Spectral Theory and Quantum Mechanics: Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation

UNITEXT 110

Valter Moretti

Spectral Theory
and Quantum
Mechanics
Mathematical Foundations
of Quantum Theories,
Symmetries and Introduction
to the Algebraic Formulation
Second Edition


UNITEXT - La Matematica per il 3+2
Volume 110

Editor-in-chief
A. Quarteroni
Series editors
L. Ambrosio
P. Biscari
C. Ciliberto
C. De Lellis
M. Ledoux
V. Panaretos
W.J. Runggaldier


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Valter Moretti

Spectral Theory
and Quantum Mechanics
Mathematical Foundations of Quantum
Theories, Symmetries and Introduction
to the Algebraic Formulation
Second Edition

123


Valter Moretti
Department of Mathematics
University of Trento
Povo, Trento
Italy
Translated by: Simon G. Chiossi, Departamento de Matemática Aplicada (GMA-IME),
Universidade Federal Fluminense

ISSN 2038-5714
ISSN 2532-3318 (electronic)
UNITEXT - La Matematica per il 3+2
ISSN 2038-5722
ISSN 2038-5757 (electronic)
ISBN 978-3-319-70705-1
ISBN 978-3-319-70706-8 (eBook)
/>Library of Congress Control Number: 2017958726
Translated and extended version of the original Italian edition: V. Moretti, Teoria Spettrale e Meccanica
Quantistica, © Springer-Verlag Italia 2010

1st edition: © Springer-Verlag Italia 2013
2nd edition: © Springer International Publishing AG 2017
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part
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The registered company is Springer International Publishing AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland


To Bianca


Preface to the Second Edition

In this second English edition (third, if one includes the first Italian one), a large
number of typos and errors of various kinds have been amended.
I have added more than 100 pages of fresh material, both mathematical and
physical, in particular regarding the notion of superselection rules—addressed from

several different angles—the machinery of von Neumann algebras and the abstract
algebraic formulation. I have considerably expanded the lattice approach to
Quantum Mechanics in Chap. 7, which now contains precise statements leading up
to Solèr’s theorem on the characterization of quantum lattices, as well as generalised versions of Gleason’s theorem. As a matter of fact, Chap. 7 and the related
Chap. 11 have been completely reorganised. I have incorporated a variety of results
on the theory of von Neumann algebras and a broader discussion on the mathematical formulation of superselection rules, also in relation to the von Neumann
algebra of observables. The corresponding preparatory material has been fitted into
Chap. 3. Chapter 12 has been developed further, in order to include technical facts
concerning groups of quantum symmetries and their strongly continuous unitary
representations. I have examined in detail the relationship between Nelson domains
and Gårding domains. Each chapter has been enriched by many new exercises,
remarks, examples and references. I would like once again to thank my colleague
Simon Chiossi for revising and improving my writing.
For having pointed out typos and other errors and for useful discussions, I am
grateful to Gabriele Anzellotti, Alejandro Ascárate, Nicolò Cangiotti, Simon G.
Chiossi, Claudio Dappiaggi, Nicolò Drago, Alan Garbarz, Riccardo Ghiloni, Igor
Khavkine, Bruno Hideki F. Kimura, Sonia Mazzucchi, Simone Murro, Giuseppe
Nardelli, Marco Oppio, Alessandro Perotti and Nicola Pinamonti.
Povo, Trento, Italy
September 2017

Valter Moretti

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 discipline
and with a curious mind, told me this story: “A great scientist named Albert Einstein

discovered that any object with a mass can't travel faster than the speed of light”. To my
bewilderment I replied, boldly: “This can't be true, if I run almost at that speed and then
accelerate 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: “That
bloke, Einstein, must've checked this thing many times … how do you say, he did many
experiments?” The answer I got was utterly unexpected: “Not even one I believe. He used
maths!”
What did numbers and geometrical figures have to do with the existence of an upper limit to
speed? How could one stand by such an apparently nonsensical statement as the existence
of 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 on
earth was it, and what did it have to do with maths? This was one of the most beguiling and
irresistible 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 a
common part that originated, in preliminary form, when I was a Physics undergraduate at the University of Genova. The third-year compulsory lecture course
called Theoretical Physics was the second exam that had us pupils seriously
climbing the walls (the first being the famous Physics II, covering thermodynamics
and classical electrodynamics).
Quantum Mechanics, taught in Institutions, elicited a novel and involved way of
thinking, a true challenge for craving students: for months we hesitantly faltered on
a hazy and uncertain terrain, not understanding what was really key among the
notions we were trying—struggling, I should say—to learn, together with a completely new formalism: linear operators on Hilbert spaces. At that time, actually, we
did not realise we were using this mathematical theory, and for many mates of
mine, the matter would have been, rightly perhaps, completely futile; Dirac's bra
vectors were what they were, and that’s it! They were certainly not elements in the
topological dual of the Hilbert space. The notions of Hilbert space and dual
topological space had no right of abode in the mathematical toolbox of the majority

ix


x

Preface to the First Edition

of my fellows, even if they would soon come back in through the back door, with
the course Mathematical Methods of Physics taught by Prof. G. Cassinelli.
Mathematics, and the mathematical formalisation of physics, had always been my
flagship to overcome the difficulties that studying physics presented me with, to the
point that eventually (after a Ph.D. in Theoretical Physics) I officially became a
mathematician. Armed with a maths’ background—learnt in an extracurricular
course of study that I cultivated over the years, in parallel to academic physics—and
eager to broaden my knowledge, I tried to formalise every notion I met in that new
and riveting lecture course. At the same time, I was carrying along a similar project
for the mathematical formalisation of General Relativity, unaware that the work put
into Quantum Mechanics would have been incommensurably bigger.
The formulation of the spectral theorem as it is discussed in x 8, 9 is the same I
learnt when taking the Theoretical Physics exam, which for this reason was a
dialogue of the deaf. Later my interest turned to Quantum Field Theory, a subject I
still work on today, though in the slightly more general framework of QFT in
curved spacetime. Notwithstanding, my fascination with the elementary formulation of Quantum Mechanics never faded over the years, and time and again chunks
were added to the opus I begun writing as a student.
Teaching this material to master’s and doctoral students in mathematics and
physics, 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 the
pile of loose notes and correct several sections, incorporating many people’s
remarks.
Concerning this, I would like to thank my colleagues, the friends from the

newsgroups it.scienza.fisica, it.scienza.matematica and free.it.scienza.fisica, and the
many students—some of which are now fellows of mine—who contributed to
improve the preparatory material of the treatise, whether directly or not, in the
course 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 am
indebted, for various reasons also unrelated to the book, to my late colleague
Alberto Tognoli. My greatest appreciation goes to R. Aramini, D. Cadamuro and
C. Dappiaggi, who read various versions of the manuscript and pointed out a
number 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 covered
in the book and for pointing out primary references.
At last, I would like to thank E. Gregorio for the invaluable and on-the-spot
technical help with the LaTeX package.
In the transition from the original Italian to the expanded English version, a
massive number of (uncountably many!) typos and errors of various kinds have
been 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,


Preface to the First Edition

xi

both mathematical and physical, including a chapter, at the end, on the so-called
algebraic formulation.
In particular, Chap. 4 contains the proof of Mercer’s theorem for positive

Hilbert–Schmidt operators. The analysis of the first two axioms of Quantum
Mechanics in Chap. 7 has been deepened and now comprises the algebraic characterisation of quantum states in terms of positive functionals with unit norm on the
C à -algebra of compact operators. General properties of Cà -algebras and à -morphisms are introduced in Chap. 8. As a consequence, the statements of the spectral
theorem and several results on functional calculus underwent a minor but necessary
reshaping 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 important
example concerning Bargmann’s theorem was added in Chap. 12 (formerly
Chap. 11). In the same chapter, after introducing the Haar measure, the Peter–Weyl
theorem on unitary representations of compact groups is stated and partially proved.
This is then applied to the theory of the angular momentum. I also thoroughly
examined the superselection rule for the angular momentum. The discussion on
POVMs in Chap.13 (ex Chap. 12) is enriched with further material, and I included a
primer on the fundamental ideas of non-relativistic scattering theory. Bell’s
inequalities (Wigner’s version) are given considerably more space. At the end
of the first chapter, basic point-set topology is recalled together with abstract
measure theory. The overall effort has been to create a text as self-contained as
possible. I am aware that the material presented has clear limitations and gaps.
Ironically—my own research activity is devoted to relativistic theories—the entire
treatise unfolds at a non-relativistic level, and the quantum approach to Poincaré’s
symmetry 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
October 2012

Valter Moretti

Contents

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1

Introduction and Mathematical Backgrounds . . . . . . . . . . . . .
1.1 On the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1
Scope and Structure . . . . . . . . . . . . . . . . . . . . .
1.1.2
Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.3
General Conventions . . . . . . . . . . . . . . . . . . . . .
1.2 On Quantum Theories . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1
Quantum Mechanics as a Mathematical Theory .
1.2.2
QM in the Panorama of Contemporary Physics .
1.3 Backgrounds on General Topology . . . . . . . . . . . . . . . . .
1.3.1
Open/Closed Sets and Basic Point-Set Topology
1.3.2
Convergence and Continuity . . . . . . . . . . . . . . .
1.3.3
Compactness . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.4
Connectedness . . . . . . . . . . . . . . . . . . . . . . . . .

1.4 Round-Up on Measure Theory . . . . . . . . . . . . . . . . . . . .
1.4.1
Measure Spaces . . . . . . . . . . . . . . . . . . . . . . . .
1.4.2
Positive r-Additive Measures . . . . . . . . . . . . . .
1.4.3
Integration of Measurable Functions . . . . . . . . .
1.4.4
Riesz’s Theorem for Positive Borel Measures . .
1.4.5
Differentiating Measures . . . . . . . . . . . . . . . . . .
1.4.6
Lebesgue’s Measure on Rn . . . . . . . . . . . . . . . .
1.4.7
The Product Measure . . . . . . . . . . . . . . . . . . . .
1.4.8
Complex (and Signed) Measures . . . . . . . . . . . .
1.4.9
Exchanging Derivatives and Integrals . . . . . . . .

2

Normed and Banach Spaces, Examples and Applications .
2.1 Normed and Banach Spaces and Algebras . . . . . . . . .
2.1.1
Normed Spaces and Essential Topological
Properties . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2
Banach Spaces . . . . . . . . . . . . . . . . . . . . . .
2.1.3

Example: The Banach Space CðK; Kn Þ, The
Theorems of Dini and Arzelà–Ascoli . . . . . .

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xiii


xiv

Contents

2.1.4
Normed Algebras, Banach Algebras and Examples .
Operators, Spaces of Operators, Operator Norms . . . . . . . . .
The Fundamental Theorems of Banach Spaces . . . . . . . . . . .
2.3.1
The Hahn–Banach Theorem and Its Immediate
Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2

The Banach–Steinhaus Theorem or Uniform
Boundedness Principle . . . . . . . . . . . . . . . . . . . . .
2.3.3
Weak Topologies. Ã -Weak Completeness of X 0 . . .
2.3.4
Excursus: The Theorem of Krein–Milman, Locally
Convex Metrisable Spaces and Fréchet Spaces . . . .
2.3.5
Baire’s Category Theorem and Its Consequences:
The Open Mapping Theorem and the Inverse
Operator Theorem . . . . . . . . . . . . . . . . . . . . . . . .
2.3.6
The Closed Graph Theorem . . . . . . . . . . . . . . . . .
2.4 Projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Equivalent Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 The Fixed-Point Theorem and Applications . . . . . . . . . . . . .
2.6.1
The Fixed-Point Theorem of Banach–Caccioppoli .
2.6.2
Application of the Fixed-Point Theorem: Local
Existence and Uniqueness for Systems of
Differential Equations . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
2.3

3

Hilbert Spaces and Bounded Operators . . . . . . . . . . . . . . . .
3.1 Elementary Notions, Riesz’s Theorem and Reflexivity . .

3.1.1
Inner Product Spaces and Hilbert Spaces . . . . .
3.1.2
Riesz’s Theorem and Its Consequences . . . . . .
3.2 Hilbert Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Hermitian Adjoints and Applications . . . . . . . . . . . . . . .
3.3.1
Hermitian Conjugation, or Adjunction . . . . . . .
Ã
-Algebras, C à -Algebras, and à -Representations
3.3.2
3.3.3
Normal, Self-Adjoint, Isometric, Unitary and
Positive Operators . . . . . . . . . . . . . . . . . . . . .
3.4 Orthogonal Structures and Partial Isometries . . . . . . . . . .
3.4.1
Orthogonal Projectors . . . . . . . . . . . . . . . . . . .
3.4.2
Hilbert Sum of Hilbert Spaces . . . . . . . . . . . . .
3.4.3
Partial Isometries . . . . . . . . . . . . . . . . . . . . . .
3.5 Polar Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1
Square Roots of Bounded Positive Operators . .
3.5.2
Polar Decomposition of Bounded Operators . . .
3.6 Introduction to von Neumann Algebras . . . . . . . . . . . . .
3.6.1
The Notion of Commutant . . . . . . . . . . . . . . .
3.6.2

Von Neumann Algebras, Also Known
as W Ã -Algebras . . . . . . . . . . . . . . . . . . . . . . .

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Contents

3.6.3
Further Relevant Operator Topologies .
3.6.4
Hilbert Sum of von Neumann Algebras
3.7 The Fourier–Plancherel Transform . . . . . . . . . . .

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4

5

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Families of Compact Operators on Hilbert Spaces and
Fundamental Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Compact Operators on Normed and Banach Spaces . . . . . .
4.1.1
Compact Sets in (Infinite-Dimensional) Normed
Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2
Compact Operators on Normed Spaces . . . . . . . .
4.2 Compact Operators on Hilbert Spaces . . . . . . . . . . . . . . . .
4.2.1
General Properties and Examples . . . . . . . . . . . . .
4.2.2
Spectral Decomposition of Compact Operators on

Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Hilbert–Schmidt Operators . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1
Main Properties and Examples . . . . . . . . . . . . . .
4.3.2
Integral Kernels and Mercer’s Theorem . . . . . . . .
4.4 Trace-Class (or Nuclear) Operators . . . . . . . . . . . . . . . . . .
4.4.1
General Properties . . . . . . . . . . . . . . . . . . . . . . .
4.4.2
The Notion of Trace . . . . . . . . . . . . . . . . . . . . . .
4.5 Introduction to the Fredholm Theory of Integral Equations .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Densely-Defined Unbounded Operators on Hilbert Spaces . . . . .
5.1 Unbounded Operators with Non-maximal Domains . . . . . . . .
5.1.1
Unbounded Operators with Non-maximal Domains
in Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2
Closed and Closable Operators . . . . . . . . . . . . . . .
5.1.3
The Case of Hilbert Spaces: The Structure of H È H
and the Operator s . . . . . . . . . . . . . . . . . . . . . . . .
5.1.4
General Properties of the Hermitian Adjoint
Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Hermitian, Symmetric, Self-adjoint and Essentially
Self-adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Two Major Applications: The Position Operator and the
Momentum Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1
The Position Operator . . . . . . . . . . . . . . . . . . . . . .
5.3.2
The Momentum Operator . . . . . . . . . . . . . . . . . . .
5.4 Existence and Uniqueness Criteria for Self-adjoint
Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1

The Cayley Transform and Deficiency Indices . . . .
5.4.2
Von Neumann’s Criterion . . . . . . . . . . . . . . . . . . .

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xvi

Contents

5.4.3
Nelson’s Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 277
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
6

7

Phenomenology of Quantum Systems and Wave Mechanics:
An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1 General Principles of Quantum Systems . . . . . . . . . . . . .
6.2 Particle Aspects of Electromagnetic Waves . . . . . . . . . .
6.2.1
The Photoelectric Effect . . . . . . . . . . . . . . . . .
6.2.2
The Compton Effect . . . . . . . . . . . . . . . . . . . .
6.3 An Overview of Wave Mechanics . . . . . . . . . . . . . . . . .
6.3.1
De Broglie Waves . . . . . . . . . . . . . . . . . . . . .
6.3.2
Schrödinger’s Wavefunction and Born’s
Probabilistic Interpretation . . . . . . . . . . . . . . . .
6.4 Heisenberg’s Uncertainty Principle . . . . . . . . . . . . . . . . .
6.5 Compatible and Incompatible Quantities . . . . . . . . . . . . .

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289
290
291

291
292
295
295

. . . . . 296
. . . . . 298
. . . . . 300

The First 4 Axioms of QM: Propositions, Quantum States
and Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1 The Pillars of the Standard Interpretation of Quantum
Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Classical Systems: Elementary Propositions and States . . . . .
7.2.1
States as Probability Measures . . . . . . . . . . . . . . . .
7.2.2
Propositions as Sets, States as Measures on Them .
7.2.3
Set-Theoretical Interpretation of the Logical
Connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.4
“Infinite” Propositions and Physical Quantities . . . .
7.2.5
Basics on Lattice Theory . . . . . . . . . . . . . . . . . . .
7.2.6
The Boolean Lattice of Elementary Propositions for
Classical Systems . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Quantum Systems: Elementary Propositions . . . . . . . . . . . . .
7.3.1

Quantum Lattices and Related Structures in Hilbert
Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.2
The Non-Boolean (Non-Distributive) Lattice of
Projectors on a Hilbert Space . . . . . . . . . . . . . . . .
7.4 Propositions and States on Quantum Systems . . . . . . . . . . . .
7.4.1
Axioms A1 and A2: Propositions, States of a
Quantum System and Gleason’s Theorem . . . . . . .
7.4.2
The Kochen–Specker Theorem . . . . . . . . . . . . . . .
7.4.3
Pure States, Mixed States, Transition Amplitudes . .
7.4.4
Axiom A3: Post-Measurement States and
Preparation of States . . . . . . . . . . . . . . . . . . . . . . .
7.4.5
Quantum Logics . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5 Observables as Projector-Valued Measures on R . . . . . . . . .
7.5.1
Axiom A4: The Notion of Observable . . . . . . . . . .

. . 303
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304
306
306
309

. . 309
. . 310
. . 312
. . 316
. . 317
. . 317
. . 318
. . 325
. . 325
. . 334
. . 335
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342

346
348
348


Contents

xvii

7.5.2

Self-adjoint Operators Associated to Observables:
Physical Motivation and Basic Examples . . . . . . . .
7.5.3
Probability Measures Associated to Couples State/
Observable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6 More Advanced, Foundational and Technical Issues . . . . . . .
7.6.1
Recovering the Hilbert Space from the Lattice: The
Theorems of Piron and Solèr . . . . . . . . . . . . . . . . .
7.6.2
The Projector Lattice of von Neumann Algebras
and the Classification of von Neumann Algebras and
Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6.3
Direct Decomposition into Factors and DefiniteType von Neumann Algebras and Factors . . . . . . .
7.6.4
Gleason’s Theorem for Lattices of von Neumann
Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6.5

Algebraic Characterisation of a State as a
Noncommutative Riesz Theorem . . . . . . . . . . . . . .
7.7 Introduction to Superselection Rules . . . . . . . . . . . . . . . . . .
7.7.1
Coherent Sectors, Admissible States and Admissible
Elementary Propositions . . . . . . . . . . . . . . . . . . . .
7.7.2
An Alternate Formulation of the Theory of
Superselection Rules . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8

Spectral Theory I: Generalities, Abstract CÃ -Algebras
and Operators in BðHÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1 Spectrum, Resolvent Set and Resolvent Operator . . . . . . .
8.1.1
Basic Notions in Normed Spaces . . . . . . . . . . . .
8.1.2
The Spectrum of Special Classes of Normal
Operators on Hilbert Spaces . . . . . . . . . . . . . . .
8.1.3
Abstract C Ã -Algebras: Gelfand–Mazur Theorem,
Spectral Radius, Gelfand’s Formula, Gelfand–
Najmark Theorem . . . . . . . . . . . . . . . . . . . . . . .
8.2 Functional Calculus: Representations of Commutative
C Ã -Algebras of Bounded Maps . . . . . . . . . . . . . . . . . . . .
8.2.1
Abstract C Ã -Algebras: Functional Calculus for
Continuous Maps and Self-adjoint Elements . . . .
8.2.2

Key Properties of à -Homomorphisms of
C Ã -Algebras, Spectra and Positive Elements . . . .
8.2.3
Commutative Banach Algebras and the Gelfand
Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.4
Abstract C Ã -Algebras: Functional Calculus
for Continuous Maps and Normal Elements . . . .
8.2.5
C Ã -Algebras of Operators in BðHÞ: Functional
Calculus for Bounded Measurable Functions . . .

. . 351
. . 356
. . 359
. . 359

. . 363
. . 370
. . 373
. . 374
. . 378
. . 378
. . 383
. . 388

. . . . 393
. . . . 394
. . . . 395
. . . . 399

. . . . 401
. . . . 407
. . . . 407
. . . . 411
. . . . 414
. . . . 420
. . . . 422


xviii

Contents

8.3

Projector-Valued Measures (PVMs) . . . . . . . . . . . . . . . .
8.3.1
Spectral Measures, or PVMs . . . . . . . . . . . . . .
8.3.2
Integrating Bounded Measurable Functions in a
PVM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.3
Properties of Operators Obtained Integrating
Bounded Maps with Respect to PVMs . . . . . . .
8.4 Spectral Theorem for Normal Operators in BðHÞ . . . . . .
8.4.1
Spectral Decomposition of Normal Operators
in BðHÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.2

Spectral Representation of Normal Operators
in BðHÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5 Fuglede’s Theorem and Consequences . . . . . . . . . . . . . .
8.5.1
Fuglede’s Theorem . . . . . . . . . . . . . . . . . . . . .
8.5.2
Consequences to Fuglede’s Theorem . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9

. . . . . 431
. . . . . 431
. . . . . 434
. . . . . 441
. . . . . 449
. . . . . 449
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Spectral Theory II: Unbounded Operators on Hilbert Spaces . .
9.1 Spectral Theorem for Unbounded Self-adjoint Operators . . . .
9.1.1
Integrating Unbounded Functions with Respect
to Spectral Measures . . . . . . . . . . . . . . . . . . . . . . .
9.1.2
Von Neumann Algebra of a Bounded Normal
Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.3
Spectral Decomposition of Unbounded Self-adjoint
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.4
Example of Operator with Point Spectrum: The
Hamiltonian of the Harmonic Oscillator . . . . . . . . .
9.1.5
Examples with Continuous Spectrum: The Operators
Position and Momentum . . . . . . . . . . . . . . . . . . . .
9.1.6
Spectral Representation of Unbounded Self-adjoint
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.7
Joint Spectral Measures . . . . . . . . . . . . . . . . . . . .
9.2 Exponential of Unbounded Operators: Analytic Vectors . . . .
9.3 Strongly Continuous One-Parameter Unitary Groups . . . . . . .
9.3.1
Strongly Continuous One-Parameter Unitary

Groups, von Neumann’s Theorem . . . . . . . . . . . . .
9.3.2
One-Parameter Unitary Groups Generated by
Self-adjoint Operators and Stone’s Theorem . . . . . .
9.3.3
Commuting Operators and Spectral Measures . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Spectral Theory III: Applications . . . . . . . . . . . . . . . . . . . . . . .
10.1 Abstract Differential Equations in Hilbert Spaces . . . . . . . .
10.1.1 The Abstract Schrödinger Equation (With Source)
10.1.2 The Abstract Klein–Gordon/d’Alembert Equation
(With Source and Dissipative Term) . . . . . . . . . .

.
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455
463
464
466

467

. . 473
. . 474
. . 474
. . 492
. . 493
. . 503
. . 507
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508
509
512
516

. . 517
. . 520
. . 529
. . 533

. . . 539

. . . 540
. . . 542
. . . 548


Contents

10.1.3 The Abstract Heat Equation . . . . . . . . . . . . . . .
10.2 Hilbert Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.1 Tensor Product of Hilbert Spaces and Spectral
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.2 Tensor Product of Operators . . . . . . . . . . . . . . .
10.2.3 An Example: The Orbital Angular Momentum . .
10.3 Polar Decomposition Theorem for Unbounded Operators .
10.3.1 Properties of Operators AÃ A, Square Roots of
Unbounded Positive Self-adjoint Operators . . . .
10.3.2 Polar Decomposition Theorem for Closed and
Densely-Defined Operators . . . . . . . . . . . . . . . .
10.4 The Theorems of Kato–Rellich and Kato . . . . . . . . . . . . .
10.4.1 The Kato–Rellich Theorem . . . . . . . . . . . . . . . .
10.4.2 An Example: The Operator ÀD þ V and Kato’s
Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xix

. . . . 557
. . . . 561
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561
567
571
574

. . . . 574
. . . . 579
. . . . 581
. . . . 581
. . . . 583
. . . . 590

11 Mathematical Formulation of Non-relativistic Quantum
Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1 Round-up and Further Discussion on Axioms
A1, A2, A3, A4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1.1 Axioms A1, A2, A3 . . . . . . . . . . . . . . . . . . . . . .
11.1.2 A4 Revisited: von Neumann Algebra of
Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1.3 Compatible Observables and Complete Sets of
Commuting Observables . . . . . . . . . . . . . . . . . . .
11.2 Superselection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2.1 Superselection Rules and von Neumann Algebra
of Observables . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2.2 Abelian Superselection Rules Induced by Central
Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2.3 Non-Abelian Superselection Rules and the Gauge
Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3 Miscellanea on the Notion of Observable . . . . . . . . . . . . . .
11.3.1 Mean Value and Standard Deviation . . . . . . . . . .
11.3.2 An Open Problem: What is the Meaning of
f ðA1 ; . . .; An Þ if A1 ; . . .; An are Not Pairwise
Compatible? . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.3 The Notion of Jordan Algebra . . . . . . . . . . . . . . .
11.4 Axiom A5: Non-relativistic Elementary Systems . . . . . . . . .
11.4.1 The Canonical Commutation Relations (CCRs) . .
11.4.2 Heisenberg’s Uncertainty Principle as a Theorem .
11.5 Weyl’s Relations, the Theorems of Stone–von Neumann
and Mackey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . 595
. . . 596

. . . 596
. . . 598
. . . 604
. . . 607
. . . 607
. . . 611
. . . 616
. . . 619
. . . 619

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622
623

624
626
627

. . . 628


xx

Contents

11.5.1
11.5.2
11.5.3
11.5.4
11.5.5
11.5.6
11.5.7
11.5.8

Families of Operators Acting Irreducibly and
Schur’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . .
Weyl’s Relations from the CCRs . . . . . . . . . . . . . .
The Theorems of Stone–von Neumann and
Mackey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Weyl à -Algebra . . . . . . . . . . . . . . . . . . . . . . .
Proof of the Theorems of Stone–von Neumann
and Mackey . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
More on “Heisenberg’s Principle”: Weakening the
Assumptions and the Extension to Mixed States . . .

The Stone–von Neumann Theorem Revisited:
Weyl–Heisenberg Group . . . . . . . . . . . . . . . . . . . .
Dirac’s Correspondence Principle, Weyl’s Calculus
and Deformation Quantisation . . . . . . . . . . . . . . . .

. . 629
. . 631
. . 639
. . 642
. . 646
. . 653
. . 655

. . 657
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661

12 Introduction to Quantum Symmetries . . . . . . . . . . . . . . . . . . . . .
12.1 Definition and Characterisation of Quantum Symmetries . . . .
12.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1.2 Symmetries in Presence of Abelian Superselection
Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1.3 Kadison Symmetries . . . . . . . . . . . . . . . . . . . . . . .
12.1.4 Wigner Symmetries . . . . . . . . . . . . . . . . . . . . . . .
12.1.5 The Theorems of Wigner and Kadison . . . . . . . . .
12.1.6 Dual Action and Inverse Dual Action of Symmetries
on Observables . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1.7 Symmetries as Transformations of Observables:
Symmetries as Ortho-Automorphisms and Segal
Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2 Introduction to Symmetry Groups . . . . . . . . . . . . . . . . . . . .

12.2.1 Projective and Projective Unitary Representations . .
12.2.2 Representations of Actions on Observables: Left and
Right Representations . . . . . . . . . . . . . . . . . . . . . .
12.2.3 Projective Representations and Anti-unitary
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2.4 Central Extensions and Quantum Group Associated
to a Symmetry Group . . . . . . . . . . . . . . . . . . . . . .
12.2.5 Topological Symmetry Groups . . . . . . . . . . . . . . .
12.2.6 Strongly Continuous Projective Unitary
Representations . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2.7 A Special Case: The Topological Group R . . . . . .
12.2.8 Round-Up on Lie Groups and Algebras . . . . . . . . .
12.2.9 Continuous Unitary Finite-Dimensional
Representations of Connected Non-compact Lie
Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . 665
. . 666
. . 667
.
.
.
.

.
.
.
.

669

670
672
674

. . 687

. . 693
. . 695
. . 696
. . 700
. . 701
. . 702
. . 705
. . 711
. . 714
. . 720

. . 730


Contents

12.2.10 Bargmann’s Theorem . . . . . . . . . . . . . . . . . . . .
12.2.11 Theorems of Gårding, Nelson, FS3 . . . . . . . . . .
12.2.12 A Few Words About Representations of Abelian
Groups and the SNAG Theorem . . . . . . . . . . . .
12.2.13 Continuous Unitary Representations of Compact
Hausdorff Groups: The Peter–Weyl Theorem . . .
12.2.14 Characters of Finite-Dimensional Group
Representations . . . . . . . . . . . . . . . . . . . . . . . .

12.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3.1 The Symmetry Group SOð3Þ and the Spin . . . . .
12.3.2 The Superselection Rule of the Angular
Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3.3 The Galilean Group and Its Projective Unitary
Representations . . . . . . . . . . . . . . . . . . . . . . . .
12.3.4 Bargmann’s Rule of Superselection of the Mass .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xxi

. . . . 732
. . . . 743
. . . . 752
. . . . 754
. . . . 768
. . . . 769
. . . . 769
. . . . 773
. . . . 774
. . . . 782
. . . . 785

13 Selected Advanced Topics in Quantum Mechanics . . . . . . . . . . .
13.1 Quantum Dynamics and Its Symmetries . . . . . . . . . . . . . . . .
13.1.1 Axiom A6: Time Evolution . . . . . . . . . . . . . . . . . .
13.1.2 Dynamical Symmetries . . . . . . . . . . . . . . . . . . . . .
13.1.3 Schrödinger’s Equation and Stationary States . . . . .
13.1.4 The Action of the Galilean Group in Position
Representation . . . . . . . . . . . . . . . . . . . . . . . . . . .

13.1.5 Basic Notions of Scattering Processes . . . . . . . . . .
13.1.6 The Evolution Operator in Absence of Time
Homogeneity and Dyson’s Series . . . . . . . . . . . . .
13.1.7 Anti-unitary Time Reversal . . . . . . . . . . . . . . . . . .
13.2 From the Time Observable and Pauli’s Theorem to
POVMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2.1 Pauli’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2.2 Generalised Observables as POVMs . . . . . . . . . . .
13.3 Dynamical Symmetries and Constants of Motion . . . . . . . . .
13.3.1 Heisenberg’s Picture and Constants of Motion . . . .
13.3.2 A Short Detour on Ehrenfest’s Theorem and Related
Mathematical Issues . . . . . . . . . . . . . . . . . . . . . . .
13.3.3 Constants of Motion Associated to Symmetry Lie
Groups and the Case of the Galilean Group . . . . . .
13.4 Compound Systems and Their Properties . . . . . . . . . . . . . . .
13.4.1 Axiom A7: Compound Systems . . . . . . . . . . . . . .
13.4.2 Independent Subsystems: The Delicate Viewpoint
of von Neumann Algebra Theory . . . . . . . . . . . . .
13.4.3 Entangled States and the So-Called “EPR
Paradox” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
.
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793
794
794
797
800

. . 808
. . 811
. . 818
. . 822
.
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826
827
828
831
831

. . 836
. . 839
. . 844
. . 844
. . 846
. . 848


xxii

Contents

13.4.4
13.4.5
13.4.6

Bell’s Inequalities and Their Experimental
Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
EPR Correlations Cannot Transfer Information . . . .
The Phenomenon of Decoherence as a Manifestation
of the Macroscopic World . . . . . . . . . . . . . . . . . . .
Axiom A8: Compounds of Identical Systems . . . . .
Bosons and Fermions . . . . . . . . . . . . . . . . . . . . . .

. . 850
. . 854

. . 857
. . 858

13.4.7
. . 860
13.4.8
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864

14 Introduction to the Algebraic Formulation of Quantum
Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.1 Introduction to the Algebraic Formulation of Quantum
Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.1.1 Algebraic Formulation . . . . . . . . . . . . . . . . . . . . .
14.1.2 Motivations and Relevance of Lie-Jordan
Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.1.3 The GNS Reconstruction Theorem . . . . . . . . . . . .
14.1.4 Pure States and Irreducible Representations . . . . . .
14.1.5 Further Comments on the Algebraic Approach
and the GNS Construction . . . . . . . . . . . . . . . . . .
14.1.6 Hilbert-Space Formulation Versus Algebraic
Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.1.7 Algebraic Abelian Superselection Rules . . . . . . . . .
14.1.8 Fell’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.1.9 Proof of the Gelfand-Najmark Theorem, Universal
Representations and Quasi-equivalent
Representations . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2 Example of a CÃ -Algebra of Observables: The
Weyl C Ã -Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2.1 Further properties of Weyl à -Algebras WðX; rÞ . . .
14.2.2 The Weyl C Ã -Algebra CWðX; rÞ . . . . . . . . . . . . .
14.3 Introduction to Quantum Symmetries Within the Algebraic
Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.3.1 The Algebraic Formulation’s Viewpoint on

Quantum Symmetries . . . . . . . . . . . . . . . . . . . . . .
14.3.2 (Topological) Symmetry Groups in the Algebraic
Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . 867
. . 867
. . 868
. . 869
. . 873
. . 880
. . 885
. . 886
. . 889
. . 894

. . 895
. . 900
. . 900
. . 904
. . 906
. . 906
. . 909

Appendix A: Order Relations and Groups . . . . . . . . . . . . . . . . . . . . . . . . 915
Appendix B: Elements of Differential Geometry . . . . . . . . . . . . . . . . . . . . 919
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937


Chapter 1

Introduction and Mathematical Backgrounds

“O frati”, dissi “che per cento milia perigli siete giunti a
l’occidente, a questa tanto picciola vigilia d’i nostri sensi ch’è
del 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 of
Quantum Mechanics (QM), and Quantum Theories in general, in a mathematically
rigorous way. This is a treatise on Mathematics (or Mathematical Physics) rather than
a text on Quantum Mechanics. Except for a few cases, the physical phenomenology
is 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 lose
touch with the world of physics.
In alternative to, and irrespective of, the physical content, the book should be
considered as an introductory text, albeit touching upon rather advanced topics, on
functional analysis on Hilbert spaces, including a few elementary yet fundamental
results on C ∗ -algebras. Special attention is given to a series of results in spectral
theory, such as the various formulations of the spectral theorem for bounded normal
operators 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 Theories
is “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 the
west, to the short evening watch which your own senses still must keep, do not choose to deny
the experience of what lies past the Sun and of the world yet uninhabited.” Dante Alighieri, The
Divine Comedy, translated by J. Finn Cotter, edited by C. Franco, Forum Italicum Publishing,

New York, 2006.)

© Springer International Publishing AG 2017
V. Moretti, Spectral Theory and Quantum Mechanics, UNITEXT - La Matematica
per il 3+2 110, />
1


2

1 Introduction and Mathematical Backgrounds

logically independent of those, although the motivations for certain mathematical
definitions 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 on
the mathematical structure of QM and Quantum Theories. These are more advanced
than what is normally encountered in quantum physics’ manuals. Many of these
aspects 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 and
Specker, the theorems of Stone–von Neumann and Mackey, Stone’s theorem and von
Neumann’s theorem about one-parameter unitary groups, Kadison’s theorem, besides
the better known Wigner, Bargmann and GNS theorems; or, more abstract results in
operator theory such as Fuglede’s theorem, or the polar decomposition for closed
unbounded operators (which is relevant in the Tomita–Takesaki theory and statistical
Quantum Mechanics in relationship to the KMS condition); furthermore, self-adjoint
properties for symmetric operators, due to Nelson, that descend from the existence
of dense sets of analytical vectors, and finally, Kato’s work (but not only his) on
the essential self-adjointness of certain kinds of operators and their limits from the
bottom of the spectrum (mostly based on the Kato–Rellich theorem).
Some chapters suffice to cover a good part of the material suitable for advanced

courses on Mathematical Methods in Physics; this is common for master’s degrees in
Physics or doctoral degrees, if we assume a certain familiarity with notions, results
and elementary techniques of measure theory. The text may also be used for a higherlevel course in Mathematical Physics that includes foundational material on QM. In
the attempt to reach out to master or Ph.D. students, both in physics with an interest
in mathematical methods or in mathematics with an inclination towards physical
applications, the author has tried to prepare a self-contained text, as far as possible: hence a primer was included on general topology and abstract measure theory,
together with an appendix on differential geometry. Most chapters are accompanied
by exercises, many of which are solved explicitly.
The book could, finally, be useful to scholars to organise and present accurately
the profusion of advanced material disseminated in the literature.
Results from topology and measure theory, much needed throughout the whole
treatise, are recalled at the end of this introductory chapter. The rest of the book is
ideally divided into three parts. The first part, up to Chap. 5, regards the general theory of operators on Hilbert spaces, and introduces several fairly general notions, like
Banach spaces. Core results are proved, such as the theorems of Baire, Hahn–Banach
and Banach–Steinhaus, as well as the fixed-point theorem of Banach–Caccioppoli,
the Arzelà-Ascoli theorem and Fredholm’s alternative, plus some elementary consequences. This part contains a summary of basic topological notions, in the belief
that it might benefit physics’ students. The latter’s training on point-set topology is at
times disparate and often presents gaps, because this subject is, alas, usually taught
sporadically in physics’ curricula, and not learnt in an organic way like students in
mathematics do.
Part two ends with Chap. 10. Beside laying out the quantum formalism, it develops
spectral theory, in terms of projector-valued measures, up to the spectral decomposition theorems for unbounded self-adjoint operators on Hilbert spaces. This includes


1.1 On the Book

3

the features of maps of operators (functional analysis) for measurable maps that are
not necessarily bounded. General spectral aspects and the properties of their domains

are investigated. A great emphasis is placed on C ∗ -algebras and the relative functional
calculus, including an elementary study of the Gelfand transform and the commutative Gelfand–Najmark theorem. The technical results leading to the spectral theorem
are stated and proven in a completely abstract manner in Chap. 8, forgetting that the
algebras in question are actually operator algebras, and thus showing their broader
validity. In Chap. 10 spectral theory is applied to several practical and completely
abstract contexts, both quantum and not.
Chapter 6 treats, from a physical perspective, the motivation underlying the theory.
The general mathematical formulation of QM concerns Chap. 7. The mathematical
starting point is the idea, going back to von Neumann, that the propositions of physical
quantum systems are described by the lattice of orthogonal projectors on a complex
Hilbert space. Maximal sets of physically compatible propositions (in the quantum
sense) are described by distributive, orthocomplemented, bounded, σ -complete lattices. From this standpoint the quantum definition of an observable in terms of a
self-adjoint operator is extremely natural, as is, on the other hand, the formulation of
the spectral decomposition theorem. Quantum states are defined as measures on the
lattice of all orthogonal projectors, which is no longer distributive (due to the presence, in the quantum world, of incompatible propositions and observables). States
are characterised as positive operators of trace class with unit trace under Gleason’s
theorem. Pure states (rays in the Hilbert space of the physical system) arise as extreme
elements of the convex body of states. Generalisations of Gleason’s statement are also
discussed in a more advanced section of Chap. 7. The same chapter also discusses
how to recover the Hilbert space starting from the lattice of elementary propositions, following the theorems of Piron and Solèr. The notion of superselection rule
is also introduced here, and the discussion is expanded in Chap. 11 in terms of direct
decomposition of von Neumann factors of observables. In that chapter the notion of
von Neumann algebra of observables is exploited to present the mathematical formulation of quantum theories in more general situations, where not all self-adjoint
operators represent observables.
The third part of the book is devoted to the mathematical axioms of QM, and more
advanced topics like quantum symmetries and the algebraic formulation of quantum
theories. Quantum symmetries and symmetry groups (both according to Wigner and
to Kadison) are studied in depth. Dynamical symmetries and the quantum version of
Noether’s theorem are covered as well. The Galilean group, together with its subgroups and central extensions, is employed repeatedly as reference symmetry group,
to explain the theory of projective unitary representations. Bargmann’s theorem on

the existence of unitary representations of simply connected Lie groups whose Lie
algebra obeys a certain cohomology constraint is proved, and Bargmann’s rule of
superselection of the mass is discussed in detail. Then the useful theorems of Gårding
and Nelson for projective unitary representations of Lie groups of symmetries are
considered. Important topics are examined that are often neglected in manuals, like
the uniqueness of unitary representations of the canonical commutation relations
(theorems of Stone–von Neumann and Mackey), or the theoretical difficulties in


4

1 Introduction and Mathematical Backgrounds

defining time as the conjugate operator to energy (the Hamiltonian). The mathematical hurdles one must overcome in order to make the statement of Ehrenfest’s theorem
precise are briefly treated. Chapter 14 offers an introduction to the ideas and methods
of the abstract formulation of observables and algebraic states via C ∗ -algebras. Here
one finds the proof of the GNS theorem and some consequences of purely mathematical flavour, like the general theorem of Gelfand–Najmark. This closing chapter also
contains material on quantum symmetries in an algebraic setting. As an example the
Weyl C ∗ -algebra associated to a symplectic space (usually infinite-dimensional) is
presented.
The appendices at the end of the book recap facts on partially ordered sets, groups
and differential geometry.
The author has chosen not to include topics, albeit important, such as the theory
of rigged Hilbert spaces (the famous Gelfand triples) [GeVi64], and the powerful
formulation of QM based on the path integral approach [AH-KM08, Maz09]. Doing
so would have meant adding further preparatory material, in particular regarding
the theory of distributions, and extending measure theory to the infinite-dimensional
case.
There are very valuable and recent textbooks similar to this one, at least in the
mathematical 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 representation theory, essential requisites are the basics of calculus in one and several real
variables, measure theory on σ -algebras [Coh80, Rud86] (summarised at the end of
this 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 of
dynamics) and electromagnetism (the key features of electromagnetic waves and the
crucial wavelike phenomena like interference, diffraction, scattering).
Lesser elementary, yet useful, facts will be recalled where needed (including
examples) to enable a robust understanding. One section of Chap. 12 will need elementary Lie group theory. For this we refer to the book’s epilogue: the last appendix
summarises tidbits of differential geometry rather thoroughly. Further details should
be 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 =.


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 numbers C, and inner product always means
Hermitian inner product.
6. The complex conjugate of a number c is denoted by c. As the same symbol is

used for the closure of a set of operators, should there be confusion we will
comment on the meaning.
7. The inner product of two vectors ψ, φ in a Hilbert space H is written as (ψ|φ) to
distinguish 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 operator
H χ → (φ|χ )ψ.
8. Complete orthonormal systems in Hilbert spaces are called Hilbert bases. When
no 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 reserved
for the case H = H .
11. By vector subspace we mean a subspace for the linear operations, even in presence of additional structures on the ambient space (e.g. Hilbert, Banach etc.).
12. For the Hermitian conjugation we always use the symbol ∗ . Note that Hermitian
operator, symmetric operator, and self-adjoint operator are not considered synonyms.
13. When referring to maps, one-to-one, 1–1 and injective mean the same, just
like 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-one
correspondence is a bijective mapping. An isomorphism, irrespective of the
algebraic structures at stake, is a 1–1 map onto its image, hence a bijective
homomorphism.
14. Boldface typeset (within a definition or elsewhere) is typically used when defining a term for the first time.
15. Corollaries, definitions, examples, lemmas, notations, remarks, propositions and
theorems are labelled sequentially to simplify lookup.
16. At times we use the shorthand ‘iff’, instead of ‘if and only if’, to say that two
statements imply one another, i.e. they are logically equivalent.
Finally, if h denotes Planck’s constant, we adopt the notation, widely used by physicists,
h
= 1.054571800(13) × 10−34 Js .

:=
2π