UNITEXT - La Matematica per il 3+2

Volume 110

Editor-in-chief

A. Quarteroni

Series editors

L. AmbrosioP. BiscariC. CilibertoC. De LellisM. LedouxV. PanaretosW.J. Runggaldier

More information about this series at http://www.springer.com/series/5418

Valter Moretti

Spectral Theoryand Quantum MechanicsMathematical Foundations of QuantumTheories, Symmetries and Introductionto the Algebraic Formulation

Second Edition

123

Valter MorettiDepartment of MathematicsUniversity of TrentoPovo, TrentoItaly

ISSN 2038-5714 ISSN 2532-3318 (electronic)UNITEXT - La Matematica per il 3+2ISSN 2038-5722 ISSN 2038-5757 (electronic)ISBN 978-3-319-70705-1 ISBN 978-3-319-70706-8 (eBook)https://doi.org/10.1007/978-3-319-70706-8

Library of Congress Control Number: 2017958726

Translated and extended version of the original Italian edition: V. Moretti, Teoria Spettrale e MeccanicaQuantistica, © Springer-Verlag Italia 20101st edition: © Springer-Verlag Italia 20132nd edition: © Springer International Publishing AG 2017This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made. The publisher remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.

Printed on acid-free paper

This Springer imprint is published by Springer NatureThe registered company is Springer International Publishing AGThe registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Translated by: Simon G. Chiossi, Departamento de Matemática Aplicada (GMA-IME),Universidade Federal Fluminense

To Bianca

Preface to the Second Edition

In this second English edition (third, if one includes the first Italian one), a largenumber of typos and errors of various kinds have been amended.

I have added more than 100 pages of fresh material, both mathematical andphysical, in particular regarding the notion of superselection rules—addressed fromseveral different angles—the machinery of von Neumann algebras and the abstractalgebraic formulation. I have considerably expanded the lattice approach toQuantum Mechanics in Chap. 7, which now contains precise statements leading upto Solèr’s theorem on the characterization of quantum lattices, as well as gener-alised versions of Gleason’s theorem. As a matter of fact, Chap. 7 and the relatedChap. 11 have been completely reorganised. I have incorporated a variety of resultson the theory of von Neumann algebras and a broader discussion on the mathe-matical formulation of superselection rules, also in relation to the von Neumannalgebra of observables. The corresponding preparatory material has been fitted intoChap. 3. Chapter 12 has been developed further, in order to include technical factsconcerning groups of quantum symmetries and their strongly continuous unitaryrepresentations. I have examined in detail the relationship between Nelson domainsand Gårding domains. Each chapter has been enriched by many new exercises,remarks, examples and references. I would like once again to thank my colleagueSimon Chiossi for revising and improving my writing.

For having pointed out typos and other errors and for useful discussions, I amgrateful to Gabriele Anzellotti, Alejandro Ascárate, Nicolò Cangiotti, Simon G.Chiossi, Claudio Dappiaggi, Nicolò Drago, Alan Garbarz, Riccardo Ghiloni, IgorKhavkine, Bruno Hideki F. Kimura, Sonia Mazzucchi, Simone Murro, GiuseppeNardelli, Marco Oppio, Alessandro Perotti and Nicola Pinamonti.

Povo, Trento, Italy Valter MorettiSeptember 2017

vii

Preface to the First Edition

I must have been 8 or 9 when my father, a man of letters but well-read in every disciplineand with a curious mind, told me this story: “A great scientist named Albert Einsteindiscovered that any object with a mass can't travel faster than the speed of light”. To mybewilderment I replied, boldly: “This can't be true, if I run almost at that speed and thenaccelerate a little, surely I will run faster than light, right?” My father was adamant: “No,it's impossible to do what you say, it's a known physics fact”. After a while I added: “Thatbloke, Einstein, must've checked this thing many times … how do you say, he did manyexperiments?” The answer I got was utterly unexpected: “Not even one I believe. He usedmaths!”What did numbers and geometrical figures have to do with the existence of an upper limit tospeed? How could one stand by such an apparently nonsensical statement as the existenceof a maximum speed, although certainly true (I trusted my father), just based on maths?How could mathematics have such big a control on the real world? And Physics ? What onearth was it, and what did it have to do with maths? This was one of the most beguiling andirresistible things I had ever heard till that moment… I had to find out more about it.

This is an extended and enhanced version of an existing textbook written in Italian(and published by Springer-Verlag). That edition and this one are based on acommon part that originated, in preliminary form, when I was a Physics under-graduate at the University of Genova. The third-year compulsory lecture coursecalled Theoretical Physics was the second exam that had us pupils seriouslyclimbing the walls (the first being the famous Physics II, covering thermodynamicsand classical electrodynamics).

Quantum Mechanics, taught in Institutions, elicited a novel and involved way ofthinking, a true challenge for craving students: for months we hesitantly faltered ona hazy and uncertain terrain, not understanding what was really key among thenotions we were trying—struggling, I should say—to learn, together with a com-pletely new formalism: linear operators on Hilbert spaces. At that time, actually, wedid not realise we were using this mathematical theory, and for many mates ofmine, the matter would have been, rightly perhaps, completely futile; Dirac's bravectors were what they were, and that’s it! They were certainly not elements in thetopological dual of the Hilbert space. The notions of Hilbert space and dualtopological space had no right of abode in the mathematical toolbox of the majority

ix

of my fellows, even if they would soon come back in through the back door, withthe course Mathematical Methods of Physics taught by Prof. G. Cassinelli.Mathematics, and the mathematical formalisation of physics, had always been myflagship to overcome the difficulties that studying physics presented me with, to thepoint that eventually (after a Ph.D. in Theoretical Physics) I officially became amathematician. Armed with a maths’ background—learnt in an extracurricularcourse of study that I cultivated over the years, in parallel to academic physics—andeager to broaden my knowledge, I tried to formalise every notion I met in that newand riveting lecture course. At the same time, I was carrying along a similar projectfor the mathematical formalisation of General Relativity, unaware that the work putinto Quantum Mechanics would have been incommensurably bigger.

The formulation of the spectral theorem as it is discussed in x 8, 9 is the same Ilearnt when taking the Theoretical Physics exam, which for this reason was adialogue of the deaf. Later my interest turned to Quantum Field Theory, a subject Istill work on today, though in the slightly more general framework of QFT incurved spacetime. Notwithstanding, my fascination with the elementary formula-tion of Quantum Mechanics never faded over the years, and time and again chunkswere added to the opus I begun writing as a student.

Teaching this material to master’s and doctoral students in mathematics andphysics, thereby inflicting on them the result of my efforts to simplify the matter,has proved to be crucial for improving the text. It forced me to typeset in LaTeX thepile of loose notes and correct several sections, incorporating many people’sremarks.

Concerning this, I would like to thank my colleagues, the friends from thenewsgroups it.scienza.fisica, it.scienza.matematica and free.it.scienza.fisica, and themany students—some of which are now fellows of mine—who contributed toimprove the preparatory material of the treatise, whether directly or not, in thecourse of time: S. Albeverio, G. Anzellotti, P. Armani, G. Bramanti, S. Bonaccorsi,A. Cassa, B. Cocciaro, G. Collini, M. Dalla Brida, S. Doplicher, L. Di Persio,E. Fabri, C. Fontanari, A. Franceschetti, R. Ghiloni, A. Giacomini, V. Marini,S. Mazzucchi, E. Pagani, E. Pelizzari, G. Tessaro, M. Toller, L. Tubaro,D. Pastorello, A. Pugliese, F. Serra Cassano, G. Ziglio and S. Zerbini. I amindebted, for various reasons also unrelated to the book, to my late colleagueAlberto Tognoli. My greatest appreciation goes to R. Aramini, D. Cadamuro andC. Dappiaggi, who read various versions of the manuscript and pointed out anumber of mistakes.

I am grateful to my friends and collaborators R. Brunetti, C. Dappiaggi and N.Pinamonti for lasting technical discussions, for suggestions on many topics coveredin the book and for pointing out primary references.

At last, I would like to thank E. Gregorio for the invaluable and on-the-spottechnical help with the LaTeX package.

In the transition from the original Italian to the expanded English version, amassive number of (uncountably many!) typos and errors of various kinds havebeen corrected. I owe to E. Annigoni, M. Caffini, G. Collini, R. Ghiloni,A. Iacopetti, M. Oppio and D. Pastorello in this respect. Fresh material was added,

x Preface to the First Edition

both mathematical and physical, including a chapter, at the end, on the so-calledalgebraic formulation.

In particular, Chap. 4 contains the proof of Mercer’s theorem for positiveHilbert–Schmidt operators. The analysis of the first two axioms of QuantumMechanics in Chap. 7 has been deepened and now comprises the algebraic char-acterisation of quantum states in terms of positive functionals with unit norm on theC�-algebra of compact operators. General properties of C�-algebras and �-morph-isms are introduced in Chap. 8. As a consequence, the statements of the spectraltheorem and several results on functional calculus underwent a minor but necessaryreshaping in Chaps. 8 and 9. I incorporated in Chap. 10 (Chap. 9 in the first edition)a brief discussion on abstract differential equations in Hilbert spaces. An importantexample concerning Bargmann’s theorem was added in Chap. 12 (formerlyChap. 11). In the same chapter, after introducing the Haar measure, the Peter–Weyltheorem on unitary representations of compact groups is stated and partially proved.This is then applied to the theory of the angular momentum. I also thoroughlyexamined the superselection rule for the angular momentum. The discussion onPOVMs in Chap.13 (ex Chap. 12) is enriched with further material, and I included aprimer on the fundamental ideas of non-relativistic scattering theory. Bell’sinequalities (Wigner’s version) are given considerably more space. At the endof the first chapter, basic point-set topology is recalled together with abstractmeasure theory. The overall effort has been to create a text as self-contained aspossible. I am aware that the material presented has clear limitations and gaps.Ironically—my own research activity is devoted to relativistic theories—the entiretreatise unfolds at a non-relativistic level, and the quantum approach to Poincaré’ssymmetry is left behind.

I thank my colleagues F. Serra Cassano, R. Ghiloni, G. Greco, S. Mazzucchi,A. Perotti and L. Vanzo for useful technical conversations on this second version.For the same reason, and also for translating this elaborate opus into English,I would like to thank my colleague S. G. Chiossi.

Trento, Italy Valter MorettiOctober 2012

Preface to the First Edition xi

Contents

1 Introduction and Mathematical Backgrounds . . . . . . . . . . . . . . . . . 11.1 On the Book. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Scope and Structure . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.3 General Conventions . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 On Quantum Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.1 Quantum Mechanics as a Mathematical Theory . . . . . 61.2.2 QM in the Panorama of Contemporary Physics . . . . . 7

1.3 Backgrounds on General Topology . . . . . . . . . . . . . . . . . . . . . 111.3.1 Open/Closed Sets and Basic Point-Set Topology . . . . 111.3.2 Convergence and Continuity . . . . . . . . . . . . . . . . . . . 141.3.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3.4 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.4 Round-Up on Measure Theory . . . . . . . . . . . . . . . . . . . . . . . . 171.4.1 Measure Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.4.2 Positive r-Additive Measures . . . . . . . . . . . . . . . . . . 201.4.3 Integration of Measurable Functions . . . . . . . . . . . . . 241.4.4 Riesz’s Theorem for Positive Borel Measures . . . . . . 281.4.5 Differentiating Measures . . . . . . . . . . . . . . . . . . . . . . 301.4.6 Lebesgue’s Measure on R

n . . . . . . . . . . . . . . . . . . . . 301.4.7 The Product Measure . . . . . . . . . . . . . . . . . . . . . . . . 341.4.8 Complex (and Signed) Measures . . . . . . . . . . . . . . . . 351.4.9 Exchanging Derivatives and Integrals . . . . . . . . . . . . 37

2 Normed and Banach Spaces, Examples and Applications . . . . . . . . 392.1 Normed and Banach Spaces and Algebras . . . . . . . . . . . . . . . . 40

2.1.1 Normed Spaces and Essential TopologicalProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.1.2 Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.1.3 Example: The Banach Space CðK;KnÞ, The

Theorems of Dini and Arzelà–Ascoli . . . . . . . . . . . . . 47

xiii

2.1.4 Normed Algebras, Banach Algebras and Examples . . . 502.2 Operators, Spaces of Operators, Operator Norms . . . . . . . . . . . 592.3 The Fundamental Theorems of Banach Spaces . . . . . . . . . . . . . 66

2.3.1 The Hahn–Banach Theorem and Its ImmediateConsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

2.3.2 The Banach–Steinhaus Theorem or UniformBoundedness Principle . . . . . . . . . . . . . . . . . . . . . . . 71

2.3.3 Weak Topologies. �-Weak Completeness of X 0 . . . . . 722.3.4 Excursus: The Theorem of Krein–Milman, Locally

Convex Metrisable Spaces and Fréchet Spaces . . . . . . 772.3.5 Baire’s Category Theorem and Its Consequences:

The Open Mapping Theorem and the InverseOperator Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 81

2.3.6 The Closed Graph Theorem . . . . . . . . . . . . . . . . . . . 842.4 Projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.5 Equivalent Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 892.6 The Fixed-Point Theorem and Applications . . . . . . . . . . . . . . . 91

2.6.1 The Fixed-Point Theorem of Banach–Caccioppoli . . . 912.6.2 Application of the Fixed-Point Theorem: Local

Existence and Uniqueness for Systems ofDifferential Equations . . . . . . . . . . . . . . . . . . . . . . . . 96

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3 Hilbert Spaces and Bounded Operators . . . . . . . . . . . . . . . . . . . . . 1073.1 Elementary Notions, Riesz’s Theorem and Reflexivity . . . . . . . 108

3.1.1 Inner Product Spaces and Hilbert Spaces . . . . . . . . . . 1083.1.2 Riesz’s Theorem and Its Consequences . . . . . . . . . . . 113

3.2 Hilbert Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1173.3 Hermitian Adjoints and Applications . . . . . . . . . . . . . . . . . . . . 131

3.3.1 Hermitian Conjugation, or Adjunction . . . . . . . . . . . . 1313.3.2 �-Algebras, C�-Algebras, and �-Representations . . . . . 1343.3.3 Normal, Self-Adjoint, Isometric, Unitary and

Positive Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 1403.4 Orthogonal Structures and Partial Isometries . . . . . . . . . . . . . . . 144

3.4.1 Orthogonal Projectors . . . . . . . . . . . . . . . . . . . . . . . . 1443.4.2 Hilbert Sum of Hilbert Spaces . . . . . . . . . . . . . . . . . . 1473.4.3 Partial Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

3.5 Polar Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1533.5.1 Square Roots of Bounded Positive Operators . . . . . . . 1533.5.2 Polar Decomposition of Bounded Operators . . . . . . . . 158

3.6 Introduction to von Neumann Algebras . . . . . . . . . . . . . . . . . . 1623.6.1 The Notion of Commutant . . . . . . . . . . . . . . . . . . . . 1623.6.2 Von Neumann Algebras, Also Known

as W�-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

xiv Contents

3.6.3 Further Relevant Operator Topologies . . . . . . . . . . . . 1663.6.4 Hilbert Sum of von Neumann Algebras . . . . . . . . . . . 169

3.7 The Fourier–Plancherel Transform . . . . . . . . . . . . . . . . . . . . . . 171Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

4 Families of Compact Operators on Hilbert Spaces andFundamental Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1974.1 Compact Operators on Normed and Banach Spaces . . . . . . . . . 198

4.1.1 Compact Sets in (Infinite-Dimensional) NormedSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

4.1.2 Compact Operators on Normed Spaces . . . . . . . . . . . 2004.2 Compact Operators on Hilbert Spaces . . . . . . . . . . . . . . . . . . . 204

4.2.1 General Properties and Examples . . . . . . . . . . . . . . . . 2044.2.2 Spectral Decomposition of Compact Operators on

Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2074.3 Hilbert–Schmidt Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

4.3.1 Main Properties and Examples . . . . . . . . . . . . . . . . . 2144.3.2 Integral Kernels and Mercer’s Theorem . . . . . . . . . . . 223

4.4 Trace-Class (or Nuclear) Operators . . . . . . . . . . . . . . . . . . . . . 2274.4.1 General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 2274.4.2 The Notion of Trace . . . . . . . . . . . . . . . . . . . . . . . . . 231

4.5 Introduction to the Fredholm Theory of Integral Equations . . . . 236Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

5 Densely-Defined Unbounded Operators on Hilbert Spaces . . . . . . . 2515.1 Unbounded Operators with Non-maximal Domains . . . . . . . . . . 252

5.1.1 Unbounded Operators with Non-maximal Domainsin Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 252

5.1.2 Closed and Closable Operators . . . . . . . . . . . . . . . . . 2535.1.3 The Case of Hilbert Spaces: The Structure of H��H

and the Operator s . . . . . . . . . . . . . . . . . . . . . . . . . . 2545.1.4 General Properties of the Hermitian Adjoint

Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2565.2 Hermitian, Symmetric, Self-adjoint and Essentially

Self-adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2595.3 Two Major Applications: The Position Operator and the

Momentum Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2635.3.1 The Position Operator . . . . . . . . . . . . . . . . . . . . . . . . 2645.3.2 The Momentum Operator . . . . . . . . . . . . . . . . . . . . . 265

5.4 Existence and Uniqueness Criteria for Self-adjointExtensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2705.4.1 The Cayley Transform and Deficiency Indices . . . . . . 2705.4.2 Von Neumann’s Criterion . . . . . . . . . . . . . . . . . . . . . 276

Contents xv

5.4.3 Nelson’s Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 277Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

6 Phenomenology of Quantum Systems and Wave Mechanics:An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2896.1 General Principles of Quantum Systems . . . . . . . . . . . . . . . . . . 2906.2 Particle Aspects of Electromagnetic Waves . . . . . . . . . . . . . . . 291

6.2.1 The Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . 2916.2.2 The Compton Effect . . . . . . . . . . . . . . . . . . . . . . . . . 292

6.3 An Overview of Wave Mechanics . . . . . . . . . . . . . . . . . . . . . . 2956.3.1 De Broglie Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 2956.3.2 Schrödinger’s Wavefunction and Born’s

Probabilistic Interpretation . . . . . . . . . . . . . . . . . . . . . 2966.4 Heisenberg’s Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . 2986.5 Compatible and Incompatible Quantities . . . . . . . . . . . . . . . . . . 300

7 The First 4 Axioms of QM: Propositions, Quantum Statesand Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3037.1 The Pillars of the Standard Interpretation of Quantum

Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3047.2 Classical Systems: Elementary Propositions and States . . . . . . . 306

7.2.1 States as Probability Measures . . . . . . . . . . . . . . . . . . 3067.2.2 Propositions as Sets, States as Measures on Them . . . 3097.2.3 Set-Theoretical Interpretation of the Logical

Connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3097.2.4 “Infinite” Propositions and Physical Quantities . . . . . . 3107.2.5 Basics on Lattice Theory . . . . . . . . . . . . . . . . . . . . . 3127.2.6 The Boolean Lattice of Elementary Propositions for

Classical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 3167.3 Quantum Systems: Elementary Propositions . . . . . . . . . . . . . . . 317

7.3.1 Quantum Lattices and Related Structures in HilbertSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

7.3.2 The Non-Boolean (Non-Distributive) Lattice ofProjectors on a Hilbert Space . . . . . . . . . . . . . . . . . . 318

7.4 Propositions and States on Quantum Systems . . . . . . . . . . . . . . 3257.4.1 Axioms A1 and A2: Propositions, States of a

Quantum System and Gleason’s Theorem . . . . . . . . . 3257.4.2 The Kochen–Specker Theorem . . . . . . . . . . . . . . . . . 3347.4.3 Pure States, Mixed States, Transition Amplitudes . . . . 3357.4.4 Axiom A3: Post-Measurement States and

Preparation of States . . . . . . . . . . . . . . . . . . . . . . . . . 3427.4.5 Quantum Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

7.5 Observables as Projector-Valued Measures on R . . . . . . . . . . . 3487.5.1 Axiom A4: The Notion of Observable . . . . . . . . . . . . 348

xvi Contents

7.5.2 Self-adjoint Operators Associated to Observables:Physical Motivation and Basic Examples . . . . . . . . . . 351

7.5.3 Probability Measures Associated to Couples State/Observable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

7.6 More Advanced, Foundational and Technical Issues . . . . . . . . . 3597.6.1 Recovering the Hilbert Space from the Lattice: The

Theorems of Piron and Solèr . . . . . . . . . . . . . . . . . . . 3597.6.2 The Projector Lattice of von Neumann Algebras

and the Classification of von Neumann Algebras andFactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

7.6.3 Direct Decomposition into Factors and Definite-Type von Neumann Algebras and Factors . . . . . . . . . 370

7.6.4 Gleason’s Theorem for Lattices of von NeumannAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

7.6.5 Algebraic Characterisation of a State as aNoncommutative Riesz Theorem . . . . . . . . . . . . . . . . 374

7.7 Introduction to Superselection Rules . . . . . . . . . . . . . . . . . . . . 3787.7.1 Coherent Sectors, Admissible States and Admissible

Elementary Propositions . . . . . . . . . . . . . . . . . . . . . . 3787.7.2 An Alternate Formulation of the Theory of

Superselection Rules . . . . . . . . . . . . . . . . . . . . . . . . . 383Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

8 Spectral Theory I: Generalities, Abstract C�-Algebrasand Operators in BðHÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3938.1 Spectrum, Resolvent Set and Resolvent Operator . . . . . . . . . . . 394

8.1.1 Basic Notions in Normed Spaces . . . . . . . . . . . . . . . . 3958.1.2 The Spectrum of Special Classes of Normal

Operators on Hilbert Spaces . . . . . . . . . . . . . . . . . . . 3998.1.3 Abstract C�-Algebras: Gelfand–Mazur Theorem,

Spectral Radius, Gelfand’s Formula, Gelfand–Najmark Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

8.2 Functional Calculus: Representations of CommutativeC�-Algebras of Bounded Maps . . . . . . . . . . . . . . . . . . . . . . . . 4078.2.1 Abstract C�-Algebras: Functional Calculus for

Continuous Maps and Self-adjoint Elements . . . . . . . . 4078.2.2 Key Properties of �-Homomorphisms of

C�-Algebras, Spectra and Positive Elements . . . . . . . . 4118.2.3 Commutative Banach Algebras and the Gelfand

Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4148.2.4 Abstract C�-Algebras: Functional Calculus

for Continuous Maps and Normal Elements . . . . . . . . 4208.2.5 C�-Algebras of Operators in BðHÞ: Functional

Calculus for Bounded Measurable Functions . . . . . . . 422

Contents xvii

8.3 Projector-Valued Measures (PVMs) . . . . . . . . . . . . . . . . . . . . . 4318.3.1 Spectral Measures, or PVMs . . . . . . . . . . . . . . . . . . . 4318.3.2 Integrating Bounded Measurable Functions in a

PVM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4348.3.3 Properties of Operators Obtained Integrating

Bounded Maps with Respect to PVMs. . . . . . . . . . . . 4418.4 Spectral Theorem for Normal Operators in BðHÞ . . . . . . . . . . . 449

8.4.1 Spectral Decomposition of Normal Operatorsin BðHÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

8.4.2 Spectral Representation of Normal Operatorsin BðHÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

8.5 Fuglede’s Theorem and Consequences . . . . . . . . . . . . . . . . . . . 4638.5.1 Fuglede’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 4648.5.2 Consequences to Fuglede’s Theorem . . . . . . . . . . . . . 466

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

9 Spectral Theory II: Unbounded Operators on Hilbert Spaces . . . . 4739.1 Spectral Theorem for Unbounded Self-adjoint Operators . . . . . . 474

9.1.1 Integrating Unbounded Functions with Respectto Spectral Measures . . . . . . . . . . . . . . . . . . . . . . . . . 474

9.1.2 Von Neumann Algebra of a Bounded NormalOperator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

9.1.3 Spectral Decomposition of Unbounded Self-adjointOperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

9.1.4 Example of Operator with Point Spectrum: TheHamiltonian of the Harmonic Oscillator . . . . . . . . . . . 503

9.1.5 Examples with Continuous Spectrum: The OperatorsPosition and Momentum . . . . . . . . . . . . . . . . . . . . . . 507

9.1.6 Spectral Representation of Unbounded Self-adjointOperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508

9.1.7 Joint Spectral Measures . . . . . . . . . . . . . . . . . . . . . . 5099.2 Exponential of Unbounded Operators: Analytic Vectors . . . . . . 5129.3 Strongly Continuous One-Parameter Unitary Groups . . . . . . . . . 516

9.3.1 Strongly Continuous One-Parameter UnitaryGroups, von Neumann’s Theorem . . . . . . . . . . . . . . . 517

9.3.2 One-Parameter Unitary Groups Generated bySelf-adjoint Operators and Stone’s Theorem . . . . . . . . 520

9.3.3 Commuting Operators and Spectral Measures . . . . . . . 529Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

10 Spectral Theory III: Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 53910.1 Abstract Differential Equations in Hilbert Spaces . . . . . . . . . . . 540

10.1.1 The Abstract Schrödinger Equation (With Source) . . . 54210.1.2 The Abstract Klein–Gordon/d’Alembert Equation

(With Source and Dissipative Term) . . . . . . . . . . . . . 548

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10.1.3 The Abstract Heat Equation . . . . . . . . . . . . . . . . . . . 55710.2 Hilbert Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

10.2.1 Tensor Product of Hilbert Spaces and SpectralProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

10.2.2 Tensor Product of Operators . . . . . . . . . . . . . . . . . . . 56710.2.3 An Example: The Orbital Angular Momentum . . . . . . 571

10.3 Polar Decomposition Theorem for Unbounded Operators . . . . . 57410.3.1 Properties of Operators A�A, Square Roots of

Unbounded Positive Self-adjoint Operators . . . . . . . . 57410.3.2 Polar Decomposition Theorem for Closed and

Densely-Defined Operators . . . . . . . . . . . . . . . . . . . . 57910.4 The Theorems of Kato–Rellich and Kato . . . . . . . . . . . . . . . . . 581

10.4.1 The Kato–Rellich Theorem . . . . . . . . . . . . . . . . . . . . 58110.4.2 An Example: The Operator �DþV and Kato’s

Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590

11 Mathematical Formulation of Non-relativistic QuantumMechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59511.1 Round-up and Further Discussion on Axioms

A1, A2, A3, A4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59611.1.1 Axioms A1, A2, A3 . . . . . . . . . . . . . . . . . . . . . . . . . 59611.1.2 A4 Revisited: von Neumann Algebra of

Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59811.1.3 Compatible Observables and Complete Sets of

Commuting Observables . . . . . . . . . . . . . . . . . . . . . . 60411.2 Superselection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607

11.2.1 Superselection Rules and von Neumann Algebraof Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607

11.2.2 Abelian Superselection Rules Induced by CentralObservables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611

11.2.3 Non-Abelian Superselection Rules and the GaugeGroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616

11.3 Miscellanea on the Notion of Observable . . . . . . . . . . . . . . . . . 61911.3.1 Mean Value and Standard Deviation . . . . . . . . . . . . . 61911.3.2 An Open Problem: What is the Meaning of

f ðA1; . . .;AnÞ if A1; . . .;An are Not PairwiseCompatible? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622

11.3.3 The Notion of Jordan Algebra . . . . . . . . . . . . . . . . . . 62311.4 Axiom A5: Non-relativistic Elementary Systems . . . . . . . . . . . . 624

11.4.1 The Canonical Commutation Relations (CCRs) . . . . . 62611.4.2 Heisenberg’s Uncertainty Principle as a Theorem . . . . 627

11.5 Weyl’s Relations, the Theorems of Stone–von Neumannand Mackey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628

Contents xix

11.5.1 Families of Operators Acting Irreducibly andSchur’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629

11.5.2 Weyl’s Relations from the CCRs . . . . . . . . . . . . . . . . 63111.5.3 The Theorems of Stone–von Neumann and

Mackey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63911.5.4 The Weyl �-Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 64211.5.5 Proof of the Theorems of Stone–von Neumann

and Mackey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64611.5.6 More on “Heisenberg’s Principle”: Weakening the

Assumptions and the Extension to Mixed States . . . . . 65311.5.7 The Stone–von Neumann Theorem Revisited:

Weyl–Heisenberg Group . . . . . . . . . . . . . . . . . . . . . . 65511.5.8 Dirac’s Correspondence Principle, Weyl’s Calculus

and Deformation Quantisation . . . . . . . . . . . . . . . . . . 657Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661

12 Introduction to Quantum Symmetries . . . . . . . . . . . . . . . . . . . . . . . 66512.1 Definition and Characterisation of Quantum Symmetries . . . . . . 666

12.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66712.1.2 Symmetries in Presence of Abelian Superselection

Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66912.1.3 Kadison Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 67012.1.4 Wigner Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 67212.1.5 The Theorems of Wigner and Kadison . . . . . . . . . . . 67412.1.6 Dual Action and Inverse Dual Action of Symmetries

on Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68712.1.7 Symmetries as Transformations of Observables:

Symmetries as Ortho-Automorphisms and SegalSymmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693

12.2 Introduction to Symmetry Groups . . . . . . . . . . . . . . . . . . . . . . 69512.2.1 Projective and Projective Unitary Representations . . . . 69612.2.2 Representations of Actions on Observables: Left and

Right Representations . . . . . . . . . . . . . . . . . . . . . . . . 70012.2.3 Projective Representations and Anti-unitary

Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70112.2.4 Central Extensions and Quantum Group Associated

to a Symmetry Group . . . . . . . . . . . . . . . . . . . . . . . . 70212.2.5 Topological Symmetry Groups . . . . . . . . . . . . . . . . . 70512.2.6 Strongly Continuous Projective Unitary

Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71112.2.7 A Special Case: The Topological Group R . . . . . . . . 71412.2.8 Round-Up on Lie Groups and Algebras . . . . . . . . . . . 72012.2.9 Continuous Unitary Finite-Dimensional

Representations of Connected Non-compact LieGroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730

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12.2.10 Bargmann’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 73212.2.11 Theorems of Gårding, Nelson, FS3 . . . . . . . . . . . . . . 74312.2.12 A Few Words About Representations of Abelian

Groups and the SNAG Theorem . . . . . . . . . . . . . . . . 75212.2.13 Continuous Unitary Representations of Compact

Hausdorff Groups: The Peter–Weyl Theorem . . . . . . . 75412.2.14 Characters of Finite-Dimensional Group

Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76812.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769

12.3.1 The Symmetry Group SOð3Þ and the Spin . . . . . . . . . 76912.3.2 The Superselection Rule of the Angular

Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77312.3.3 The Galilean Group and Its Projective Unitary

Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77412.3.4 Bargmann’s Rule of Superselection of the Mass . . . . . 782

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785

13 Selected Advanced Topics in Quantum Mechanics . . . . . . . . . . . . . 79313.1 Quantum Dynamics and Its Symmetries . . . . . . . . . . . . . . . . . . 794

13.1.1 Axiom A6: Time Evolution . . . . . . . . . . . . . . . . . . . . 79413.1.2 Dynamical Symmetries . . . . . . . . . . . . . . . . . . . . . . . 79713.1.3 Schrödinger’s Equation and Stationary States . . . . . . . 80013.1.4 The Action of the Galilean Group in Position

Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80813.1.5 Basic Notions of Scattering Processes . . . . . . . . . . . . 81113.1.6 The Evolution Operator in Absence of Time

Homogeneity and Dyson’s Series . . . . . . . . . . . . . . . 81813.1.7 Anti-unitary Time Reversal . . . . . . . . . . . . . . . . . . . . 822

13.2 From the Time Observable and Pauli’s Theorem toPOVMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82613.2.1 Pauli’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82713.2.2 Generalised Observables as POVMs . . . . . . . . . . . . . 828

13.3 Dynamical Symmetries and Constants of Motion . . . . . . . . . . . 83113.3.1 Heisenberg’s Picture and Constants of Motion . . . . . . 83113.3.2 A Short Detour on Ehrenfest’s Theorem and Related

Mathematical Issues . . . . . . . . . . . . . . . . . . . . . . . . . 83613.3.3 Constants of Motion Associated to Symmetry Lie

Groups and the Case of the Galilean Group . . . . . . . . 83913.4 Compound Systems and Their Properties . . . . . . . . . . . . . . . . . 844

13.4.1 Axiom A7: Compound Systems . . . . . . . . . . . . . . . . 84413.4.2 Independent Subsystems: The Delicate Viewpoint

of von Neumann Algebra Theory . . . . . . . . . . . . . . . 84613.4.3 Entangled States and the So-Called “EPR

Paradox” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 848

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13.4.4 Bell’s Inequalities and Their ExperimentalViolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 850

13.4.5 EPR Correlations Cannot Transfer Information . . . . . . 85413.4.6 The Phenomenon of Decoherence as a Manifestation

of the Macroscopic World . . . . . . . . . . . . . . . . . . . . . 85713.4.7 Axiom A8: Compounds of Identical Systems . . . . . . . 85813.4.8 Bosons and Fermions . . . . . . . . . . . . . . . . . . . . . . . . 860

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864

14 Introduction to the Algebraic Formulation of QuantumTheories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86714.1 Introduction to the Algebraic Formulation of Quantum

Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86714.1.1 Algebraic Formulation . . . . . . . . . . . . . . . . . . . . . . . 86814.1.2 Motivations and Relevance of Lie-Jordan

Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86914.1.3 The GNS Reconstruction Theorem . . . . . . . . . . . . . . 87314.1.4 Pure States and Irreducible Representations . . . . . . . . 88014.1.5 Further Comments on the Algebraic Approach

and the GNS Construction . . . . . . . . . . . . . . . . . . . . 88514.1.6 Hilbert-Space Formulation Versus Algebraic

Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88614.1.7 Algebraic Abelian Superselection Rules . . . . . . . . . . . 88914.1.8 Fell’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89414.1.9 Proof of the Gelfand-Najmark Theorem, Universal

Representations and Quasi-equivalentRepresentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895

14.2 Example of a C�-Algebra of Observables: TheWeyl C�-Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90014.2.1 Further properties of Weyl �-Algebras WðX; rÞ . . . . . 90014.2.2 The Weyl C�-Algebra CWðX; rÞ . . . . . . . . . . . . . . . 904

14.3 Introduction to Quantum Symmetries Within the AlgebraicFormulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90614.3.1 The Algebraic Formulation’s Viewpoint on

Quantum Symmetries . . . . . . . . . . . . . . . . . . . . . . . . 90614.3.2 (Topological) Symmetry Groups in the Algebraic

Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 909

Appendix A: Order Relations and Groups . . . . . . . . . . . . . . . . . . . . . . . . 915

Appendix B: Elements of Differential Geometry . . . . . . . . . . . . . . . . . . . . 919

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937

xxii Contents