spectral measure of the operator a = a∗ ∈ Eω≥t in the state n−1λ Tr(Πλ(t)(·)).

Then

0 ≤ ω((a− a)2) =∑λ∈Λ

pλ(t)nλ1

nλTr(Πλ(t)(a− αλ)2) =

∑λ∈Λ

pλ(t)nλ

∫dµλ(α)(α− αλ)2.

Thus,pλ(t)nλ

∫dµλ(α)(α− αλ)2 ≤ ω((a− a)2) ≤ δ2.


for every λ ∈ Λ. We conclude that if, for some λ ∈ Λ,

1

pλ(t)nλω((a− a)2) < ε2,

for some ε > 0, then a has spectrum at a distance less than ε from αλ. In particular, if a hasdiscrete spectrum then a has at least one eigenvalue αλ, with

|αλ − αλ| < ε. (5.1.36)

Note that the maximal uncertainty δ admissible in statement (2) above depends on the spec-trum of the operator a.

Remark 5.1.3.2. Let a ∈ OS be an operator representing some potential property of S. Thena(t) := τt,t0(a) ∈ Eω≥t, t0 being the fiducial time at which ω is given. One may then argue thatif ω((a(t) − a)2) is very small, and if a measurement or observation of a(t) ∈ OS at a time≈ t indicates that it has a value α ≈ αλ, then one may use the state

ωλ :=1

nλTr(Πλ(t)(·)) (5.1.37)

to predict the behavior of the system S at times later than t. This idea, reminiscent of "statecollapse", will be further discussed in Section 5.3.

Remark 5.1.3.3. In Local Relativistic Quantum theories, the algebras E≥t, t ∈ R, are replacedby algebras, EP , of "observables" localized inside the forward light cone of a point P (themomentary position of an observer) on a time-like curve in space-time, (the obsever’s worldline). If the theory describes a massless photon and if ω is a state normal to the vacuumthen the von Neumann algebras EωP are all isomorphic to the hyperfinite factor of type III1,as discussed in [31]. Hence the algebras EωP do not have any pure states, and the principle ofLoss of Information (LoI) is a fundamental feature of the theory.

5.2 Illustration of decoherence and information loss: Study ofan exactly solvable model

To avoid confusion between annihilation and creation operators on Fock space (also standardlydenoted by a,a∗), and observables on a system S, we denote observables on S with capitalletters A, B,... in this paragraph.

We do not claim any originality concerning the model studied in this section; see e.g. [25]or [102] for similar models. However, the process of information loss was not discussed inthese works, and we feel that it is important to provide some explicit model to illustrate thisconcept. Even if the calculations are fairly easy and standards, we choose to give detailedproofs to render this section self-content.

5.2.1 The model

We consider a finite dimensional system, P , interacting with a quantized bosonic field, E. Purestates of the composed system S = P ∨E are unit rays in the Hilbert space HS = HP ⊗HE ,


where dim(HP ) = n0 ∈ N and HE = F+(L2(R3)) is the bosonic Fock space over L2(R3). Weintroduce the essentially self-adjoint operator

Φ(f) := a(f) + a∗(f) (5.2.1)

defined on the core DS (see the section devoted to notations at the beginning of this thesis),for all f ∈ L2(R3). We identify Φ(f) with its self-adjoint extension. Easy calculations (seee.g. [27]) show that

[Φ(f),Φ(g)]ψ = 2i=(〈f |g〉)ψ, ∀ψ ∈ D(N), (5.2.2)

eiΦ(f)Φ(g)e−iΦ(f)ψ = Φ(g)ψ − 2=(〈f |g〉)ψ, ∀ψ ∈ D(Φ(g)) (5.2.3)

eiΦ(f)eiΦ(g) = e−i=(〈f |g〉)ei(Φ(f)+Φ(g)), (5.2.4)

for all f, g ∈ L2(R3). These formulae are used in our proofs.We want to measure an observable

A = A∗ =n∑j=1

αjΠj ∈ B(HP ), (5.2.5)

where Πjnj=1 is a family of mutually orthogonal projections on HP such that

n∑j=1

Πj = 1HP .

One designs an experiment where the dynamics of the coupled system S is the one-parametergroup of unitary transformations on HS generated by the self-adjoint operator

HS := HP ⊗ 1HE + 1HP ⊗HE +HP,E , (5.2.6)

where the system, field, and interaction Hamiltonians are given by

HP :=n∑j=1

ejΠj , (5.2.7)

HE :=

∫R3w(k)a∗(k)a(k) d3k, (5.2.8)

HP,E :=n∑j=1

Πj ⊗ Φ(fj), (5.2.9)

respectively, and where

Φ(fj) :=

∫R3d3kÄfj(k)a∗(k) + f j(k)a(k)

ä.

The functions fj are in L2(R3), for all j = 1, ..., n, and a(k), a∗(k) are the annihilation andcreation operators in Fock space. The one-particle dispersion relation of the field, w(k), isequal to |k| for all k ∈ R3. To ensure that HS is self-adjoint on D(HP +HE), we impose thatfj/√w ∈ L2(R3) for all j = 1, ..., n. As already discussed in previous sections, self-adjointness

of HS on D(HP +HE) is a direct application of Kato-Rellich theorem. The dynamics cannotcause any transition between states that belong to two distinct subspaces (Πj ⊗ 1HE )HS


and (Πl ⊗ 1HE )HS because HS commutes with the projections Πj ⊗ 1HE , j = 1, ..., n. Thisassumption is idealistic and a more general model where the commutator [HS , A ⊗ 1HE ] isnon-zero is presented in Paragraph 5.2.5. We rewrite

HS =n∑j=1

Πj ⊗Hj , (5.2.10)

with

Hj = ej +

∫R3d3kÄfj(k)a∗(k) + f j(k)a(k)

ä+

∫R3w(k)a∗(k)a(k) d3k. (5.2.11)

The one-parameter group

U(t) =n∑j=1

Πj ⊗ e−itHj (5.2.12)

can be explicitly calculated using Bogoliubov transformations.

5.2.2 Dynamics of observables in B(HP )

Lemma 5.2.2.1. Let B ∈ B(HP ). Then

eitHS (B ⊗ 1HE )e−itHS =n∑

j,l=1

ei(sj(t)−sl(t)+ujl(t))ΠjBΠl ⊗Wjl(t), (5.2.13)

where

sj(t) :=

∫R3d3k|fj(k)|2t|k|

(sin(|k|t)|k|t

− 1)

+ tej , (5.2.14)

ujl(t) := 2

∫R3d3k=(f j(k)fl(k))

|k|2(1− cos(|k|t)

), (5.2.15)

Wjl(t) := eiΦ(i(fl−fj) eiwt−1w

), (5.2.16)

for all t ∈ R.

Proof. The proof of Lemma 5.2.3.1 is fairly standard. A similar argument (but not usingBogoliubov transformations and treating a different model) can be found, for instance, in [25].The idea is to “diagonalize" the Hamiltonians Hj by making a Bogoliubov transformationa(k) → a(k) + fj(k)/|k| for all j = 1, ..., n. Let j ∈ 1, ..., n. We assume that fj/w, fj/

√w

and fj are in L2(R3), where fj is the form factor of index j in the interaction HamiltonianHP,E , and where w(k) = |k| is the one-particle dispersion relation of the field. We will lift therestriction that fj/w ∈ L2(R3) at the end of the proof.

We introduce the unitary operator

Uj := eiΦ(ifj/w) (5.2.17)

onHE . Using the canonical commutation relations [a(k), a∗(k′)] = δ(k−k′), [a](k), a](k′)] = 0,we obtain that

UjHEU∗j = Hj + 〈fj/w|fj〉 − ej (5.2.18)


on the domain of HE . To prove (5.2.18) rigorously, just remark that HE is C1(Φ(ifj/w)).Indeed, we observe that Ffin(D(|~k|)) is a common core for HE and Φ(ifj/w), and that (HE +

1)−1 preserves Ffin(D(|~k|)). Recall that we use the notation

Ffin(V ) = Φ = (Φ(n))n∈N0 ∈ HE , Φ(n) ∈ ⊗nsV for all n, Φ(n) = 0 for all but finitely many n′s.

The commutator îHE ,Φ(ifj/w)

ó= iΦ(fj) (5.2.19)

is HE-relatively bounded and maps D(HE) to HE . Therefore, HE is C1(Φ(ifj/w)) (see [4,Theorem 6.2.10]), and using [73, Lemma 2], eisΦ(ifj/w)D(HE) ⊂ D(HE) for all s ∈ R. Wededuce that for any ϕ,ψ ∈ D(HE), the function

s 7→ 〈e−isΦ(ifj/w)ϕ|HEe−isΦ(ifj/w)ψ〉

is of class C1. Integrating its derivative from s = 0 to s = 1, it is straightforward to see thatEq. (5.2.18) holds true in the sense of the quadratic forms on D(HE)×D(HE). Eq. (5.2.18) isthen an equality between operators on D(HE) because both sides of (5.2.18) are well-definedas operators on D(HE).Carrying a few straightforward calculations based on (5.2.2)-(5.2.4), we get that

e−itHj = Uje−itHEU∗j e

it(〈fj/w|fj〉−ej) = eit(〈fj/w|fj〉−ej) e−itHE (Uj(t)U∗j ), (5.2.20)

whereUj(t) := eitHEUje

−itHE = eiΦ(ifjeiwt/w),

andUj(t)U

∗j = eiΦ(ifj(e

iwt−1)/w)ei=(〈fjeiwt/w|fj/w〉). (5.2.21)

Therefore,e−itHj = e−isj(t) e−itHEeiΦ(ifj(e

iwt−1)/w). (5.2.22)

We remark that both sides of (5.2.22) are well-defined for any element fj ∈ L2(R3) withfj/√w ∈ L2(R3) (but so far we have established this equation with the additional assumption

that fj/w ∈ L2(R3)). Eq. (5.2.22) holds actually for all fj ∈ L2(R3) with fj/√w ∈ L2(R3).

Indeed, let us introduce the sequence of functions (fj;n)n∈N, where fj;n(k) := χ(|k| > 1/n)fj(k)for all k ∈ R3. If we denote by Hj;n the new Hamiltonian obtained from Hj by replacing Φ(fj)with Φ(fj;n), we remark that s 7→ e−i(t−s)Hj;ne−isHj is strongly differentiable on D(HE).Carrying out straightforward calculations, we get that

‖(e−itHj − e−itHj;n)ψ‖ ≤∫ t

0‖Φ(fj − fj;n)e−isHjψ‖ds (5.2.23)

for all ψ in the domain of HE . As (fj − fj;n)/√w ∈ L2(R3), fj − fj;n is H1/2

E -bounded, and

‖(e−itHj − e−itHj;n)ψ‖ ≤ C(∥∥∥χ(w ≤ 1/n)fj√

w

∥∥∥+ ‖χ(w ≤ 1/n)fj‖) ∫ t

0‖(HE +1)1/2e−isHjψ‖ds.

The right side converges to zero as n tends to infinity. We therefore deduce that (5.2.22) isvalid for all fj with fj ∈ L2(R3) and fj/

√w ∈ L2(R3), using that eiΦ(f) is strongly continuous

in f (see e.g. [27]).


Next we use (5.2.10) and (5.2.12), and we deduce that

eitHS (B ⊗ 1HE )e−itHS =n∑

j,l=1

ΠjBΠl ⊗ eitHje−itHl .

for all B ∈ B(HP ). Using finally (5.2.14)-(5.2.16), we obtain that

eitHje−itHl = ei(sj(t)−sl(t))e−iΦ(ifj(eiwt−1)/w)eiΦ(ifl(e

iwt−1)/w),

= ei(sj(t)−sl(t)+ujl(t))eiΦ(i(fl−fj) eiwt−1w

),

for all t ∈ R.

5.2.3 Decoherence if the interaction is infrared singular

Lemma 5.2.3.1. Let α ∈ (−1,−1/2]. We assume that

|fj(k)− fl(k)| ∼k→0

cjl|k|α, (5.2.24)

where cjl > 0 for all j, l ∈ 1, ..., n with j 6= l. Let ρS be the initial state of S at time t = 0.Then

limt→+∞

Tr(ρSeitHS (B ⊗ 1HE )e−itHS ) =n∑j=1

Tr(ρPΠjBΠj), (5.2.25)

for all B ∈ B(HP ), where ρP = TrE(ρS) is the reduced density matrix of ρ to the subsystemP .

Remark 5.2.3.1. Hypothesis (5.2.24) concerns the infrared behavior of the interaction Hamil-tonian HP,E. In particular, if α ∈ (−1,−1/2], and if fj ∼ cj |k|α for all j = 1, ..., n, thecondition that cj 6= cl for all j, l ∈ 1, ..., n with j 6= l implies (5.2.24). The behavior of theform factors fj in the infrared regime is responsible for the so-called infrared catastrophe whenα ≤ −1/2: the interaction between P and the field creates an infinite number of field modesof very low energy; see e.g. [25].

Remark 5.2.3.2. (5.2.25) shows that the observable A =∑nj=1 αjΠj becomes empirical as

t→∞ for all initial normal states ρ. Choosing a specific initial state, the criterion stated in(5.2.24) can be soften and one can only require (5.2.24) to hold for the indices j, l such thatΠjρSΠl 6= 0.

Proof. Using (5.2.13), we deduce that

Tr(eitHS (B ⊗ 1HE )e−itHSρS) =n∑

j,l=1

TrS(ΠjBΠlTrE(Wjl(t)ρS)).

Full decoherence takes place if the operators Wjl(t) defined in (5.2.16) converge weakly tozero for all l, j ∈ 1, ..., n with j 6= l. Our proof is similar to [25]. Let m, p ∈ N andft, g1, ..., gm, g

′1, ..., g

′p ∈ L2(R3). It follows from a direct application of the CCR commutation

relations thateiΦ(ft)a∗(gk)e

−iΦ(ft) = a∗(gk) + i〈ft|gk〉


on D(Φ(gk)) ∩ D(Φ(igk)), for all k ∈ 1, ...,m. Therefore,

〈eiΦ(ft)a∗(g1)...a∗(gk)Ω|a∗(g′1)...a∗(g′p)Ω〉

= 〈eiΦ(ft)Ω|(a(gm)− i〈ft|gm〉)...(a(g1)− i〈ft|g1〉)a∗(g′1)...a∗(g′p)Ω〉.(5.2.26)

Next we remark that

〈eiΦ(ft)Ω|a∗(gj1)...a∗(gjk)Ω〉 = (−i)ke−‖ft‖2/2k∏l=1

〈ft|gjl〉,

where we have used that 〈eiΦ(ft)Ω|Ω〉 = e−‖ft‖2/2. Introducing x = max(maxmi=1 ‖gi‖,maxpl=1 ‖g′l‖),

we carry out some easy combinatorics to bound the right side of (5.2.26). This yields

|〈eiΦ(ft)a∗(g1)...a∗(gm)Ω|a∗(g′1)...a∗(g′p)Ω〉|

≤ 2max(m,p) min(m!, p!)xm+pe−‖ft‖2/2(1 + ‖ft‖2)min(m,p)‖ft‖max(m,p)−min(m,p).

(5.2.27)

Let l, j ∈ 1, ..., n, with j 6= l. We choose

ft = i(fl − fj)eiwt − 1

w. (5.2.28)

If we show that ‖ft‖ → ∞ as t tends to infinity, then we can conclude with the estimate(5.2.27) that the operator Wjl(t) defined in (5.2.16) converges weakly to zero as t→∞, sincethe linear span of the set a∗(g1)...a∗(gm)Ω | gi ∈ L2(R3),m ∈ N is dense in F+(L2(R3)).Let t > 0. We have that

‖ft‖2 = 2

∫R3

|fl(k)− fj(k)|2

|k|2(1− cos(|k|t)

)d3k

We split the integral in two parts and we single out the ball B1(0) ⊂ R3 of radius 1 centered atthe origin. The integral on R3 \B1(0) is finite and can be bounded uniformly in the parametert because fj ∈ L2(R3). To estimate the integral over B1(0), we carry out a change of variablesand we use the infrared behavior of |fl(k)− fj(k)|; see (5.2.24). This yields∫ 1

0d|k|

∫S2dΘ |fl(k)− fj(k)|2

(1− cos(|k|t)

)=

1

t1+2α

∫ t

0d|k|

∫S2dΘ |k|2αρjl

(kt

)(1− cos(|k|)

),

(5.2.29)

where ρjl(k) is a bounded positive and measurable function that converges to c2jl > 0 as k

tends to 0. If α ∈ (−1,−1/2), the factor t−1−2α diverges, and the integral in the second lineof Eq. (5.2.29) converges to some non-zero limit as t tends to infinity (by Lebesgue dominatedconvergence). Therefore,

‖ft‖2 −→t→+∞

+∞.

If α = −1/2, the integral ∫ t

0d|k|1− cos(|k|)

|k|d|k| (5.2.30)

behaves as ln(t) for large positive values of t, and we conclude that ‖ft‖2 diverges logarithmi-cally fast at infinity.


5.2.4 Information loss if S is non-autonomous

We now assume that the system S is non-autonomous, and that the interaction between P andE can be turned on and off. In that case, time evolution is generated by the time dependentHamiltonian

HS(t) := HP ⊗ 1HE + 1HP ⊗HE + λ(t)HP,E , (5.2.31)

where λ(t) depends on t, and HP , HE and HP,E are defined as in (5.2.7)-(5.2.9). The functiont 7→ λ(t) is piecewise constant and is defined by

λ(t) :=

®0 if t ∈ (−∞, t0) ∪ [T,+∞),λ0 if t ∈ [t0, T ),

(5.2.32)

for all t ∈ R, where λ0 ∈ R, and where t0, T ∈ R with T > t0. Using (5.2.13), one can calculatethe algebras E≥t. We denote by U(s′, s) the family of unitary operators generated by the timedependent Hamiltonian H(t).

To simplify matters, we assume that the state ω of the system S is specified at the fiducialtime t = T (and not at t = 0). We assume that OS is the set of all self-adjoint elements ofthe form B ⊗ 1HE , where B = B∗ ∈ B(HP ). We denote by B(HP )⊗ 1HE the C∗-algebra

B(HP )⊗ 1HE := B ⊗ 1HE | B ∈ B(HP ). (5.2.33)

The choice of our fiducial time implies that the algebra E≥t is equal to B(HP ) ⊗ 1HE for alltimes t ≥ T , because

U∗(t, T )(B ⊗ 1HE )U(t, T ) = ei(t−T )HSBe−i(t−T )HS ⊗ 1HE .

Let t ∈ [t0, T ]. We introduce the set

Ξt :=¶ n∑j,l=1

ΠjBΠl ⊗ Vjl(s) | B = B∗ ∈ B(HP ), s ∈ [t, T ]©,

whereVjl(s) := eiλ0Φ(i(fl−fj) e

iw(s−T )−1w

). (5.2.34)

We remind the reader that if the set Ξ is contained in B(HP ⊗HE), we denote by 〈Ξ〉 thesmallest C∗-subalgebra of B(HP ⊗HE) containing Ξ. Lemma (5.2.4.1) is a direct applicationof (5.2.13). It shows that information is lost in the sense of (5.1.7).

Lemma 5.2.4.1.

• Let t ≤ t0. ThenE≥t = 〈Ξt0 ∨ (B(HP )⊗ 1HE )〉. (5.2.35)

• Let t ∈ (t0, T ). ThenE≥t = 〈Ξt ∨ (B(HP )⊗ 1HE )〉. (5.2.36)

• Let t ≥ T . ThenE≥t = B(HP )⊗ 1HE . (5.2.37)


Lemma 5.2.4.1 shows that information is lost into the field as time goes on, since E≥T ( E≥tfor all t < T .

Let now ω be a normal state of the system S, specified at the time t0 (t0 < T is the time atwhich the coupling between P and E is turned on). We denote by ρω its corresponding densitymatrix. For any time t ≥ T , the restriction of ω to E≥t, ωt, is a density matrix, denoted byρω≥t. As B(HP ) ⊗ 1HE is a type In-von Neumann algebra, we can apply Lemma 5.1.3.2, andwe deduce that


pλ(t)Πλ(t)⊗ 1HE , (5.2.38)

for all t ≥ T , where Λ is a finite set. We have that

ω≥T (B ⊗ 1HE ) = Tr(ρωU∗(T, t0)(B ⊗ 1HE )U(T, t0)), (5.2.39)

and, therefore, ∣∣∣ω≥T (B ⊗ 1HE )−n∑i=1

TrP (ρωPΠiBΠi)∣∣∣ = r(T − t0)‖B‖, (5.2.40)

for all B⊗1HE ∈ E≥T , where the rest term r(T − t0) ≥ 0 is very close to zero if T − t0 is largeand if the hypotheses of Lemma 5.2.3.1 are satisfied.

Let ε > 0. We choose T − t0 such that r(T − t0) < ε. Eq. (5.2.40) and (5.2.38) imply that

∣∣∣TrP Ä( n∑i=1

ΠiρωPΠi −

∑λ∈Λ

pλ(T )Πλ(T ))Bä∣∣∣ < ε‖B‖, (5.2.41)

for all B ∈ B(HP ). Choosing B = B∗ =∑ni=1 Πiρ

ωPΠi −

∑λ∈Λ pλ(T )Πλ(T ) in (5.2.40), we

deduce that the Hilbert-Schmidt norm of the operator∑ni=1 Πiρ

ωPΠi−

∑λ∈Λ pλ(T )Πλ(T ) is of

order ε1/2. This shows that the norm difference between∑ni=1 Πiρ

ωPΠi and

∑λ∈Λ pλ(T )Πλ(T )

can be made arbitrarily small by an appropriate tuning of T − t0. By a perturbative argument(using Riesz projection and the second resolvent Formula), we deduce that each projectionΠi can be made arbitrarily close in norm to a sum of projections of the form

∑λ∈Λi Πλ(T ),

where ∪iΛi = Λ. Therefore, A can be made arbitrarily close to Aω defined in (5.1.28), and Ais empirical as soon as T − t0 is sufficiently large.

5.2.5 Some refinements to the model treated in Sections 5.2.1-5.2.3.

To complete our study of decoherence, we assume in this paragraph that HP,E does notcommute with HP . More precisely, we replace HP,E , defined in (5.2.9), by

HP,E :=n∑j=1

Πj ⊗ Φ(fj) +G⊗ Φ(f), (5.2.42)

and we replace HS by HS = HP + HE + HP,E (remark that the Hamiltonian is time inde-pendent). We suppose that [Πj , G] 6= 0 for all j = 1, ..., n, and that f and f/

√w belong to

L2(R3). As [Πj , G] 6= 0, the eigenvalues ej , for j > 1, dissolve into the continuous spectrum,and there is a "competition" between decoherence and the instability of the excited states. Weestimate the effect of the interaction Hamiltonian (5.2.42) on a finite time scale by a simple


"Cook" argument. To simplify Notations, we assume that the total system S is initially in apure state of the form

Ψ =1√N

N∑m=1

ϕm ⊗ ψm,

where 〈ψm|ψp〉 = 〈ϕm|ϕp〉 = δmp and ψm ∈ D(HE), for all m, p = 1, ..., N .

Corollary 5.2.5.1. Let α ∈ (−1,−1/2). We assume that

|fj(k)− fl(k)| ∼k→0

cjl|k|α, (5.2.43)

where cjl > 0 for all j, l ∈ 1, ..., n with j 6= l. There is a positive constant C > 0 (see(5.2.29)) such that

∣∣∣〈e−itHSΨ|(B ⊗ 1HE )e−itHSΨ〉 −n∑j=1

N∑m=1

1

N〈ϕm|ΠjBΠjϕm〉

∣∣∣ = O(t‖G‖) +O(e−Ct−1−2α

),

(5.2.44)for all B ∈ B(HP ) and for all t ≥ 0.

Proof. The proof relies on a “Cook argument". As D(HS) = D(HS) = D(HP + HE), thefunction s 7→ e−isHSe−i(t−s)HS is strongly differentiable on the domain of HS , and we write(the equalities between operators being meant on the domain of HS)

e−itHS = e−itHS +

∫ t

0

d

ds

(e−isHSe−i(t−s)HS

)ds

= e−itHS − i∫ t

0e−isHS (G⊗ Φ(f))e−i(t−s)HSds.

Next we use the expression for e−itHj derived in (5.2.20). Using the CCR relations, we getthat

Φ(f)e−itHj = e−isj(t) Φ(f)e−itHEeiΦ(ifj(eiwt−1)/w)

= e−isj(t) e−itHEΦ(feiwt)eiΦ(ifj(eiwt−1)/w)

= e−isj(t) e−itHEeiΦ(ifj(eiwt−1)/w)

ÄΦ(feiwt) + 2=〈ifj(eiwt − 1)/w|feiwt〉

ä.

Therefore, for any ψ ∈ D(HE),

‖Φ(f)e−i(t−s)Hjψ‖ ≤ ‖Φ(feiw(t−s))ψ‖+ 2‖ψ‖ |〈fj(eiw(t−s) − 1)/w|feiw(t−s)〉|.

The operator Φ(feiw(t−s)) is H1/2E -bounded, and a direct application of the Cauchy-Schwarz

inequality for the second term leads to the upper bound

‖Φ(f)e−i(t−s)Hjψ‖ ≤ 4‖ψ‖‖fj/√w‖‖f/

√w‖+ α‖HEψ‖+ β(α)‖ψ‖,

for all α > 0 and for some β(α) > 0. We deduce that

‖(e−itHS − e−itHS )Ψ‖ = O(t‖G‖).


5.3 When does an observation or measurement of a physicalquantity take place?

Let a = a∗ ∈ OS represent a potential property of a quantum-mechanical system S, whichis assumed to be prepared in a state ω on the algebra ES . We propose to analyze whetherand when a corresponds to an empirical property of S, in the sense that, given the timeevolution τt,st,s∈R of S and the state ω, a is measurable (i.e., the value of a can be measuredor observed) at some finite time. Let t0 be the fiducial time at which ω is specified, andlet a(t) = τt,t0(a) =

∑i αiΠi(t). Definition 5.1.3.2 and the discussion thereafter suggest to

introducet 7→ ∆ω

t a(t) := infω((a(t)− a)2) | a = a∗ ∈ Zω≥t. (5.3.1)

Let δ be some non-negative number below the resolution threshold of the equipment used tomonitor S. Let t∗ be defined as the smallest time such that

∆ωt∗a(t∗) ≤ δ. (5.3.2)

Then it is reasonable to say that a is observed/measured – put differently, a becomes an em-pirical property of S within an uncertainty of size δ – at a time ? t∗. Next, we analyze repeatedobservations/measurements. It suffices to consider only two subsequent measurements. Leta = a∗ ∈ OS represent a potential property of S, and let δ ≥ 0 be a measure for the resolutionof the equipment E used to monitor S in a measurement of a.

Definition 5.3.0.1. For a = a∗ ∈ OS, δ ≥ 0, and a time t∗ > −∞, we define a subset ofstates on AS (or on ES ⊂ AS) by

S(a, δ, t∗) := ω ∈ SS | inft≥t∗

∆ωt a(t) < δ, (5.3.3)

where δ is below any reasonable experimental threshold.

Apparently, S(a, δ, t∗) is the set of states of S with the property that, given the timeevolution τt,st,s∈R, the operator a corresponds to an empirical property of S, within anuncertainty of size δ, that is measurable at some time after t∗.

Next, we consider two potential properties of S represented by two self-adjoint operators,a1 and a2, and we suppose that, first, a1 and, afterwards, a2, are measured. For simplicity wesuppose that the spectra of a1 and a2 consist of finitely many eigenvalues α(i)

j , j = 1, ..., ki <∞, i = 1, 2. We assume that the state, ω, of S before the measurement of a1, belongs toS(a1, δ1, t1∗), for a sufficiently small number δ1 (below a threshold of resolution), and weassume that the Hypotheses of Lemma 5.1.3.2 are fulfilled.

A successful measurement of a1 around some time t1 ≥ t1∗ results in the assignment of avalue α(1)

j ≈ a1,λ(t1), λ ∈ Λ(j)ω , to the physical quantity represented by a1, where

Λ(j)ω := λ ∈ Λω | |a1,λ(t1)− α(1)

j | < δ1. (5.3.4)

(For consistency, we assume that minj 6=l|α(1)j −α

(1)l | > 2δ1.) The probability of this measurement

outcome is given by

P(1)j (t1) =

∑λ∈Λ

(j)ω

ω(Πωλ(t1)) =

∑λ∈Λ

(j)ω

pωλ(t1)nωλ = ω(Π(1)j (t1)) +O(δ1), (5.3.5)


where pλ(t1) ≡ pωλ(t1), nλ ≡ nωλ = dim(Πωλ(t1)), and Πλ(t1) ≡ Πω

λ(t1) are as defined in Eqs.(5.1.32) and (5.1.33), (the superscript "ω" is supposed to highlight the dependence on the stateω), and Π

(1)j (t1) is the eigenprojection of the operator a1(t1) corresponding to the eigenvalue

α(1)j . If P (1)

j (t1) is very small one can ignore the possibility that, for a system S prepared in

the state ω, an observation/measurement of a1 will yield a value ≈ α(1)j . Let ωj denote the

state

ωj(b) =

∑λ∈Λ

(j)ωω(Πω

λ(t1)bΠωλ(t1))

P(1)j (t1)

=ω(Π

(1)j (t1)bΠ

(1)j (t1))

ω(Π(1)j (t1))

+O(δ1), (5.3.6)

for an arbitrary operator b ∈ Eω≥t, with t ≥ t1; (recall that Eω≥t ⊆ Eω≥t1 , for t ≥ t1.) Let ussuppose that, for all j ∈ 1, ..., k1 for which P (1)

j (t1) > δ2 > 0,

ωj ∈ S(a2, δ2, t(j)2∗ ), (5.3.7)

for some time t(j)2∗ > t1. If δ2 is chosen small enough one may expect to be able to successfullymeasure the quantity represented by a2 at a time t2 ≥ t(j)2∗ , assuming that, at a time t1 < t

(j)2∗ ,

a1 was found to have a value ≈ α(1)j . The joint probability to find a value ≈ α

(1)j in a

measurement of a1 around some time t1 and, in a subsequent measurement around a timet2 > t1, a value ≈ α(2)

l of the quantity represented by a2, (with l ∈ 1, ..., k2), is given by

ProbωΠ(1)j (t1),Π

(2)l (t2) = P

(1)j (t1)

∑λ∈Λ

(l)ωj

ωj(Πωjλ (t2))

= ω(Π(1)j (t1)Π

(2)l (t2)Π

(1)j (t1)) +O(δ1 ∨ δ2),

(5.3.8)

where Λ(l)ωj = λ ∈ Λωj | |a2,λ(t2)− α(2)

l | < δ2, and δ1 ∨ δ2 = maxδ1, δ2.The definitions of centralizers, Cω≥t1 , etc., and of the variance ∆ω

t a(t) readily imply that

k1∑j=1

ω(Π(1)j (t1)Π

(2)l (t2)bΠ

(2)l (t2)Π

(1)j (t1)) = ω(Π

(2)l (t2)bΠ

(2)l (t2)) +O(δ1), (5.3.9)

and if ωj ∈ S(a2, δ2, t(j)2∗ ) then

k2∑l=1

ω(Π(1)j (t1)Π

(2)l (t2)bΠ

(2)l (t2)Π

(1)j (t1)) = ω(Π

(1)j (t1)bΠ

(1)j (t1)) +O(δ1 ∨ δ2), (5.3.10)

for an arbitrary operator b ∈ Eω≥t, with t > maxj t(j)2∗ . It is clear how to extend our discussion

to an arbitrary chronological (time-ordered) sequence of measurements of quantities a1, ..., an,(ai ∈ OS ,∀i). Moreover, the mathematical relationship between Eqs. (5.3.9) and (5.3.10), onone side, and δ−consistent families of histories – see (5.1.9) and (5.1.10) – on the other side,is easy to unravel.

Remark 5.3.0.1 (POVM’s). It may happen that, given that a quantity represented by anoperator a1 has been observed/measured, the quantity represented by the operator a2 can bemeasured, subsequently, only for certain, but not all, outcomes of the measurement of a1. More


precisely, it may happen that, for some eigenvalues α(1)j , j ∈ G, of a1, ωj ∈ S(a2, δ2, t

(j)2∗ ),

while, for i ∈ B := 1, ..., k1 \G,

ωi /∈ S(a2, δ2, t2∗), (5.3.11)

for any t2∗ < ∞; (δ1 and δ2 being chosen appropriately, depending on the resolution of thecorresponding measurements, as discussed above).

If B 6= ∅ then one must take the position that the observations of a1 and a2 representone single measurement, which must be described using "positive operator-valued measures"(POVM’s).

X = Xjl, Xi | j ∈ G, l = 1, ..., k2, i ∈ B (5.3.12)

where, for j ∈ G,

Xjl =∑

λ1∈Λ(j)ω

∑λ2∈Λ

(l)ωj

Πωjλ2

(t(j)2 )Πω

λ1(t1) ≈ Π

(2)l (t

(j)2 )Π

(1)j (t1), (5.3.13)

(up to a small perturbation of O(δ1 ∨ δ2)), while, for i ∈ B,

Xi =∑

λ1∈Λ(i)ω

Πωλ1

(t1) ≈ Π(1)i (t1), (5.3.14)

where t1 and t(j)2 are the times of measurement of a1 and a2, respectively. Then

∑j∈G

k2∑l=1

X∗jlXjl +∑i∈B

X∗iXi = 1. (5.3.15)

Simple examples showing why one needs to introduce POVM’s are encountered in the analysisof repeated Stern-Gerlach measurements of atomic spins (followed by detectors sensitive to thearrival of the atoms).

Remark 5.3.0.2. We have made the simplifying assumptions that Eω≥t was type I. It is,however, not very hard to develop our ideas in full generality. For this purpose, we mustreturn to formula (5.1.30): The space Λ = Λω appearing in (5.1.30) is the spectrum of thecenter, Zω≥t, of the centralizer, Cω≥t, of the state ω, viewed as a state on the algebra Eω≥t.The theory of "conditional expectations" [122] enables us (under fairly general hypotheses) toconstruct a conditional expectation ε≥t : Eω≥t → Zω≥t, which permits us to associate with everyoperator a ∈ Eω≥t an operator aω ∈ Zω≥t, as for the type I case discuss above. The map a 7→ aω

is linear, and (aω)ω = aω. (In the special case where Eqs. (5.1.31) hold it is given by formula(5.1.35).)

5.4 Non-demolition measurements and quantum trajectories

We review in Section 5.4.1 some recent results concerning the theory of "indirect (non-demolition) measurements"; see e.g. [2], [22], [101]. A new result concerning the purificationon the spectrum of an observable is proven in Section 5.4.2. To simplify matters, we work inthe realm of Hilbert spaces, and we denote observables by capital letters.


5.4.1 Measuring A on P with an infinite series of probes

The model

We consider a physical system P (e.g. the quantized electromagnetic field in a cavity). Wewish to measure a physical quantity represented by an operator A = A∗ ∈ B(HP ) (e.g. abounded function of the photon number inside the cavity) with the help of "non-demolitionmeasurements". For this purpose, we bring P into contact with a sequence E1, E2, E3,...,of identical "probes" (e.g., excited atoms going through the cavity); the interaction betweenEk and P takes place in the time interval [k − 1, k] and it is turned off for all other times.Actually, after some direct measurement of a property Xk = X∗k ∈ B(HEk) at a time laterthan k – as described in Section 5.3 – the probe Ek "gets lost for ever", in the sense that nofurther information about Ek can be retrieved anymore.

Let ρ denote the initial state of P and ψ(k) := ψ the initial state of the probe Ek, (the samefor all k). For simplicity, we assume that the spectrum of the operator A representing thephysical property of P to be measured is finite pure-point spectrum. We denote the spectralprojection corresponding to an eigenvalue α of A by Πα = Π∗α. Then

ΠαΠβ = δαβΠα,∑

α∈σ(A)

Πα = 1HP .

Next, we specify the time-evolution of the composed system P ∨ E1 ∨ E2 ∨ ... : Up to timek = 1, 2, 3, ...,, the time evolution of Ej is assumed to be trivial, for all j > k. For thesubsystem P ∨ E1 ∨ .... ∨ Ek it is specified as follows : Let Dα,α′ be an arbitrary operatorin B(HP ) mapping Ran Πα′ to Ran Πα, with ΠβDα,α′Πβ′ = δαβδα′β′Dα,α′ . Let Bj be anoperator in B(HEj ) = B(Hpro), j ≤ k. Then the time-evolution of Dα,α′ ⊗B1 ⊗ ...⊗Bk fromtime 0 to time k in the Heisenberg picture is given by

τk,0(Dα,α′ ⊗B1 ⊗ ...⊗Bk) := Dα,α′ ⊗ UαB1U∗α′ ⊗ ...⊗ UαBkU∗α′ ,

where Uα is a unitary operator in B(Hpro), for all α ∈ σ(A). Defining

U(i, i− 1) :=∑

α∈σ(A)

Πα ⊗ 1Hpro ⊗ ...⊗ Uα ⊗ 1Hpro ⊗ ...,

with Uα inserted in the (i+ 1)st factor of the tensor product, we have that

τk,0(Dα,α′ ⊗B1 ⊗ ...⊗Bj) =1∏i=k

U(i, i− 1)(Dα,α′ ⊗B1 ⊗ ...⊗Bj)k∏i=1

U(i, i− 1)∗ =

τk,0(Dα,α′ ⊗B1 ⊗ ...⊗Bk)⊗Bk+1 ⊗ ...⊗Bj ,(5.4.1)

for arbitrary j ≥ k. This is a typical example of time-evolution in a non-demolition measure-ment. We remark that [A,U(i, i − 1)] = 0 for all i = 1, 2, ....: time evolution cannot causetransitions between the subspaces Ran Πα and Ran Πα′ , α 6= α′.

Decoherence on the spectrum of A

Let Ψ := ρ⊗ ψ ⊗ ψ ⊗ ... denote the initial state of the composed system, P ∨E1 ∨E2 ∨ .... Ifwe set

B1 = B2 = ... = Bk0 = 1Hpro ,


for some k0 <∞ then

ΨÄτk,0(Dα,α′ ⊗ 1Hpro ⊗ ...⊗ 1Hpro ⊗Bk0+1 ⊗ ...⊗Bk0+l)

ä=

ρ(Dα,α′)ψ(UαU∗α′)

k0

k∏i=k0+1

ψ(UαBiU∗α′)

k0+l∏i=k+1

ψ(Bi),(5.4.2)

for k0 ≤ k ≤ k0 + l. Because Uα is unitary, for all α ∈ σ(A),

|Ψ(UαU∗α′)| ≤ 1, for all α, α′,

by the Cauchy-Schwarz inequality. We assume that

|Ψ(UαU∗α′)| ≤ κ < 1, for α 6= α′. (5.4.3)

Then|ΨÄτk,0(Dα,α′ ⊗ 1Hpro ⊗ ...⊗ 1Hpro ⊗Bk0+1 ⊗ ...⊗Bk0+l)

ä| ≤ κk0 , (5.4.4)

which, by (5.4.3), tends to 0 exponentially fast, as k0 →∞, for arbitraryDα,α′ , Bk0+1,...,Bk0+l,with ‖Dα,α′‖, ‖Bk0+1‖,..., ‖Bk0+l‖ bounded by 1. This is "decoherence" over the spectrum ofthe operator A representing the quantity to be measured:

Ψ|E≥k0−→

∑α∈σ(A)

Ψ(Πα(·)Πα)|E≥k0, (5.4.5)

as k0 →∞, where E≥k0 is the algebra introduced in Definition 5.1.1.1.

Non-demolition measurement of A

We assume that a direct measurement of a physical quantity represented by an operatorX = X∗ ∈ B(Hpro) is carried out on every probe Ek ' E, after it has interacted with P . Weassume that the spectrum of X is pure-point, with eigenvalues denoted by ξ and correspondingspectral projections written as πξ. Then πξ = π∗ξ and

πξπξ′ = δξξ′πξ,∑

ξ∈σ(X)

πξ = 1Hpro . (5.4.6)

The probability, µ(ξk|α), of a history

ξk

:= πξ1 , ..., πξk (5.4.7)

of possible outcomes of those direct measurements in the state Ψα defined by

Ψα := Πα ⊗ ψ ⊗ ψ ⊗ ...,

is given by

µ(ξk|α) =

k∏i=1

p(ξi|α), (5.4.8)

wherep(ξ|α) := ψ(UαπξU

∗α). (5.4.9)


Note that∑ξ∈σ(X) p(ξ|α) = 1, by (5.4.6) and the unitarity of Uα. In the following, we

identify πξ with ξ and use the notation ξk

= (ξk−1

, ξk). Without loss of generality, we assumethe the projections Πα are one-dimensional and that ψ = Tr(|ψ〉〈ψ|·). In the initial stateΨ = ρ⊗ ψ ⊗ ψ ⊗ ..., the probability of the history ξ

kis then given by

µ(ξk) =

∑α∈σ(A)

ρααµ(ξk|α). (5.4.10)

The reduced state of the system P is actualized at each iteration step; see also Section 5.3.We denote by ρ(k) the reduced state of P after k iterations. Using (5.4.1), we get that

ρ(k+1)(ξk, ξ) =

∑α,β TrHpro

Ä(1HP ⊗ πξ)[Παρ

(k)(ξk)Πβ ⊗ Uα|ψ〉〈ψ|U∗β ](1HP ⊗ πξ)

ä∑α′,β′ Tr

Ä(1HP ⊗ πξ)[Πα′ρ(k)(ξ

k)Πβ′ ⊗ Uα′ |ψ〉〈ψ|U∗β′ ](1HP ⊗ πξ)

ä=

∑α,β Παρ

(k)(ξk)Πβ 〈ψ|U∗βπξUαψ〉∑

α′∈σ(A)ρ

(k)α′α′(ξk)p(ξ|α

′).

(5.4.11)

Looking only at the diagonal components, it follows from (5.4.9) that

ρ(k+1)αα (ξ

k, ξ) =

ρ(k)αα(ξ

k) p(ξ|α)∑

α′∈σ(A)ρ

(k)α′α′(ξk) p(ξ|α

′)(5.4.12)

for any α ∈ σ(A). An easy induction shows that

ρ(k+1)αα (ξ

k, ξ) =

ραα p(ξ|α)∏ki=1 p(ξi|α)

µ(ξk, ξ)

. (5.4.13)

Equation (5.4.13) implies that (ρ(k)αα)k is a bounded martingale for all α ∈ σ(A). Indeed, if we

denote by Ek the conditional expectation given ξk, we obtain that

(Ekρ(k+1)αα )(ξ

k) :=

∑ξ∈σ(X)

ρ(k+1)αα (ξ

k, ξ)

µ(ξk, ξ)

µ(ξk)

=∑

ξ∈σ(X)

ρααp(ξ|α)∏ki=1 p(ξi|α)

µ(ξk, ξ)

µ(ξk, ξ)

µ(ξk)

= ρ(k)αα(ξ

k).

(5.4.14)

Furthermore, ρ(k)αα ∈ [0, 1] for all α and for all k. The Martingale Convergence Theorem (see

e.g., [106]) then implies thatρ(k)αα(ξ) −→

k→∞ρ(∞)αα (ξ)

almost surely, where ξ = ξ∞ = (ξ1, ξ2, ....), and ρ(k)αα(ξ) does not depend on ξk+1, ξk+2, ....

Taking the limit in (5.4.12) for every ξ∞ ∈ σ(X),

ρ(∞)αα (ξ) =

ρ(∞)αα (ξ) p(ξ∞|α)∑

α′∈σ(A)ρ

(∞)α′α′(ξ) p(ξ∞|α′)

. (5.4.15)


If for every α 6= β ∈ σ(A), there is ξ ∈ σ(X) such that p(ξ|α) 6= p(ξ|β), then (5.4.15) obviouslyimplies that

ρ(∞)αα (ξ) = δαα0 , (5.4.16)

for some α0 (depending on ξ).Thus, for almost every history ξ = ξ∞ of outcomes of "von Neumann measurements"

of the probes E1, E2, ...., the state Ψ τk,0, conditioned on ξ∞, converges on B(HP ) to aneigenstate of the operator A ∈ B(HP ) representing the physical quantity to be measured, ask → ∞. The probability (with respect to the histories ξ∞) of convergence to an eigenstatecorresponding to the eigenvalue α of A is given by ραα. Indeed, E(ρ(k)(α | ·)) = ραα for allk ∈ N. Stated differently, the range of values of the functions ρ(∞)

αα on the space of historiesconsists of 0, 1.

5.4.2 A scenario where free evolution on P alternates with many quantumnon demolition measurements

The model

We consider the same model as in Section 5.4.1, but we modify the dynamics: we allow thesystem P to evolve freely between N non-demolition measurements, where N ∈ N is large.Let ∆ > 0. The reduced dynamics of P is an alternation of free evolutions, of duration ∆,e−i∆ad(HP ), and N non-demolition measurements as described in Section 5.4.1.

1 2 3 4 5 6 7 8 9 10 11 12 time axis...

∆∆ ∆

Figure 5.1: Sequence of non-demolition measurements followed by free evolutions on P . The freeevolution takes place after N consecutive non-demolition measurements. Here N = 4.

The recursive expression of the reduced state of the system P after k+ 1-iteration is givenby (5.4.11) if k 6= Np, with p ∈ N. It is given by

ρ(k+1)(ξk, ξ) =

∑α,β TrHpro

Ä(1HP ⊗ πξ)[Πα(e−i∆ad(HP )ρ(k)(ξ

k))Πβ ⊗ Uα|ψ〉〈ψ|U∗β ](1HP ⊗ πξ)

ä∑α,β Tr

Ä(1HP ⊗ πξ)[Πα(e−i∆ad(HP )ρ(k)(ξ

k))Πβ ⊗ Uα|ψ〉〈ψ|U∗β ](1HP ⊗ πξ)

ä=

∑α,β Παe

−i∆HP ρ(k)(ξk)ei∆HPΠβ 〈ψ|U∗βπξUαψ〉∑

α′∈σ(A)〈α′|e−i∆HP ρ(k)(ξ

k)ei∆HPα′〉 p(ξ|α′)

.

(5.4.17)

if k = Np, with p ∈ N.We assume that the projections πξ = |ξ〉〈ξ| are one-dimensional (and hence that the vectors

|ξ〉 form an orthonormal basis of the Hilbert space of the probe), and that dim(HP ) < ∞.The reduced density matrix of P at step k + 1 can be rewritten into the form

ρ(k+1)(ξk, ξ) =

(C(k)ξ )∗ρ(k)(ξ

k)C

(k)ξ

Tr((C(k)ξ )∗ρ(k)(ξ

k)C

(k)ξ )

, (5.4.18)


where, for all k ∈ N,

C(k≡0(N))ξ :=

∑α∈σ(A)

〈ξ|Uαψ〉ei∆HP Πα, (5.4.19)

C(k 6≡0(N))ξ :=

∑α∈σ(A)

〈ξ|Uαψ〉 Πα, (5.4.20)

are operators on HP (here "≡ 0(N)" means "0 modulo N") . It is easy to check that∑ξ∈σ(X)

C(k)ξ (C

(k)ξ )∗ =

∑ξ∈σ(X)

(C(k)ξ )∗C

(k)ξ = 1HP ,

for all k ∈ N.As we explained in the introduction, this property is important because it implies (using

Kolmogorov’s extension theorem) that the functional

µ(ξ1, ..., ξk) := Tr((C

(k)ξk

)∗...(C(1)ξ1

)∗ ρ C(1)ξ1...C

(k)ξk

)generates a probability measure (denoted by the same symbol) on the measurable space (Ω,Σ)of all infinite sequences ξ = (ξ1, ξ2, ...), where ξi ∈ σ(X) for all i ∈ N. Σ is the σ-algebragenerated by the cylinder sets

Λx1,...,xk := ξ | ξ1 = x1, ..., ξk = xk,

k ∈ N, and x1, ..., xk ∈ σ(X); see also [101]. In particular,

µ(Λx1,...,xk) = Tr((C(k)

xk)∗...(C(1)

x1)∗ ρ C(1)

x1...C(k)

xk

).

The expectation of random variables on (Ω,Σ, µ) is denoted below (and as usual) by the symbolE. As before, to simplify matters, we assume that the projections Πα are one-dimensional.Due to mixing coming from the free evolution on P , the full sequence of diagonal terms(ρ

(k)αα)k∈N is not a martingale. Actually, we do not expect ρ(k)

αα to converge as k tends toinfinity. Under some non-degeneracy assumptions, one can show that the density matrix ρ(k)

purifies as k tends to infinity; see e.g. [101, 21]. The main idea of the present proof is to showthat Tr((ρ(k))2) converges to 1 almost surely as k tends to +∞. An alternative proof, whichalso holds for more general models, will be presented in a forthcoming work.

If for all α 6= β, there is ξ ∈ σ(X) such that p(ξ|α) 6= p(ξ|β), we prove here that thereduced state of the system P purifies on the spectrum of A for large enough values of N .

Purification on the spectrum of A for large values of N

We assume that there is δ > 0 such that

infα∈σ(A),ξ∈σ(X)

p(ξ|α) ≥ δ

and such that for all α 6= β, there is ξ ∈ σ(X) with |p(ξ|α)− p(ξ|β)| ≥ δ. The following resultholds:


Lemma 5.4.2.1. Let ε ∈ (0, δ2/4). If N is large enough (depending on ε), then there is aconstant pε(N) ∈ N such that

µ(ξ | maxα∈σ(A)

(ρ(Np-1)αα (ξ)) ≥ 1− dim(HP )ε1/4) > 1− ε, (5.4.21)

for all p ≥ pε(N).

Proof. We first use that decoherence occurs between kp := Np + 1 and k′p := N(p + 1) − 1,for all p ∈ N. Let k ∈ kp, ..., k′p. An easy recursion shows that

ρ(k)αβ (ξ

k) =

ρ(Np+1)αβ (ξ

Np+1)∏ki=Np+2〈ξi|Uαψ〉〈ψ|U∗βξi〉∑

α′∈σ(A)ρ

(Np+1)α′α′ (ξ

Np+1)∏ki=Np+2 p(ξi|α′)

. (5.4.22)

The hypothesis that δ > 0 implies that

κ := maxα 6=β

∑ξ′∈σ(X)

|〈ξ′|Uαψ〉| |〈ψ|U∗βξ′〉| < 1. (5.4.23)

Indeed, using Cauchy-Schwarz inequality,∑ξ′∈σ(X)

|〈ξ′|Uαψ〉| |〈ψ|U∗βξ′〉| ≤Ä ∑ξ′∈σ(X)

|〈ξ′|Uαψ〉|2ä1/2Ä ∑

ξ′∈σ(X)

|〈ξ′|Uβψ〉|2ä1/2

= 1,

and equality cannot occur if α 6= β because this would imply that p(ξ′|α) = p(ξ′|β) for allξ′ ∈ σ(X), contradicting the fact that δ > 0. Therefore, if α 6= β,

E(|ρ(k)αβ |) = E

ÄENp+1|ρ(k)

αβ |ä≤ E

(|ρ(Np+1)αβ |

k∏i=Np+2

Ä ∑ξi∈σ(X)

|〈ξi|Uαψ〉〈ψ|U∗βξi〉|ä)

≤ κk−Np−1.

This bound holds uniformly in p ∈ N.Let ε > 0. If N is large enough, we deduce from the above calculations that µ(Σ

(p)ε ) < ε/2 for

all p ∈ N, whereΣ(p)ε :=

ξ | max

α 6=β|ρ(k′p)

αβ (ξ)| ≥ ε

4 dim(HP )

. (5.4.24)

Let N > Nε where Nε is the critical integer such that the sets Σ(p)ε have measure smaller

than ε/2. We now show that Tr((ρ(k))2) is a submartingale. The probability of observing ξ′

at step k + 1 knowing ρ(k) (i.e. knowing ξk) is given by

ν(k)ξ′ := Tr((C(k)

ξ′ )∗ρ(k)C(k)ξ′ ). (5.4.25)

We remark that ν(k)ξ′ > 0 for all k ∈ N and for all ξ′ ∈ σ(X), because

ν(k)ξ′ =

∑α′ρ

(k)α′α′p(ξ

′|α′)1k 6≡0(N) +∑α′〈α′|e−i∆HP ρ(k)ei∆HPα′〉 p(ξ′|α′)1k≡0(N)

≥ δ > 0.


Therefore, conditioning on ξk,

EkTr((ρ(k+1))2) =∑

ξ′∈σ(X)

ν(k)ξ′ Tr((ρ

(k+1))2)(·, ξ′)

=∑

ξ′∈σ(X)

Tr(Ä

(C(k)ξ′ )∗ρ(k)C

(k)ξ′

ä2) 1

ν(k)ξ′

.

Subtracting Tr((ρ(k))2) on both sides and using that∑ξ′ ν

(k)ξ′ = 1, we deduce that

EkTr((ρ(k+1))2)− Tr((ρ(k))2) =∑

ξ′∈σ(X)

Tr(ÄC

(k)ξ′ )∗ρ(k)C

(k)ξ′

ä2 1

ν(k)ξ′

− ν(k)ξ′ (ρ(k))2

)

=∑

ξ′∈σ(X)

ν(k)ξ′ Tr

[ρ(k)

(C(k)ξ′ (C

(k)ξ′ )∗

ν(k)ξ′

− 1HP)ρ(k)

(C(k)ξ′ (C

(k)ξ′ )∗

ν(k)ξ′

− 1HP)]

=∑

ξ′∈σ(X)

ν(k)ξ′ Tr

[θ(k)

(C(k)ξ′ (C

(k)ξ′ )∗

ν(k)ξ′

− 1HP)θ(k)θ(k)

(C(k)ξ′ (C

(k)ξ′ )∗

ν(k)ξ′

− 1HP)θ(k)

]≥ 0,

where we have introduced θ(k) := (ρ(k))1/2. As Tr((ρ(k))2) ≤ 1, the sequence (uk)k∈N definedby uk := Tr((ρ(k))2) for all k ∈ N is a bounded submartingale. Therefore, it converges almostsurely as k tends to +∞; see e.g. [92]. Since (uk)k∈N is bounded, it also converges in mean.Consequently,

E(uk+1 − uk) = E(Ek(uk+1 − uk)) = E(|Ek(uk+1 − uk)|)→ 0 (5.4.26)

as k tends to infinity, and Ek(uk+1 − uk) converges to 0 in mean. It also converges to 0 inprobability. The convergence of Ek(uk+1 − uk) to zero in probability implies that there iskε(N) > 0 (that depends a priori on N), such that, for any k > kε(N),

∑ξ′∈σ(X)

ν(k)ξ′ (ξ)Tr

[θ(k)(ξ)

(C(k)ξ′ (C

(k)ξ′ )∗

ν(k)ξ′ (ξ)

−1HP)θ(k)(ξ)θ(k)(ξ)

(C(k)ξ′ (C

(k)ξ′ )∗

ν(k)ξ′ (ξ)

−1HP)θ(k)(ξ)

]≤ δÅε

2

ã2

,

for all ξ in a set S(k)ε (depending on k) of measure 1 − ε/2. All the terms under the sum in

the formula above are ≥ 0, and ν(k)ξ′ (ξ) ≥ δ for all ξ′ ∈ σ(X).

Let k > kε(N). The Frobenius norm ‖A‖2 :=»Tr(A∗A) is sub-multiplicative, and we

deduce that∣∣∣〈α|ρ(k)(ξ)ÄC

(k)ξ′ (C

(k)ξ′ )∗−ν(k)

ξ′ (ξ)1HPäρ(k)(ξ)β〉

∣∣∣ ≤ ∥∥∥ρ(k)(ξ)ÄC

(k)ξ′ (C

(k)ξ′ )∗ − ν(k)

ξ′ (ξ)1HPäρ(k)(ξ)

∥∥∥2

≤∥∥∥θ(k)(ξ)θ(k)(ξ)

ÄC

(k)ξ′ (C

(k)ξ′ )∗ − ν(k)

ξ′ (ξ)1HPäθ(k)(ξ)θ(k)(ξ)

∥∥∥2

≤ ‖θ(k)(ξ)‖22∥∥∥θ(k)(ξ)

ÄC

(k)ξ′ (C

(k)ξ′ )∗ − ν(k)

ξ′ (ξ)1HPäθ(k)(ξ)

∥∥∥2≤ ε

2

for all α, β ∈ σ(A), for all ξ ∈ S(k)ε , and for all ξ′ ∈ σ(X). Moreover, if k 6≡ 0(N), then

〈α|ρ(k)(C

(k)ξ′ (C

(k)ξ′ )∗−ν(k)

ξ′ 1HP

)ρ(k)β〉 =

∑α′∈σ(A)

ρ(k)αα′ρ

(k)α′βp(ξ

′|α′)−∑

α′,β′∈σ(A)

p(ξ′|α′)ρ(k)α′α′ρ

(k)αβ′ρ

(k)β′β.

(5.4.27)


Let now p ∈ N with Np > kε(N), and let ξ ∈ (Σ(p)ε )c ∩ S(k′p)

ε . The set (Σ(p)ε )c ∩ S(k′p)

ε hasmeasure larger than 1− ε. Taking α = β in (5.4.27), we deduce from (5.4.24) that

(ρ(k′p)αα (ξ))2|p(ξ′|α)−

∑α′∈σ(A)

p(ξ′|α′)ρ(k′p)

α′α′(ξ)| ≤ ε, (5.4.28)

for all ξ ∈ (Σ(p)ε )c ∩ S(k′p)

ε and for all ξ′ ∈ σ(X). If ρ(k′p)αα (ξ) > ε1/4, then

|p(ξ′|α)−∑α′p(ξ′|α′)ρ(k′p)

α′α′(ξ)| <√ε.

By the triangular inequality, this implies that |p(ξ′|α) − p(ξ′|β)| < 2√ε for all α, β ∈ σ(A)

with ρ(k′p)αα (ξ), ρ

(k′p)

ββ (ξ) > ε1/4, for all ξ′ ∈ σ(X). Since ε < δ2/4 we deduce that one, and onlyone α(ξ) ∈ σ(A) can satisfy

ρ(k′p)

α(ξ)α(ξ)(ξ) > ε1/4.

The trace of ρ(k′p)(ξ) is equal to one, and therefore, for all ξ ∈ (Σ(p)ε )c ∩ S(k′p)

ε ,

max(ρ(k′p)αα (ξ)) ≥ 1− dim(HP )ε1/4.

Remark 5.4.2.1. If the initial state ρ of P is pure, then ρ(k) stays pure for all k ∈ N becausethe recursion relation preserves pure states. In that case, (5.4.27) is equal to zero for all k,and we deduce that Lemma 5.4.2.1 is valid for all p ∈ N, uniformly on the set of pure states(because the speed of decoherence is the only ingredient used to control the dynamics).

Transition rates

We calculate the transition rates. The transition rates in this particular model are the prob-abilities of the reduced state of the system P to be in α at time N(p + 1)− 1 if it were in βat time Np − 1. As we will see, these transition rates are constant up to a small error. Thecalculation follows from (5.4.12). Indeed, (5.4.12) holds for all k ∈ Np+ 1, ..., N(p+ 1)− 1,and

ρ(k)αα(ξ

k) = ρ(Np+1)

αα (ξNp+1

)

∏ki=Np+2 p(ξi|α)∑

α′∈σ(A)ρ

(Np+1)α′α′ (ξ

Np+1)∏ki=Np+2 p(ξi|α′)

(5.4.29)

for all k ∈ Np+ 1, ..., N(p+ 1). Conditioning on ξNp+1

, we deduce that

ENp+1ρ(k)αα = ρ(Np+1)

αα . (5.4.30)

Using (5.4.17),

ρ(Np+1)αα (ξ

Np, ξ) =

〈α|e−i∆HP ρ(Np)(ξNp

)ei∆HPα〉〈ξ|Uαψ〉〈ψ|U∗αξ〉∑α′∈σ(A)

〈α′|e−i∆HP ρ(Np)(ξNp

)ei∆HPα′〉 p(ξ|α′),


and conditioning on ξNp

,

ENpρ(Np+1)αα =

∑ξ∈σ(X)

ρ(Np+1)αα (·, ξ)

Ä∑α′〈α′|e−i∆HP ρ(Np)ei∆HPα′〉 p(ξ|α′)

ä= 〈α|e−i∆HP ρ(Np)ei∆HPα〉.

(5.4.31)

Therefore,Eρ(N(p+1)−1)

αα = E 〈α|e−i∆HP ρ(Np)ei∆HPα〉.

Using the same notations as in the proof of Lemma 5.4.2.1 (see (5.4.24)), we deduce that if Nand p are large enough, then there is a set Σ

(p)ε defined as in (5.4.24) but with k′p replaced by

N(p+ 1), and such that∣∣∣Eρ(N(p+1)−1)αα −

∑β∈σ(A)

|〈α|e−i∆HP β〉|2Eρ(Np−1)ββ

∣∣∣=∣∣∣ ∑ξ∈(Σ

(p−1)ε )c

µ(ξ)〈α|e−i∆HP ρ(Np)(ξ)ei∆HPα〉+∑

ξ∈Σ(p−1)ε

µ(ξ)〈α|e−i∆HP ρ(Np)(ξ)ei∆HPα〉

−∑

β∈σ(A)


∣∣∣≤∣∣∣ ∑ξ∈(Σ

(p−1)ε )c

µ(ξ)∑

β∈σ(A)

|〈α|e−i∆HP β〉|2 ρ(Np)ββ (ξ)−

∑β∈σ(A)


∣∣∣+ (1 + dim(HP ))ε

≤ (2 + dim(HP )ε,

where we have used that Eρ(Np)ββ = Eρ(Np−1)

ββ in the last step. Furthermore, using the result ofLemma 5.4.2.1, we have that

ρ(Np−1)ββ (ξ) = δfp(ξ)β +O(ε1/4), ρ(N(p+1)−1)

αα (ξ) = δfp+1(ξ)α +O(ε1/4)

for all ξ in a set of measure 1−2ε. We can consequently reinterpret the expectations Eρ(Np−1)ββ

and Eρ(N(p+1)−1)αα as the probability to be in the state β at step k = Np−1, and the probability

to be in the state α at step k = N(p+1)−1, respectively, up to a small error of order O(ε1/4).We therefore deduce that |〈α|e−i∆HP β〉|2 is the transition rate between α at step N(p+ 1)−1and β at step Np− 1, for each p ∈ N, up to an error of order O(ε1/4).

Remark 5.4.2.2. We claim that Lemma 5.4.2.1 can be extended to the less simplistic casewhere the reduced density matrix at each step k is given by (5.4.18) with C(k)

ξ as in (5.4.19). Inthis setting, free motion on P takes place between two successive non-demolition measurements.

1 2 3 4 5 6 7 8 9 time axis...

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

Figure 5.2: Free evolution on P takes place between any two consecutive non-demolition measure-ments.


The proof is essentially the same, under the restriction that ∆ must be sufficiently small.The only difference concerns the control of decoherence. The idea is to use that

Ek−q(|ρ(k)αβ |) ≤ κ

q + 2q∆‖H‖ dim(HP )2q+1/2

for all α 6= β and all k > q. Therefore, if k is large enough, and if ∆ is small enough, we cancontrol uniformly in k the expectation value of |ρ(k)

αβ |. We then get the lemma

Lemma 5.4.2.2. If ε ∈ (0, δ2/4), then there exist ∆ε > 0 and Nε > 0 such that, for all0 < ∆ < ∆ε, for all k > Nε,

µ(ξ | maxα

(ρ(k)αα(ξ)) ≥ 1− dim(HP )ε1/4) > 1− ε. (5.4.32)

5.4.3 Open systems and quantum trajectories: a simple result in "contin-uous time"

Heuristic derivation of the stochastic differential equation

We first review general features of open quantum systems undergoing indirect measurements;see e.g. [19, 108, 41]. We then state an easy result concerning purification on the spectrum ofan observable when the system P is "continuously" indirectly measured. The results exposedin this section are almost all well-known, and we do not claim to give any new original anddeep insight with respect to the present state of the art concerning quantum trajectories incontinuous time; see e.g. [20]. However, we do feel that the property of "purification" on thespectrum of an observable was not discussed before, and we aim to partially fill out this gapwith this simple result.

We assume that P is a finite dimensional quantum system with dim(HP ) = n0. Ob-servables on P are self-adjoint operators in B(HP ). The effective dynamics of P (when theenvironment is traced out) is a completely positive map generated by a Lindblad operator (see[99]), L, that has the general expression

L := −i[H, ·]− 1

2

∑k∈ICkC∗k , ·+

∑k∈I

C∗k · Ck. (5.4.33)

H, in (5.4.33), is the Hamiltonian of P , the operators Ck are bounded, and I is a finite subsetof R. To simplify matters, we do assume that the operators Ck are bounded invertible. Weset

L0 := −i[H, ·]− 1

2

∑k∈ICkC∗k , · (5.4.34)

andMk := C∗k · Ck, (5.4.35)

for all k ∈ I. Let ρ ∈ B(HP ) be a density matrix. The Duhamel expansion

etLρ = etL0ρ+∞∑n=1

∑k1,...,kn∈I

∫ t

0dtn...

∫ t2

0dt1e

t1L0Mk1e(t2−t1)L0Mk2 ... Mkne

(t−tn)L0ρ (5.4.36)

is norm convergent for all t ≥ 0. One can interpret the operatorsMki as indirect measurementsof a property ki of the system P (see e.g. [41, 45]), and the right side of (5.4.36) can bereinterpreted as a sum over all possible trajectories. Indeed, let us introduce the sets

Ωt :=ω = (k1, t1), ..., (kn, tn) | n ∈ N, ki ∈ I, ti ∈ [0, t]

(5.4.37)


andΩ := ∪

t≥0Ωt. (5.4.38)

Elements ω ∈ Ω are called trajectories. We then rewrite the Duhamel expansion in (5.4.36) as

ρ(t) := etLρ =

∫Ωt

dω Zt(ω)ρ, (5.4.39)

where

Zt((t1, k1), ..., (tn, kn)) := et1L0Mk1e(t2−t1)L0Mk2 ... Mkne

(t−tn)L0 (5.4.40)

is an operator on B(HP ), and where we use the shorthand∫Ωt

dω F (ω) :=∞∑n=0

∑k1,...,kn∈I

∫ t

0dtn...

∫ t2

0dt1F ((t1, k1), ..., (tn, kn)) (5.4.41)

for every function F : Ω → B(HS) such that (5.4.41) is norm convergent. The probability ofa trajectory ω is given by the trace

Tr(Zt(ω)ρ). (5.4.42)

The operators Ck being bounded invertible, we can rewrite that

ρ(t) =

∫Ωt

dω Tr(Zt(ω)ρ)Zt(ω)ρ

Tr(Zt(ω)ρ)(5.4.43)

for all times t ≥ 0. The operator

ρ(t, ω) :=Zt(ω)ρ

Tr(Zt(ω)ρ)(5.4.44)

is a density matrix for all pairs (t, ω) because Zt(ω) is positivity preserving. Eq. (5.4.43) canbe interpreted as a sum over all possible trajectories between [0, t], weighted by their relativeprobability to occur. To get a stochastic differential equation from (5.4.43), we make use ofa heuristic argument: in real life experiments, a counter device is usually used to register theevents "ki is measured at time ti". This counter device has a finite time resolution and it canonly register "clicks" regularly at every time ti = i∆, where ∆ > 0 is very close to zero. Letti, ti+1 ≥ 0 with ti+1− ti = ∆. We only consider histories ω such that the time delay betweentwo consecutive events is a multiple of ∆. We deduce from (5.4.43) that the recursive equationdescribing the state of P in an experiment where the counter has a finite time resolution ∆ isgiven by

ρ(ti+1)(ω) =∑k∈I

χ(k)i+1(ω)

Mke∆L0ρ(ti)(ω)

Tr(Mke∆L0ρ(ti)(ω))

+ (1−∑k∈I

χ(k)i+1(ω))

e∆L0ρ(ti)(ω)

Tr(e∆L0ρ(ti)(ω)),

(5.4.45)

where χ(k)i+1(ω) = 1 if the event "k" is measured at time ti+1, and zero otherwise. The proba-

bility of measuring "k" at time ti+1 knowing ρ(ti) at time ti is given by ∆Tr(Mke∆L0ρ(ti)(ω)).

On the contrary to the models considered in Section 5.4.1 and 5.4.2, there may not be any event


recorded between ti and ti+1. This happens with probability 1 −∆∑k Tr(Mke

∆L0ρ(ti)(ω)).The discrete process ρ(ti) is equivalent (in the sense that they have the same probabilitydistribution) to a process ρ(i) defined on the space of all infinite sequences k = (k1, k2, ....),where ki ∈ I∪β, the letter β corresponding to the event "nothing is measured". If ∆ = 1/nfor some n ∈ N, this process is recursively defined by

ρ(i+1)(k) =∑k∈I

χi+1;k(k)Mke

1nL0ρ(i)(k)

Tr(Mke1nL0ρ(i)(k))

+ (1−∑k∈I

χi+1;k(k))e

1nL0ρ(i)(k)

Tr(e1nL0ρ(i)(k))

,

(5.4.46)

where χi+1;k(k) = 1 if the event ki+1 = k, and zero otherwise, and where ρ(0) = ρ0 is someinitial density matrix. For any fixed time t ≥ 0, one can investigate the convergence ofthe process ρ(bntc)n∈N as n tends to infinity. Given an initial state ρ0, one can show thatthe process ρ(bntc)n∈N converges in distribution to the solution of a stochastic differentialequation. The rigorous proof is long, and we refer the reader to [108] for similar arguments.The main idea is to view the function χi+1;k as the "infinitesimal" increment of a countingstochastic process, N (k)(t, ·). To construct this stochastic process, one considers a filteredprobability space (Ω,F ,F, P ), where F is a σ-algebra, F is a filtration, and P is a probabilitymeasure. We assume that this space is chosen such that it supports a random poisson measureon R+ × R, with intensity dt ⊗ dx. A random poisson measure µ on R+ × R is a family ofmeasures µ(ω, ·) | ω ∈ Ω on the space (R+×R,Σ), where Σ is given by Σ := Σ(R+)⊗Σ(R)( Σ(R) is the Borel sigma algebra of R), that satisfies the following properties (see [108] formore details):

• for all σ ∈ Σ, µ(·, σ) is N ∪ +∞ valued,

• µ(ω, t × R) ≤ 1 for all t ∈ R+ and all ω ∈ Ω,

• the measure m(σ) := E(µ(σ)) on Σ is non-atomic,

• m(0 × R) = 0,

• if t ∈ R+, and if the sets Ai ∈ Σ([t,+∞))⊗Σ(R) are all disjoints, the random variablesµ(·, Ai) are independent. Furthermore, they do not depend on Ft (where F = Ftt∈R+).

Such measures can be constructed from the space of poisson point processes; see [108].The measure m is the stochastic intensity of µ. In our case of interest, m is nothing else thandt⊗dx. Once µ is defined, the rest of the proof goes as follows: one first introduces a discretestochastic process ρ(i) on the space (Ω,F ,F, P ), whose jumps are encoded by the randompoisson measure µ, and that is equivalent to ρ(i) in (5.4.45) ( in the sense that ρ(i) and ρ(i)

have the same probability distributions). One then shows that ρ converges in distribution tothe solution ρ(t) of the stochastic differential equation

dρ(t) = Lρ(t)dt+∑k∈I

ÇMkρ(t)

Tr(Mkρ(t))− ρ(t)

åÄdN (k)(t)− Tr(Mkρ(t))dt

ä(5.4.47)

with initial condition ρ(0) = ρ0, as n→∞, where

N (k)(t) :=

∫ t

0

∫R+

10≤x≤Tr(Mkρ(s)) µ(ds, dx) (5.4.48)


for all t ∈ R+. The properties of the random measure µ imply that

E( ∫ t

0X(s) dN (k)(s)

)=

∫ t

0E[X(s)Tr(Mkρ(s))]ds (5.4.49)

for every bounded cadlag process X.Detailed proofs can be found in [108]. Specifying an initial state ρ0 at time t = 0, it is

shown in [108] that the solution ρ(t) to (5.4.47) with initial condition ρ0 exists and is unique,in the case where dim(HP ) = 2. Moreover, the unique solution ρ(t) is a density matrix forall times t ≥ 0. The proof presented there can be adapted to the case where dim(HP ) isarbitrary but finite if

∑k∈I C

∗kCk = 1HP , which is the assumption that will be made further

below. Therefore, we will not worry about existence and unicity of a density matrix ρ(t)solution to (5.4.47) given a specific initial condition at time t = 0.

Well-known results

We state two interesting results concerning (5.4.47).

Lemma 5.4.3.1 (Barchielli, Paganoni, [21]).

• Eq. (5.4.47) preserves pure states.

• We assume that for every time t ≥ 0, there is no projection Pt in HP with dim(RanPt) ≥2 such that

PtMk(1HP )Pt = qk(t)Pt, qk(t) ∈ C, ∀k ∈ I. (5.4.50)

Then, for any initial state ρ0, the solution ρ(t) to (5.4.47) with initial condition ρ(0) = ρ0

purifies asymptotically:

limt→+∞

Tr(ρ(t)(1− ρ(t))→ 0 a.s.. (5.4.51)

Remark 5.4.3.1. Assumption (5.4.50) is similar to the non-degeneracy condition imposedby Maassen and Kümmerer in [101] to show asymptotic purification of the state of P in the"discrete time" setting. The proofs are similar.

Another interesting lemma has been derived by T.Benoist and C.Pellegrini in [24]. Theystudied the behavior of the solution of (5.4.47) under the condition that the operators C ′ksand the Hamiltonian H are diagonalizable in a common orthonormal basis. In that case, thestochastic differential equation describes a non-demolition measurement, similar to the one weconsidered in Section 5.4. They proved:

Lemma 5.4.3.2 (Benoist, Pellegrini, [24]). Let (|α〉)α=1,...,n0 be an orthonormal basis of HPsuch that

Ck =n0∑α=1

ckα|α〉〈α| (5.4.52)

for all k ∈ I. We assume that H =∑n0α=1Hαα|α〉〈α| and that, for all k ∈ I, there are αk and

βk with|ckαk | 6= |ckβk |. (5.4.53)


Then, for every solution ρ(t) to (5.4.47) with ρ(0) ≥ 0 and Tr(ρ(0)) = 1, there is a vectorvalued random variable Ω 3 ω 7→ |Γ(ω)〉 ∈ HP with values in the set |α〉 | α = 1, ..., n0 suchthat

limt→∞

ρ(t) = |Γ〉〈Γ| a.s.. (5.4.54)

Furthermore, P (ω ∈ Ω | Γ(ω) = α) = ραα(0).

Remark 5.4.3.2. Eq. (5.4.53) is similar to the condition imposed in Section 5.4 to provepurification on the spectrum of A in the non-demolition case.

Remark 5.4.3.3. One can show that the solution ρ(t) converges exponentially fast to |Γ〉〈Γ|as t tends to infinity; see [24]. However, the speed of convergence of ρ(t) to |Γ〉〈Γ| for finitetimes and for a particular choice of initial condition is harder to estimate. The rate of decaya priori depends on the initial state ρ(0) and it is not uniform in the variable ω ∈ Ω.

An estimate concerning purification on the spectrum of A in continuous time

Let A = A∗ ∈ B(HP ) with A =∑n0α=1 α|α〉〈α|, where (|α〉)α=1,...,n0 is the orthonormal basis

introduced in Lemma 5.4.3.2. If the Hamiltonian H is diagonal in this basis, Lemma 5.4.3.2shows that ρ(t) purifies asymptotically to an eigenvector of the observable A. As in Section5.4.2, we analyze purification on the spectrum of A if the commutator between H and A isnon-zero. We cannot prove a result as strong as in Section 5.4.2, for the following reason:there are, a priori, many trajectories ω that exhibit only a few number of counts, or that aresuch that the frequency of counts is not very high: for a given time T > 0,

∑kNk(T, ω)/T

can be quite small for all ω in a non-negligible set. This problem was absent in the discretesetting presented in Section 5.4.2, because the frequency of counts was fixed and could bemade arbitrarily large. One can show here purification on the spectrum of A only on a subsetof trajectories with sufficiently regular counts. To do so, one can introduce stopping times,and the problem is almost the same as the one treated in Section 5.4.2. There are two typesof evolution: non-demolition measurements take place at stopping times T (k)

i ,

ρ(T(k)i ) =

Mkρ((T(k)i )−)

Tr(Mkρ((T(k)i )−))

, (5.4.55)

and free evolution takes place between two consecutive stopping times,

dρ(t) =Ä− i[H, ρ(t)] +

∑k∈I

Tr(Mkρ(t))ρ(t)− 1

2CkC∗k , ρ(t)

ädt = −i[H, ρ(t)]dt, (5.4.56)

where we used here that∑k CkC

∗k = 1HP .

We present here another estimate by comparing the non-demolition solution of Lemma5.4.3.2 with the solution where the Hamiltonian H of Lemma 5.4.3.2 is replaced by

H = H +HND, (5.4.57)

where HND is non-diagonal in the basis (|α〉). We set

L := −i[H, ·]− 1

2

∑k∈ICkC∗k , ·+

∑k∈I

Mk, (5.4.58)


and we denote by ρ the solution (given any initial state) of (5.4.47) with L replaced by L andwith N (k) replaced by N (k), where

dρ(t) = Lρ(t)dt+∑k∈I

ÇMkρ(t)

Tr(Mkρ(t))− ρ(t)

åÄdN (k)(t)− Tr(Mkρ(t))dt

ä, (5.4.59)

andN (k)(t) :=

∫ t

0

∫R+

10≤x≤Tr(Mkρ(s)) µ(ds, dx), (5.4.60)

for all t ∈ R+. In the rest of this Section, we always assume that∑k∈I

C∗kCk =∑k∈I

CkC∗k = 1HP . (5.4.61)

Lemma 5.4.3.3. Let ρ0 be a state and let ρ and ρ be the solutions of (5.4.47) and (5.4.59)on [t0,∞) with initial conditions ρ(t0) = ρ(t0) = ρ0, respectively. We define the functiong : [t0,∞)→ R+ by

g(t) := E(‖ρ(t)− ρ(t)‖2) (5.4.62)

for all t ∈ [t0,∞), where ‖ · ‖2 is the Hilbert Schmidt norm in B(HP ). We assume that theoperators Ck’s are all invertible and that the operators Ck and the Hamiltonian H satisfythe assumptions of Lemma 5.4.3.2. Then g(t) is Lipschitz continuous. Moreover, there is aconstant C > 0 such that

g(t) ≤ 2‖HND‖(t− t0) +2‖HND‖

C(eC(t−t0) + C(t0 − t)− 1). (5.4.63)

for all t ≥ t0.

Before discussing the proof of Lemma 5.4.3.3, we formulate an immediate corollary.

Corollary 5.4.3.1. Let ρ0 be a state and let ρ be the solution of (5.4.59) with initial conditionρ(0) = ρ0. Let t1, ..., tn be a finite set of positive times and let η > 0. There is εη,n > 0 andTη,n > 0, such that, if ‖HND‖ < εη,n, then

En∑i=1

‖ρ(ti + Tη,n)− |Γi〉〈Γi|‖2 < η. (5.4.64)

For every i = 1, ..., n, the vector valued random variable |Γi〉〈Γi| in (5.4.64) is the asymptoticlimit of the stochastic process ρi(t) defined on [ti,+∞), that satisfies (5.4.47) with the initialcondition ρ(ti) = ρ(ti); see Lemma 5.4.3.2.

Corollary 5.4.3.1 is a direct consequence of Lemmas 5.4.3.2 and 5.4.3.3, using that almostsure convergence implies L1 convergence for bounded stochastic processes.

Proof. (Lemma 5.4.3.3) We introduce ∆ρ(t) := ρ(t)− ρ(t). We have that

ρ(t) = ρ0 +

∫ t

t0

Lρ(s)ds+∑k∈I

∫ t

t0

ÇMkρ(s)

Tr(Mkρ(s))− ρ(s)

åÄdN (k)(s)− Tr(Mkρ(s))ds

ä,

ρ(t) = ρ0 +

∫ t

t0

Lρ(s)ds+∑k∈I

∫ t

t0

ÇMkρ(s)


åÄdN (k)(s)− Tr(Mkρ(s))ds

ä.


Collecting the terms in front of dN (k) and using (5.4.61), we easily get that

ρ(t) = ρ0 − i∫ t

t0

[H, ρ(s)]ds+∑k∈I

∫ t

t0

ÇMkρ(s)


ådN (k)(s). (5.4.65)

A similar formula holds for ρ(t), and we deduce that

∆ρ(t) = i

∫ t

t0

(−[H, ρ(s)] + [H, ρ(s)] + [HND, ρ(s)])ds

+∑k∈I

∫ t

t0

ÇMkρ(s)


ådN (k)(s)−

∑k∈I

∫ t

t0

ÇMkρ(s)


ådN (k)(s).

(5.4.66)

Then we bound the Hilbert Schmidt norm of the operator [H, ρ(s)]− [H, ρ(s)]. We have that

‖[H, ρ]− [H, ρ]‖22 = −Tr((H(ρ− ρ) + (ρ− ρ)H)2) = 2Tr(H2(∆ρ)2 −H∆ρH∆ρ)

= 2∑α,β

|∆ραβ|2(H2ββ −HββHαα) ≤ 4‖H‖2‖∆ρ‖22.

To bound the norm of the integrand on the second line of (5.4.66), we introduce the map

Fk : ρ 7→ Mkρ

Tr(Mkρ)

for every density matrix ρ. We have that

Fk(ρ)− Fk(ρ) =∑α,β

ckβckα( ραβνk(ρ)

− ραβνk(ρ)

)|α〉〈β|

in the basis (|α〉)α=1,...,n0 , where νk(ρ) := Tr(Mkρ) =∑α′ |ckα′ |2ρα′α′ ≥ δ > 0 because the

operators Ck are bounded invertible. Therefore,

‖Fk(ρ)− Fk(ρ)‖22 =∑α,β

|ckβ|2|ckα|2∣∣∣∣ ραβνk(ρ)

− ραβνk(ρ)

∣∣∣∣2

=∑α,β

|ckβ|2|ckα|2

νk(ρ)2νk(ρ)2

∣∣∣∣(ραβ − ραβ)νk(ρ) + ραβ(νk(ρ)− νk(ρ))

∣∣∣∣2

≤ 2

δ2(‖ρ− ρ‖22 +

‖ρ‖2

δ2|νk(ρ− ρ)|2),

where we have used that |ckα| ≤ 1 for all α ∈ 1, ..., n0; see (5.4.61). Moreover,

|νk(ρ− ρ)|2 = Tr(CkC∗k(ρ− ρ))2 ≤ dim(HP )‖ρ− ρ‖22.

Collecting the previous estimates and taking the norm of ∆ρ in (5.4.66), we deduce that

‖∆ρ(t)‖2 ≤ 2‖HND‖(t− t0) + 2

∫ t

t0

‖H‖‖∆ρ(s)‖2ds+ C∑k∈I

∫ t

t0

‖ρ(s)− ρ(s)‖2 dN (k)(s)

+

∥∥∥∥ ∫ t

t0

(Fk(ρ(s))− ρ(s))(dN (k)(s)− dN (k)(s))

∥∥∥∥2,

(5.4.67)


whereC :=

2

δ

((1 + dim(HP )/δ2)1/2 + δ

).

As ∫ t

t0

f(s)(dN (k)(s)− dN (k)(s)) =

∫ t

t0

∫ 1

0f(s)(10≤x≤νk(ρ(s)) − 10≤x≤νk(ρ(s)))µ(ds, dx),

we take the expectation in (5.4.67) and we finally find the upper bound

g(t) ≤ 2‖HND‖(t− t0) + C

∫ t

t0

g(s)ds, (5.4.68)

where C = C + 2‖H‖+ 2 dim(HP )1/2. Similar calculations lead to

|g(t)−g(s)| = |E(‖∆ρ(t)‖2−‖∆ρ(s)‖2)| ≤ E‖∆ρ(t)−∆ρ(s)‖2 ≤ 2‖HND‖(t−s)+C∫ t

sg(s′)ds′.

Since 0 ≤ g(t) ≤ 2 for all t, we deduce that g(t) is Lipschitz continuous. Eq.(5.4.63) thenfollows directly from (5.4.68) using Gronwall Lemma; see e.g. [81].


5.5 Appendix to Chapter 5

5.5.1 Expectations and Tomita-Takesaki theory

The purpose of this appendix is to describe some mathematical structure useful to imbed thematerial in Subsections 5.1.3 into a more general context. In particular, we do not wish toassume that the algebras Eω≥t are type-I von Neumann algebras; (i.e., we do not start fromEqs. (5.1.31) – (5.1.32)). To begin with, we summarize some further basic facts concerningvon Neumann algebras; (see also Subsection 5.5.2). Let M be a von Neumann algebra, andlet ω be a normal state onM. Then (πω,Hω,Ω) stands for the representation, πω, ofM onthe Hilbert space Hω, with Ω the cyclic unit vector in Hω (unique up to a phase) such that

ω(a) = 〈Ω, πω(a)Ω〉Hω . (5.5.1)

This is the GNS construction applied to (M, ω). We say that ω is separating for M iff, forany a ∈M,

ω(ba) = 0, ∀b ∈M =⇒ a = 0; (5.5.2)

or, equivalently, πω(a)Ω = 0 (in Hω) implies that a = 0; (it is assumed that πω is faithful, andwe will henceforth write a for πω(a)).

Given a separating state, ω, on a von Neumann algebraM, Tomita-Takesaki theory [122,27] guarantees that there is a one-parameter unitary group ∆iσ

ω σ∈R, where ∆ω > 0 isa self-adjoint operator on Hω (the Tomita-Takesaki modular operator) and an anti-unitaryinvolution, Jω, on Hω, with the properties

∆iσω a∆−iσω ∈M, JωaJω ∈M′, (5.5.3)

for all a ∈M and for all σ ∈ R, (M′ is the commutant ofM),

∆iσω Ω = Ω, JωΩ = Ω, (5.5.4)

for all σ, and〈Ω, abΩ〉Hω = 〈Ω, b∆ωaΩ〉Hω , (5.5.5)

for arbitrary a, b ∈M; (KMS condition). If ϕ is a linear functional onM we define

‖ϕ‖ := supb∈M

|ϕ(b)|‖b‖

(5.5.6)

Eqs. (5.5.1) and (5.5.5) then show that if ω is separating forM,

‖a, ω]M‖ < ε⇐⇒ ‖(∆ωa− a)Ω‖Hω < ε, (5.5.7)

for any a ∈M; (recall that a, ω]M(b) = ω([a, b]), b ∈M – see (5.1.22), Subsection 5.1.3). In(5.1.23), we have defined the centralizer of ω to be the subalgebra ofM given by

CωM := a ∈M | a, ω]M = 0. (5.5.8)

We recall that ω defines a trace on CωM. By (5.5.7),

CωM = a ∈M | ∆ωaΩ = aΩ, (5.5.9)


assuming that ω is separating forM. The following claim is easy to verify (using Liouville’stheorem for analytic functions of one complex variable, and (5.5.9)): If ω is separating forM

a, ω]M = 0⇐⇒ ∆iσω a∆−iσω = a, ∀σ ∈ R, (5.5.10)

for any a ∈ M; (see, e.g., [5]). The group, ασσ∈R, of ∗automorphisms of M definedby ασ(a) = ∆iσ

ω a∆−iσω is called the Tomita-Takesaki modular automorphism group. Theequivalence in (5.5.10) together with (5.5.8) show that if ω is separating for M then thecentralizer, CωM, is nothing but the subalgebra ofM of fixed points under the Tomita-Takesakimodular automorphism group. The following result is due to Takesaki, [122]: Let N be a vonNeumann subalgebra ofM, and let ω be a faithful, normal, separating state onM. Then thefollowing statements are equivalent:

(i) N is invariant under the modular automorphism group ασσ∈R associated with (M, ω).

(ii) There exists a (σ-weakly) continuous projection, ε, of norm 1 (a "conditional expecta-tion") ofM onto N such that

ω(a) = ω|N (ε(a)), (5.5.11)

for all a ∈M.

Remark 5.5.1.1. For a, b in N and x ∈M, we have that

ε(x∗x) ≥ ε(x)∗ε(x) ≥ 0,ε(axb) = aε(x)b.

´(5.5.12)

As a corollary of Takesaki’s result on conditional expectations, we have that if ω is separatingforM then

(a) there is a conditional expectation, ε = εω, fromM onto the centralizer CωM of ω satisfying(5.5.11); and

(b) there is a conditional expectation, εω, from M onto the center, ZωM, of CωM satisfying(5.5.11).

Definition 5.5.1.1. The variance of an operator a ∈M in the state ω is defined by

∆ωMa :=

»ω((a− aω)), (5.5.13)

where aω := εω(a).

These general results can be applied to the considerations in subsections 5.1.3-5.3, withthe following identifications:

M→ Eω≥t, Cωω → Cω≥t, ZωM → Zω≥t. (5.5.14)

We then use the notations εω → εω≥t, εω → εω≥t and ∆ω

Ma → ∆ωt a; (see Subsect. 5.1.3). Con-

cerning the special case introduced in Eqs. (5.1.31)-(5.1.32), we remark that ω is separatingfor Eω≥t iff all eigenvalues of the density matrix ρω≥t introduced in (5.1.32) are strictly positive(which is generically the case).



First step . Let P be a bounded selfadjoint operator on a Hilbert space H, and let 0 < ε < 14 .

We prove that if ‖P 2 − P‖ < ε, then there there exists an orthogonal projection, P , on Hsuch that

‖P − P‖ < 2ε.

We introduce the function f(x) := x2 − x. Because P is a selfadjoint bounded operator,σ(f(P )) = f(σ(P )), and ‖f(P )‖ = supλ∈σ(P ) |λ2 − λ| < ε, by hypothesis. We considerthe polynomials Qε(X) := X2 −X − ε and Q′ε(X) := −X2 +X − ε. The real roots of thesepolynomials are given by x±(ε) = 1/2±

√1 + 4ε/2, and x′±(ε) = 1/2±

√1− 4ε/2, respectively.

Introducing the open intervals ∆0 := (x−(ε), x′−(ε)) and ∆1 := (x′+(ε), x+(ε)), we see thatσ(P ) ⊂ ∆0 ∪∆1. By the spectral theorem,

P =

∫σ(P )

λ dEP (λ),

and we can defineP :=

∫σ(P )∩∆1

dEP (λ).

Clearly, P is an orthogonal projection. Moreover,

P − P =

∫σ(P )∩∆1

(λ− 1) dEP (λ) +

∫σ(P )∩∆0

λ dEP (λ)

For λ ∈ σ(P )∩∆1, |λ−1| < 2ε(1+√

1− 4ε−1, and, for λ ∈ σ(P )∩∆0, |λ| < 2ε(1+

√1− 4ε)−1.

Consequently, every element in the spectrum of P − P is smaller, in absolute value, than2ε

1+√

1−4ε, and thus

‖P − P‖ ≤ 2ε

1 +√

1− 4ε< 2ε.

Second step. We only prove Lemma 5.1.2.1 for one history α = (α1, ..., αn). A generalizationfor a family of history can be deduced easily. We set Πi = Π

(i)αi . We construct the orthogonal

projectors Πi := Π(i)αi inductively. We first consider the case where n = 2. Let Π1 and Π2 be

two orthogonal projections satisfying the hypotheses of Lemma 5.1.2.1. We set Π2 := Π2. Toconstruct Π1, we define operators Q := Π⊥2 Π1Π⊥2 and Q′ := Π2Π1Π2. Clearly, Q and Q′ areselfadjoint bounded operators. Moreover,

Q2 −Q = Π⊥2 Π1Π⊥2 Π1Π⊥2 − Π⊥2 Π1Π⊥2

= Π⊥2 Π1

îΠ⊥2 ,Π1

óΠ⊥2

and hence‖Q2 −Q‖ < ε,

by hypothesis. The same holds for (Q′)2 −Q′. According to the first step of the proof, thereis an orthogonal projection Q commuting with Q and an orthogonal projection Q′ commutingwith Q′ such that ‖Q−Q‖ < 2ε and ‖Q′ −Q′‖ < 2ε. We define

Π1 := QΠ⊥2 + Q′Π2


which is easily seen to be a projection commuting with Π2. Moreover,

‖Π1 − Π1‖ ≤ ‖Π1 −Q−Q′‖+ ‖Q− QΠ⊥2 ‖+ ‖Q′ − Q′Π2‖< ‖Π2Π1Π⊥2 + Π⊥2 Π1Π2‖+ 4ε

< 6ε

using that Π2Π1Π⊥2 + Π⊥2 Π1Π2 = Π2

îΠ1, Π

⊥2

ó+îΠ⊥2 ,Π1

óΠ2. We have thus constructed two

commuting projections Π1 and Π2 with the properties claimed to hold in the lemma, withC2 = 6. It follows that Π1Π2Π1 is an orthogonal projection, as well.

Let n ∈ N and let j > 1 be an integer smaller than n. We suppose that we have alreadyconstructed projections Πj , ..., Πn, starting from Πn, such that

îΠk,Π

ni=k+1Πi

ó= 0, for k =

j, ..., n − 1, and ‖Πk − Πk‖ < Cn−k+1ε, for k = j, ..., n, with ε < (4(4∑ni=j Cn−i+1 + 1))−1.

We proceed to construct a projection Πj−1 close to Πj−1 and commuting with the operatorHj−1 := (Πn

i=jΠi)(Πji=nΠi), using the ideas used above to prove the lemma in the special case

where n = 2. Then Hj−1 takes the role of Π2 and Πj−1 the role of Π1 in the argument to provethe special case where n = 2. Indeed, define Qj−1 = H⊥j−1Πj−1H

⊥j−1, Q′j−1 = Hj−1Πj−1Hj−1.

Q2j−1 −Qj−1 = H⊥j−1Πj−1

îH⊥j−1,Πj−1

óH⊥j−1 (5.5.15)

and thus,

‖Q2j−1 −Qj−1‖ ≤ ‖

îH⊥j−1,Πj−1

ó‖ ≤ ‖

îHj−1 −Hj−1,Πj−1

ó‖+ ‖ [Hj−1,Πj−1] ‖

≤ 2‖Hj−1 −Hj−1‖+ ε ≤ 4εn∑i=j

Cn−i+1 + ε

where the last inequality follows from the use of Πk = (Πk − Πk) + Πk in the expression ofHj−1. The same holds for Q′j−1. As we assumed that ε < (4(4

∑ni=j Cn−i+1 + 1))−1, we can

find Qj−1 and Q′j−1 obeying the conditions of the first step of our proof, i..e. Qj−1 and Q′j−1

are orthogonal projectors commuting respectively with Qj−1 and Q′j−1, with ‖Qj−1−Qj−1‖ <8ε∑ni=j Cn−i+1 + 2ε. Defining

Πj−1 := Qj−1H⊥j−1 + Q′j−1Hj−1

one finds that

‖Πj−1 −Πj−1‖ < 6εÄ4

n∑i=j

Cn−i+1 + 1ä

i.e. Cn−j+2 = 6Ä4∑ni=j Cn−i+1 + 1

ä.

Appendix

Some basic notions from the theory of operator algebras

We summarize some basic definitions and notions from the theory of operator algebras; forfurther details see e.g. [121, 123].

Basic definitions

An algebra, A, over the complex numbers is a complex vector space equipped with a multi-plication: If a and b belong to A, then

• λa+ µb ∈ A, λ, µ ∈ C,

• a · b ∈ A,

where "·" denotes multiplication in A. We focus in this thesis on a particular type of algebras:

Definition .0.2.1. Banach algebraThe algebra A is a Banach algebra if it comes with a norm ‖(·)‖ : A → [0,∞[, satisfying, forall λ ∈ C, for all a, b ∈ A,• ‖a‖ = 0, for a ∈ A =⇒ a = 0,• ‖λa‖ = |λ|‖a‖, and ‖a+ b‖ ≤ ‖a‖+ ‖b‖,• ‖ab‖ ≤ ‖a‖‖b‖,• A is complete in ‖(·)‖, i.e., every Cauchy sequence in A converges to an element of A.

One says that A is an involutive Banach algebra if there exists an anti-linear involution,∗, on A, i.e., ∗ : A → A, with (a∗)∗ = a, for all a ∈ A, such that

(λa+ µb)∗ = λa∗ + µb∗,

where λ is the complex conjugate of λ ∈ C,

(a · b)∗ = b∗ · a∗,

and‖a∗‖ = ‖a‖.

In this thesis, we only consider unital involutive Banach algebras, i.e. algebras that possesa unit element 1 for the multiplication law ” · ”. This is reasonable since every non-unitalinvolutive Banach Algebra (or C∗-algebra, see the definition below) can be imbeded into aunital involutive Banach algebra ( or C∗-algebra); see e.g. [123].

196

197

Definition .0.2.2. C∗-algebraAn involutive Banach algebra, A, is called a C∗−algebra if

‖a∗ · a‖ = ‖a · a∗‖ = ‖a‖2, ∀a ∈ A. (.0.16)

Examples of C∗-algebras: The Banach space B(H) of bounded operator on a Hilbert spaceH; The set of continuous functions on a compact space Ω equipped with the supremum norm.

Given a set P = aii∈I of operators in a C∗-algebra A, we define 〈P〉 to be the C∗-subalgebra of A generated by P, i.e., the norm-closure of arbitrary finite complex-linearcombinations of arbitrary finite products of elements in the set ai, a∗i i∈I , where ∗ is the∗-operation on A.

Definition .0.2.3. ∗-automorphismA ∗-automorphism, α, of a C∗-algebra A is a linear isomorphism from A onto A with theproperties

α(a · b) = α(a) · α(b),

α(a∗) = (α(a))∗,(.0.17)

for all a, b ∈ A.

It is easy to check that every ∗automorphism of A is an isometry: ‖α(a)‖ = ‖a‖ for alla ∈ A.

Definition .0.2.4. StateA state, ω, on a C∗-algebra A (with unit) is a linear functional ω : A → C with the propertiesthat

ω(a∗) = ω(a), ω(a∗a) ≥ 0, (.0.18)

for all a ∈ A, andω(1) = 1. (.0.19)

The fact that ω(a∗a) ≥ 0 is referred to as the "positivity" of the functional ω; the factthat ω(1) = 1 is referred to as the "normalisability" of the functional ω. Moreover, states arecontinuous functions; see e.g. [123].

Definition .0.2.5. RepresentationA representation, π, of a C∗-algebra A on a complex Hilbert space, H, is a ∗homomorphismfrom A to the algebra, B(H), of all bounded linear operators on H; i.e., π is linear, π(a · b) =π(a) ·π(b), π(a∗) = (π(a))∗, and ‖π(a)‖ ≤ ‖a‖, (where ‖A‖ is the operator norm of a boundedlinear operator A on H).

Well-known theorems

We recall a well-known theorem due to I.M. Gel’fand. Let A be an abelian C∗-algebra.The spectrum, M , of A is the space of all non-zero ∗homomorphisms from A into C (the"characters" of B); M is a locally compact topological (Hausdorff) space. If A contains anidentity, 1, then M is compact.

198

Theorem .0.2.1 (Gel’fand). If A is an abelian C∗−algebra, then it is ∗-isomorphic to theC∗-algebra, C0(M), of continuous functions on M vanishing at ∞, i.e.,

B ' C0(M). (.0.20)

Furthermore, every state, ω, on B is given by a unique (Borel) probability measure, dµω, onM (and conversely).

Another very useful theorem is the Gel’fand-Naimark-Segal (GNS) construction.

Theorem .0.2.2 (GNS construction, see e.g. [123]). Let A be a C∗-algebra and let ω be astate on A. Then, there is a representation (πω,Hω) (unique up to unitary equivalence) of Aand a unit vector Ω ∈ Hω such that πω(a)Ω | a ∈ A is dense in Hω (i.e. Ω is cyclic forπω(A)), and

ω(a) = 〈Ω|πω(a)Ω〉, (.0.21)

for all a ∈ A, where 〈·|·〉 is the scalar product on Hω.

A theorem due to Gel’fand and Naimark says that every C∗−algebra, A, can is isometri-cally isomorphic to a norm-closed subalgebra of B(H), closed under ∗, for some Hilbert spaceH.

Von-Neumann algebras

Using this theorem, consider a C∗-algebra A ⊂ B(H), for some Hilbert space H. We definethe commuting algebra, or commutant, A′, of A by

A′ := a ∈ B(H) | a · b = b · a,∀b ∈ A. (.0.22)

The double commutant of A, A′′, is defined by

A′′ ≡ (A′)′ = a ∈ B(H) | a · b = b · a,∀b ∈ A′ ⊇ A. (.0.23)

It turns out that A′ and A′′ are closed in the so-called weak ∗ topology of B(H); i.e., if aii∈Iis a sequence (net) of operators in A′ (or in A′′), with

〈ϕ|aiψ〉 → 〈ϕ|aψ〉, as i→∞,

for all ϕ,ψ ∈ H, where a ∈ B(H), then a ∈ A′ (or a ∈ A′′, respectively).

Definition .0.2.6. von Neumann algebraLetM be a ∗-subalgebra of B(H) that is closed in the weak ∗-topology and such that aϕ |a ∈M, ϕ ∈ H is dense in H. ThenM is called a von Neumann algebra (or W ∗-algebra).

Another equivalent characterization of von Neumann algebras comes from the followingfamous theorem:

Theorem .0.2.3 (von Neumann double commutant theorem, see [123]). Let M be a ∗-subalgebra of B(H). ThenM is a Von Neumann algebra iffM′′ =M.

199

Thus, if A is a C∗-algebra contained in B(H), for some Hilbert space H, then A′ and A′′are von Neumann algebras. A von Neumann algebra M ⊆ B(H) is called a factor iff itscentre, ZM, consists of multiples of the identity operator 1.

A von Neumann factorM is said to be of type I iffM is isomorphic to B(H0), for someHilbert space H0. A general von Neumann algebra, N , is said to be of type I iff N is a directsum (or integral) over its centre, ZN , of factors of type I. A C∗-algebra A is called a type-IC∗-algebra, iff, for every representation π, of A on a Hilbert space H,

π(A) := π(a) | a ∈ A

has the property that π(A)′′ is a von Neumann algebra of type I. (For mathematical propertiesof type-I C∗-algebra see [76], and for examples relevant to quantum physics see [30]).

We will need a few further notions in Section 5:

Definition .0.2.7. Relative commutantLet A ⊂ B be two C∗-algebras. We define

A′ ∩ B := b ∈ B | b · a = a · b, ∀a ∈ A, (.0.24)

the "relative commutant" of A in B.

Definition .0.2.8. TraceA trace τ :M+ → [0,∞] on a von Neumann AlgebraM is a function defined on the positivecone,M+, of positive elements ofM (i.e., elements x ∈M of the form x = y∗y, y ∈M) thatsatisfies the properties

(i) τ(x+ y) = τ(x) + τ(y), x, y ∈M+

(ii) τ(λx) = λτ(x), λ ∈ R+, x ∈M+

(iii) τ(x∗x) = τ(xx∗), x ∈M.

A trace τ is said to be finite if τ(1) < +∞. It can then be uniquely extended by linearityto a state τ onM. Conversely, any state τ onM enjoying the property

τ(a · b) = τ(b · a), ∀a, b ∈M, (.0.25)

defines a finite trace on M. We say that τ is faithful if τ(x) > 0 for any non-zero elementx ∈M+. A trace τ is said to be normal if τ(supxi) = sup τ(xi) for every bounded net (xi)i∈Iof positive elements in M, and semifinite, if, for any x ∈ M+, x 6= 0, there exists y ∈ M+,0 < y ≤ x, such that τ(y) < ∞. Traces play an important role in the classification of vonNeumann algebras. It can be shown that a von Neumann algebraM is a direct sum (or directintegral) of factors of type In and type II1 if and only if it admits a faithful finite normal trace;see [123]. Similarly, M is a direct sum (or direct integral) of type I, type II1 and type II∞factors iff it admits a faithful semifinite normal trace. We use these results in Section 5 tocharacterize the centralizer of a state ω.

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In Copyright - Non-Commercial Use Permitted Rights ......1 _S 2, oftwosystemsS 1 andS 2,characterizedbythesetsO S 1 andO S 2,andthealgebrasA S 1 and A S 2,respectively,canbedeﬁnedbychoosing