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Geometry and Physics

Jürgen Jost
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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“Physics is much too hard for physicists.” At least this is what David Hilbert asserted in connection with the inauguration of some of his own studies in the area of theoretical physics; the exact details of the quote’s context are to be found in the pages of what is still probably the best biography ever written about a mathematician, Constance Reid’s Hilbert. As I recall, the thrust of Hilbert’s damning utterance was that the mathematical messes quantum mechanics and relativity were in required a mathematician’s conscientiousness and respect for rigor to rectify properly. Alas, not even Hilbert managed to reach this goal: mathematicians and physicists are manifestly still separated by the same sort of gulf of misunderstanding that would separate us from our neighbors across the Atlantic “pond,” i.e., two peoples separated by a common language. To be sure, it is excruciatingly painful for us to witness what physicists do to our art: just think of where they put their measures in their notation for our integrals, or how they deal with infinities of various flavors as per what transpired in quantum electrodynamics with the renormalization affair.

This lament having been spoken, there is another side to the coin. The physicists’ pragmatism with mathematics and their predisposition for expediency has made for a lot of “the good stuff” (to quote Richard Feynman). In particular, they pointed the way to marvelous new mathematical playgrounds. Apparently we are only beginning these explorations.

There’s a lot of physical meat and meaning to it, too: there is a good deal of evidence in place already to support the proposition that God has indeed chosen the same mathematics for not only His design of the universe but in fact for nothing less than the theory of numbers. My beloved undergraduate professor, V. S. Varadarajan, has published a number of articles rather recently on this theme, e.g., “Weyl systems, Heisenberg groups, and arithmetic physics,” and “Has God made the quantum world p-adic?” (both are available on Varadarajan’s web-page). The former of these two, featuring Heisenberg groups, in point of fact harkens back to André Weil’s magnificent 1964 paper, “Sur certains groupes d’opérateurs unitaires” (Acta Math 111), which features what is now called the Segal-Shale-Weil (projective) representation as well as the oscillator representation. Weil was concerned with Carl Ludwig Siegel’s analytic theory of quadratic forms (and revolutionized the whole subject) and found that one of the main players in the enterprise was a projective representation that went back (in another context) to David Shale’s work in quantum mechanics, following his advisor, I. E. Segal. The latter two worked in space-time; Weil locally and adelically over number or function fields. So it’s all a question of topology (as it always is, isn’t it?), and to make things even more fun, Varadarajan talks in his other paper about p-adic analysis vis à vis physical space.

The point of all this is that it’s a bonanza these days for cross fertilization between mathematics and physics. It would be foolish for us not to take an ecumenical view as far as the physicists’ recent explorations are concerned; particularly as regards what they do with, or to, mathematics.

And this brings me to the book of review.

It’s an extremely unusual book on a number of counts. The tone is set already with the Preface, which starts with rather provocative epigraphs and then hits the reader with a bizarre but tantalizing first paragraph. The epigraphs are “The aim of physics is to write down the Hamiltonian of the universe. The rest is mathematics...” and “Mathematics wants to discover and investigate universal structures. Which of them are realized in nature is left to physics.” Unfortunately Jost gives no attributions for these quotes (I have heard the former before, but where?), but they hit the nail on the head — talk about two sides of a coin (or complementarity, if I may be so perverse).

The subsequent opening paragraph is worth quoting in full: “Perhaps, this is a bad book [!!!]. As a mathematician, you will not find a systematic theory with complete proofs, and, even worse, the standards of rigor for mathematical writing will not always be maintained. As a physicist you will not find coherent computational schemes for arriving at predictions.” Wow!

Well, what do we find then? The answer is a lot, a heck of a lot. Says Jürgen Jost: “The aim of the present book is to present some basic aspects of [the] powerful interplay between physics and geometry that should serve for a deeper understanding of either of them.” And Jost then touches on nothing less than the physicists’ “everything,” which is to say, their TOEs (and, yes, feel free to make up your own pun).

Jost takes us, in the first half of the book titled “Geometry,” from Riemannian and Lorentzian manifolds (e.g., spacetime) to Riemann surfaces, moduli spaces, and supermanifolds (culminating in a section on super Minkowski space). Next, in the second half, “Physics,” he starts off with a comparison of classical and quantum physics, the Lagrangian formalism, and “variational aspects” leading up to Emmy Nöther’s theorem on symmetries (and conservation laws); he then goes on to a smørgasbord of “good stuff” such as the sigma model, symmetry breaking (and supersymmetry breaking), Feynman’s functional integrals, field theory (à la physics, not algebra, of course), and, finally, string theory.

I find the book eminently readable, inpact spellbinding. Jost is a fine writer: see my review of another one of his books: Compact Riemann Surfaces. To be sure, the emerging connections between number theory and physics are of huge interest to me professionally, and historically (gratia Professor Varadarajan), so I am certainly well-disposed to the whole undertaking beforehand. But I am happy to recommend Jost’s Geometry and Physics to any one with enough mathematics under his belt not to wince at a pair of phrases such as “A Riemann surface is a finite algebraic extension of the field of rational functions C(x) in one variable over C,” followed two pages later by: “A Riemann surface is a conformal structure on S, that is, a possibility to measure angles. Equivalently, it is an isometry class of Riemannian metrics modulo conformal factors. The moduli space is obtained by dividing the space of all Riemann metrics on S by isometries and conformal changes.” Jost certainly presents more than adequate preludes to such austerities, but he should be taken at his word when he warns us that he is not giving us a mathematics text properly so-called. He leads us through the landscape, but we should be sure to fasten our safety-belts.

The mathematics is there, however, and so is the physics (although I feel less than qualified to judge the latter). Read and enjoy this fascinating book, delve deeper where you wish, but be aware that Jost has a specific destination in mind, providing a bird’s eye view of the interface of modern physics and geometry. Jost takes his passenger for a wild and wonderful ride and the better prepared the passenger is the more he’ll enjoy the trip and the better he’ll be able to discern the vistas at the end of the voyage.

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