An Introduction to Hilbert Space and Quantum Logic

by David William Cohen

Historically, nonclassical physics developed in three stages. First came a collection of ad hoc assumptions and then a cookbook of equations known as "quantum mechanics". The equations and their philosophical underpinnings were then collected into a model based on the mathematics of Hilbert space. From the Hilbert space model came the abstaction of "quantum logics". This book explores all three stages, but not in historical order. Instead, in an effort to illustrate how physics and abstract mathematics influence each other we hop back and forth between a purely mathematical development of Hilbert space, and a physically motivated definition of a logic, partially linking the two throughout, and then bringing them together at the deepest level in the last two chapters. This book should be accessible to undergraduate and beginning graduate students in both mathematics and physics. The only strict prerequisites are calculus and linear algebra, but the level of mathematical sophistication assumes at least one or two intermediate courses, for example in mathematical analysis or advanced calculus. No background in physics is assumed.

Genres: Physics, Quantum Mechanics

149 pages, Hardcover

First published May 1, 1989

Reviews

[William Bies · Rating 3 out of 5]

As Max Jammer discusses in his work on the philosophy of quantum mechanics (which we have just reviewed, here), during the 1950’s and 1960’s a small band of researchers – including inter alia Reichenbach, Jauch, Popper, Mackey and Gleason – elaborated John von Neumann’s schematic suggestions for a quantum logic into a discipline in its own right. If one wishes to break into this discourse, it helps to have a guide. David W. Cohen undertakes to provide just this in his unassuming problem book in mathematics, entitled An introduction to Hilbert space and quantum logic (Springer-Verlag, 1989). Here, unlike Jammer, Cohen eschews any attempt to ground the subject in its history and adopts an unabashedly ahistorical approach.

Presupposes advanced calculus and introductory analysis at the intermediate level – Walter Rudin’s Principles of Mathematical Analysis (cf. our review here) or one of the more recent contenders to displace it would do just fine. Supposedly, the author assumes, his reader need not have any background in physics per se but if so, one would miss the point entirely – in line with the lamentable trend in the teaching of mathematics for undergraduates, in which one divorces from their context what, in themselves, are perfectly appropriate subjects and develops them in isolation from the rest of mathematics. The student may indeed acquire facility in a narrow domain, provided he can work up the motivation to master something he doesn’t know why is being taught, but lacks all perspective. Not a problem for us! It may be presumed that readers of this review either are themselves physicists or will have picked up enough background on physics elsewhere to appreciate the reasons behind what Cohen does.

On to the text. Cohen prefaces his exposition with a lightning-fast review of basics of measure theory and Hilbert spaces [pp. 1-20]. Everyone will have seen the material in the prerequisite course on real analysis, so these chapters serve mainly to fix notation. Chapter three launches into the real matter at hand, to explain the logic of non-classical physics and to capture it in a precise mathematical formalism. Keywords include manuals of experiments, weights, events, propositions, orthogonality and compatibility, refinement and implication, operational logic and orthocomplementation. The idea of a manual does help to organize our intuitions about experiments without presupposing classicality, as was ordinarily the case before one was confronted with quantum phenomena in the atomic domain. These notions are illustrated by means of examples, not just electron spin of course. What this reviewer finds most instructive are a handful of contrived examples involving a firefly in a maze which one can peer into via various slits from different vantage points, each of which may be obstructed by the walls of the maze. The observable is whether one sees the firefly in a given chamber or not.

Cohen adverts to the existence of more generally defined logics that are not lattices [p. 37]. What is not so evident, to this reviewer at least, is whether there are any significant examples of quantum logics more general than the usual one associated with orthodox quantum mechanics, which does form a lattice – or does the possibility of a non-lattice quantum logic arise merely from a drive to formal generality without much substance behind it? One is faced with this issue all the time in mathematics. Usually, however, when one does relax a major axiom it is for the sake of a significant model system that demands one do so and therefore possesses intrinsic interest: for instance, the way matrices lead naturally to non-commutative algebras. Whether the same applies in quantum logic remains undecided. In any case, though, Cohen’s text is too concise and introductory in spirit for him to go into any depth on this topic; one wishes he had mentioned his views on it, though, for otherwise what motivates study of the whole field of quantum logic in the first place?

Chapter four [pp. 43-48] returns again to Hilbert space theory, basic but not quite trivial in infinitely many dimensions. The point of the chapter is to interpret the subspace structure in Hilbert space (with its notion of orthogonality) according to the logic of the frame manual with its orthocomplementation.

The first half of chapter five [pp. 49-55] is devoted to basic operator theory (up to the Riesz representation theorem). 5A10 provides an interesting example of a dual. Cohen next discusses the question as to why one employs Hilbert space in quantum mechanics without really answering it (just because John von Neumann’s presentation was elegant, see his 1932 work on the mathematical foundation of quantum mechanics reviewed by us here); even though the assumption of a Hilbert space model is restrictive it is not so clear that it is also unphysical.

Chapter six is central to the entire text. Here Cohen sets forth the elements of a general quantum logic. It is not however immediately clear whether this abstraction of spectral theory to arbitrary logics will be fruitful. Nevertheless, the idea has its merit in that the formal definitions lead to a theory with a certain degree of richness to it. The primary result is Gleason’s theorem, which in Hilbert spaces of dimension greater than two allows one to reconstruct an arbitrary state as a mixture of unit vectors.

For quantum logic to have any grounding in reality, it has to connect to observables one can measure. This leads to a central role for spectral theory, for the spectral measure associated with a self-adjoint operator provides just the link sought. The exposition in the first half of chapter seven is somewhat inefficient since he elects to do everything first in the finite-dimensional case (presumably Cohen does this because spectral theory in finite dimensions admits of elementary proofs). The second half of chapter seven [pp. 78-84] summarizes spectral theory in infinite dimensions without proofs; Barbara MacCluer’s Elementary Functional Analysis (see our review here) might be a more ample and lucid reference on this subject.

At last Cohen gets to physics in chapter eight, in which he outlines the Hilbert space model of quantum mechanics from the point of view of quantum logic. The early sections can be anticlimactic since everyone will probably already be familiar with the basic axioms of quantum mechanics but at least Cohen’s perspective offers clarity. For instance, he states clearly the incompatibility of position and momentum. De Broglie’s matter-wave hypothesis has always troubled this reviewer since the wave amplitude of a particle in configuration space is not the same thing as a wave in classical field theory. Thus, when one speaks of matter waves and their interference, constructive or destructive, the analogy to classical fields may be something of a red herring; one suspects that what is really involved in quantum mechanics goes deeper than this. Let us remark on a pointed comment by the author:

We now leap to the Schrödinger equation, which is the mathematical equation at the heart of the quantum mechanical description of our moving particle. We give no pretense of a logical development leading to this equation. The literature abounds with heuristic explanations [of Schrödinger’s equation], but every one of them contains at some point a leap of faith based on a blend of classical physics and quantum assumptions. Even Schrödinger abandoned his original rationalization for his equation very soon after he published it. [p. 87]

Maybe so, but it shouldn’t be if quantum mechanics were to have the status of a true theory of physics, which is to say that its mathematical apparatus could be derived from foundational principles constructively, just like the derivations in Euclid’s Elements. Merely to postulate Schrödinger’s equation doesn’t count! For that would not be a principle, properly speaking: a principle consists rather in a verbal description of a feature deemed to hold of the real natural world – it can very well motivate an explicit mathematical model, but is not after all the same thing. After a digression on trivialities about spin-1/2 in quantum mechanics [pp. 94-100], Cohen closes by reprising David Mermin’s version of the EPR thought experiment [pp. 100-104]: ingenious, though somewhat distant from the original EPR paper which involves continuous classicial observables not discrete spin. Bohr’s rebuttal of EPR’s original version of their argument was fairly clear in any event; in Cohen’s hands, Mermin’s version of it becomes a tour de force by which to elucidate quantum logic, but, to this reviewer at least, doesn’t necessarily aid philosophical comprehension all that much.

Cohen’s is a conceptually clean presentation of the beginning steps in quantum logic that opens the path to alternative logics for quantum theory. The heart of it is Gleason’s theorem; not very physical though. Nothing on decoherence. Contrast Cohen’s approach with Griffiths/Omnès’? All the latter really do is to add the dimension of time (history) – are inconsistent histories nothing but incompatible multi-time-valued observations in Cohen’s sense? In any case, recommended as light reading for culture’s sake. There will be time enough later on to get down to brass tacks.