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Fifty Years of Mathematical Physics: Selected Works of Ludwig Faddeev

Molin Ge and Antti J. Niemi, editors
World Scientific
Publication Date: 
Number of Pages: 
World Scientific Series in 21th Century Mathematics 2
[Reviewed by
Michael Berg
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This is an age of wondrous things for both geometry and physics, not the least of which is their fecund interplay. In the wake of by now rather famous work by, e.g., Atiyah and Witten (independently) a great deal of interest has arisen over the last few decades in, for example, the area of topological (or even non-topological) quantum field theory, and a marvelous synergy has evolved bringing forth results of surpassing interest to both mathematicians and physicists. Perhaps the most well-known publication illustrating this détente between geometers and quantum physicists — well, it’s obviously a lot more: it’s a downright joining of forces — is the imposing two-volume set of books, Quantum Fields and Strings: A Course for Mathematicians, published in 1999 by the AMS; see also the very informative review by William Faris. One of the notable contributors to this enterprise is the scholar whose selected works comprise the book under review here, namely, the Russian academician, Ludwig Faddeev. On pp. 513–550 of the first volume of this set of books we find his article, “Elementary Introduction to Quantum Field Theory,” which includes a truly superb presentation of the construction of the Feynman path integral, of such central importance to the whole enterprise.

Certainly this testifies to Faddeev’s scientific ecumenism. That in itself is a strong recommendation for the book under review. But more is true: Faddeev is an excellent expositor, and is concerned about reaching a large audience with his message, as we may already conclude from the thrust of his well-known elementary text, Lectures on Quantum Mechanics for Mathematics Students, co-authored with O. A. Yakubovskiĭ and based on over 30 years of lecturing on this material at the University of Leningrad. Faddeev, in true Russian form, is deeply concerned not just with his science but with its proper communication, at all levels. More about this below.

Thus, in the book under review, we not only encounter rather specialized and sophisticated material aimed primarily at physicists (e.g. “Scattering theory for a three-particle system” (p. 37, ff.), “The unravelling of the quantum group structure in the WZNW theory” (p. 202, ff.), or “Spectrum and scattering excitations in the one-dimensional isotropic Heisenberg model” (p. 296, ff.)), but also a rather large number of papers of interest to mathematicians as such. We might mention in this connection, “Formulas for traces for a singular Sturm-Liouville differential operator” (p. 3, ff.), “Scattering theory and automorphic forms” (p. 75, ff.), “A nonarithmetic derivation of the Selberg trace formula” (p. 102, ff.), “Quantum groups” (p. 455, ff.), “Quantization of Lie groups and Lie algebras” (p. 469, ff.), “Modular double of a quantum group” (p. 523, ff.), and the final article in the book, “Modern mathematical physics: What it should be” (p. 549, ff.).

So, then, in light of the last-names article, what does this broad and deep scholar say modern mathematical physics should be? Well, on p. 555, Faddeev quotes P. MacPherson as follows: “The goal is to create an understanding, in terms congenial to mathematicians, of some fundamental notions of physics, such as quantum field theory. The emphasis will be on developing the intuition stemming from functional integrals … the plan is [not] to consider fundamental new constructions within mathematics that were inspired by physics … [n]or is the aim to discuss how to provide mathematical rigor for physical theories. Rather, the goal is to develop the sort of intuition common among physicists for those who are used to thought processes stemming from geometry and algebra …” Faddeev then goes on to say, “I propose to call the intuition to which MacPherson refers that of mathematical physics … The union of these two groups constitutes an enormous intellectual force. In the next century [i.e. the present one] we will learn if this force is capable of substituting for the traditional experimental base of the development of fundamental physics and pertinent physical intuition.” This is evidently a rather dramatic and revealing statement: it presents Faddeev as neither a pure mathematician (nor even an applied one, in the usual sense) — well, no surprise there — nor a traditional physicist. Perhaps it is proper to say that such a scholar as Faddeev is a hypermodern phenomenon, and the rest of the closing article in the present book, replete with fascinating autobiographical elements, bears this out. It is truly a wonderful essay to read, and a wonderful comment of the history of physics in the 20th century, with a lot of welcome emphasis placed on the most recent developments. Faddeev was, and is, a major figure in this connection and it is a marvelous experience to read his views on these interesting matters, affording those of us whose “thought processes [stem] from geometry and algebra” a glimpse into how a mathematical physicist of Faddeev’s ilk sees his subject.

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

  • Scattering Theory
  • Automorphic Functions
  • Field Theory
  • Theory of Solitons
  • Quantum Groups
  • Knots
  • General Questions