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Quantum Mechanics for Mathematicians

American Mathematical Society
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What does an analytic number theorist do when he is caught in the tractor-beam of non-commutative geometry with its end-of-the-rainbow promises of magical new number theory methods, and methods with roots in quantum field theory at that? What does he do when he hears rumors of future wild and reckless new attacks on the fortresses of the primes where the zeta function of Riemann stands guard, still seemingly invincible after all these years? What does he do when no less than Alain Connes promises to lead the charge — peut être, peut être — and irresistible books begin to appear on the scene with titles like Lectures on Arithmetic Noncommutative Geometry, by Matilde Marcolli, Non-commutative Geometry, Quantum Fields and Motives, by Connes and Marcolli (yes, these are Grothendieck’s near-mythical motives), and Quantum Fields and Strings: A Course for Mathematicians, by Deligne, Etinghof, Freed, Jeffreys, Kazhdan, Morgan, Morrison, and Witten?

Well, the keenly excited but dizzy and disoriented analytic number theorist starts frantically to look for a royal road to quantum mechanics and thereafter to quantum field theory, things he has shunned ever since his youthful all-consuming discovery of (for instance) modular forms and theta functions. [Yes, there were times over the years when he got suspicious about physics popping up out of number theoretic shadows (akin to spontaneous symmetry breaking, it would seem), what with Heisenberg groups and Schrödinger representations finding their way even into the austere purity of André Weil’s work on the analytic theory of quadratic forms, but he always managed to sweep this all pushy QM stuff under the carpet. No more, however: those days are gone…]

And so, full of hope, he looks toward the two-volume tome by Deligne, Etinghof, et al, for an entry into this burgeoning new way of things: it’s a course for mathematicians, after all. But there are all these physicists in the mix! Even Deligne and Kazhdan, certainly the purest of mathematicians, appear to be unable to resist the physicists’ ways of doing things, from writing integrals with differentials glued to Leibniz’ big “ess” to Dirac’s bra’s and ket’s perverting linear operators in Hilbert space. Aaargh! What can be done to ease the shock delivered by the physicists’ patois, now that it must be learned because of the arithmetical poetry it supports?

Well, consider Quantum Mechanics for Mathematicians, by Leon Armenovich Takhtajan, who offers this wonderful book in the spirit of his teacher, Ludwig Dmitrievich Faddeev. This text, from now on referred to as QM 4 Us, is heaven-sent to the aforementioned analytic number theorist, i.e., me, because it is mathematics, not physics: the exposition is peppered with definitions and theorems, and proofs!, proofs!, proofs!: no weird discussions of weird experiments that end up as spring-boards for dancing Wu-Li masters (recall Gary Zukav’s book by that name?) or cats that are neither here nor there — just mathematics!

I think I have found an entry into the desired realms of current quantum physics which removes the pain and shear culture shock of full immersion in the patois of those folks in physics who are somehow always geographically close to us but all too alien in their attitude toward mathematics. After all, what can one say when none other than Richard Feynman was once heard to say something to the effect that “mathematics is trivial but I need it in my work”? Clearly, what we have here is a failure to communicate… (with apologies to Strother Martin).

Of course, there are great books on the mathematics attending QM, written by fine mathematicians. Right off, Weyl, von Neumann, Varadarajan, and Mackey come to mind. But I think it is fair to say that their goals must be considered as pedagogically different from Takhtajan’s. Weyl’s object in Theory of Groups and Quantum Mechanics was ultimately unitary representation theory vis à vis QM; von Neumann aimed at presenting QM’s mathematical framework mathematically correctly, and his approach through functional analysis and representation theory was legendary for its success; Mackey’s objective was also foundational, so to speak, in his well-known book by the same title as von Neumann’s (Mathematical Foundations of Quantum Mechanics); and Varadarajan’s famous book on The Geometry of Quantum Theory addresses specifically geometric aspects of QM in statu quo.

But, unlike QM 4 Us, none of these fine books are graduate texts per se, their eminent readability notwithstanding; it is fair to say that they are more in the way of scholarly monographs. Moreover, I dare say that a pure mathematician reading any of these three gems finds himself largely sheltered from the physicists’ mathematical unheimlichkeit, leaving him in the lurch when it comes to finding out about the hypermodern stuff Connes is doing. Indeed, we run up against the question of chronological location: Weyl’s book’s English version dates back to the 1930s, von Neumann wrote in the 1950s, and Varadarajan and Mackey wrote in the 1960s, when such things as quantum field theory and supersymmetry were still leading a sub rosa existence among theoretical physicists. Specifically the path integral formulation of QM, due to Feynman, so much a part of quantum field theory (QFT), and featured in spades in the aforementioned book by Connes-Marcolli, is absent from the classical mathematician-friendly treatments of QM mentioned above. And this fact alone already suffices to make Takhtajan’s book indispensable. In fact, all of Part Two of QM 4 Us is devoted to “Functional Methods and Supersymmetry,” with Chapter 5 tantalizingly titled “Path integral formulation of quantum mechanics.” Perfect!

By the way, in 2004 Varadarajan published Supersymmetry for Mathematicians: An Introduction, and this is clearly a source to be considered carefully in the present context. Varadarajan is a master, after all. But I would guess that this book, as certainly the books mentioned above by Connes, Marcolli, and Deligne and his co-authors, only becomes truly accessible once prerequisites at the graduate student level are settled, and QM 4 Us looks to be just the ticket.

So it is, then, that the book is split into two parts, the first dealing with “Foundations,” the second with the yoga of path integration and supersymmetry (which, I gather, the physicists call “SUSY”). Lagrangian and Hamiltonian mechanics start things off, followed by classical QM: Heisenberg and Schrödinger (all of Chapter 3) and the first part finishes with a discussion of spin and identical particles.

I am personally keen on Takhtajan’s treatment of the Stone-von Neumann theorem, which is given a thorough going-over. (I am reminded of a remark by Mumford in his Tata Lectures on Theta, to the effect that the only treatment of this material he found palatable was that by Varadarajan in The Geometry of Quantum Theory. Takhtajan’s discussion also looks excellent.)

And it appears that Part Two of QM 4 Us has it all, including a very careful treatment of Gaussian integrals and measures (where we encounter Wiener’s work as adapted by Kac so as to yield the Feynman-Kac formula) and a discussion of fermions (mass carriers, as opposed to bosons, which are force carriers: I just learned this last week at a physics colloquium (!)). The last thing discussed in the book is nothing less than the Atiah-Singer index formula, illustrating the fact that Takhtajan’s graduate courses on this material must be marvelous experiences indeed.

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

Date Received: 
Thursday, August 21, 2008
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Leon A. Takhtajan
Graduate Studies in Mathematics 95
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Michael Berg


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Monday, September 22, 2008