Operators and Representation Theory: Canonical Models for Algebras of Operators Arising in Quantum Mechanics

Quantum mechanics is both a blessing and a curse. Or maybe I should say that to many of us it at first appears as a curse that, as we grow used to it, begins to look like a blessing. This probably describes the experience of pure mathematicians rather than physicists (or, in any case, the experience of at least one such mathematician). And this may well have to do with the fact that mathematicians are after all fundamentally disposed to a certain Platonism while physicists style themselves empiricists or at least pragmatists.

A propos, I can recommend an excellent article in this regard, Bokulich’s “Open or Closed? Dirac, Heisenberg, and the relation between classical and quantum mechanics,” where, perhaps ironically, Heisenberg is shown to be the wilder of the two figures, philosophically speaking. In any event, with quantum physicists leading the way, the modern attitude in physics is best exemplified by a pair of quotes which Pelle E. T. Jorgensen, the author of the book under review, juxtaposes (on p.2): “… Heisenberg removed the conceit that the workings of Nature should necessarily accord with common sense …” (from Brian Cox and Jeff Forshaw, The Quantum Universe) and the famous observation, “Ach, die Physik! Die is ja für die Physiker viel zu schwer!” attributed to the Göttingen School. I first read the latter assertion (in the pithy and peppery form “Physics is much too difficult for physicists”) in Reid’s Hilbert, attributed to David Hilbert himself. I guess the point is that what Hilbert saw in early and developing quantum mechanics was, and possibly still is, responsible for the profound discomfort some mathematicians feel when confronted with QM: it’s surely not complete, nor is it quite rigorous. Famously, even the physicists’ cappo di tutti capi himself, Albert Einstein, would never let go of his deep reservations, which he expressed this way in a letter to Max Born: “Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the ‘Old One.’ I, at any rate, am convinced that He does not throw dice.”

But I am starting to digress: it is not Einstein’s objections I wish to highlight here, but rather the (not unconnected) fact that the mathematics of quantum mechanics is, shall we say, somewhat unsettling. It imbues probability with an excess of authority, and the subject’s axiomatization is, accordingly, if not elusive, then certainly of a very different philosophical flavor than that of the axiom systems accepted in mathematical logic. Indeed, this axiomatization (by Paul Dirac and John von Neumann) is not really of the same flavor as what Gödel and Tarski, or Turing and Church, cooked up in mathematics per se (let alone the more classical axiom systems by, e.g. Hilbert and Ackermann or Zermelo and Fraenkel). This is in large part due to the intrinsic nature of what that is posited in QM, the most prominent example being the uncertainty principle, of course, the centerpiece of the Copenhagen interpretation replete with the Unheimlichkeit of probabilities delineating the veracity of results in the theory. By now we’re at least pretty much used to it — we’d better be, given that the theory is vindicated again and again by its success as a predictor of how nature behaves, but it still chafes a bit and one continually hears echoes of Einstein’s complaint that these are not “the thoughts of the Old One.”

All right, with this polemical backdrop in place, there are in fact a variety of different, if ultimately equivalent, ways of doing QM, ranging from the Copenhagen-Göttingen way to Richard Feynman’s sortilege. Regarding the former, the most mathematically satisfying approach is, not surprisingly, that of von Neumann, founded on his theory of unbounded (and therefore not everywhere continuous) operators densely defined on a Hilbert space; in physics, the latter is the famous space of quantum mechanical states. The best book I have ever seen on this material is Prugovecki’s classic Quantum Mechanics in Hilbert Space, which is in many ways a paean to functional analysis; another excellent source is Thomas Jordan’s Linear Operators for Quantum Mechanics.

Now, in the present book by Jorgensen, this theme is again heavily represented, and rightly so, of course. But Jorgensen adds something else to the program — something very welcome and very important — namely, the perspective of Lie algebras, from early on. It is roughly on target to say that his Part 2 contains the critical functional analysis for quantum mechanical purposes, but it is apposite to note that in the middle of all this he discussed the subject of operators in enveloping algebras. Right after this, it’s on to the familiar (and centrally important) theme of spectral theory: here Jorgensen covers the discrete, continuous, and Lebesgue spectra, and singles out the trace formula for a special discussion of high energy behavior of QM systems. After this, with Part 3 of the book, titled, “Covariant Representations and Connections,” the perspective of Lie algebras takes center stage in no uncertain terms, taking the reader from “Infinite-Dimensional Lie Algebras” to “Vector Fields on the Circle” and “Noncommutative Tori.” This focus on Lie theory is possibly the book’s most marvelous feature.

Additionally, the book is positively charming and a pleasure to read. It is exceptionally well-written, and contains a lot of useful and enlightening Remarks, Examples, and so on. The theorems and propositions are clear and come furnished with nice proofs. And, to boot, Jorgensen has taken the time to add some pithy and piquant quotes throughout, supplementing his cogent presentation of the material (see e.g. his excellent presentation of Campbell-Baker-Hausdorff on p. 174 ff.). I like this book very much, and here are two of his chosen quotations to close with:

A good deal of my research in physics has consisted in not setting out to solve some particular problem, but simply examining mathematical equations of a kind that physicists use, and trying to fit them together in an interesting way, regardless of any application the work may have. It is simply a search for pretty mathematics. It may turn out later to have applications. (Paul Dirac, cf. p. 21)

The universe is an enormous direct product of representations of symmetry groups. (Hermann Weyl, cf. p. 30)

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