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Quantum Field Theory: A Tourist Guide for Mathematicians

American Mathematical Society
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The relationship between mathematics and physics has, over the centuries, not so much resembled a stable marriage as stormy liaison between passionate lovers whose compatibility leaves something to be desired. Starting with the heroic origins of analysis in the 17th and 18th centuries, the great classical developments in mathematics have often been tied to groundbreaking investigations in natural science. Newton, Euler, the Bernoullis, Laplace, Fourier, Poisson, Cauchy, and even Gauss himself, manifestly saw no real incompatibility or disharmony between pure and applied mathematics or between mathematics and physics. Indeed, singling out Euler and Gauss among these titans of science (for parochial reasons), it is noteworthy that analytic number theory was born with Euler’s investigations in arithmetic, while Gauss, princeps mathematicorum, famously crowned number theory “the queen of mathematics”: already at this historically early stage these two mathematical prophets chose to devote their great gifts both to the concrete workings of nature and to the abstract play of numbers.

So what happened? It is difficult if not impossible to nail down coordinates and specify parameters for the fissure that now separates mathematics from physics, or even pure mathematics from applied mathematics, but it seems at least approximately accurate to identify the issue of rigor as the crux of the problem. Analysis itself became the battleground for these ensuing developments (already foreshadowed by Gauss) and its eventual arithmetization indicates an orbit away from its physical origins toward what we now characterize as a pure and safe mathematical haven. It is undeniable that as the 19th century unfolded, mathematicians grew to embrace rigor with missionary and religious zeal — and rightly so, given the historical evidence of what trouble an absence of rigor can cause.

Still, at this point in history there was certainly no overt enmity between pure mathematicians and physicists; in fact at many European faculties it was the rule for students to specialize in both mathematics and physics. Recall in this connection the German and Dutch curricula of even the early 20th century. So, again, what set off the alienation that in due course led Feynman to quip that “mathematics is trivial, but I can’t do my work without it”? Perhaps the answer here is in fact not all that mysterious: surely it’s quantum mechanics that started all the trouble…

There are two complementary ways to regard what happened in this connection in the first decades of the last century: Bohr and Heisenberg crafted a theoretical framework asking for the suspension of established conventions and entrenched intuitions to such a degree that the accompanying mathematics was, to put it mildly, contorted. At the same time, as Wigner put it, it was mathematics’ “unreasonable effectiveness” that permitted it to be enlisted in the unprecedented natural philosophy that was coming into being.

Superposing these two states, to coin (or, rather, abuse) a phrase, the reality is that while pragmatists like Heisenberg and, later, Feynman might genuinely regard mathematics as, in the plaintive phrase of Eric Temple Bell, “the scullery slut of science,” other physicists saw it somewhat differently: it was Dirac who insisted that the mathematics in physics must be beautiful since, after all, God chose it for His creation.

But even a fellow-traveler like Dirac, whom Bohr characterized with the phrase that “of all the physicists [he had] the purest soul,” ultimately had no time for the kind of rigor that pure mathematics insists on, as a famous episode involving Harish-Chandra amply illustrates. Harish-Chandra was for a while Dirac’s research student and had been set the task of investigating the unitary representations of the Lorentz group; at some point Harish-Chandra reported to the great physicist that he had determined a certain classification to hold true but could not prove it yet. Dirac’s response was that he was interested only in the truth, not in proofs. Harish-Chandra apparently realized then that he was meant for mathematics not physics. Even the purest of physicists was not a mathematician.

Here it is then, out in the open for every one to see: it’s really all about proof. And it is certainly the case that all of us in mathematics experience a certain lack of rapport with physicists in this connection. But the physicists’ discoveries are often so marvelous that it is surely devoutly to be wished that a way be found to couch these findings, and the ad hoc mathematical tools accompanying them, in a mathematically sound framework. Historically speaking, both Weyl and von Neumann made major contributions to this cause. And, to be sure, their work both imparted the necessary rigor for the mathematics that physics was using at the time and produced new mathematics. Differential geometry, representation theory, and functional analysis, for example, all experienced new growth because these wonderful mathematicians entered the physicists’ sanctum sanctorum with their own unique gifts to present.

It is also interesting to observe that both Weyl and von Neumann were closely associated with the Göttingen of David Hilbert, where, simultaneously (at least for as while), Max Born held court in theoretical physics, making for a particularly fecund environment for collaboration. Constance Reid, in her incomparable biography, Hilbert, cites (on p. 182) Ed Condon’s ever-so-revealing anecdote of Hilbert’s appraisal of what the physicists were up to. Evidently, shortly after Heisenberg’s discovery of matrix mechanics, Born, his research supervisor, arranged a pilgrimage to Hilbert’s office to ask the éminence grise of all things mathematical what he thought of this bizarre new algebra. Not missing a beat, Hilbert suggested that they should look for a partial differential equation having these matrices Heisenberg discovered “as a sort-of by-product of the eigenvalues of the [attendant] boundary-value problem” (in Condon’s words). Born and Heisenberg ignored the old man to their eventual embarrassment, given what Schrödinger published some six months later. Hilbert offered appropriate Homeric laughter, of course. And here is Reid again (p. 127): “Walther Lietzmann, one of [Hilbert’s] students … recalled ‘what discomfort we mathematicians felt when sometimes this, sometimes that principle, without proof, was placed before us and all sorts of propositions and conclusions derived from it …’” Well, there it is in a nutshell, of course. Said Hilbert: “Physics is much too hard for physicists” (loc. cit.).

But we must take careful note of the fact that this man whom Reid pithily described as a pied piper for the mathematicians of his time also devoted a considerable time of his working life to physics. Indeed, Göttingen’s famous style of Naturforschung was decidedly ecumenical, embracing both mathematics and physics, and Hilbert’s lead was followed by almost every one at the Georg August Universität, at least to some extent. Recall, for instance, Emmy Nöther’s work, done at Hilbert’s request, on the correspondence between conservation laws and symmetries in physics, leading to what physicists now call Nöther’s theorem.

Nonetheless, Hilbert’s work in physics never matched his Olympian achievements in mathematics properly so-called, from the Zahlbericht and his proof of Waring’s Conjecture to his groundbreaking work in integral equations and functional analysis (is there any space more beautiful than Hilbert space?). And it is arguable, then, that with even Hilbert failing to introduce axiomatic rigor into physics as recently as less than a century ago, and physics marching on at break-neck speed on its own, it is a sense of methodological alienation that accounts for the unwillingness of the vast majority of recent mathematicians to touch physics.

But should we not say that now, in 2008, this disposition actually characterizes yesterday’s mathematicians more than today’s? For is it not true that, for the last decade or so, increasing numbers of mathematicians have begun to turn their attention to the avant garde mathematical methods that have emerged in the wake of such things as supersymmetry, string theory, mirror symmetry, and the recent advances in quantum field theory? All one has to do is sneak a peak at the spectacular work of Ed Witten, Maxim Kontsevich, or Alain Connes to realize that there are very exciting prospects on the scene today for the utilization of physics in the exploration of the purely mathematical frontier, and mathematicians have begun to take note. It is Connes’ recent work on what he terms non-commutative geometry, for example, that has created a clear need for analytic number theorists to learn quantum field theory: Connes’ vision includes tailoring the according methods from physics in order to reveal new, or long-suspected but never proven, aspects of the inner life of primes, all in the magisterial presence of the Riemann Hypothesis.

All of this provides the background and the setting for Gerald Folland’s brand new book, Quantum Field Theory: A Tourist Guide for Mathematicians. The book starts out with the passage: “This book is an attempt to present the rudiments of quantum field theory in general and quantum electrodynamics in particular, as actually practiced by physicists for the purpose of understanding subatomic particles, in a way that will be comprehensible to mathematicians … It is, therefore, not an attempt to develop quantum field theory in a mathematically rigorous fashion” (the italics are Folland’s). The author goes on to say that “[p]eople with mathematical training are entitled to ask for a deeper … understanding of what is going on here. They may feel optimistic about attaining it from their experience with the older areas of fundamental physics that have proved very congenial to mathematical study: the differential equations of classical mechanics, the geometry of Hamiltonian mechanics, and the functional analysis of quantum mechanics. But … [with] quantum field theory, they are likely to feel like they have run up against a solid wall.”

Folland also mentions that the best-known attempt at “cross-cultural communication” along these lines, the huge two-volume IAS and AMS production, Quantum Fields and Strings: A Course for Mathematicians, leaves a lot to be desired “as an introduction to quantum field theory for ordinary mortals.” Accordingly it is in this climate that he offers us this tourist’s guide with the comment: “…not that it is easy reading (it’s not) but … the intended audience consists of people who approach physics as tourists approach a foreign country, as a place to enjoy and learn from but not to settle in permanently.”

And here is Folland’s travel itinerary: Ch.1, “Prologue,” contains a précis of relevant Lie theory (and other stuff); Ch. 2, “Review of Quantum Physics,” takes us from Newton to Einstein (special relativity) and electromagnetism; Ch. 3 is “Basic Quantum Mechanics;” Ch. 4 is “Relativistic Quantum Mechanics;” in Ch. 5, “Free Quantum Fields,” Lagrangians and Hamiltonians take center stage; in Ch. 6, “Quantum Fields with Interactions,” the dial is set (much) higher: this is where we encounter the scattering matrix, Feynman diagrams, and quantum electrodynamics; and this naturally sets the stage for Ch. 7, “Renormalization,” which is obviously central to the book; next we get, in Ch. 8, “Functional Integrals,” such irresistible topics as “functional integrals and quantum mechanics” and “Gaussian processes;” the book closes with Ch. 9, “Gauge theories,” the last two sections of which are particularly tantalizing: “broken symmetries” and “the electroweak theory.” Evidently reading Folland’s book is a fine way to start, if one’s goal is to do serious mathematics in this physicist’s culture: it is wise first to visit a land one may want to live in for a while.

Thus, even factoring in the reality that true mathematical rigor is still lacking in certain places, and the author’s own warnings as mentioned above, the voyage Folland has booked for his mathematical tourists is full of great sights to see and places to visit. Pack the right luggage: Folland suggests “Fourier analysis, distributions …, linear operators on Hilbert spaces, together with a couple of more advanced results … most notably the spectral theorem,” and enjoy the trip. I will!

Michael Berg is professor of mathematics at Loyola Marymount University in Los Angeles, CA.

Date Received: 
Thursday, September 4, 2008
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Gerald B. Folland
Mathematical Surveys and Monographs 149
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Michael Berg
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Thursday, October 9, 2008