1. Preface
I must have been 8 or 9 when my father, a man of letters but well-read in every dis-
cipline 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 real-
ise we were using this mathematical theory, and for many mates of mine the matter
2. 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 back-
ground – 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 form-
alise 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 in-
commensurably 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 dia-
logue 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 ma-
terial, 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 LTEX the pile
A
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 news-
groups 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. Tes-
saro, 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. Cada-
muro 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 tech-
nical help with the LTEX package.
A
3. Preface VII
In the transition from the original Italian to the expanded English version a mas-
sive 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 math-
ematical 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 operat-
ors. 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 func-
tional 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 rep-
resentations 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 funda-
mental ideas of non-relativistic scattering theory. Bell’s inequalities (Wigner’s ver-
sion) 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
