The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement

It’s a brave new world for people like me, (late?) middle aged mathematicians schooled in the old ways, when boundaries between subjects were well-defined and apparently slated to stay that way. Of course such an appraisal betrays a certain parochial naiveté and a lack of awareness of the lessons of history, but I think it’s correct to say that when I was in school not only were there far fewer mathematical subdivisions listed in the AMS Subject Classification scheme but also there was little thought given to, say, any putative real-world applications number theory might afford. And then cyberspace and an increased need for cryptography burst onto the scene, of course, and everything changed radically.

But the last few decades have evinced a change, an evanescence of boundaries in a deeper sense than what is suggested by direct applications of such things as the theory of elliptic curves to coding theory, fascinating and important though these are. We are seeing possible synergies between, for example, number theory and quantum mechanics, or between broad-based algebraic topology and quantum field theory. Consider along the latter lines the 1999 AMS publication, Quantum Fields and Strings, for instance. It is a huge collection of lecture notes in two-volumes, originating in IAS workshops in which physicists (e.g. Witten) taught physics to mathematicians (e.g. Deligne) and mathematicians taught mathematics to physicists, all in the service of string theory. Yes, it’s indeed a brave new world.

Well, not really, actually… If you look carefully enough, you see the tracks of some prescient scholars from the past, such as Segal, Mackey, and Varadarajan. Of course, going back on the order of a full century (to when the world was genuinely very different), there are such players as Hilbert and Poincaré who famously refused to acknowledge the existence of the aforementioned borders. Going back even further, there is Riemann of course, and who better to exemplify this ecumenism than the man who gave us his Hypothesis as the ne plus ultra of number theory and whose revolutionary differential geometry anticipated Einstein’s relativity?

To be sure, the more things change, the more they stay the same. Perhaps what we are currently witnessing is a return to something like a status quo. Despite G. H. Hardy’s comments in his justly famous A Mathematician’s Apology, not even a pure number theorist can claim these days that his work has no chance (or risk, as Hardy might put it) of practical application. Theoretical physics, too, in the wake of the Manhattan Project, has no business claiming any such ivory tower aloofness from “the world.” We are all physicists now, or at least we shouldn’t shun physics anymore, even if we were programmed that way when we were kids.

There is a yang to this yin. Mathematicians are temperamentally not physicists and have, generally speaking, no desire whatsoever to be anything like physicists. All that hand-waving pragmatism flying in the face of rigor, heuristics masquerading as proofs, and the prevailing chauvinism that would treat mathematics as, well, just a tool kit, is nothing short of infuriating to every one of us, I’m sure. Even the most mathematically sympathetic of the physicists tends to lapse into what to us is within ε/5 of blasphemy. The stories are many, of course, but I think the best one is told by Harish-Chandra in connection with his discovery that his heart belonged not to physics, as he had previously thought, but to mathematics. When a thesis student, Harish-Chandra was tasked by his advisor, Dirac, to produce the list of irreducible representations of a transformation group of physical interest. In due course the latter presented the list to Dirac saying that he had them all but couldn’t prove the result yet. Dirac replied that he was not interested in the proof, just in the truth of the matter. According to Harish-Chandra that was when he realized he was not meant to be a physicist.

Thus, it is indeed a question of two cultures with very different mores and traditions. If we mathematicians are going to learn the mathematics God chose for the design of the universe, we had better express it all in our own argot, instead of in the patois the physicists tend to employ. Happily, this massive enterprise of translation and interpretation (and rigorization) is well underway at this point in (hyper-modern) history.

The book under review is, as it were, a converse contribution to the cause. Its title conveys the authors’ objective: it’s all about the mathematics — our language — and the proper way for physicists to learn their discipline expressed in this language. In a way it’s a book about mathematics aimed at physicists. (Well, again, not really: see below.)

Its scope is both structurally modest (“…an introduction to the basic structure of quantum theory”) and sweeping (“…[m]any textbooks on quantum physics concentrate either on finite- or infinite-dimensional Hilbert spaces. In this book the idea has been to treat finite- or infinite-dimensional Hilbert-space formalisms on the same footing”), and is mathematically appealing. It devotes its opening hundred pages or so to a refresher course on Hilbert spaces and the central notions of states and effects. Following this, the authors hit the fundamental notion appearing next in the queue: the notion of observables. This discussion affords a good example of how, when all is said and done, the book under review is physics, not mathematics (my earlier ecumenism notwithstanding): on pp.116–117 we read that

…[i]n Section 2.1 we discussed mixtures of states. The convex structure of the set of states results from the possibility of alternating between preparation procedures. Similarly, we can think of an experiment where the preparation procedure is kept fixed but different measurement apparatuses are alternated. In this way, we can mix two measurements and get a third.

The authors then go on to present a formulaic approach to computing observables with an eye to the physicist’s business of getting numbers out:

…we randomly switch between two observables A and B, so that their relative frequencies are λ and 1-λ … [W]e assume that the outcome sets are Ω_A and Ω_B … We define a new outcome set Ω_C=Ω_AUΩ_B and then extend A and B to this new outcome set by writing A(X)=A(X∩ Ω_A), B(X)=B(X∩ Ω_B), for all XCΩ_C … An observable C with outcome set Ω_C is now defined by C(X)=λA(X)+(1-λ)B(X)…

For comparison, if we check what the aforementioned Quantum Fields and Strings says on the subject of observables and states, we find (p.515, the article “Basic quantum mechanics and canonical quantization in Hilbert space,” by Ludwig Faddeev):

I decided to start my lecture by presenting a suitable and reasonably natural framework into which QM naturally fits. The framework itself is an abstraction [!] of the thoughts and work of many people … [e.g.] P. Dirac, H. Weyl, J. [v.] Neumann, I. Segal, and G. Mackey … Each state ω assigns to each observable A its probability distribution ω_A(λ) on the real line. The pairing (mean value) <ω|A>=∫λd ω_A(λ) … defines a duality between A and Ω …

Faddeev then goes on to introduce such mainstays as one-parameter groups of automorphisms of the algebra of observables of the QM system in question, and soon Poisson brackets enter the game: it’s really functional analysis with a vengeance, and a comfortable place for us mathematicians to dwell.

(A propos, as far as semi-classical pedagogical works on quantum mechanics for mathematicians are concerned, which is to say works not aiming at string theory — just at QM itself, there is no better source than Prugovečki’s1971 monograph Quantum Mechanics in Hilbert Space, where the subject of observables is treated at great length and in marvelously functional analytic terms: Fadeev’s aforementioned pairing is presented immediately as the spectral measure of A, soon after this Prugovečki gets at the notion of “determin[ing] the value of A … [with a certain probability P^A(B)] to have as outcome a value within the set B.”)

Note that, ultimately, what the authors of the book under review are saying is readily fitted into the explications given by Fadeev and by Prugovečki, but that the former are emphatically dealing with different pedagogical goals: again, physicists need actual numbers as measurements, while we need, well, structure.

Thus, an important caveat regarding The Mathematical Language of Quantum Theory is that the authors’ treatment of mathematics per se, while by no means cavalier, is more pragmatically oriented than what we mathematicians are used to. As a matter of fact, after Heinosaari and Ziman are done with their chapter on observables, they take up the theme of operations and channels, and after a brief discussion of the Schrödinger and Heisenberg pictures of quantum mechanics (roughly speaking, wave mechanics vs. matrix mechanics), they proceed to address the subject of “physical models of quantum channels” — Mathematician beware!

After this, in the book’s penultimate chapter, it’s on to the ostensibly concrete business of “measurement models and instruments,” where, nonetheless, we do encounter such abstract themes as “von Neumann’s measurement model” and, in the area of repeatable measurements, the “Wigner-Araki-Yanase theorem” (with its proof: see p.252).

Finally, the authors take up the theme of entanglement, which they present with the following introductory characterization: “As a result of its tensor product structure, quantum theory has embedded into it the phenomenon of entanglement, which is often seen as the foremost quantum feature.” Heinosaari and Ziman note the problems that both Schrödinger and Einstein had with this notion, as well as Bell’s famous 1964 theorem regarding the Einstein-Podolsky-Rosen experiment.

The Mathematical Language of Quantum Theory qualifies as a physics text properly so-called, which, however, is to an extent rather mathematician-friendly. It’s not too fast-paced, and lays out the attendant mathematics well, in the notation and dialect favored by the physicists (Dirac’s notation is ubiquitous, but this is of course itself a ubiquity). The book has the added feature being both introductory, as it advertises itself to be, at least modulo the usual requirement that the reader be active and energetic with his marginalia, and very much up to date. This latter feature is consonant with the authors’ statement in the Introduction to the effect that “the book has a quantum information flavor although it is not a quantum information textbook.” It is in any case a very interesting book indeed.

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