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Spectral Theory and Quantum Mechanics

Valter Moretti

Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation

Second Edition

UNITEXT 110

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

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 Rn . . . . . . . . . . . . . . . . . . . . 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

xviii Contents

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

xx Contents

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

Contents xxi

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

Chapter 1Introduction and Mathematical Backgrounds

“O frati”, dissi “che per cento milia perigli siete giunti al’occidente, a questa tanto picciola vigilia d’i nostri sensi ch’èdel rimanente non vogliate negar l’esperienza, di retro al sol,del mondo sanza gente”.

Dante Alighieri, the Divine Comedy, Inferno, canto XXVI1

1.1 On the Book

1.1.1 Scope and Structure

One of the aims of the present book is to explain the mathematical foundations ofQuantum Mechanics (QM), and Quantum Theories in general, in a mathematicallyrigorousway. This is a treatise onMathematics (orMathematical Physics) rather thana text on Quantum Mechanics. Except for a few cases, the physical phenomenologyis left in the background in order to privilege the theory’s formal and logical aspects.At any rate, several examples of the physical formalism are presented, lest one losetouch with the world of physics.

In alternative to, and irrespective of, the physical content, the book should beconsidered as an introductory text, albeit touching upon rather advanced topics, onfunctional analysis on Hilbert spaces, including a few elementary yet fundamentalresults on C∗-algebras. Special attention is given to a series of results in spectraltheory, such as the various formulations of the spectral theorem for bounded normaloperators and not necessarily bounded, self-adjoint ones. This is, as a matter of fact,one further scope of the text. The mathematical formulation of Quantum Theoriesis “confined” to Chaps. 6, 7, 11–13 and partly Chap. 14. The remaining chapters are

1(“Brothers” I said, “who through a hundred thousand dangers have reached the channel to thewest, to the short evening watch which your own senses still must keep, do not choose to denythe experience of what lies past the Sun and of the world yet uninhabited.” Dante Alighieri, TheDivine Comedy, translated by J. Finn Cotter, edited by C. Franco, Forum Italicum Publishing,New York, 2006.)

© Springer International Publishing AG 2017V. Moretti, Spectral Theory and Quantum Mechanics, UNITEXT - La Matematicaper il 3+2 110, https://doi.org/10.1007/978-3-319-70706-8_1

1

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2 1 Introduction and Mathematical Backgrounds

logically independent of those, although the motivations for certain mathematicaldefinitions are to be found in Chaps. 7, 10–14.

A third purpose is to collect in one place a number of rigorous and useful results onthe mathematical structure of QM and Quantum Theories. These are more advancedthan what is normally encountered in quantum physics’ manuals. Many of theseaspects have been known for a long time but are scattered in the specialistic literature.We should mention Solèr’s theorem, Gleason’s theorem, the theorem of Kochen andSpecker, the theorems of Stone–von Neumann and Mackey, Stone’s theorem and vonNeumann’s theorem about one-parameter unitary groups,Kadison’s theorem, besidesthe better known Wigner, Bargmann and GNS theorems; or, more abstract results inoperator theory such as Fuglede’s theorem, or the polar decomposition for closedunbounded operators (which is relevant in the Tomita–Takesaki theory and statisticalQuantumMechanics in relationship to the KMS condition); furthermore, self-adjointproperties for symmetric operators, due to Nelson, that descend from the existenceof dense sets of analytical vectors, and finally, Kato’s work (but not only his) onthe essential self-adjointness of certain kinds of operators and their limits from thebottom of the spectrum (mostly based on the Kato–Rellich theorem).

Some chapters suffice to cover a good part of the material suitable for advancedcourses onMathematical Methods in Physics; this is common for master’s degrees inPhysics or doctoral degrees, if we assume a certain familiarity with notions, resultsand elementary techniques of measure theory. The text may also be used for a higher-level course in Mathematical Physics that includes foundational material on QM. Inthe attempt to reach out to master or Ph.D. students, both in physics with an interestin mathematical methods or in mathematics with an inclination towards physicalapplications, the author has tried to prepare a self-contained text, as far as possi-ble: hence a primer was included on general topology and abstract measure theory,together with an appendix on differential geometry. Most chapters are accompaniedby exercises, many of which are solved explicitly.

The book could, finally, be useful to scholars to organise and present accuratelythe profusion of advanced material disseminated in the literature.

Results from topology and measure theory, much needed throughout the wholetreatise, are recalled at the end of this introductory chapter. The rest of the book isideally divided into three parts. The first part, up to Chap.5, regards the general the-ory of operators on Hilbert spaces, and introduces several fairly general notions, likeBanach spaces. Core results are proved, such as the theorems of Baire, Hahn–Banachand Banach–Steinhaus, as well as the fixed-point theorem of Banach–Caccioppoli,the Arzelà-Ascoli theorem and Fredholm’s alternative, plus some elementary con-sequences. This part contains a summary of basic topological notions, in the beliefthat it might benefit physics’ students. The latter’s training on point-set topology is attimes disparate and often presents gaps, because this subject is, alas, usually taughtsporadically in physics’ curricula, and not learnt in an organic way like students inmathematics do.

Part two endswith Chap.10. Beside laying out the quantum formalism, it developsspectral theory, in terms of projector-valued measures, up to the spectral decomposi-tion theorems for unbounded self-adjoint operators on Hilbert spaces. This includes

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1.1 On the Book 3

the features of maps of operators (functional analysis) for measurable maps that arenot necessarily bounded. General spectral aspects and the properties of their domainsare investigated.Agreat emphasis is placed onC∗-algebras and the relative functionalcalculus, including an elementary study of the Gelfand transform and the commuta-tive Gelfand–Najmark theorem. The technical results leading to the spectral theoremare stated and proven in a completely abstract manner in Chap. 8, forgetting that thealgebras in question are actually operator algebras, and thus showing their broadervalidity. In Chap.10 spectral theory is applied to several practical and completelyabstract contexts, both quantum and not.

Chapter6 treats, from a physical perspective, themotivation underlying the theory.The general mathematical formulation of QM concerns Chap.7. The mathematicalstarting point is the idea, goingback to vonNeumann, that the propositions of physicalquantum systems are described by the lattice of orthogonal projectors on a complexHilbert space. Maximal sets of physically compatible propositions (in the quantumsense) are described by distributive, orthocomplemented, bounded, σ -complete lat-tices. From this standpoint the quantum definition of an observable in terms of aself-adjoint operator is extremely natural, as is, on the other hand, the formulation ofthe spectral decomposition theorem. Quantum states are defined as measures on thelattice of all orthogonal projectors, which is no longer distributive (due to the pres-ence, in the quantum world, of incompatible propositions and observables). Statesare characterised as positive operators of trace class with unit trace under Gleason’stheorem. Pure states (rays in theHilbert space of the physical system) arise as extremeelements of the convex body of states. Generalisations ofGleason’s statement are alsodiscussed in a more advanced section of Chap. 7. The same chapter also discusseshow to recover the Hilbert space starting from the lattice of elementary proposi-tions, following the theorems of Piron and Solèr. The notion of superselection ruleis also introduced here, and the discussion is expanded in Chap. 11 in terms of directdecomposition of von Neumann factors of observables. In that chapter the notion ofvon Neumann algebra of observables is exploited to present the mathematical for-mulation of quantum theories in more general situations, where not all self-adjointoperators represent observables.

The third part of the book is devoted to the mathematical axioms of QM, andmoreadvanced topics like quantum symmetries and the algebraic formulation of quantumtheories. Quantum symmetries and symmetry groups (both according to Wigner andto Kadison) are studied in depth. Dynamical symmetries and the quantum version ofNoether’s theorem are covered as well. The Galilean group, together with its sub-groups and central extensions, is employed repeatedly as reference symmetry group,to explain the theory of projective unitary representations. Bargmann’s theorem onthe existence of unitary representations of simply connected Lie groups whose Liealgebra obeys a certain cohomology constraint is proved, and Bargmann’s rule ofsuperselection of the mass is discussed in detail. Then the useful theorems of Gårdingand Nelson for projective unitary representations of Lie groups of symmetries areconsidered. Important topics are examined that are often neglected in manuals, likethe uniqueness of unitary representations of the canonical commutation relations(theorems of Stone–von Neumann and Mackey), or the theoretical difficulties in

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defining time as the conjugate operator to energy (the Hamiltonian). The mathemati-cal hurdles one must overcome in order to make the statement of Ehrenfest’s theoremprecise are briefly treated. Chapter14 offers an introduction to the ideas and methodsof the abstract formulation of observables and algebraic states via C∗-algebras. Hereone finds the proof of the GNS theorem and some consequences of purely mathemat-ical flavour, like the general theorem of Gelfand–Najmark. This closing chapter alsocontains material on quantum symmetries in an algebraic setting. As an example theWeyl C∗-algebra associated to a symplectic space (usually infinite-dimensional) ispresented.

The appendices at the end of the book recap facts on partially ordered sets, groupsand differential geometry.

The author has chosen not to include topics, albeit important, such as the theoryof rigged Hilbert spaces (the famous Gelfand triples) [GeVi64], and the powerfulformulation of QM based on the path integral approach [AH-KM08, Maz09]. Doingso would have meant adding further preparatory material, in particular regardingthe theory of distributions, and extending measure theory to the infinite-dimensionalcase.

There are very valuable and recent textbooks similar to this one, at least in themathematical approach. One of the most interesting and useful is the far-reaching[BEH07].

1.1.2 Prerequisites

Apart from a firm background on linear algebra, plus some group theory and repre-sentation theory, essential requisites are the basics of calculus in one and several realvariables, measure theory on σ -algebras [Coh80, Rud86] (summarised at the end ofthis chapter), and a few notions on complex functions.

Imperative, on the physics’ side, is the acquaintance with undergraduate physics.More precisely, analytical mechanics (the groundwork of Hamilton’s formulation ofdynamics) and electromagnetism (the key features of electromagnetic waves and thecrucial wavelike phenomena like interference, diffraction, scattering).

Lesser elementary, yet useful, facts will be recalled where needed (includingexamples) to enable a robust understanding. One section of Chap.12 will need ele-mentary Lie group theory. For this we refer to the book’s epilogue: the last appendixsummarises tidbits of differential geometry rather thoroughly. Further details shouldbe looked up in [War75, NaSt82].

1.1.3 General Conventions

1. The symbol := means “equal, by definition, to”.2. The inclusion symbols ⊂,⊃ allow for equality =.

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1.1 On the Book 5

3. The symbol⊔

denotes the disjoint union.4. N is the set of natural numbers including zero, and R+ := [0,+∞).5. Unless otherwise stated, the field of scalars of a normed, Banach or Hilbert

vector space is the field of complex numbersC, and inner product always meansHermitian inner product.

6. The complex conjugate of a number c is denoted by c. As the same symbol isused for the closure of a set of operators, should there be confusion we willcomment on the meaning.

7. The inner product of two vectorsψ, φ in a Hilbert spaceH is written as (ψ |φ) todistinguish it from the ordered pair (ψ, φ). The product’s left entry is antilinear:(αψ |φ) = α(ψ |φ).If ψ, φ ∈ H, the symbols ψ(φ| ) and (φ| )ψ denote the same linear operatorH � χ �→ (φ|χ)ψ .

8. Complete orthonormal systems in Hilbert spaces are called Hilbert bases. Whenno confusion arises, a Hilbert basis is simply referred to as a basis.

9. The word operator tacitly implies it is linear.10. An operator U : H → H′ between Hilbert spaces H and H′ that is isometric and

surjective is called unitary, even if elsewhere in the literature the name is reservedfor the case H = H′.

11. By vector subspace we mean a subspace for the linear operations, even in pres-ence of additional structures on the ambient space (e.g. Hilbert, Banach etc.).

12. For the Hermitian conjugation we always use the symbol ∗. Note that Hermitianoperator, symmetric operator, and self-adjoint operator are not considered syn-onyms.

13. When referring to maps, one-to-one, 1–1 and injective mean the same, justlike onto and surjective. Bijective means simultaneously one-to-one and onto,and invertible is a synonym of bijective. One should beware that a one-to-onecorrespondence is a bijective mapping. An isomorphism, irrespective of thealgebraic structures at stake, is a 1–1 map onto its image, hence a bijectivehomomorphism.

14. Boldface typeset (within a definition or elsewhere) is typically used when defin-ing a term for the first time.

15. Corollaries, definitions, examples, lemmas, notations, remarks, propositions andtheorems are labelled sequentially to simplify lookup.

16. At times we use the shorthand ‘iff’, instead of ‘if and only if’, to say that twostatements imply one another, i.e. they are logically equivalent.

Finally, if h denotes Planck’s constant, we adopt the notation, widely used by physi-cists,

� := h2π

= 1.054571800(13) × 10−34 Js .


1.2 On Quantum Theories

1.2.1 Quantum Mechanics as a Mathematical Theory

From a mathematical point of view Quantum Mechanics represents a rare blend ofmathematical elegance and descriptive insight into the physical world. The theoryessentially makes use of techniques of functional analysis mixed with incursions andoverlaps with measure theory, probability and mathematical logic.

There are (at least) two possible ways to formulate precisely (i.e. mathemati-cally) elementary QM. The eldest one, historically speaking, is due to von Neumann([Neu32]) in essence, and is formulated using the language of Hilbert spaces and thespectral theory of unbounded operators. A more recent and mature formulation wasdeveloped by several authors in the attempt to solve quantum field theory problems inmathematical physics. It relies on the theory of abstract algebras (∗-algebras and C∗-algebras) that are built mimicking the operator algebras defined and studied, again,by von Neumann (nowadays known as W ∗-algebras or von Neumann algebras), butfreed from the Hilbert-space structure (for instance, [BrRo02] is a classic on operatoralgebras). The core result is the celebrated GNS theorem (after Gelfand, Najmarkand Segal) [Haa96, BrRo02], that we will prove in Chap. 14. The newer formulationcan be considered an extension of the former one, in a very precise sense that weshall not go into here, also by virtue of the novel physical context it introduces and bythe possibility of treating physical systems with infinitely many degrees of freedom,i.e. quantum fields. In particular, this second formulation makes precise sense of thedemand for locality and covariance of relativistic quantum field theories [Haa96],and allows to extend quantum field theories to a curved spacetime.

The algebraic formulation of elementary QM, even though it can be achievedand despite its utmost finesse, is not a strict necessity (see for example [Str05a]and parts of [DA10]). Given the relatively basic nature of our book we shall treatalmost exclusively the first formulation, which displays an impressive mathematicalcomplexity together with amanifest formal elegance.Wewill introduce the algebraicformulation in the last chapter only, and stay within the general framework ratherthan consider QM as a physical object.

A crucial mathematical tool to develop a Hilbert-space formulation for QM is thespectral theorem for self-adjoint operators (unbounded, usually) defined on densesubspaces of a Hilbert space. This theorem, which can be extended to normal oper-ators, was first proved by von Neumann in [Neu32] apropos the mathematical struc-ture of QM: this fundamental work ought to be considered a XX century milestoneof mathematical physics and pure mathematics. The definition of abstract Hilbertspaces and much of the corresponding theory, as we know it today, are also due tovon Neumann and his formalisation of QM. Von Neumann built the modern,axiomatic notion of an abstract Hilbert space [Neu32, Sect. 1] by considering thetwo approaches to QM known at that time: the one relying on Heisenberg matrices,and the one using Schrödinger’s wavefunctions.

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1.2 On Quantum Theories 7

The relationship between QM and spectral theory depends upon the followingfact. The standard way of interpreting QM dictates that physical quantities that aremeasurable over quantum systems can be associated to unbounded self-adjoint oper-ators on a suitable Hilbert space. The spectrum of each operator coincides with thecollection of values the associated physical quantity can attain. The construction ofa physical quantity from easy properties and propositions of the type “the value ofthe quantity falls in the interval (a, b]”, which correspond to orthogonal projectorsin the mathematical scheme one adopts, is nothing else but an integration proce-dure with respect to an appropriate projector-valued spectral measure. In practice,then, the spectral theorem is just a means to construct complicated operators startingfrom projectors or, conversely, decompose operators in terms of projector-valuedmeasures.

The contemporary formulation of spectral theory is certainly different from thatof von Neumann, although the latter already contained all basic constituents. VonNeumann’s treatise (dating back to 1932) discloses an impressive depth still today,especially in the most difficult parts of the physical interpretation of the QM formal-ism. If we read that book it becomes clear that von Neumann was well aware of theseissues, as opposed to many colleagues of his. It would be interesting to juxtapose hisopus to the much more renowned book by Dirac [Dir30] on QM’s fundamentals, acomparison that we leave to the interested reader. At any rate, the great interpretativestrength von Neumann gave to QM begins to be recognised by experimental physi-cists as well, in particular those concerned with quantum measurements [BrKh95].

The so-called quantum logics arise from the attempt to formalise QM from themost radical stand: endowing the same logic used to treat quantum systems withproperties different from those of ordinary logic, and modifying the semantic theory.For example, more than two truth values are allowed, and the Boolean lattice ofpropositions is replaced by a more complicated non-distributive structure. In the firstformulation of quantum logic, known as standard quantum logic and introduced byVon Neumann and Birkhoff in 1936, the role of the Boolean algebra of propositionsis taken by an orthomodular lattice: this is modelled, as a matter of fact, on the set oforthogonal projectors on a Hilbert space, or the collection of closed projection spaces[Bon97], plus some composition rules.Despite its sophistication, thatmodel is knownto contain many flaws when one tries to translate it in concrete (operational) physicalterms. Beside the various formulations of quantum logic [Bon97, DCGi02, EGL09],there are also other foundational formulations based on alternative viewpoints (e.g.,topos theory).

1.2.2 QM in the Panorama of Contemporary Physics

Quantum Mechanics and General and Special Relativity (GSR) represent the twoparadigms by which the physics of the XX and XXI centuries developed. QM is,roughly speaking, the physical theory of the atomic and sub-atomicworld,whileGSRis the physical theory of gravity, the macroscopic world and cosmology (as recently


as 2016, GSR received thunderous experimental confirmation with the detection ofgravitational waves). These two paradigms coalesced, in several contexts, to giverise to relativistic quantum theories. Relativistic Quantum Field Theory [StWi00,Wei99], in particular, haswitnessed a striking growth and a spectacular predictive andexplanatory success relative to the theory of elementary particles and fundamentalinteractions. Two examples for all. In the so-called standard model of elementaryparticles, that theory predicted the unification of theweak and electromagnetic forceswhich was confirmed experimentally at the end of the 1980s during a memorableexperiment at C.E.R.N., in Geneva, where the particles Z0 and W ±, expected byelectro-weak unification, were first observed. More recently, another prediction wasconfirmed: the existence of theHiggs boson, suspected since the 1960s and eventuallydetected 50 years after. On March 14, 2013, referring to the newly observed particle,C.E.R.N. confirmed that: ‘CMS and ATLAS have compared a number of options forthe spin-parity of this particle, and these all prefer no spin and even parity. This,coupled with the measurements of the interactions with other particles, stronglyindicates that the new particle is a Higgs boson.’

The best-ever accuracy in the measurement of a physical quantity in the wholehistory of physics was predicted by quantum electrodynamics. The quantity is theso-called gyro-magnetic ratio g of the electron, a dimensionless number. The valueexpected by quantum electrodynamics for a := g/2 − 1 was

0.001159652359 ± 0.000000000282 ,

and the experimental result turned out to be

0.001159652209 ± 0.000000000031 .

Many physicists believe QM to be the fundamental theory of the universe (more thanrelativistic theories), also owing to the impressive range of linear scales at which itholds: from 1m (Bose–Einstein condensates) to at least 10−16 m (inside nucleons, atquark level). QM has had an enormous success, both theoretical and experimental,in materials’ science, optics, electronics, with several key repercussions: every tech-nological object of common use that is complex enough to contain a semiconductor(childrens’ toys, mobile phones, remote controls…) exploits the quantum propertiesof matter.

Going back to the twomajor approaches of the past century –QMandGSR – thereremain a number of obscure pointswhere these paradigms seem to clash. In particular,the so-called “quantisation of gravity” and the structure of spacetime atPlanck scales(∼10−35 m, ∼10−43 s, the length and time scales obtained from the fundamentalconstants of the two theories: the speed of light, the universal constant of gravity andPlanck’s constant). The necessity of a discontinuous spacetime at ultra-microscopicscales is also reinforced by certainmathematical (and conceptual) hurdles that the so-called theory of quantumRenormalisation has yet to overcome, as consequence of theinfinite values arising in computing processes due to the interaction of elementaryparticles. From a purely mathematical perspective the existence of infinite values

1.2 On Quantum Theories 9

is actually related to the problem, already intrinsically ambiguous, of defining theproduct of two distributions: infinites are not the root of the problem, but a meremanifestation of it.

These issues, whether unsolved or partially solved, have underpinned importanttheoretical advancements of late, which in turn influenced the developments of puremathematics itself. Examples include (super-)String theory, the various Noncommu-tative Geometries, first of all Alain Connes’ version, and Loop Quantum Gravity.The difficulty in deciding which of these theories makes any physical sense and isapt to describe the universe at very small scales is also practical: today’s technologyis not capable of preparing experiments that enable to distinguish among all avail-able theories. However, it is relevant to note that recent experimental observationsof the so-called γ -bursts, conducted with the telescope “Fermi Gamma-ray”, havelowered the threshold for detecting quantum-gravity phenomena (e.g. the violationof Lorentz’s symmetry) well below Planck’s length [Abd09]. Other discrepanciesbetween QM and GSR, about which the debate is more relaxed today than it wasin the past, have to do with QM verses the notions of locality of relativistic nature(Einstein–Podolsky–Rosen paradox [Bon97]) in relationship to QM’s entanglementphenomena.This is due in particular toBell’swork in the late 1960s, and to the famousexperiments of Aspect. Both disproved Einstein’s expectations, and secondly theyconfirmed the Copenhagen interpretation, eventually proving that nonlocality is acharacteristic of Nature, independent of whether one accepts the standard interpre-tation of QM or not. The vast majority of physicists seems to agree that the existenceof nonlocal physical processes, as QM forecasts, does not imply any concrete viola-tion of the core of Relativity (quantum entanglement does not involve superluminaltransmission of information, nor the violation of causality [Bon97]).

In the standard interpretation of QM that is called the Copenhagen interpretationthere are parts that remain physically and mathematically unintelligible, yet stillvery interesting conceptually. In particular, and despite several appealing attempts,it still not clear how standard mechanics may be seen as a limit subcase of QM,nor how to demarcate (even roughly, or temporarily) the two worlds. Further, thequestion remains about the physical and mathematical description of the so-calledprocess of quantum measurement, of which more later, which is strictly related tothe classical limit of QM. From this fact, as well, other interpretations of the QMformalisms were born that differ deeply from the Copenhagen interpretation. Amongthese more recent interpretations, once considered heresies, Bohm’s interpretationrelies on hidden variables [Bon97, Des99] and is particularly intriguing.

Doubts are sometimes raised about the formulation of QM and about it being nottruly clear, but just a list of procedures that “actually work”, whereas its true nature issomething inaccessible, sort of “noetic”. In the author’s opinion a dangerous episte-mological mistake hides behind this point of view. Themisconception is based on thebelief that “explaining” a phenomenon means reducing it to the categories of dailylife, as if everyday experience reached farther than reality itself. Quite the contrary:those categorieswere built upon conventionalwisdom, and hencewithout any allegedmetaphysical insight. There could be a deep philosophical landscape unfolding onthe other side of that simple “actually works”, and it may draw us closer to reality


rather than pushing us away from it. Quantum Mechanics has taught us to think ina different fashion, and for this reason it has been (is, actually) an incredible oppor-tunity for humanity. To turn our backs on QM and declare we do not understand itbecause it refuses to befit our familiar mental categories means locking the door thatseparates us from something huge. This is the author’s stance, who does indeed con-sider Heisenberg’s uncertainty principle (a theorem in today’s formulation, despitethe name) one of the highest achievements of the human enterprise.

Mathematics is the most accurate of languages invented by man. It allows tocreate formal structures corresponding to worlds that may or may not exist. Theplausibility of these hypothetical realities is found solely in the logical or syntac-tical coherence of the corresponding mathematical structure. In this way semantic“chimeras”might arise, that turn out to be syntactically coherent nevertheless. Some-times these creatures are consistent with worlds or states that do exist, although theyhave not been discovered yet. A feature that is attributable to an existing entity canonly either be present or not, according to the classical ontological view. QuantumMechanics, in particular, leads to say that any such property may not simply obeythe true/false pattern, but also be “uncertain”, despite being inherent to the objectitself. This tremendous philosophical leap can be entirely managed within the math-ematical foundations of QM, and represents the most profound philosophical legacyof Heisenberg’s principle.

At least two general issues remain unanswered, both of gnoseological nature,essentially, and common to the entire formulation of modern science. The first isthe relationship between theoretical entities and the objects we have experience of.The problem is particularly delicate in QM, where the notion of what a measuringinstrument is has not yet been fully clarified. Generally speaking, the relationship ofa theoretical entity with an experimental object is not direct, and still based on oftenunderstated theoretical assumptions. But this is also the case in classical theories,when one, for example, wants to tackle problems such as the geometry of the phys-ical space. There, it is necessary to identify, inside the physical reality, objects thatcorrespond to the idea of a point, a segment, and so on, and to do that we use otherassumptions, like the fact that the geometry of the straightedge is the same as wheninspecting space with light beams. The second issue is the hopelessness of trying toprove the syntactic coherence of a mathematical construction. We may attempt toreduce the latter to the coherence of set theory, or category theory. That this reduc-tion should prove the construction’s solidity has more to do with psychology thanwith it being a real fact, due to the profusion of well-known paradoxes disseminatedalong the history of the formalisation of mathematics, and eventually due to Gödel’sfamous theorem.

In spite of all, QM (but also other scientific theories) has been – and is – capableof predicting new facts and not yet observed phenomena that have been confirmedexperimentally.

In this sense Quantum Mechanics must contain elements of reality.

1.3 Backgrounds on General Topology 11

1.3 Backgrounds on General Topology

For the reader’s sake we collect here notions of point-set topology that will be usedby and large in the book. All statements are elementary and classical, and can befound easily in any university treatise, so for brevity we will prove almost nothing.The practiced reader may skip this section completely and return to it at subsequentstages for reference.

1.3.1 Open/Closed Sets and Basic Point-Set Topology

Open and closed sets are defined as follows [Ser94II], in the greatest generality.

Definition 1.1 The pair (X,T ), where X is a set and T a collection of subsets ofX, is called a topological space if:

(i) ∅,X ∈ T ,(ii) the union of (arbitrarily many) elements of T is an element of T ,(iii) the intersection of a finite number of elements of T belongs to T .

T is called a topology on X and the elements of T are the open sets of X.

Definition 1.2 On a topological space (X,T ):(a) a basis for the topology of (X,T ) is a subsetB ⊂ T such that each element inT is the union of elements of B;(b) an open neighbourhood of p ∈ X is an element A ∈ T such that p ∈ A;(c) x ∈ S ⊂ X is an interior point of S if there exists an open neighbourhood A ofx contained in S.The interior of a set S ⊂ X is the set:

I nt (S) := {x ∈ X | x is an interior point o f S} .

The exterior of a set S ⊂ X is the set:

Ext (S) := {x ∈ X | x is an interior point o f X \ S} .

The frontier of a set S ⊂ X is the difference set:

∂S := X \ (I nt (S) ∪ Ext (S)) .

(d) C ⊂ X is called closed if X \ C is open.A subset S ⊂ X in a topological space (X,T ) inherits the structure of a topologicalspace from (X,T ) by defining a topology on S as TS := {S ∩ A | A ∈ T }. Thistopology (the definition is easily satisfied) is called the induced topology on S by(X,T ).


Most of the topological spaces we will see in this text are Hausdorff spaces, in whichopen sets “separate” points.

Definition 1.3 A topological space (X,T ) and its topology are called Hausdorffif they satisfy the Hausdorff property: for every x, x ′ ∈ X there exist A, A′ ∈ T ,with x ∈ A, x ′ ∈ A′, such that A ∩ A′ = ∅.Remark 1.4 (1) Both X and ∅ are open and closed sets.(2) Closed sets satisfy properties that are “dual” to open sets, as follows straightfor-wardly from their definition. Hence:

(i) ∅,X are closed,(ii) the intersection of (arbitrarily many) closed sets is closed,(iii) the finite union of closed sets is a closed set.

(3) The simplest example of a Hausdorff topology is the collection of subsets of Rcontaining the empty set and arbitrary unions of open intervals. This is a basis forthe topology in the sense of Definition1.1. It is called the Euclidean topology orstandard topology of R.(4) A slightly more complicated example of Hausdorff topology is the Euclideantopology, or standard topology, of Rn and Cn . It is the usual topology one refers toin elementary calculus, and is built as follows. If K := R or C, the standard normof (c1, . . . , cn) ∈ Kn is, by definition:

||(c1, . . . , cn)|| :=√√√√

n∑

k=1|ck |2 , (c1, . . . , cn) ∈ Kn . (1.1)

The set:Bδ(x0) := {x ∈ Kn | ||x − x0|| < δ} (1.2)

is, hence, the usual open ball of Kn of radius δ > 0 and centre x0 ∈ Kn . The opensets in the standard topology of Kn are, empty set aside, the unions of open balls ofany given radius and centre. These balls constitute a basis for the standard topologyof Rn and Cn . �

Here are notions that will come up often in the sequel.

Definition 1.5 If (X,T ) is a topological space, the closure of S ⊂ X is the set:

S := ∩{C ⊃ S , C ⊂ X | C is closed} . (1.3)

The subset S is called dense in X if S = X.The space (X,T ) is said to be separable if there exists a dense and countable subsetS ⊂ X.From the definition these properties follow.


Proposition 1.6 If (X,T ) is a topological space and S ⊂ X:(a) S is closed.(b) S = S.(c) If T ⊂ X, then S ⊂ T implies S ⊂ T .(d) S is closed if and only if S = S.Definition 1.7 A topological space (X,T ) has a countable basis, or is second-countable, if there is a countable subset T0 ⊂ T (the “countable basis”) such thatevery open set is the union of elements of T0.

If (X,T ) has a countable basis then Lindelöf’s lemma holds:

Theorem 1.8 (Lindelöf’s lemma) Let (X,T ) be a second-countable topologicalspace. Then any open covering of a given subset in X admits a countable sub-covering: if B ⊂ X and {Ai }i∈I ⊂ T with ∪i∈I Ai ⊃ B, then ∪i∈J Ai ⊃ B for somecountable subset J ⊂ I .

Remark 1.9 Rn andCn , equipped with the standard topology, are second-countable:for Rn , T0 can be taken to be the collection of open balls with rational radii andcentred at rational points. The generalisation to Cn is obvious. �

In conclusion, we recall the definition of product topology.

Definition 1.10 If {(Xi ,Ti )}i∈F is a collection of topological spaces indexed by afinite set F , the product topology on ×i∈FXi is the topology whose open sets are ∅and the unions of Cartesian products ×i∈F Ai , with Ai ∈ Ti for any i ∈ F .If F has arbitrary cardinality, the previous definition cannot be generalised directly.If we did so in the obvious way we would not maintain important properties, suchas Tychonoff’s theorem, that we will discuss later. Nevertheless, a natural topol-ogy on ×i∈FXi can be defined, still called product topology because is extendsDefinition1.10.

Definition 1.11 If {(Xi ,Ti )}i∈F is a collection of topological spaces with F of arbi-trary cardinality, the product topology on ×i∈FXi has as open sets ∅ and the unionsof Cartesian products×i∈F Ai , with Ai ∈ Ti for any i ∈ F , such that on each×i∈F Aiwe have Ai = Xi but for a finite number of indices i .Remark 1.12 (1) The standard topology of Rn is the product topology obtained byendowing the single factors R with the standard topology. The same happens for Cn .(2) Either in case of finitely many factors, or infinitely many, the canonical projec-tions:

πi : × j∈FX j � {x j } �→ xi ∈ Xiare clearly continuous if we put the product topology on the domain. It can be provedthat the product topology is the coarsest among all topologies making the canonicalprojections continuous (coarsest means it is contained in any such topology). �


1.3.2 Convergence and Continuity

Let us pass to convergence and continuity. First of all we need to recall the notionsof convergence of a sequence and limit point.

Definition 1.13 Let (X,T ) be a topological space.(a) A sequence {xn}n∈N ⊂ X converges to a point x ∈ X, called the limit of thesequence:

x = limn→+∞ xn and also wri t ten xn → x as n → +∞

if, for any open neighbourhood A of x there exists NA ∈ R such that xn ∈ Awhenevern > NA.(b) x ∈ X is a limit point of a subset S ⊂ X if any open neighbourhood A of xcontains a point of S \ {x}.Remark 1.14 It should be patent from the definitions that in a Hausdorff space thelimit of a sequence is unique, if it exists. �

The relationship between limit points and closure of a set is sanctioned by the fol-lowing classical and elementary result:

Proposition 1.15 Let (X,T ) be a topological space and S ⊂ X.S coincides with the union of S and the set of its limit points.

The definition of continuity and continuity at a point is recalled below.

Definition 1.16 Let f : X → X′ be a function between topological spaces (X,T ),(X′,T ′).(a) f is called continuous if f −1(A′) ∈ T for any A′ ∈ T ′.(b) f is said continuous at p ∈ X if, for any open neighbourhood A′f (p) of f (p),there is an open neighbourhood Ap of p such that f (Ap) ⊂ A′f (p).(c) f is called a homeomorphism if:

(i) f is continuous,(ii) f is bijective,(iii) f −1 : X′ → X is continuous.

In this case X and X′ are said to be homeomorphic (under f ).(d) f is called open (respectively closed) if f (A) ∈ T ′ for A ∈ T (resp. X′ \f (C) ∈ T ′ for X \ C ∈ T ).Remark 1.17 (1) It is easy to check that f : X → X′ is continuous if and only if itis continuous at every point p ∈ X.(2)The notion of continuity at p as of (b) reduces to themore familiar “�-δ” definitionwhen the spaces X and X′ are Rn (or Cn) with the standard topology. To see this bearin mind that: (a) open neighbourhoods can always be chosen to be open balls of radiiδ and �, centred at p and f (p) respectively; (b) every open neighbourhood of a pointcontains an open ball centred at that point. �


Let us mention a useful result concerning the standard real line R. One defines thelimit supremum (also superior limit, or simply limsup) and the limit infimum(inferior limit or just liminf) of a sequence {sn}n∈N ⊂ R as follows:

lim supn∈N

sn := infk∈N

supn≥k

sn

(

= limk→+∞ supn≥k

sn

)

, lim infn∈N

sn := supk∈N

infn≥k sn

(

= limk→+∞ infn≥k sn

)

.

Note how these numbers exist for any given sequence {sn}n∈N ⊂ R, possibly beinginfinite, as they arise as limits of monotone sequences, whereas the limit of {sn}n∈Nmight not exist (neither finite nor infinite). However, it is not hard to prove thefollowing elementary fact.

Proposition 1.18 If {sn}n∈N ⊂ R, then limn→+∞ sn exists, possibly infinite, if andonly if

lim supn∈N

sn = lim infn∈N

sn .

In such a case:lim

n→+∞ sn = lim supn∈N sn = lim infn∈N sn .

1.3.3 Compactness

Let us briefly recall some easy facts about compact sets.

Definition 1.19 Let (X,T ) be a topological space and K ⊂ X a subset.(a) K is called compact if any open covering of it admits a finite sub-covering: if{Ai }i∈I ⊂ T with ∪i∈I Ai ⊃ K then ∪i∈J Ai ⊃ K for some finite subset J ⊂ I .(b) K is said relatively compact if K is compact.(c) X is locally compact if any point in X has a relatively compact open neighbour-hood.

Compact sets satisfy a host of properties [Ser94II] and we will not be concernedwith them much more (though we shall return to them in Chap. 4). Let us recall afew results, at any rate.We begin with the relationship to calculus on Rn . If X is Rn (or Cn identified withR

2n), the celebrated Heine–Borel theorem holds [Ser94II].

Theorem 1.20 (Heine–Borel) If Rn is equipped with the standard topology, K ⊂R

n is compact if and only if K is simultaneously closed and bounded (meaningK ⊂ Bδ(x) for some x ∈ Rn, δ > 0).In calculus, the Weierstrass theorem, which deals with continuous maps definedon compact subsets of Rn (or Cn), can be proved directly without the definitionof compactness. Actually one can prove a more general statement on Rn-valued(Cn-valued) continuous maps defined on compact subsets.

http://dx.doi.org/10.1007/978-3-319-70706-8_4


Proposition 1.21 If K = C or R, let || || denote the standard norm of Kn as in (1.1),and endow Kn with the standard topology.If f : K → Kn is continuous on the compact subset K of a topological space, thenit is bounded, i.e. there exists M ∈ R such that || f (x)|| ≤ M for any x ∈ K .Proof Since f is continuous at any point p ∈ K , we have || f (x)|| ≤ Mp ∈ R forall x in an open neighbourhood Ap of p. As K is compact, we may extract afinite sub-covering {Apk }k=1,...,N from {Ap}p∈K that covers K . The number M :=maxk=1,...,N Mk satisfies the request. �Remark 1.22 (1) It is easily proved that if X is a Hausdorff space and K ⊂ X iscompact then K is closed.(2) Similarly, if K is compact in X, then every closed subset K ′ ⊂ K is compact.(3) Continuous functions map compact sets to compact sets.(4)By definition of compactness and of induced topology it is clear that a set K ⊂ Y ,with the induced topology on Y ⊂ X, is compact in Y if and only if it is compact inX. �The properties of being compact and Hausdorff bear an interesting relationship. Onesuch property is expressed by the following useful statement.

Proposition 1.23 Let f : M → N be a continuous map from the compact space Mto the Hausdorff space N. If f is bijective then it is a homeomorphism.

On locally compact Hausdorff spaces an important technical result, known as Ury-sohn’s lemma, holds. To state it, we first need to define the support of a mapf : X → C:

supp( f ) := {x ∈ X | f (x) �= 0} ,

where the bar is the topological closure in the space X.

Theorem 1.24 (Urysohn’s lemma) If (X,T ) is a Hausdorff, locally compact space,for any compact K ⊂ X and any open set U ⊃ K there exists a continuous mapf : X → [0, 1] such that:

(i) supp( f ) ⊂ U,(ii) supp( f ) is compact,(iii) f (x) = 1 if x ∈ K .

Eventually, the following key theorem relates the product topology to compactness.

Theorem 1.25 (Tychonoff) The Cartesian product of (arbitrarily many) compactspaces is compact in the product topology.

1.3.4 Connectedness

Definition 1.26 A topological spaceX

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