spectral theory and quantum mechanics; with an introduction to the algebraic formulation

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UNITEXT – La Matematica per il 3+2
Volume 64

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Valter Moretti

Spectral Theory and
Quantum Mechanics
With an Introduction to the Algebraic
Formulation

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Valter Moretti
Department of Mathematics
University of Trento
Translated by: Simon G. Chiossi, Department of Mathematics, Politecnico di Torino
Translated and extended version of the original Italian edition: V. Moretti, Teoria

Spettrale e Meccanica Quantistica, © Springer-Verlag Italia 2010

UNITEXT – La Matematica per il 3+2
ISSN 2038-5722
ISBN 978-88-470-2834-0
DOI 10.1007/978-88-470-2835-7

ISSN 2038-5757 (electronic)
ISBN 978-88-470-2835-7 (eBook)

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Preface

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: “No, not even one I think, he used maths!”.
What did numbers and geometrical figures have to do with the existence of a
limit speed? How could one stand behind 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 Institutions
of 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 that course, 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

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VI

Preface

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 of my fellows, even if
they would soon come back in throught 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 § 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 topic I still work
on today, though in the slightly more general framework of quantum field theory 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 Master’s and doctoral students in mathematics and physics this material, 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 of not, in the course of time: S.
Albeverio, 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,
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, suggestions on many topics covered and
for pointing out primary references.
Lastly I would like to thank E. Gregorio for the invaluable and on-the-spot technical help with the LATEX package.

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Preface

VII

In the transition from the original Italian to the expanded English version a massive number of (uncountably many!) typos and errors of various kind have been
amended. 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, both mathematical and physical, including a chapter, at the end, on the so-called algebraic
formulation.
In particular, Chapter 4 contains the proof of Mercer’s theorem for positive
Hilbert–Schmidt operators. The now-deeper study of the first two axioms of Quantum
Mechanics, in Chapter 7, comprises the algebraic characterisation of quantum states
in terms of positive functionals with unit norm on the C∗ -algebra of compact operators. General properties of C∗ -algebras and ∗ -morphisms are introduced in Chapter 8.
As a consequence, the statements of the spectral theorem and several results on functional calculus underwent a minor but necessary reshaping in Chapters 8 and 9.
I incorporated in Chapter 10 (Chapter 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 Chapter 12 (formerly Chapter 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 Chapter 13 (formerly
Chapter 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, A. Perotti and
L. Vanzo for useful technical conversations on this second version. For the same
reason, and also for translating this elaborate opus to English, I would like to thank
my colleague S.G. Chiossi.
Trento, September 2012

Valter Moretti

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Contents

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 Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 σ -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 . . . . . . . . . . . . . . . . . . . .

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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 C(K; Kn ), the theorems of Dini
and Arzelà–Ascoli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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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. ∗ -weak completeness of X . . . . . . . . . . . . .
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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53
60
60

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 . . . . . . . . . . . . . . . . . . . .
3.3.2 ∗ -algebras and C∗ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Normal, self-adjoint, isometric, unitary and
positive operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Orthogonal projectors and partial isometries . . . . . . . . . . . . . . . . . . . .

3.5 Square roots of positive operators and polar decomposition
of bounded operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 The Fourier-Plancherel transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.2
2.3

3

4

Families of compact operators on Hilbert spaces and fundamental
properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Compact operators in normed and Banach spaces . . . . . . . . . . . . . . . .
4.1.1 Compact sets in (infinite-dimensional) normed spaces . . . . .
4.1.2 Compact operators in normed spaces . . . . . . . . . . . . . . . . . . . .
4.2 Compact operators in Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 General properties and examples . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Spectral decomposition of compact operators
on Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6

7

XI

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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184
187
187
191
195
202

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: H ⊕ H and the operator τ . . . . . . .
5.1.4 General properties of Hermitian adjoints . . . . . . . . . . . . . . . . .
5.2 Hermitian, symmetric, self-adjoint and essentially selfadjoint 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.3 Nelson’s criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

209
209

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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The first 4 axioms of QM: propositions, quantum states and

observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
7.1 The pillars of the standard interpretation
of quantum phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

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7.2

Classical systems: elementary propositions and states . . . . . . . . . . . .
7.2.1 States as probability measures . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.2 Propositions as sets, states as measures . . . . . . . . . . . . . . . . . .
7.2.3 Set-theoretical interpretation of the logical connectives . . . . .
7.2.4 “Infinite” propositions and physical quantities . . . . . . . . . . . .
7.2.5 Intermezzo: the theory of lattices . . . . . . . . . . . . . . . . . . . . . . .
7.2.6 The distributive lattice of elementary propositions
for classical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Quantum propositions as orthogonal projectors . . . . . . . . . . . . . . . . . .
7.3.1 The non-distributive lattice of orthogonal projectors
on a Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.2 Recovering the Hilbert space from the lattice . . . . . . . . . . . . .
7.3.3 Von Neumann algebras and the classification of factors . . . .
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 Superselection rules and coherent sectors . . . . . . . . . . . . . . . .
7.4.6 Algebraic characterisation of a state as a noncommutative
Riesz theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5 Observables as projector-valued measures on R . . . . . . . . . . . . . . . . .
7.5.1 Axiom A4: the notion of observable . . . . . . . . . . . . . . . . . . . .
7.5.2 Self-adjoint operators associated to observables:
physical motivation and basic examples . . . . . . . . . . . . . . . . .
7.5.3 Probability measures associated to state/observable couples .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8

Spectral Theory I: generalities, abstract C∗ -algebras and operators
in B(H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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
in Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.3 Abstract C∗ -algebras: Gelfand-Mazur theorem, spectral
radius, Gelfand’s formula, Gelfand–Najmark theorem . . . . .
8.2 Functional calculus: representations of commutative C∗ -algebras
of bounded maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.1 Abstract C∗ -algebras: continuous functional calculus for
self-adjoint elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.2 Key properties of ∗ -homomorphisms of C∗ -algebras,
spectra and positive elements . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.3 Commutative Banach algebras and the Gelfand transform . .

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Abstract C∗ -algebras: continuous functional calculus for
normal elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.5 C∗ -algebras of operators in B(H): functional calculus for
bounded measurable functions . . . . . . . . . . . . . . . . . . . . . . . . .
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 integrals of bounded maps in PVMs . . . . . . . . .
8.4 Spectral theorem for normal operators in B(H) . . . . . . . . . . . . . . . . . .
8.4.1 Spectral decomposition of normal operators in B(H) . . . . . .
8.4.2 Spectral representation of normal operators in B(H) . . . . . . .
8.5 Fuglede’s theorem and consequences . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5.1 Fuglede’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5.2 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8.2.4

9

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359
359
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371
372
374
375

Spectral theory II: unbounded operators on Hilbert spaces . . . . . . . . .
9.1 Spectral theorem for unbounded self-adjoint operators . . . . . . . . . . . .
9.1.1 Integrating unbounded functions in 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 with pure point spectrum: the Hamiltonian of the
harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.5 Examples with pure 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

379
379
380
392
393

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) . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1.3 The abstract heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Hilbert tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.1 Tensor product of Hilbert spaces and spectral properties . . . .

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10.2.2 Tensor product of operators (typically unbounded) and
spectral properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.3 An example: the orbital angular momentum . . . . . . . . . . . . . .
10.3 Polar decomposition theorem for unbounded operators . . . . . . . . . . .
10.3.1 Properties of operators A∗ A, square roots of unbounded
positive self-adjoint operators . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.2 Polar decomposition theorem for closed and denselydefined operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4 The theorems of Kato-Rellich and Kato . . . . . . . . . . . . . . . . . . . . . . . .
10.4.1 The Kato-Rellich theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4.2 An example: the operator −Δ +V and Kato’s theorem . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11

Mathematical formulation of non-relativistic Quantum Mechanics . .
11.1 Round-up on axioms A1, A2, A3, A4 and superselection rules . . . . .
11.2 Axiom A5: non-relativistic elementary systems . . . . . . . . . . . . . . . . .
11.2.1 The canonical commutation relations (CCRs) . . . . . . . . . . . . .
11.2.2 Heisenberg’s uncertainty principle as a theorem . . . . . . . . . . .
11.3 Weyl’s relations, the theorems of Stone–von Neumann and Mackey
11.3.1 Families of operators acting irreducibly and Schur’s lemma .
11.3.2 Weyl’s relations from the CCRs . . . . . . . . . . . . . . . . . . . . . . . .
11.3.3 The theorems of Stone–von Neumann and Mackey . . . . . . . .
11.3.4 The Weyl ∗ -algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.5 Proof of the theorems of Stone–von Neumann and Mackey .
11.3.6 More on “Heisenberg’s principle”: weakening the
assumptions and extension to mixed states . . . . . . . . . . . . . . .
11.3.7 The Stone–von Neumann theorem revisited, via the
Heisenberg group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.8 Dirac’s correspondence principle and Weyl’s calculus . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12 Introduction to Quantum Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1 Definition and characterisation of quantum symmetries . . . . . . . . . . .
12.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1.2 Symmetries in presence of superselection rules . . . . . . . . . . .
12.1.3 Kadison symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1.4 Wigner symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1.5 The theorems of Wigner and Kadison . . . . . . . . . . . . . . . . . . .
12.1.6 The dual action of symmetries on observables . . . . . . . . . . . .
12.2 Introduction to symmetry groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2.1 Projective and projective unitary representations . . . . . . . . . .
12.2.2 Projective unitary representations are unitary or antiunitary .
12.2.3 Central extensions and quantum group associated to a

symmetry group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2.4 Topological symmetry groups . . . . . . . . . . . . . . . . . . . . . . . . . .

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12.2.5
12.2.6
12.2.7
12.2.8

XV

Strongly continuous projective unitary representations . . . . .
A special case: the topological group R . . . . . . . . . . . . . . . . . .
Round-up on Lie groups and algebras . . . . . . . . . . . . . . . . . . .
Symmetry Lie groups, theorems of Bargmann, Gårding,
Nelson, FS3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2.9 The Peter–Weyl theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.3.1 The symmetry group SO(3) 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

557
559
564

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 Review of scattering processes . . . . . . . . . . . . . . . . . . . . . . . . .
13.1.6 The evolution operator in absence of time homogeneity and
Dyson’s series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.1.7 Antiunitary time reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2 The time observable and Pauli’s theorem. POVMs in brief . . . . . . . .
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 Detour: Ehrenfest’s theorem and related 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 Entangled states and the so-called “EPR paradox” . . . . . . . . .
13.4.3 Bell’s inequalities and their experimental violation . . . . . . . .
13.4.4 EPR correlations cannot transfer information . . . . . . . . . . . . .
13.4.5 Decoherence as a manifestation of the macroscopic world . .
13.4.6 Axiom A8: compounds of identical systems . . . . . . . . . . . . . .
13.4.7 Bosons and Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

607
608
608
610
613
621
623

14 Introduction to the Algebraic Formulation of Quantum Theories . . . .
14.1 Introduction to the algebraic formulation of quantum theories . . . . .
14.1.1 Algebraic formulation and the GNS theorem . . . . . . . . . . . . .
14.1.2 Pure states and irreducible representations . . . . . . . . . . . . . . .

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14.1.3 Hilbert space formulation vs algebraic formulation . . . . . . . .
14.1.4 Superselection rules and Fell’s theorem . . . . . . . . . . . . . . . . . .
14.1.5 Proof of the Gelfand–Najmark theorem, universal
representations and quasi-equivalent representations . . . . . . .
14.2 Example of a C∗ -algebra of observables: the Weyl C∗ -algebra . . . . .
14.2.1 Further properties of Weyl ∗ -algebras . . . . . . . . . . . . . . . . . . .
14.2.2 The Weyl C∗ -algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.3 Introduction to Quantum Symmetries within the algebraic
formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.3.1 The algebraic formulation’s viewpoint on quantum
symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.3.2 Symmetry groups in the algebraic formalism . . . . . . . . . . . . .

677
680
683
686
686
690
691
692
694

Appendix A. Order relations and groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697
A.1 Order relations, posets, Zorn’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . 697
A.2 Round-up on group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698
Appendix B. Elements of differential geometry . . . . . . . . . . . . . . . . . . . . . . . .
B.1 Smooth manifolds, product manifolds, smooth functions . . . . . . . . . .
B.2 Tangent and cotangent spaces. Covariant and contravariant vector
fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

B.3 Differentials, curves and tangent vectors . . . . . . . . . . . . . . . . . . . . . . .
B.4 Pushforward and pullback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

701
701
705
707
708

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715

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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 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, and Quantum Theories in general, in a mathematically rigorous way. That said, 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, 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 Chapters 6, 7, 11, 12, 13 and partly 14. The remaining chapters are
logically independent of those, although the motivations for certain mathematical
definitions are to be found in Chapters 7, 10, 11, 12, 13 and 14.
A third purpose is to collect in one place a number of rigorous and useful results on the mathematical structure of Quantum Mechanics and Quantum Theories.
These are more advanced than what is normally encountered in quantum physics’
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.

Moretti V.: Spectral Theory and Quantum Mechanics
Unitext – La Matematica per il 3+2
DOI 10.1007/978-88-470-2835-7_1, © Springer-Verlag Italia 2013

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2

1 Introduction and mathematical backgrounds

manuals. Many of these aspects have been known for a long time but are scattered
in the specialistic literature. We should mention 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 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 Matematical Physics that includes foundational material on Quantum
Mechanics. 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 on general topology and abstract measure theory
was included, together with an appendice on differential geometry. Most chapters are
accompanied by exercises, many of which solved explicitly.
The book could, finally, be useful to scientists when organising and presenting
accurately the profusion of advanced material disseminated in the literature.
At the end of this introductory chapter some results from topology and measure
theory are recalled, much needed throughout the whole treatise. The rest of the book is
ideally divided into three parts. The first part, up to Chapter 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. In this part basic topological notions are summarised, in the belief that

this might benefit physics’ students. The latter’s training on general 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 of the book ends in Chapter 10. Beside setting 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 the features of maps of operators (functional analysis) for measurable
maps that are not necessarily bounded, whose general spectral aspects and domain
properties are investigated. A great emphasis is placed on the structure of C∗ -algebras
and the relative functional calculus, including an elementary study of Gelfand’s transform and the commutative Gelfand–Najmark theorem. The technical results leading

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1.1 On the book

3

to the spectral theorem are stated and proven in a completely abstract manner in
Chapter 8, forgetting that the algebras in question are actually operator algebras, and
thus showing their broader validity. In Chapter 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 Quantum Mechanics concerns
Chapter 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 and orthocomplemented, bounded and 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). Using Gleason’s theorem states are characterised as
positive operators of trace class with unit trace. Pure states (rays in the Hilbert space

of the physical system) arise as extreme elements of the convex body of states.
The third part of the book is devoted to formulating axiomatically the mathematical foundations of Quantum Mechanics and investigating more advanced topics like
quantum symmetries and the algebraic formulation of quantum theories. A comprehensive study is reserved to the notions of quantum symmetry and symmetry group
(both Wigner’s and Kadison’s definitions are discussed). Dynamical symmetries and
the quantum version of Nöther’s theorem are covered as well. The Galilean group
is employed repeatedly, together with its subgroups and central extensions, as reference symmetry group, to explain the theory of projective unitary representations.
Bargmann’s theorem on the existence of unitary representations of simpy 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 unit ary representations of Lie groups
of symmetries are considered. Important topics are examined that are often neglected
in manuals, like the formulation of the uniqueness of unitary representations of the
canonical commutation relations (theorems of Stone–von Neumann and Mackey),
or the theoretical difficulties in 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 to the GNS theorem
and some consequences of purely mathematical flavour, like the general theorem of
Gelfand–Najmark. This closing chapter contains also material on quantum symmetries in an algebraic setting. As example the notion of C∗ -algebra of Weyl associated
to a symplectic space (usually infinite-dimensional) is discussed.
The appendices at the end of the book recap elementary notions about partially
ordered sets, group theory and differential geometry.

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The author has chosen not to include topics, albeit important, such as the theory
of rigged Hilbert spaces (the famous Gelfand triples); doing so would have meant
adding further preparatory material, in particular regarding the theory of distributions.

1.1.2 Prerequisites
Essential requisites to understand the book’s contents (apart from firm backgrounds
on linear algebra, plus some group- and representation theory) are the basics of calculus in one and several real variables, measure theory on σ -algebras [Coh80, Rud82]
(summarised at the end of the chapter), and a few notions about 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, useful facts will be recalled where needed (examples included)
to allow for a solid understanding. One section of Chapter 12 will use the notion of
Lie group and elemental facts from the corresponding theory. For these we refer to
the book’s epilogue: the last appendix summarises some useful differential geometry
rather thoroughly. More details should be seeked in [War75, NaSt82].

1.1.3 General conventions
1.
2.
3.
4.
5.

6.

7.

8.
9.
10.

The symbol := means “equal, by definition, to”.
The inclusion symbols ⊂, ⊃ allow for equality =.
The symbol denotes the disjoint union.
N is the set of natural numbers including nought, and R+ := [0, +∞).
Unless otherwise stated, the field of scalars of a normed/Banach/Hilbert vector
space is the field of complex numbers C. Inner product always means Hermitian
inner product, unless specified differently.
The complex conjugate of a number c will be denoted by c. The same symbol
is used for the closure of a set of operators; should there be confusion, we will
comment on the meaning.
The inner/scalar product of two vectors ψ , φ in a Hilbert space H will be indicated by (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 .
Complete orthonormal systems in Hilbert spaces will be called Hilbert bases, or
bases for short.
The word operator tacitly implies linear operator, even though this will be often
understated.
A linear operator U : H → H between Hilbert spaces H and H that is isometric
and onto will be called unitary, even if elsewhere in the literature the name is
reserved for the case H = H .

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1.2 On Quantum Mechanics

5

11. By vector subspace we will mean a subspace for the vector-space operations,
even in presence of additional structures on the ambient space (e.g. Hilbert,
Banach etc.).
12. For the Hermitian conjugation we will always use the symbol ∗ . Hermitian operator, symmetric operator, and self-adjoint operator will not be considered synonyms.
13. One-to-one, 1-1, and injective are synonyms, just like onto and surjective. Bijective means simultaneously one-to-one and onto. Beware that a one-to-one correspondence is a bijective mapping. An isomorphism, irrespective of the algebraic
structure 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 all labelled sequentially to simplify their retrieval.
16. At times we will use the shorthand “iff”, instead of ‘if and only if’, to say that
two statements imply one another, i.e. they are logically equivalent.

1.2 On Quantum Mechanics
1.2.1 Quantum Mechanics as a mathematical theory
From a mathematical point of view Quantum Mechanics (QM) represents a rare blend
of mathematical elegance and descriptive insight into the physical world. The theory
essentially makes use of techniques of functional analysis mixed with incursions and
overlaps with measure theory, probability and mathematical logic.
There are (at least) two possible ways to formulate precisely (i.e. mathematically) elementary QM. The eldest one, historically speaking, is due to von Neumann
(1932) in essence, and is formulated using the language of Hilbert spaces and the
spectral theory of unbounded operators. A more recent and mature formulation was
developed by several authors in the attempt to solve quantum-field-theory problems
in mathematical physics; it relies on the theory of abstract algebras (∗ -algebras and
C∗ -algebras) built mimicking operator algebras that were defined and studied, again,
by von Neumann (nowadays known as W ∗ -algebras or von Neumann algebras), but
freed from the Hilbert-space structure (for instance, [BrRo02] is a classic on operator
algebras). The core result is the celebrated GNS theorem (from Gelfand, Najmark and

Segal) [Haa96, BrRo02] that we will prove in Chapter 14. The newer formulation can
be considered an extension of the former one, in a very precise sense that we shall
not go into here, also by virtue of the novel physical context it introduces and by the
possibility of treating physical systems with infinitely many degrees of freedom, i.e.
quantum fields. In particular, this second formulation makes precise sense of the demand for locality and covariance of relativistic quantum field theories [Haa96], and
allows to extend quantum field theories to curved spacetime.
The algebraic formulation in elementary QM is not strictly necessary, even though
it can be achieved and is very elegant (see for example [Str05a] and parts of [DA10]).

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Given the relatively basic nature of our book we shall treat almost exclusively the first
formulation, which displays an impressive mathematical complexity together with a
manifest formal elegance. We will introduce the algebraic formulation in the last
chapter only, and stay within the general framework rather than consider QM as a
physical object.
A crucial mathematical tool to develop a Hilbert-space formulation for QM is the
so-called spectral theorem for self-adjoint operators (unbounded, usually) defined on
dense subspaces of a Hilbert space. This theorem, which can be extended to normal
operators, was first proved by von Neumann in [Neu32] apropos the mathematical
structure of QM: this fundamental work ought to be considered a XX century milestone of mathematical physics and pure mathematics. The definition of abstract Hilbert spaces and much of the relative general theory, as we know it today, are also due
to von Neumann and his formalisation of QM. Von Neumann built the modern and axiomatic notion of an abstract Hilbert space by considering, in [Neu32, Chapter 1], the
two approaches to QM known at that time: the one relying on Heisenberg matrices,
and the one using Schrödinger’s wavefunctions.
The relationship between QM and spectral theory depends upon the following

fact. The standard way of interpreting QM dictates that physical quantities that are
measurable over quantum systems can be associated to unbounded self-adjoint operators on a suitable Hilbert space. The spectrum of each operator coincides with the
collection of values the associated physical quantity can attain. The construction of
a physical quantity from easy properties and propositions of the type “the value of
the quantity falls in the interval (a, b]”, which correspond to orthogonal projectors in
the adopted mathematical scheme, is nothing else but an integration procedure with
respect to an appropriate projector-valued spectral measure. In practice the spectral
theorem is just a means to construct complicated operators starting from projectors,
or conversely, decompose operators in terms of projector-valued measures.
The contemporary formulation of spectral theory is certainly different from von
Neumann’s, although the latter already contained all basic constituents. Von Neumann’s treatise (dating back 1932) discloses still today an impressive depth, especially in the most difficult sides of the physical interpretation of QM’s formalism:
by reading that book it becomes clear that von Neumann was well aware of these
issues, as opposed to many colleagues of his. It would be interesting to juxtapose his
opus to the much more renowned book by Dirac [Dir30] on QM’s fundamentals, a
comparison that we leave to the interested reader. At any rate, the great interpretative profundity given to QM by von Neumann begins to be recognised by experimental physicists as well, in particular those concerned with quantum measurements
[BrKh95].
The so-called Quantum Logics arise from the attempt to formalise QM from the
most radical position: endowing the same logic used to treat quantum systems with
properties different from those of ordinary logic, and modifying the semantic theory. For example, more than two truth values are possible, and the Boolean lattice of
propositions is replaced by a more complicated non-distributive structure. In the first
formulation of quantum logic, known as Standard Quantum Logic and introduced by
Von Neumann and Birkhoff in 1936, the role of the Boolean algebra of propositions

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1.2 On Quantum Mechanics

7

is taken by an orthomodular lattice: this is modelled, as a matter of fact, on the set of
orthogonal projectors on a Hilbert space, or the collection of closed projection spaces
[Bon97], plus some composition rules. Despite its sophistication, that model is known
to contain many flaws when one tries to translate it in concrete (operational) physical
terms. Beside the various formulations of Quantum Logic [Bon97, DCGi02, EGL09],
there are also other foundational fomulations based on alternative viewpoints (e.g.,
topos theory).

1.2.2 QM in the panorama of contemporary Physics
Quantum Mechanics, roughly speaking the physical theory of the atomic and subatomic world, and General and Special Relativiy (GSR) – the physical theory of
gravity, the macroscopic world and cosmology, represent the two paradigms through
which the physics of the XX and XXI centuries developed. These two paradigms
coalesced, in several contexts, to give rise to relativistic quantum theories. Relativistic Quantum Field Theory [StWi00, Wei99], in particular, has witnessed a striking
growth and a spectacular predictive and explanatory success in relationship to the
theory of elementary particles and fundamental interactions. One example for all: regarding the so-called standard model of elementary particles, that theory predicted
the unification of the weak and electromagnetic forces which was confimed experimentally at the end of the ‘80s during a memorable experiment at the C.E.R.N., in
Geneva, where the particles Z0 and W ± , expected by electro-weak unification, were
first observed.
The best-ever accuracy in the measurement of a physical quantity in the whole
history of Physics was predicted by quantum electrodynamics. The quantity is the
so-called gyro-magnetic ratio of the electron g, a dimensionless number. The value
expected 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
than relativistic theories), also owing to the impressive range of linear scales where
it holds: from 1m (Bose-Einstein condensates) to at least 10−16 m (inside nucleons:
quarks). QM has had an enormous success, both theoretical and experimental, in
materials’ science, optics, electronics, with several key technological repercussions:

every technological object of common use that is complex enough to contain a semiconductor (childrens’ toys, mobiles, remote controls . . . ) exploits the quantum properties of matter.
Going back to the two major approaches of the past century – QM and GSR –
there remain a number of obscure points where the paradigms seem to clash; in particular the so-called “quantisation of gravity” and the structure of spacetime at Planck
scales (10−33 cm, 10−43 s, the length and time intervals obtained from the fundamental

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constants of the two theories: the speed of light, the universal constant of gravity and
Planck’s constant). The necessity of a discontinuous spacetime at ultra-microscopic
scales is also reinforced by certain mathematical (and conceptual) hurdles that the
so-called theory of quantum Renormalisation has yet to overcome, as consequence
of the infinite values arising in computing processes due to the interaction of elementary particles. From a purely mathematical perspective the existence of infinite
values is actually related to the problem, already intrinsically ambiguous, of defining
the product of two distributions: infinites are not the root of the problem, but a mere
manifestation of it.
These issues, whether unsolved or partially solved, have underpinned important
theoretical advancements of late, which in turn influenced the developments of pure
mathematics itself. Examples include (super-)string theory, and the various Noncommutative Geometries, first of all A. Connes’ version and the so-called Loop Quantum
Gravity. The difficulty in deciding which of these theories makes any physical sense
and is apt to describe the universe at very small scales is also practical: today’s technology is not capable of preparing experiments that enable to distinguish among all
available theories. However, it is relevant to note that recent experimental observations of the so-called γ -bursts, conducted with the telescope “Fermi Gamma-ray”,
have lowered the threshold for detecting quantum-gravity phenomena (e.g. the violation of Lorentz’s symmetry) well below Planck’s length2 . Other discrepancies between QM and GSR, about which the debate is more relaxed today than it was in the
past, have to do with QM vs the notions of locality of relativistic nature (EinsteinPodolsky-Rosen paradox [Bon97]) in relationship to QM’s entanglement phenomena. This is due in particular to Bell’s study of the late ‘60s, and to the famous experiments of Aspect that first disproved Einstein’s expectations, secondly confirmed
the Copenhagen interpretation, and eventually proved that nonlocality is a character
istic of Nature, independent of whether one accepts the standard interpretation of QM

or not. The vast majority of physicists seems to agree that the existence of nonlocal
physical processes, as QM forecasts, does not imply any concrete violation of Relativity’s core (quantum entanglement does not involve superluminal transmission of
information nor the violation of causality [Bon97]).
In the standard interpretation of QM that is called of Copenhagen there are parts
that remain physically and mathematically unintelligible, yet still very interesting
conceptually. In particular, and despite several appealing attempts, it it still not clear
how standard mechanics may be seen as a subcase, or limiting case, of QM, nor
how to demarcate (even roughly, or temporarily) the two worlds. Further, the question remains of the physical and mathematical description of the so-called process
of quantum measurement, of which more later, that is strictly related to the classical
limit of QM. From this fact, as well, other interpretations of the QM formalisms
were born that differ deeply from Copenhagen’s interpretation. Among these more
recent interpretations, once considered heresies, Bohm’s interpretation relies on hidden variables [Bon97, Des99] and is particularly intriguing.
2

Abdo A.A. et al.: A limit on the variation of the speed of light arising from quantum gravity
effects. Nature 462, 331–334 (2009).

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