Miguel Carrion Alvarez- Loop quantization versus Fock quantization of p-form electromagnetism on static spacetimes

8/3/2019 Miguel Carrion Alvarez- Loop quantization versus Fock quantization of p-form electromagnetism on static spacetimes

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Loop quantizationversus

Fock quantizationof p-form electromagnetism

on static spacetimes

by

Miguel Carrion Alvarez

B.Sc. Physics (Complutense University of Madrid, Spain) 1998B.Sc. Mathematics (Complutense University of Madrid, Spain) 2000

M.Sc. Mathematics (University of California at Riverside) 2002

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

in

Mathematics

in the

GRADUATE DIVISION

of the

UNIVERSITY OF CALIFORNIA, RIVERSIDE

Committee in charge:Professor John C. Baez, Chair

Professor Michel L. LapidusProfessor Xiao-Song Lin

Fall 2004


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The dissertation of Miguel Carrion Alvarez is approved:

Chair Date

Date

Date

University of California, Riverside

Fall 2004


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Copyright 2004by

Miguel Carrion Alvarez


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Modify the copyright page to include a free documentation licence.


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Abstract




by

Miguel Carrion AlvarezDoctor of Philosophy in Mathematics

University of California, Riverside

Professor John C. Baez, Chair

Write this in a style appropriate for the arXiv

Professor John C. Baez

Dissertation Committee Chair


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A mis padres,

Pedro Carrion Lopez y Pilar Alvarez Ura.

que la sabran apreciaren su justa medida


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Contents

List of Figures vi

List of Tables vii

1 Introduction 1

I Electromagnetism 7

2 Classical vacuum electromagnetism 112.1 Geometric setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Technical definitions . . . . . . . . . . . . . . . . . . . . . . . 132.1.2 Spacetime geometry and topology . . . . . . . . . . . . . . . . 14

2.1.3 Issues of analysis on noncompact spaces . . . . . . . . . . . . 162.2 Maxwells theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.1 Overview of covariant mechanics . . . . . . . . . . . . . . . . . 222.2.2 Kinematical phase space . . . . . . . . . . . . . . . . . . . . . 24

Lagrangian formulation . . . . . . . . . . . . . . . . . . . . . . 24Hamiltonian formulation with a Lagrange multiplier . . . . . . 25Hamiltonian formulation without Lagrange multipliers . . . . 26

2.2.3 Dynamical phase space . . . . . . . . . . . . . . . . . . . . . . 262.2.4 Physical phase space . . . . . . . . . . . . . . . . . . . . . . . 28

2.3 Free and oscillating modes . . . . . . . . . . . . . . . . . . . . . . . . 292.3.1 The space of pure-gauge potentials . . . . . . . . . . . . . . . 30

2.3.2 AharonovBohm modes . . . . . . . . . . . . . . . . . . . . . 312.3.3 Decomposition into free and oscillating modes . . . . . . . . . 312.3.4 The oscillating sector . . . . . . . . . . . . . . . . . . . . . . . 322.3.5 The free sector . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 p-form electromagnetism in n + 1 dimensions 383.1 Spacetime Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 41


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3.2 p-Form electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . 423.3 Mathematical details . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 Hodgede Rham theory on noncompact manifolds 524.1 Cohomologies galore . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Known results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.3 Twisted L2 cohomologies . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3.1 (2 + 1)-dimensional axisymmetric spacetimes . . . . . . . . . . 584.3.2 Twisted L2 cohomology . . . . . . . . . . . . . . . . . . . . . 59

Square-integrable functions . . . . . . . . . . . . . . . . 60Square-integrable 1-forms . . . . . . . . . . . . . . . . . 60Square-integrable 2-forms . . . . . . . . . . . . . . . . . 61

The case of cylindrical spaces . . . . . . . . . . . . . . . . . . 61Old hyperbolic plane example . . . . . . . . . . . . . . . . . . 62

4.3.3 Simply-connected (2 + 1)-dimensional spacetime . . . . . . . . 63Cylindrical spaces . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.3.4 The Schwarzchild solution . . . . . . . . . . . . . . . . . . . . 65

II p-form Electromagnetism as a Free Boson Field 66

5 Coherent-state quantization of linear systems 705.1 The general boson field . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.1.1 Linear phase spaces . . . . . . . . . . . . . . . . . . . . . . . . 725.1.2 Quantizing a linear phase space . . . . . . . . . . . . . . . . . 745.1.3 Heisenberg and Weyl systems . . . . . . . . . . . . . . . . . . 745.1.4 Constructing Weyl and Heisenberg systems . . . . . . . . . . . 775.1.5 Representations of the general boson field . . . . . . . . . . . 83

5.2 Properties of the characteristic functional . . . . . . . . . . . . . . . . 865.2.1 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . 865.2.2 Probabilistic interpretation . . . . . . . . . . . . . . . . . . . . 875.2.3 The characteristic functional as generating functional . . . . . 875.2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.3 The free boson field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.3.1 Single-particle Hilbert space . . . . . . . . . . . . . . . . . . . 915.3.2 Wick products . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.3.3 Quasioperators . . . . . . . . . . . . . . . . . . . . . . . . . . 945.3.4 Quantizing linear dynamics . . . . . . . . . . . . . . . . . . . 95

5.4 Digression: general linear quantization . . . . . . . . . . . . . . . . . 965.4.1 Schrodinger quantization . . . . . . . . . . . . . . . . . . . . . 975.4.2 Fock quantization . . . . . . . . . . . . . . . . . . . . . . . . . 975.4.3 BargmannSegal quantization . . . . . . . . . . . . . . . . . . 97


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5.4.4 Coherent state representation . . . . . . . . . . . . . . . . . . 975.5 Fock quantization of linear systems . . . . . . . . . . . . . . . . . . . 97

5.5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97One-particle Hilbert space . . . . . . . . . . . . . . . . . . . . 98Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Field operators . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Field quantization . . . . . . . . . . . . . . . . . . . . . 101Time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.5.2 Schrodinger quantization . . . . . . . . . . . . . . . . . . . . . 103Real Hilbert space structure on configuration space . . . . . . 103Description of Fock representation . . . . . . . . . . . . . . . . 104Description of Schrodinger representation . . . . . . . . . . . . 104

5.5.3 Fock quantization . . . . . . . . . . . . . . . . . . . . . . . . . 105Phase space as a complex Hilbert space . . . . . . . . . . . . . 105Second quantization . . . . . . . . . . . . . . . . . . . . . . . 106

A representation of the Weyl algebra . . . . . . . . . . 106BargmannSegal representation . . . . . . . . . . . . . . . . . 107

5.5.4 Free part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.5.5 Isomorphism between Fock and Schrodinger representations . 109

6 Quantum electromagnetism 1106.1 C-algebraic quantization . . . . . . . . . . . . . . . . . . . . . . . . 112

6.1.1 Schrodinger quantization . . . . . . . . . . . . . . . . . . . . . 112

6.1.2 Loop quantization . . . . . . . . . . . . . . . . . . . . . . . . 1126.2 Fock quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.2.1 The space of one-photon states . . . . . . . . . . . . . . . . . 112Construction of the one-photon state space . . . . . . . . . . . 114Quantizing the oscillating modes of the electromagnetic field . 115

6.2.2 Electromagnetic Field Operators . . . . . . . . . . . . . . . . 1186.3 Coherent state quantization . . . . . . . . . . . . . . . . . . . . . . . 1196.4 Field quasioperators . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.4.1 Quantizing the vector potential . . . . . . . . . . . . . . . . . 1206.5 Vacuum maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . 122

6.6 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.7 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.8 Loop observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.8.1 Wick powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124


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III Loop Quantization ofp-Form Electromagnetism 127

7 Wilson Loops and the Loop representation 1297.1 Generalized measures on A/G . . . . . . . . . . . . . . . . . . . . . . 1317.1.1 Measures and linear functionals . . . . . . . . . . . . . . . . . 1317.1.2 Measures and C-algebras . . . . . . . . . . . . . . . . . . . . 1317.1.3 Generalized measures . . . . . . . . . . . . . . . . . . . . . . . 132

7.2 The U(1) holonomy C-algebra . . . . . . . . . . . . . . . . . . . . . 1327.3 States on A/G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347.4 The Fock sector ofA/G . . . . . . . . . . . . . . . . . . . . . . . . 134

7.4.1 Smearing map . . . . . . . . . . . . . . . . . . . . . . . . . . . 1367.4.2 Pushing forward the measure . . . . . . . . . . . . . . . . . . 1387.4.3 Equivalence of the representations . . . . . . . . . . . . . . . . 139

8 Comparison with the free boson field approach 1408.1 Gambini and Pullin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1408.2 Varadara jan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

9 Conclusion 141

Bibliography 142

A Comments 156A.1 July 28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

A.2 July 29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157A.3 August 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157A.4 August 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158A.5 August 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158A.6 September 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160


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List of Figures


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List of Tables


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Acknowledgments

[By way of preface]I am indebted in one way or another to the following people.Pilar Alvarez, porque madre no hay mas que una; Pedro Carrion, who taught

me to count (the rest just follows); Coral Duro, who introduced me to the FeynmanLectures on Physics at a tender age; Petra Solera, who sent me to the Math Olympiad;Luis Vazquez, who sent me on my Erasmus exchange; Guillermo GarcaAlcaine, whotaught me quantum mechanics; Antonio Dobado, who taught me high-energy physics;Lee Smolin, who suggested that I study with John Baez; Jose Gaite, who supervisedme on my first serious research; Fernando Bombal, who taught me functional analysis;Miguel Martn Daz, who taught me probability theory; Ignacio Sols, who tried toteach me things about algebra that I had to rediscover on my own years later; JohnBaez, the best advisor this side of the Virgo cluster; Fotini Markopoulou, who wasinterested enough in my research to invite me to PI; and Barbara Helisov a, who isjust wonderful, and wonderfully patient too.

Without them I would never have come this far.Question: can I have a preface (by which I mean a personal statement

about the meaning of it all)?


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Chapter 1

Introduction

Write this, but only when youre almost done.Following ideas of M. Varadarajan, we construct a loop representation for the

quantized vacuum Maxwell equations on any globally hyperbolic static spacetime,and set up an equivalence between the resulting Hilbert space and the usual Fockspace of photon states. (UNDER CERTAIN CONDITIONS). We do this for boththe version of Maxwell theory with gauge group R and the version with gauge groupU(1). In the process we encounter a number of interesting subtleties concerning theBohmAharanov effect. In the 3+1-dimensional case our work mainly uses the L2

de Rham complex, but in other dimensions we need a generalization which involvesthe gravitational potential, that is, the function appearing in the spacetime metric

g = e2dt2 + gS.Following ideas of M. Varadarajan, we construct a loop representation for the

quantized vacuum Maxwell equations on any globally hyperbolic static spacetime,and set up an equivalence between the resulting Hilbert space and the usual Fockspace of photon states. We do this for both the version of Maxwell theory withgauge group R and the version with gauge group U(1). In the process we encountera number of interesting subtleties concerning the BohmAharanov effect. In the(3 + 1)-dimensional case our work mainly uses the L2 de Rham complex, but in otherdimensions we need a twisted version of this complex, in which the differential involvesthe gravitational potential. Our account is pedagogical and designed in part to fill

some gaps in the literature.Mention [AERss]Long introduction, in English!Whats new:

1. Using twisted L2 cohomology to understand p-form EM on arbitrary noncom-pact spaces.

(a) BohmAharonov

(b) Concept of gauge equivalence


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2. Careful treatment of infrared divergences

3. Wilson loops as quasioperators; wilson loop quasistates.

4. Focus on coherent states, down with creation/annihilator operators!

Everything you think you know about Electromagnetism is wrong.Every interesting equation is a lie. In physics, the most pervasive such lie is

that vector spaces are equal to their duals.The great virtue of loop quantum gravity is that it is a manifestly background-

free theory. In other words, it does not describe gravitational degrees of freedom asperturbations about a fixed background metric. Instead, the formalism is explicitlydiffeomorphism-invariant at every step. Unfortunately, it is difficult to say precisely

how the notion of graviton arises in this formalism. At least at the kinematicallevel, the fundamental description of quantum states is not in terms of gravitons,but rather spin networks. Still, one wishes to see how gravitons emerge as a usefulconcept in the semiclassical limit. A general strategy for doing this has been knownever since 1992 [ARS92], when Ashtekar, Rovelli and Smolin formulated the idea ofweave states which approximate solutions of classical general relativity. Since thenthere has been a great deal of work on states of this sort [coh]. However, a fullsolution has not yet been achieved, mainly because the problem involves not onlykinematical issues, but also dynamics. Ideally one would like exact solutions of theHamiltonian constraint which serve as approximations to n-graviton states, but thiswould require a better understanding of the Hamiltonian constraint.

Indeed, until recently we were in the embarrassing situation of not even knowingthe precise relation between the loop representation of electromagnetism and the usualFock representation. Here, of course, the theory is linear and formulated on a fixedbackground metric, which drastically simplifies the situation. The technical problemis that n-photon states do not lie in the Hilbert space L2(A/G, ), where A/G is theappropriate space of connections modulo gauge transformations and is the naturaldiffeomorphism-invariant generalized measure on this space [AI92]. The reason isthat while this Hilbert space is designed for background-free theories, the photonFock space depends crucially on a background metric. In particular, with respect tothe Fock vacuum, the photon 2-point function blows up at short distances at such a

rate that Wilson loops are not well-defined operators on Fock space. However, theyare well-defined operators on L2(A/G,). This shows the drastic difference betweenthe loop representation and the Fock representation. Indeed, it almost seems tosuggest an unbridgeable gap between the two.

Recently M. Varadarajan [vara] solved this problem by finding a family of gen-eralized measures c such that L

2(A/G, c) is unitarily equivalent to the usual Fockspace for vacuum electromagnetism on Minkowski spacetime. These measures dependon the metric and also on a smearing parameter c > 0. To interpret a Wilson loopas an operator on L2(A/G, c), Varadarajan first convolves this loop with a Gaussian


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whose variance depends on c. This smearing eliminates the ultraviolet divergencesdue to the blow-up of the photon 2-point function. This trick does not set up a uni-

tary equivalence between Fock space and the diffeomorphism-invariant Hilbert spaceL2(A/G, ); that is impossible. However, it does put the two in a common framework,permitting a precise comparison between them (???).

In this paper we extend these ideas to an arbitrary static globally hyperbolicspacetime that is, any manifold M = R S with a metric

gM = e2dt2 + gS = e2(dt2 + g)

such that no lightlike geodesics run off to infinity in a finite amount of coordinate time.This is equivalent to g making S complete. For any constant c > 0, we construct ageneralized measure c on A/

Gand a unitary equivalence between L2(A/

G, c) and

the Fock space for electromagnetism. While our work is of limited relevance to thedeeper problem of describing gravitons in loop quantum gravity, we hope it shedssome light on the relation between loop quantization and the Fock representation.

Technically, the subtlest aspects of our work arise from the function . Thisfunction measures the time dilation due to the gravitational field, and reduces to theNewtonian gravitational potential in the limit 0. When = 0, our constructionsuse rather familiar mathematics, mainly this portion of the L2 de Rham complex:

L20d L21 d L22

where L

2

p

stands for the Hilbert space of square-integrable p-forms on S. In par-ticular, the right generalization of Gaussian convolution is heat-kernel regularization,where the heat equation is defined using the Laplacian on 1-forms, dd+dd. The case = 0 requires some less familiar mathematics except in 3+1 dimensions, wherethe conformal invariance of Maxwells equations allows us to eliminate by rescalingthe metric. In general, the most efficient approach is to replace the L2 de Rhamcomplex by one involving the function , as follows:

L20d L21 d L2+2

where L2

p is the Hilbert space of p-forms on S with (

|) 1, a > |n 2p| and a 1, thereare complete Riemannian manifolds diffeomorphic to Rn, with curvature boundedby a2 K 1 and such that their pth square-integrable cohomology Hp2 is infinite-dimensional.

For rotationally symmetric n-dimensional manifolds with metric

ds2 = dr2 + f(r)2d2,


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where d is the standard metric on Sn1, the square-integrable cohomology is Hk2 ={0} if k = 0, n/2, n. As we know, when k = 0, n the cohomology depends on thevolume of spacetime. Finally, when k = n/2, H

k2 = {0} if dsf(s) = , and infinite-

dimensional otherwise. This is because of conformal invariance, and the fact thatconvergence of the integral correlates with conformal compactness [Dod79].

explain "conformally compact", and introduce MMazzeo [Maz88] proves that a complete conformally compact n-dimensional Rie-

mannian manifold has finite-dimensional cohomology except possiblly for the middledimensions, and gives a topological interpretation of them:

Hk2

Hk(M,M,R) k < (n 1)/2Hk(M,R) k > (n + 1)/2

where M is the conformally equivalent compact manifold, M is its boundary, playingthe role of an ideal boundary at infinity of the original noncompact manifold.Moreover, if a2 is the maximum limiting curvature at infinity, then the essentialspectrum of the Laplacian k is

ess(k) =

[a2(n 2k 1)2, ) k < n/2{0} [a2/4, ) k = n/2[a2(n 2k + 1)2, ) k > n/2

In particular, if n = 2k, the kth cohomology group is infinite-dimensional and, if|n 2k| 1, the essential spectrum extends all the way to 0. For hyperbolicmanifolds which are geometrically finite (i.e., having no tubular ends), Mazzeo andPhillips [MP90] prove that the cohomology of the middle dimensions k = (n 1)/2 isfinite-dimensional and has a topological interpretation. These results are extended byLott [Lot97] to the case of hyperbolic 3-manifolds which are diffeomorphic to the in-terior of a compact manifold with boundary and geometrically infinite. In particular,Lott proves that, if such a space is nice (has incompressible ends and its injectivityradius does not go to zero at infinity), the kernel of the Laplacian on 1-forms is finite-dimensional. He also provides a variety of results on the spectrum of the Laplacianon 1-forms.

These results have a direct physical interpretation when p-form electromagnetismis conformally invariant, as otherwise one has to consider an appropriately twisted L2

cohomology complex for which there are no known general results. It is reasonable toassume, though, that the behaviour of the twisted cohomology will be at least as richas that of the ordinary L2 cohomology. In the conformally invariant cases, we have

the massless scalar field (0-form electromagnetism) in 1 + 1 dimensions. In thiscase, since the space manifold Sis assumed to be noncompact, it is diffeomorphicto R. Global hyperbolicity then requires that the optical metric give S infinitelength, and so H02 = {0} because the constant field is not square integrable. Inother words, square-integrable fields must go to zero at infinity.


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ordinary (1-form) electromagnetism in 3 + 1 dimensions. If space is spher-ically symmetric there are no harmonic, square-integrable 1-forms according

to [Dod79]. This is not a surprise since the first de Rham cohomology is alsotrivial. In more general cases, if the space manifold S is conformally compactthe spectrum of the Laplacian reaches all the way to 0 (physically, the photondoes not acquire a topological mass), but the dimension of the kernel of theLaplacian is not known in general. Andersons example [And85] shows that itis possible for this to be infinite-dimensional.

when p-form electromagnetism is conformally invariant the dimension of spaceis p = (n1)/2, and we are always in one of the middle dimension cases wherethe dimension of the space of harmonic vector potentials remains unresolved,although for a large class of manifolds it is known that the essential spectrum

of the Laplacian is all of [0, ) and so there is no topological mass gap.

4.3 Twisted L2 cohomologies

In this section we study the twisted L2 cohomologies for various spherically sym-metric spacetimes. We work out in detail the cases of 2-dimensional disks and annuli,and of the 3-dimensional spatial geometry of the Schwarzschild black hole.

summarize conclusions

4.3.1 (2 + 1)-dimensional axisymmetric spacetimesFor this family of examples we consider static, globally hyperbolic, spacetimes

with noncompact, rotationally symmetric space manifold. In other words, we assumethat the metric can be put in the form

gM = e2(r)[dt2 + dr2 + e2(r)d],

where t R, [0, 2) is an angular coordinate and the range of r and the form of are restricted by global hyperbolicity, which is equivalent to the completeness ofthe space metric

g = dr2 + e2(r)d2 with r < r < r+.

Assuming completeness of this metric, the space S can have the topology of

a disk, if one of the limits of r is finite and the other infinite (without loss ofgenerality we can assume r = 0 and r+ = ), and

limr0+

1

re(r) = 1;

or


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a spherical shell, if r = .In first case we have a contractible space whose first cohomology vanishes in thesmooth case, and is possibly infinite-dimensional in the L2 case (as in the case ofthe hyperbolic plane); in the second case, we have a nontrivial first de Rham co-homology generated by d, but the first L2 cohomology may be either trivial orinfinite-dimensional depending on whether harmonic forms of middle dimension blowup at infinity.

(but not finite-dimensional [Dod79])The situation for the twisted cohomologies is slightly more complicated.

4.3.2 Twisted L2 cohomology

There are essentially two questions we need to answer in order to determinethe space of D-harmonic 1-forms on S. First, is the non-exact f(r)d closed andsquare-integrable for any f? Second, are there any functions f which are not square-integrable but such that Df is square-integrable? The answer to these questionsdepends on the interplay of and at spacelike infinity.

Because the dimension of space is N = 2, we define D = e/2de/2. Also,vol = edr d, and

g(1, 1) = g(dr, dr) = e2g(d,d) = e2g(dr d,dr d) = 1.

We write all differential forms on Sas linear combinations (over kZ) of the standard

forms

fk = eik/2f

k = eik(e/2rdr + e/2d)

k = eik+/2+kdr d

with all functions depending only on r. Then,

Dfk = eik/2(rf dr + ikf d)

Dk = eik(e/2r

ike/2r)dr

d

and

Dk = eik(e/2rr + ike/2)Dk = eik+/2(ikekdr erkd)


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k = 0, in which case ik = Deik/2. These 1-forms do not contribute tothe twisted de Rham cohomology, but they do contribute to the 1st twisted L2

cohomology if

(eik/2 | eik/2) = 2

e||2dr

diverges and

(Deik/2 | Deik/2) = (e/2deik | e/2deik) = 2

[e|r|2+k2e

converges.

Square-integrable 2-forms A 2-form eik(r)dr d has norm

(dr d | dr d) = 2

e||2dr.

The question is whether there are any square-integrable 2-forms of the form Deik/2[r(r)dr+(r)d] = e

ik/2[r ikr]dr d such that eik/2[r(r)dr + (r)d] is notnormalizable. In other words, we need

2

[e|r|2 + e||2]dr

to diverge, but

2

e|r ikr|2dr

to converge.

The case of cylindrical spaces

In this case S homeomorphic to a cylinder, and the spacetime metric is

gM = e2[

dt2 + dr2 + e2d2], with t

R, r

R,

[0, 2),

The optical metric g = dr2 + e2d2 is also conformally flat. In this case we needto solve du = edr to get

g = e2[du2 + d2]. with u (u, u+).

The range ofu is finite or infinite depending on the behaviour of the integral e

(r)drat either limit.


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Old hyperbolic plane example

is an interesting example because the hyperbolic plane is contractible so it hastrivial de Rham cohomology but, according to the quotation above, its 1st L2 coho-mology is infinite-dimensional. This means that ker(d + d) is infinite-dimensional.

The hyperbolic plane H is conformal to the open unit disk D, with metric

ds2H =ds2D

(1 r2)2 on x21 + x

22 = r

2 < 1

and volume form

volH =volD

(1 r2)2

(ds2

D and volD denote the usual metric and volume form on the unir disk ofR2

).Differential forms are independent of the metric, so we can write them as differentialforms on the unit disk. The exterior derivative is also independent of the metric,dH = dD, so the conformal rescaling between D and H only comes in through theinner product ( | ) and the codifferentials H and D. We have

((1 r2)f | (1 r2)f)H = (f | f)D for f 0(S)( | )H = ( | )D for 1(S)( | )H = ((1 r2), (1 r2))D for 2(S).

It follows that

(H, )D = (H, )H = (, dH)H = ((1 r2), (1 r2)dD)D = (D(1 r2)2, )D, and(H | f)H = ( | dHf)H = ( | dDf)D = (D | f)D = ((1 r2)2D | f)H

soH = (1 r2)2D: 1 0 and D(1 r2)2 = H: 2 1.

Therefore, the conditions dHA = 0 and HA = 0 on a 1-form A are equivalent todDA = 0 and DA = 0.

Assume now that A is closed. Then, 0 = dHA = dDA, so A = dDf = dHf, sinceD is contractible. If A is also coclosed, 0 = HA = HdHf or (1 r2)2DdDf = 0, sof is harmonic. Now, any harmonic function f has (f

|f)H =

. [Sketch of proof:

since f is harmonic is has the mean-value property and the maximum principle.This means that

|x|=r f

2 is monotonic in r. But f2H = f1r2 2D, so f2Hdiverges like

rdr

(1r2)2 .]In conclusion, the relative (compactly-supported) smooth cohomology and the

absolute (de Rham) smooth cohomology of the hyperbolic plane are both trivial,just like the cohomology of the closed unit disk, but the L2 cohomology is infinite-dimensional because closed 1-forms are the differentials of non-normalizable 0-forms(so they are de Rham-exact but not L2-exact).


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4.3.3 Simply-connected (2 + 1)-dimensional spacetime

In the case where S homeomorphic to a disk, the spacetime metric is

gM = e2[dt2 + dr2 + e2d2], with t R, r R+, [0, 2),

and

limr0+

1

re = 1.

We will exploit the fact that the optical metric on space g = dr2+e2d2 is conformalto a flat disk D of radius u+. Indeed, solving the differential equation

du

u= edr with u = 0 at r = 0,

we get

g =e2

u2[du2 + u2d2] with u [0, u+)

the function satisfies the conditions

limu0+

1

ue = 1 and

u+0

edu

u= ,

which determines the value of u+ (if this is finite, the optical metric is actuallyconformal to a flat, compact disk). We denote gD = du

2 + u2d2 = dx2 + dy2 (D for

disk). The volume form associated to the optical metric is

vol = edr d = e2

udu d = e

2

x2 + y2volD.

For the purposes of p-form electromagnetism, we define

Dk = e 12 (2p1)dke

12 (2p1).

according to equation (3.3). We now calculate the twisted and untwisted relative L2

cohomologies. The untwisted cohomology corresponds formally to p = 12

.

The only nonzero solutions ofD0f = 0 are proportional to e12 (12p), which shows

that the zeroth de Rham cohomology is one-dimensional. Now, this function hasnorm

(e12 (12p), e

12 (12p)) =

e(12p)vol =

e(12p)+2

u2volD

If this integral diverges, the zeroth twisted L2 cohomology is trivial.The first de Rham cohomology of S is trivial, since S is contractible, but the L2

cohomology may be nontrivial. Elements of the first twisted L2 cohomology are given


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by square-integrable 1-forms Df such that f is not square-integrable. In other words,they can be obtained from functions f = e

12 (12p)g such that

(f, f) =g2e(12p)+2

u2vol =

and

(Df,Df) =

e(12p)g(dg, dg)vol =

e(12p)gD(dg, dg)volD < .

Cylindrical spaces

Consider a metric of the form

ds2 = (f(r)dr)2 + (g(r)d)2, r0 < r < r1,

where is an angular coordinate, and the manifold is complete if, and only if,r1r0

|f(r)|dr diverges at both limits. The change of variables dv = f(r)drg(r)

yields

ds2 = k(v)2(dv2 + d2), v0 < u < v1,

where k(v) = g(r), and v1v0

|k(v)|dv =r1r0

|f(r)|dr.

The volume form of this space is vol = k(v)2dv d = f(r)g(r)dr d, so the totalarea of the manifold is

2

v1v0

k(v)2dv = 2

r1r0

f(r)g(r)dr.

Note that for the manifold to be complete we need h to diverge at any limit which isnot infinite. This means that the area will be infinite unless v0 = and v1 = .But then the space is conformal to an infinite straight cylinder, and this implies thatthere are no square-integrable harmonic 1-forms in a complete cylindrical space of

finite area. If [v0, v1] is a finite interval this is a conformally compact 2-dimensionalmanifold of infinite area, and we are in the same situation as in Example 4.3.3, wherethe L2 cohomology is infinite-dimensional. The surprising conclusion is that, if dis square-integrable, then there is an infinite-dimensional space of square-integrableharmonic 1-forms. It is impossible to have just the representative of the nontrivialfirst de Rham cohomology class.


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4.3.4 The Schwarzchild solution

The Schwarzschild metric can be put in the isotropic form

gM = 1 M/2

1 + M/2

2dt2 + (1 + M/2)4(d2 + 2(d2 + sin2 d2)) =

=1 M/2

1 + M/2

2dt2 +

(1 + M/2)31 M/2

2(d2 + 2(d2 + sin2 d2))

,

for > M/2, in which the constant-time surfaces are explicitly shown to be confor-mally flat. It is easy to check completeness by verifying that the integral

M/2

(1 + M/2)3d

1 M/2diverges at both limits.

In 3+1 dimensions we do not need to twist the cohomology complex by powers ofthe lapse function 1M/2

1+M/2, so D = d. The zeroth L2 cohomology consists of constant

square-integrable functions, but there are none since the integral

(1 | 1) = 4M/2

(1 + M/2)31 M/2

32d = 4

0

(u + M)9du

u3(u + M/2)4

diverges at both limits. Therefore L2H0 = {0}.12M/

(1+2M/)p

The first de Rham cohomology is trivial, so the only question is whether there arefunctions f such that df is square-integrable but f isnt. Assume f depends only on. Then, we require

(f | f) = 4M/2

(1 + M/2)31 M/2

3f22d =

but

(df|

df) = (f d|

f d) = 4

M/2

(1 + M/2)3

1 M/2(f)

22d = 4

0

(u + M)3

u(uf)

2du


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Part II

p-form Electromagnetism as a FreeBoson Field


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Improve the introduction to this part, but only after youre done writing

it.

The apparent truism that a quantum mechanical theory needs to be cast in clas-sical language in order to correlate its predictions with our experience, a point thatNiels Bohr made into a cornerstone of his philosophy of quantum mechanics, has prac-tical consequences for the development of quantum descriptions of physical systems.This is because a physical system will be described operationally or geometrically ininevitably classical terms, and this information needs to be fashioned into a quantumtheory whose predictions need to be, again, reexpressed in classical terms. In addi-tion, the process of constructing a classical theory from operational or geometric datais so well-understood that it is convenient to construct the quantum theory by firstconstructing a classical theory from the data and then quantizing it.

Quantization is a catch-all term for any process taking as input a classical me-chanical system, and producing as output a quantum mechanical system reducing tothe original classical system in an appropriate limit. Quantization would ideally bealgorithmic or functorial, but it turns out to be neither, although formulating quan-tization in algebraic language seems to bring it closest to the goal of functoriality.

In algebraic terms, a classical mechanical system is defined by specifying a Poissonalgebra of observables, while any associative algebra can play the role of algebraof observables for a quantum system. The Dirac quantization prescription [Dir57,Chapter IV] promotes the commuting classical observables to operators satisfyingthe Heisenberg commutation relations

[f , g] = i{f, g},where {f, g} is the Poisson bracket of the classical observables f and g, [f , g] is thecommutator of their quantum counterparts, and Plancks constant measures thedeparture from classical behaviour (where observables commute). It is not hard toconvince oneself that, because the algebra of quantum observables is nonabelian, theoperation f f cannot be an algebra homomorphism. That is, f g = fg in general.Physicists call this fact operator ordering ambiguities.

An operator algebra of quantum observables realizing the canonical commutationrelations achieves quantization in a kinematical sense, but the physical and dynam-ical content of the theory comes about by means of a specific representation of the

quantum observables as an algebra of (unbounded) linear operators on a Hilbertspace of quantum states. Each representation is associated to a choice of vacuumexpectation on the algebra of observables and it is known that, for systems withinfinitely many degrees of freedom, different states may lead to unitarily inequivalentrepresentations. The choice of representation can be narrowed down by the need torecover an appropriate classical limit, and by requiring that physical symmetries beimplemented unitarily.

The classical limit is encoded in the correspondence principle, by which we meanthe following. The Poisson algebra of classical observables consists of smooth func-


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tions on a symplectic manifold (phase space) playing the role of state space for theclassical theory. The correspondence principle requires that, for any phase space

point x P and any observable f, there should be a quantum state |x such that theexpected value of f in the state |x equals the classical value f(x), if not exactly, atleast in the limit 0. That is,

x| f|x = f(x) + O().

There is one last requirement that a sensible quantization must satisfy, and thatis that physical symmetries be represented by unitary operators on the Hilbert spaceof quantum states of the system.

In the case where the classical phase space is a vector space, the linear observablescan be identified with the points of the phase space itself, and so the Heisenberg

commutation relations can be implemented on the phase space. In chapter 5 wedevelop the quantization of an abstract linear system, and in chapter 6 we apply thisto Maxwells equations for the electromagnetic field.

In Section 6.2 we do the Fock quantization of vacuum electromagnetism. We alsoconsider Schrodinger quantization, which is equivalent, but serves as a halfway housebetween Fock and loop quantization. In Section 7 we do the loop quantization of thetheory and show that this is equivalent to the Schrodinger quantization. Finally, inSection ?? we describe how the results are affected by using the gauge group U(1)instead ofR.

Dimock [Dim92] also constructs the Weyl C-algebra, but does not exhibit any

states or Hilbert-space representations. He notes in passing that a Fock space repre-sentation of the canonical commutation relations is assumed to exist.

Because Dimock describes the classical theory in the covariant canonical formal-ism, he is forced to focus on the algebraic structure of the theory, not in the spec-ification of particular states. In our terms, Dimock quantizes the electromagneticfield as a General Boson Field. Dimock also constructs a Poisson bracket, and hisquantization procedure is equivalent to our general linear quantization.

The limitations of Dimocks work include:

He claims that in any case such [Hilbert-space] representations exist, say by aFock space construction. We discuss the ways in which this may fail.

Dimock shows that different Hilbert-space representations lead to -isomorphic C-algebas of observables. This form of equivalence obviates the consequences ofunitary inequivalence of Hilbert-space representations, and for this reason Di-mocks paper suffers from what Earman and coauthors critically term algebraicimperialism in [AERss].

Dimock also stating without proof or reference that for globally hyperbolicmanifolds, the usual classical linear field equations will have global solutions


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if they are well-behaved locally. We quote the result for scalar fields. Werepair this defect by reference to Chernoffs work in chapter ??. In the proof

of existence of solutions with given Cauchy data Dimock states The equation[above] has principal part g and thus is strictly hyperbolic, which easilyfollows from Chernoffs work.

Dimock assumes a trivial U(1)-bundle saying presumably our results can beextended to non-trivial bundles for which A is only defined locally, while wetake the more drastic step of assuming an R-bundle.

The phase space constructed by Dimock does not have a topology other thanthat induced by imposing the continuity of the symplectic structure. Therefore,it is not a real inner-product space like ours is. It is questionable whether that

is strictly necessary.

Dimock does not show that the classical canonical transformations associated tochanges in Cauchy surface are implemented unitarily on the C-algebras of quantumobservables, because that is simply not true. In fact, Torre and Varadarajan [TV]show that, even in the case of free scalar fields on a flat spacetime of dimensionhigher than two, there is no unitary transformation between the Fock representationsassociated to the initial and final Cauchy surfaces. They point out that unitaryimplementability is easily obtained if the Cauchy surfaces are related by a spacetimeisometry, though. They mention related results of Helfer (no unitary implementationof the S-matrix if the in and out states are Hadamard states) [Hel96], and of

van Hove (only a small subgroup of the classical canonical transformations is unitarilyimplementable) [vH51].

Another paper addressing specifically the quantization of the electromagnetic fieldis the one by Corichi [Cor]. Corichi stresses that Fock quantization depends cruciallyon the linear structure of phase space, and characterizes the Fock quantization pro-cedure as completely elementary.

Here we perform Fock quantization of Maxwells equations on a static, globallyhyperbolic spacetime with a trivial R bundle on it. Presumably this can be extendedto stationary spacetimes, but not beyond that because of the need for a nontrivialgroup of isometries. The treatment of nontrivial or U(1) bundles should require only

straightforward modifications, but one of the lessons of our work is that sometimesthere are surprises in store even for topics as well-understood as electromagnetism.

Because of the appearance of negative powers of in the sequel, we will be forcedto assume that ker = {0}, especifically for on L21. Although, for mathematicalconvenience, one often assumes that > 0 for some , we will not exclude thepossibility that the spectrum of reach all the way to 0 because that is the case inphysically interesting situations such as Minkowski space.


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Chapter 5

Coherent-state quantization oflinear systems

In this chapter we present a rigorous framework for quantization of linear dynamicsbased on the ideas of Irving Segal.

Segal pioneered the idea of of formalizing quantum mechanics in terms of algebrasof observables, making Hilbert spaces play the subordinate role of supporting linearrepresentations of them. These Hilbert spaces can, in fact, be constructed from theabstract algebra of observables by means of the GelfandNamarkSegal constructionusing a single state or, in physics parlance, vacuum expectation.

Implicit in the work of segal is a concept of general boson field associated to any

linear phase space, which formalizes the Heisenberg commutation relations amongfield operators in terms of exponentiated field operators, using the so-called Weylrelations . This has the advantage of avoiding the technicalities of unbounded opera-tors. In addition, physical symmetries are readily implemented as automorphisms ofthe Weyl algebra.

Segal introduced the related concept of free boson field, which can be constructedfrom a phase space equipped with a compatible complex structure. Segals free bosonfield axiomatizes the properties of the usual of Fock space, and the axiomatic approachmakes it transparent that the Fock, Schrodinger and BargmannSegal representationsof linear quantum fields are all unitarily equivalent. Within this framework, Segal

also studied the problem of representing time evolution unitarily on Fock space, andthe stability the generator of unitary time evolution, the quantum Hamiltonian.Here we put together both ideas, and the result is a new construction of the free

boson field based on coherent states . In this construction we not only associate toeach linear functional on phase space a field operator but, given a choice of vacuumstate, we can associate to each point in phase space a coherent state. The collectionof all coherent states indexed by points of phase space spans the Hilbert space ofquantum states of the theory, and the result is what Segal called the general bosonfield. The free boson field, which as we have mentioned is unitarily equivalent to the


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Fock representation, is obtained by means of a GNS state with Gaussian statistics.We find that the mathematical process of quantization can be understood with

reference to three physical guiding principles: the canonical commutation relations,the correspondence principle, and the unitary implementation of physical symmetries.

We proceed as follows: we first construct the Weyl algebra of observables asso-ciated to a linear phase space, and then choose a compatible complex structure onthe Phase space (actually, and equivalently, select a vacuum expectation on the Weylalgebra) with the help of the correspondence principle and the requirement that timeevolution be unitarily and stably implemented.

Coherent states are great because many classical equations hold exactly betweenexpectation values on coherent states. Thus, by using coherent states, our quanti-zation procedure never loses sight of the correspondence principle. In addition, thevacuum expectation value acts as a generating function of the matrix elements of fieldoperators between coherent states, not only for ordinary field operators but also fortheir Wick powers (called normal-ordered operators in physics). As an unexpectedbonus, using matrix elements between coherent states one can define normal-orderedWilson loops as quasioperators without the need for regularization.

Segals treatment of the free boson field is presented in [BSZ92]. A comprehensivephysical treatment of the coherent states of the electromagnetic field can be foundin [MW95, Chapter 11].


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5.1 The general boson field

The development that follows may seem idiosyncratic to those familiar with thetraditional quantization methods and the notations used in physics. In particular,we insist on distinguishing the phase space P from its dual P. There are somegood reasons for this. At the present stage of development of mathematical physics,the most compelling reason for studying the quantization of a linear systems is as aspringboard for quantization of nonlinear systems, or as a testing ground for ideassuggested by the study of nonlinear systems. Our approach is motivated by the factthat the ordinary quantization of linear systems makes use of several identificationsthat can only be made for a linear system. Adopting the view that a classical me-chanical system is characterized by its Poisson algebra of observables, the cotangentspace at each point of phase space acquires a symplectic structure. When the phasespace P is linear, the following identifications can be made: the dual P can be iden-tified with the linear observables, and the restriction of the Poisson bracket to P isa symplectic structure. Also, the cotangent spaces to each point of phase space arecanonically isomorphic to each other and to P, and the globally-defined symplecticstructure on P makes P isomorphic to P and also edows it with a symplectic struc-ture. All of these identifications, and even the possibility of considering P to be asymplectic space, are accidents of linearity. Accordinly, we will avoid making use ofthese features as much as possible. Every time we are forced to make use of one ofthese identifications, it will be a sign that the procedure cannot be readily generalizedto nonlinear situations.

5.1.1 Linear phase spaces

We start by formalizing the notion of linear phase space, which is the necessaryclassical input of our quantization procedure.

Definition 3 (linear phase space). A linear phase space is a reflexive real topo-logical vector space P whose dual P is a symplectic vector space. That is, P isa topological vector space equipped with a symplectic structure: a continuous, skew-symmetric bilinear form which is weakly nondegenerate in the sense that the dualitymap

: P Pf f = (f, )

is injective.

Notes. Without the assumption that P is reflexive, the duality map would be : P P. This would have a bearing on the definition of the Hilbert space of quantumstates below.


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A finite-dimensional vector space has a unique Hausdorff topology, and any infinite-dimensional vector space can be topologized algebraically1 [BSZ92, 1.2]; in eithercase the continuity of is vacuously true. In general, the dual P of a topologicalvector space is itself naturally a topological vector space, with the weak- topologymaking every element of P a continuous linear functional on P. If P has a normedtopology, P can also be given the (normed) strong operator topology. In eithercase, P P is a continuous inclusion.

The phase space is the space of states (gauge equivalence classes of solutions ofthe equations of motion) of a classical system, and the smooth functions on it arethe (gauge-invariant) observables of the system. The phase space P is naturally aPoisson Manifold. Symmetries of the physical system are represented by smooth mapspreserving the Poisson structure of P. Continuous symmetries can be generated by

Poisson brackets with appropriate observables, the conserved quantities associated tothe symmetries via Noethers theorem. A symmetry of the field configuration x P isa Poisson map leaving x fixed, and induces a linear symplectic transformation ofTxP.When the equations of motion are linear, one can take P = P in definition 3, andrestrict ones attention to linear observables and symmetry transformations. In thenonlinear case, it is still possible to obtain from P a linear phase space satisfying thedefinition, but there are some subtleties involved.

Suppose, then, that the phase space P is a Poisson manifold, meaning that CPis a Poisson algebra with Poisson bracket { , }. The Poisson bracket defines a bivec-tor : 2P R given by

(df, dg) = {f, g} for all f, g C(P).

For all x P, this bivector defines a symplectic structure x on TxP. In physicalterms, x is a field configuration and TxP is the space of linear observables in thevicinity of this field configuration. This is the only symplectic vector space that canbe constructed in a natural way from the phase space P, and definition 3 applieswith P = TxP and = x.

Identifying all the TxP amounts to putting a connection on P, and this identifi-cation is natural only if P is a linear space admitting a canonical flat connection. Inthat case, each of the TxP is canonically isomorphic to P

itself.

1Why does Segal define topologized algebraically by saying that a set is open iff its intersectionwith every finite-dimensional subspace is open?. This way of defining it is a little awkward to me,although I am sure Segal must have had a reason for it. Is this not just the weak topology obtainedby demanding that every linear functional be continuous? I have a nagging suspicion that it mightactually be a stronger topology than the weak one.

The reason was probably just that he wanted to define it by saying what the open sets are. Itsounds plausible, but I dont instantly know. Hmm, maybe we should lay that suspicion to rest.

FIGURE IT OUT


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5.1.2 Quantizing a linear phase space

Linear quantization is a process promoting each x

P to a unit vector|x

in asuitable Hilbert space K, and each f P to a self-adjoint operator f on K, in sucha way that the Heisenberg commutation relations

[f , g] = i(f, g)1K for all f, g P. (5.1)

hold. Equation 5.1 is a restricted form of the Dirac quantization prescription, since itis applies only to linear observables on P, and not to arbitrary ones it was originallyformulated. In addition, the correspondence principle is required to hold in the form

x| f|x = f(x) for all x P, f P. (5.2)

Finally, one would hope to represent every physical symmetries T: P P as a uni-tary operator UT: K K in such a way that USUT = UST for all linear symplecticmaps S, T: P P. As we shall see, in general this is only possible for a subgroupof symplectic transformations of P and, in fact, choosing a small subgroup of physi-cal symmetries that must be unitarily implemented can be enough to determine K,sometimes uniquely. Time evolution is always required to be a physical symmetryand, in this sense, the dynamics determine the quantization.

5.1.3 Heisenberg and Weyl systems

The Heisenberg relations cannot be implemented on an algebra of bounded op-erators [Rud91, 13.6], and so equation 5.1 must be understood as holding on the(hopefully) dense domain of [f , g] in K. This is only the first of a long list of nui-sances that arise from necessarily dealing with unbounded operators, but all the samewe encode it as a definition.

Definition 4 (Heisenberg system). A Heisenberg system on a symplectic vectorspace (P, ) is a real-linear map : f (f) from P to the self-adjoint operatorson some complex Hilbert space K, satisfying the Heisenberg commutation relations

[(f), (g)] = i(f, g)1K. for all f, g

P

as an operator equation holding on the common domain of (f)(g) and (f)(g),which is assumed to be dense. The operator (f) is called the Heisenberg operatorassociated to f P.

In other words, linear quantization is partially achieved by constructing a Heisen-berg system on the space of linear observables (P, ). However, there are lots ofHeisenberg systems that have nothing to do with physics, examples of which canbe found in [MR80, BSZ92], so for honest quantum physics one needs to impose


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Sketch of proof. Differentiating the Weyl relation

W(tf)W(tg) = e

t2(f,g)/2i

W(t(f + g))

twice and setting t = 0, one obtains that f (f) is linear and satisfies the Heisen-berg commutation relations

[(f), (f)] = i(f, f)1K.

The proof of the closure properties of the Heisenberg operators is in [BSZ92, 1.2].At this point, a theorem of von Neumann [MR80, VIII.5] guarantees that all Weyl

systems on a finite-dimensional phase space are unitarily equivalent. At any rate, wesee that Weyl systems are the right formalization of equation (5.1), the Heisenbergcommutation relations. The following lemma shows why one must insist that physicalsymmetries be represented by linear symplectic maps on P.

Lemma 18. Suppose that : W(P, ) W(P, ) is a -algebra endomorphismsuch that

for every f P, (W(f)) = W(g) for some g P,

and suppose furthermore that the map F: (P, ) (P, ) given by F f = g iscontinuous. Then, F is in fact linear and preserves the symplectic structure . If, inaddition, is an automorphism, then F is invertible.

What this means is that the formalization of quantization using Weyl systems isbest suited to linear dynamics.

Proof. Assuming is a -algebra endomorphism,

(W(f))(W(h)) = (W(f)W(h))

so, applying the definition of F on the left-hand side and the Weyl relations on theright-hand side,

W(F f)

W(F h) = (e(f,h)/2i

W(f + h)).

Now, the Weyl relations on the left-hand side and the properties ofon the right-handside imply

e(Ff,Fh)/2iW(F f + F h) = e(f,h)/2iW(F(f + h)).Since all the {W(f)}fP are linearly independent by construction, it follows that Fis additive and preserves . Finally, continuous additive functions are linear.

The converse of this result is also true.


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Lemma 19. If F: (P, ) (P, ) is a continuous, invertible, linear, symplectictransformation, there exists a unique automorphism (F): W(P, ) W(P, )determined by

(F): W(f) W(F1f) for all f P

and such that (F G) = (F)(G).

In other words, is the unique representation of the group of symplectic auto-morphisms of (P, ) as -algebra automorphisms ofW(P, ).Proof. This result is [BSZ92, Corollary 5.1.1].

Definition 7 (general boson field). If (P, ) is a symplectic vector space, thegeneral boson field over it is the pair(

W, ) where

W: f

W(f) is the map from P

to W(P, ), and is the representation of physical symmetries by automorphismsof W(P, ) mentioned in Lemma 19.Notes. This definition is implicit in [BSZ92, 5.3].

So, what do we have? Given any symplectic vector space (P, ) one can constructthe associated Weyl algebra W(P, ), which supports a representation of thesymplectic automorphisms of (P, ) as -algebra automorphisms of W(P, ). Inaddition, any Weyl system on (P, ), that is, any strongly continuous representationofW(P, ) as unitary operators on a complex Hilbert space K provides a realizationof the Heisenberg commutation relations. This is the general boson field on (P, ).

5.1.4 Constructing Weyl and Heisenberg systems

The general boson field realizes the canonical commutation relations and the phys-ical symmetries of a linear system, but it does not provide a complete quantizationof a linear phase space, as there are a few lingering issues. The first is how to actu-ally construct Weyl systems. The second is whether the correspondence principle issatisfied. The third is whether physical symmetries are implemented unitarily on thesupporting Hilbert space of the Weyl system. It turns out that all three are related.In this section we will first use the GelfandNamarkSegal construction to produce

Weyl systems, and then use the correspondence principle and unitary implementabil-ity of physical symmetries to select the Weyl systems that produce physically sensiblequantizations.

The following example constructs the so-called Schrodinger representation of theHeisenberg commutation relations in one dimension.

Example. Let K = L2(R) and, for each f = (q, p) R2, define

W(f)(s) = eip(sq/2)(s q) for all K,


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which clearly makes W(f) a unitary operator on K. Also,

W(f)W(f)(s) = e(pqpq)/2iW(f + f)(s),

so W is a Weyl system on the symplectic vector space (P, ) with

P = {f = (p,q) R2} and (f; f) = pq pq.The Heisenberg operators are given by

(f)(s) = ps(s) iqs(s).This representation of the Weyl relations is called the Schrodinger representation.Given that is linear, it might seem odd that the momentum coordinate p appearsas the coefficient of the operator of multiplication by s, which we would usually

denote q. This crossing is related to the skew-symmetry of the symplectic structure,as we will now show.

The configuration space is R with coordinate function s:RtoR, and the phase spaceis P = R2 with coordinate functions s, k:R2 R (k being a momentum coordinatefunction). Then, s and k are a basis ofP, and (q, p) are coordinates on P. We nowidentify f = (q, p) with f = ps + qk. This is the correct pairing despite what whatour intuition might suggest, namely pairing s with q since they refer to the samequantity, because ps + qk is homogeneous under changes in the units of position andmomentum in a way that qs + pk is not. Then, we have

(s)(s) = s(s) and (k)(s) =

is(s)

as expected, and(f) = p(s) + q(k).

It is clear how this representation can be extended to any finite number of di-mensions, and by the theorem of von Neumann alluded to after Lemma (17), theserepresentations are unique up to unitary equivalence. For the infinite-dimensionalcase relevant to field theories, though, one needs to use the GelfandNamarkSegalconstruction, which is based on the concept of a state and leads to possibly unitarily

inequivalent representations.Definition 8 (GNS state). A state on a -algebra A is a linear functional

: A Cwhich is nonnegative

aa 0 for all a A,and normalized

1 = 1.


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Notes. The usage here is completely analogous to that for linear functionals on vectorspaces. A purely algebraic definition of linear functional on a vector space requires

that it be defined everywhere, but when a topology is introduced one finds it useful toconsider discontinuous, densely-defined linear functionals.. In the same vein, as longas A is not assumed to have a topology one must require that states be defined on allof A. However, if A has a topology making addition and multiplication continuous,then one can talk about continuous or bounded states, and also about discontinuous,densely-defined states. At this point, W(P, ) does not have a topology definedon it so states on it should be defined everywhere. On the other hand, the Weylsystem W(P, ) on K is given the strong operator topology, and so densely-definedstates make sense on it. In fact, we will use a state on W(P, ) to construct K, andit is not guaranteed that the state will be everywhere defined on it.

A state on W(P, ) defines a nonnegative-definite sesquilinear form | on W(P, ) by means of

W | W: = WW for all W, W W(P, )with the associated seminorm

|W|2 = W|W.Note that

W(f)

| W(g)

= ei(f,g)/2

W(g

f)

for all f, g

P, (5.3)

so|W(f)| = 1 for all f P

and | | is finite on all of W(P, ). By the standard procedurenamely, taking thequotient ofW(P, ) by the null subspace of| | and completing the result with respectto | | (which is a norm after quotienting by the null subspace)one can construct acomplex Hilbert space K with inner product | .

This is a version of the GelfandNamarkSegal construction. We now show thatwe can give a description of K in terms of the phase space P. For this, we draw thefollowing definition from [BSZ92, 5.3].Definition 9 (characteristic functional). If is a state on the Weyl alge-braW(P, ), its characteristic functional : P C is given by

(f): = W(f) for all f P. (5.4)The state is said to be regular if, for every f P, the function

t (tf) (t R)is twice differentiable at t = 0.


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Theorem 20. Let(P, ) be a symplectic vector space. Then, given a regular state on W(P, ) with characteristic function , there is an x P such that

it|t=0 (tf) = f(x) for all f P.Let = {|x + f : f P}. Then,

1. the sesquilinear form

x + f | x + g = e(g,f)/2i(g f)makes the span of into a complex pre-Hilbert space whose Hilbert space com-pletion is denoted K

2. there is a Weyl system W: P U(K) on (P, ), given by

W(f) |x + g = e(f,g)/2i

|x + f + g for all f, g P3. the unit vector |x K is a cyclic vector of the Weyl system W(P, )4. the associated Heisenberg system : P L(K) satisfies

x + f| (g) |x + f = g(x + f) for all f, g P.The physical interpretation of the vector x P is that of a background field

configuration. Clearly any vector in K can be used to define a state leading to aunitarily equivalent Weyl system, possibly with a different background field configu-ration. Since we have not assumed that : P P is onto, it may not be possible toeliminate the background altogether by a unitary change of representation.

The fact that the span of is dense in the Hilbert space K will be used consistentlyin the sequel to obtain or characterize densely defined linear operators and sesquilinearforms on K.

Proof. This proof has a curious way of pulling itself up by its own bootstraps: themain conceptual difficulty is that, in order to show that f it|t=0 (tf) is a con-tinuous linear functional on P one needs to have the Weyl system W in place. Weproceed by constructing and K before the names |x + f are available, and thenrenaming the vectors after x is shown to have the advertised properties.

We will temporarily denote by f

K the image of

W(f) under the GNS con-

struction described immediately before definition 9. We denote = {f: f P}. Itfollows immediately from equation (5.3) that

f | g = ei(f,g)/2(g f) for all f, g P. (5.5)The span of , which consists of unit vectors, is dense in K with respect to this innerproduct.

We are now ready to construct a Weyl system W: P U(K). The follow-ing lemma shows that the Hilbert space K automatically supports a Weyl sistemon W(P, ).


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Lemma 21. Suppose that a regular state is given on the Weyl algebraW(P, ) andthe GNS construction is performed resulting in the Hilbert space K, as just described.

Then, the formula

W(f)g = e(f,g)/2if+g for all f, g P.

defines a map W: P U(K) which is a Weyl system on (P, ). In addition, theunit vector 0 K is a cyclic vector of the Weyl system W: P U(K).Proof. First, we need to show that W(f) U(K) for all f P. Indeed, observethat W(f) maps to itself and that, for all f , g , h P,W(f)g | W(f)h = e(f,hg)/2if+g | f+h = e(f,hg)/2iei(f+g,f+h)/2(h g) =

= ei(g,h)/2

(h g) = g | hbecause f(x) = (f, x) for all x P and f P. This implies that W(f) is aninvertible isometry on the span of . Then, by density of the span of in K andlinearity, it follows that W(f) is unitary on K.

Now, we need to show that, for all f , g , h P,

W(f)W(g)h = e(f,g)/2iW(f + g)h.

The left-hand side is equal to

W(f)e(g,h)/2ig+h

= e(g,h)/2ie(f,g+h)/2if+g+h

,

and the right-hand side is equal to

e(f,g)/2iW(f + g)h = e(f,g)/2ie(f+g,h)/2if+g+h.

To show strong continuity of the Weyl system W we need to show that, iffn fin P, then W(fn) W(f) in the strong operator topology on U(K). To this end,we consider

[W(f) W(g)]h = e(f,h)/2if+h e(g,h)/2ig+h.Then,

[W(f) W(g)]h2 = 2Re[1 ei(fg,h)+i(f,g)/2(g f)],which indeed vanishes as f g 0 because of the continuity of and and theantisymmetry of .

Finally, the unit vector 0 K is a cyclic vector of the Weyl system W: P U(K) because W(f)0 = f for all f P, and the collection of all f is densein K.

We now move study the Heisenberg system associated to the Weyl system definedin Lemma 21.


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Lemma 22. In the hypotheses of Lemma 21, g | (f)g and (f)g are bothfinite for all f, g P. Moreover,

g | (f)g = (f, g) + 0 | (f)0and

(f)g2 (f)02 = g | (f)g2 0 | (f)02.Proof. Observe that, if g is in the domain of (f), then

g | (f)g = i t|t=0 g | W(tf)gand

(f)g2 = g | (f)2g = 2tt=0

g | W(tf)g.

Conversely, since (f) is a closed operator, the finiteness of 2

t |t=0 g | W(tf)gwould imply that g is in the domain of (f). We now show this.First we compute the matrix elements of the unitary operator W(f) between

arbitrary elements of :

g | W(f)h = e(f,g+h)/2i+i(g,h)/2(f g + h) for all f , g , h P

which, when f = 0, reduces to equation (5.5) for g | h. It also follows that

g | W(f)h = e(f,g+h)/2i(f + h g)(h g) g | h.

Differentiating the matrix element g | W(tf)g twice with respect to t andsetting t = 0 one obtains(f)g2 = [(f, g)]2 + 2i(f, g) t|t=0 (tf) 2t

t=0

(tf),

which is finite by the assumption that (tf) is twice-differentiable. Particularizingto g = 0 we obtain

2tt=0

(tf) = (f)02.One obtains the matrix elements of the Heisenberg operator (f) by differentiating

the matrix element g | W(tf)h with respect to t and setting t = 0, namely:

g | (f)h = 12(f, g + h) + it|t=0 (tf + h g)(h g) g | h.If, in particular, h = g,

g | (f)g = (f, g) + it|t=0 (tf).The case g = 0 shows that

i t|t=0 (tf) = 0 | (f)0,and the result follows by elementary algebraic manipulations.


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At this point, we can assert that

i t|t=0 (tf) = 0 | (f)0 = f(x)for some x P (had we not assumed that P is reflexive, we could only deducethat x P). This takes care of the first conclusion of the theorem. If we now makethe identification f |x + f, the result follows.

Insert a picture of x + P as an affine subspace of P.

Definition 10 (coherent states). Given a regular state on W(P, ) such that

i t|t=0 (tf) = f(x) for allf

P,

the image of W(f) inside K by the GNS construction, denoted by |x + f, is calleda coherent state relative to the background x. We denote the set of coherent statesby = {|x + f : f P}.Notes. One of the conclusions of lemma 22 is that

Varx+f(g) = x + f| (g)2 |x + f x + f| (g) |x + f2

is independent of f P. In other words, all the relative coherent states have thesame covariant matrix for all observables. Note that we are not claiming that the

common covariance saturates the Heisenberg uncertainty principle.

The problem of quantizing a linear phase space (P, ) can thus be partly solvedby finding a state on the Weyl algebra W(P, ). This leads to a Weyl systemon (P, ) and so to Heisenberg operators (f) satisfying the canonical commutationrelations and the correspondence principle, albeit possibly with a nontrivial back-ground field configuration.

Still, the canonical commutation relations and the correspondence principle to-gether are far from sufficient to uniquely determine the quantization and, unless Pis finite-dimensional, different states may lead to unitarily inequivalent Weyl sys-tems. The problem remains how to construct or identify representations suitable forparticular physical applications. In the next section we investigate the implicationsof requiring that physical symmetries, in particular time evolution, be implementedunitarily.

5.1.5 Representations of the general boson field

Having quantized the phase space itself, we now consider the quantization ofdynamics and, more generally, any physical symmetries, that is, maps T: P P


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preserving the symplectic structure. The ultimate goal is to represent physical sym-metries as unitary operators on the quantum state space K. Note, however, that we

have used the Weyl algebra on W(P, ) to define the Hilbert space K, and the Weylsystem W: P U(K). Thus, given a physical symmetry T: P P, we need toknow its related action on P. Defining T: P P by

(Tf)(x) = f(T x) for all x P, f P,it easily follows that F preserves , and that TS = (ST).

Putting together Lemmas 19 and 21 and Definition 10, we obtain the followingresult.

Lemma 23. Assume that T: P P is a continuous, invertible linear map and

that T: P P preserves the symplectic structure . Given any regular state on W(P, ), there is a densey defined linear map (T): K K such that(T) |T x + T f = |x + f . (5.6)

This map intertwines the unitary operators W(f), that is,

W(Tf) = (T)W(f)(T1) for all f P. (5.7)Proof. By Lemma 19, there is a unique automorphism (T) of the -algebra W(P, )such that

(T)W(Tf) = W(f) for all f

P.

The GNS construction preceding Definition 10 produces a unique linear operator (T): K K defined on the dense subset (P) of K by equation (5.6). This map extends bylinearity and density from (P) to all of K. does it, or does it stay denselydefined?

As for the intertwining of the Weyl operators, note that ((Tf), y) = (Tf)yimplies that (Tf) = T1f, so

W(Tf)y = ef(Ty)T1f+y.

Letting y = T1x,

W(Tf)T1

x = e

f(x)

T1

(f

+x)

and, from equation (5.6), we get

W(Tf)(T)x = (T)W(f)x for all x P, f P.Now, the x are dense in K and (T) is invertible because T is, and equation (5.7)follows.

Perhaps surprisingly, (F) is not necessarily an isometry of K despite the factthat it preserves the norm of all the coherent states x.


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Lemma 24. If T: P P is invertible, linear and symplectic, then (T) is unitaryon K if, and only if (T) preserves in the sense that

(T)W(x) = W(x) for all x P.In other words, the invertible operator (T) is unitary on K if, and only if, the

characteristic functional is constant on orbits of T.

Proof. From equation (5.5), since T is symplectic and by definition of(T), it followsthat

T1x | T1y = ei(x,y)/2(T)W(y x)so (T) is an isometry on the span of P if, and only if, (T)W(x) = W(x) forall x

P. An isometry is unitary if and only if it is invertible.

It follows that, for a whole subgroup G of symplectic transformations on (P, ) tobe unitarily implemented on K, it is necessary and sufficient that be constant onthe orbits of the whole subgroup. Now, if a G is a continuous group generated by apoisson algebra g of classical observables on P, this is equivalent to the characteristicfunctional having vanishing Poisson brackets with all the elements of g, but thisimplies that must be a function of the physical observables making up g. It isin this precise sense that the dynamics determine the quantization. In particular,if time evolution is to be implemented unitarily, the characteristic functional of thestate must be a function of the Hamiltonian of the system, or of any number of otherconserved quantities.

Putting Lemmas 2124 together with the GNS construction outlined after Defi-nition 8 we obtain the following theorem listing the properties of representations ofthe general boson field.

Theorem 25. Let(P, ) be a linear phase space, let be a GNS state on the Weylalgebra W(P, ) with characteristic function . Let P = {x | x P}. Then,

1. the sesquilinear form

x | y = e(y,x)/2i(y x)

makes the span ofP into a complex pre-Hilbert space whose Hilbert-space com-pletion is denoted K

2. there is a Weyl system W: P U(K) on (P, ), given byW(f)x = e

f(x)/2if+x for all x P, f P

3. the associated Heisenberg system : P L(K) satisfiesx | (f)x = f(x) + 0 | (f)0 for all x P, f P


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4. there is a continuous2 representation from the group of continuous, linear,symplectic transformations of (P, ) to the group of invertible linear operators

on K, given by(T)x = T1x for all x P

and satisfying

(T)W(f)(T)1 = W(Tf) for all f P

5. the unit vector 0 K is invariant under (T) for all linear symplectic trans-formations T of (P, ), and a cyclic vector of W(P, )

6. (T) is unitary if, and only if, is constant on orbits of T.

Since is defined on symplectic and not unitary transformations the generatorsare not self-adjoint and it is not clear that there is a meaningful notion of positivityof , in contrast with the free boson field. In other words, there seems to be no wayto define what a stable representation of the general boson field is.

Proof. 1. This follows from equations (5.5) and (5.4). The Hilbert space K is con-structed in the paragraph preceding Definition 10. [This probably indicatesthe need for an additional lemma, possibly merged with Definition 10.]

2. This is the content of Lemma 21.

3. This is by Lemma 22.

4. This is Lemma 23.

5. This is part of the conclusions of Lemmas 21 and 23.

6. This is the content of Lemma 24.

5.2 Properties of the characteristic functional

5.2.1 CharacterizationIn terms of the characteristic functional (x) = W(x), the inner product on K

isx | y = ei(x,y)/2(y x).

The positivity and normalization conditions on the state can be expressed in termsof as

(0) = 1

2weakly?


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and

i,jai aje

i(xi,xj)/2(xj xi) 0. for all xi P, ai C.

Any functional : P C satisfying these conditions uniquely determines a so-calledregular state [BSZ92, 5.3].

5.2.2 Probabilistic interpretation

If the xi are restricted to a Lagrangian subspace P0 of P (namely, a subspaceon which the restriction of the symplectic structure vanishes), then the positivitycondition becomes

i,j aiaj

(xj

xi)

0. for all xi

P0, a

i C.

By Bochners theorem [Bre92, problem 8.17]3 the function : P0 C is the charac-teristic functional of a probability distribution. That is, there is a random variable Xwith values in P0 whose characteristic functional is E[e

if(X)] = (f). This randomvariable X has the property that

E[(f(X))n] = 0 | (f)n0.

Hence, the natural interpretation of the random variable X is that it is the value

of (f) in the vacuum state 0. The state can, therefore, be aptly named the vac-uum expectation. In probability theory, the characteristic functional is the Fouriertransform of the probability density. Accordingly, the characteristic functional onthe full phase space P is the Fourier transform of the Wigner function [Fol89, 1.8].

5.2.3 The characteristic functional as generating functional

The vacuum expectation W(x) is enough to determine the matrix elements ofany polynomial of Heisenberg operators between arbitrary coherent states. The reasonfor this is that

y | (f1) (fn)x = in t1 tn |ti=0 W(y)W(t1f1 ) W (tnfn)W(x)for all x, y P and fi P, and the expectation in the right-hand side is a functionof x,y,fi times W(t1f1 + + tnfn + x y).

3IfX is a random variable with values in a vector space V and f(u) = E[eiuX ] for all u V,then

f(ui uj)ij 0 for all (i) Cn and (ui) Vn. The converse is Bochners theorem. See

also [BSZ92, 5.3]


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Successive logarithmic derivatives of the generating functional provide informationabout the matrix elements of Heisenberg operators and their powers. Indeed,

x, (f)yx, y = f

x + y2

+ if ln (y x)

and

x, (f)(g)yx, y

x, (f)yx, y

x, (g)yx, y =

i

2(f, g) + fg ln (y x).

In particular, if y = x,

x, (f)x

= f(x) + if ln (0)

and

x, (f)(g)x x, (f)xx, (g)x = i2

(f, g) + fg ln (0).

On the other hand, if g = f then

x, (f)2yx, y

x, (f)yx, y

2= 2f ln (y x)

and if, in addition, y = x,

x, (f)2x x, (f)x2 = 2f ln (0).

Finally, if y = x = 0,0, (f)0 = if ln (0)

and

0, (f)(g)0 0, (f)00, (g)0 = i2

(f, g) + fg ln (0);

in particular,

0, (f)20 0, (f)02 = 2f ln (0).

5.2.4

[Additive renormalization and generalized operators as in [BSZ92, ch. 7],

but for the general boson field.]


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5.3 The free boson field

The following example develops the case where the values of all the Heisenbergoperators in coherent states are jointly Gaussian random variables.4

Is W(x) the Wigner function? Im pretty sure its the Fourier transformof the Wigner function! To be sure, take a peek at Follands wonderful

book "Harmonic analysis on phase space". You should browse this book and

refer to it at suitable points! It has lots of cool stuff in it.

Example. Suppose now that Q is a nonnegative-definite quadratic form on P. Then,one possible state on W(P, ) is given by

W(x) = eQ(x)/4 for all x P,In this case, the matrix elements of W(f) are given by

y | W(f)x = ef(x+y)/2i+i(y,x)/2Q(f+xy)/4 for all x, y P, f P.In particular, if f = 0,

y | x = ei(y,x)/2Q(xy)/4,and so

y | W(f)x = ef(x+y)/2ieQ(xy)/4Q(f+xy)/4y | x.We can define a real inner product on P by polarizing Q,

h(x, y): = 14

[Q(x + y) Q(x y)],

so Q(x) = h(x, x) and, using Q(x) Q(y) = h(x + y, x y),y | W(f)x = e(f,x+y)/2ih(xy,f)/2eQ(f)/4y | x.

In particular, if x = y,

x | W(f)x = eif(x)Q(f)/4

which means, according to the probabilistic interpretation of the state

, that in

a coherent state x the values of Heisenberg operators (f) are jointly Gaussianrandom variables with mean and covariance

Ex(f) = f(x) and Covx(f, g) =1

2h(f, g).

4maybe the free boson field

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