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Spectral Theory and Quantum Mechanics

Valter Moretti
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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This big book is a laudable attempt at teaching large chunks of quantum mechanics (QM) in a mathematically desirable fashion, i.e. in a way that mathematicians would desire it to be presented. The most troublesome part of QM from the point of view of a pure mathematician is its absence of an axiomatic basis, most especially so in the way it is done in physics, its natural habitat. It is of course unreasonable to ask for an axiomatization of physics, or even just QM, along similar lines to what one can do for mathematical subjects properly so-called, i.e. axiomatic theories in the wake of Hilbert, even given the snakes in the grass coming from the work of Gödel. Not only is physics in a sense too broad for this sort of thing, it should be borne in mind that the working physicist has a far, far different view of mathematics than we have. Feynman’s notorious comments comparing physics and mathematics are illustrations of this bifurcation, and his position is by no means atypical among our physicist friends: to them mathematics is indeed a tool, and such matters as axiomatic formulations are looked upon as all but irrelevant.

There are exceptions: mathematicians of the finest water have strayed into QM, most notably Hermann Weyl and John von Neumann, to go back to the middle of the last century. In their works, physics actually starts to look like mathematics. Von Neumann in fact offered axioms for quantum mechanics, and what they lead to is nothing less than his famous formulation of QM in functional analytic terms: unitary operators on a Hilbert space (of “states”) and spectral analysis rule the roost; measure theory dominates the question of what can be “observed” and there needs to be agreement between what certain eigenvalue problems yield and what is observed experimentally. So, even if the physicists eschew our philosophical views of mathematics, they are face-to-face with one of our most beautiful and fecund theoretical developments, functional analysis.

But there’s so much more. There are intrinsic connections with (unitary) group representation theory, of course, as well as Lie theory and, of course, algebra even beyond the inner life of groups. The book under review takes this breadth as one of its foci in that all these aspects are given their due, while at all times the connections with “feet on the ground” physics are maintained.

As far as axiomatic formulations go, Moretti properly situates his discussion under the heading of phenomenology, this taking up the book’s seventh chapter; phenomenology vis à vis Schrödinger’s wave mechanics is in chapter six. What is particularly nice about this arrangement of topics is that Moretti’s first five chapters do a great job in setting the mathematical stage for QM in no uncertain terms (if you’ll pardon the pun), with a thorough treatment of analysis, functional analysis, operators on a Hilbert space — including those that are unbounded and densely defined: the life’s blood of this approach to QM — and with proper emphasis placed on themes that are important in the indicated physical context.

With this background in place, and after his treatment of the axiomatic background done, the focus falls dramatically on spectral theory: three beefy chapters. In the next chapter, on non-relativistic QM per se, Moretti very virtuously gives a thorough treatment of the theorem of Stone and Von Neumann, a pillar of the attendant unitary representation theory. Mackey’s work in this connection is also dealt with — another indication of Moretti’s thoroughness. In this context Hermann Weyl also features rather prominently, and this is apposite since the next topic the author hits (quantum symmetries) involves Lie theory; he takes the reader all the way to the Peter-Weyl theorem (sans Beweis).

The last part of the book proper is devoted to advanced topics in QM (and we meet Einstein, Podolsky, and Rosen, as well as Bell; the discussion closes with “bosons and fermions”), followed by an introduction to algebraic formulations of QM, meaning that we encounter such things as Weyl C*-algebras of observables “involved in the description of several systems, among which non-interacting Bosonic quantum systems (cf. p.686).

Thus, as the preceding remarks are intended to convey, Moretti’s book covers a great deal of ground regarding QM, does so in a mathematically attractive manner, and does so accessibly: the book is easy to read, modulo the proposition that one should go through the proofs carefully — there’s a lot in them. Moretti succeeds in presenting a titanic development of his subject proper to a broad spectrum of readers, from students to, say, mathematicians having occasion to work with QM. This all seems to fit with an autobiographical disclosure Moretti presents in his Preface: “… after a PhD in theoretical physics … I officially became a mathematician.”  It shows.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.