The Ubiquitous Heat Kernel

Jay Jorgenson and Serge Lang

A chapter in Mathematics Unlimited — 2001 and Beyond, 2001, pp 655-683 from Springer

Abstract: Abstract Like others, we came to the heat kernel via one direction of mathematics. However, as we progressed in that direction, we realized that the heat kernel plays a central role in almost all directions we can think of. That it bears a name related to physics only indicates that it was originally discovered in connection with heat. But even in physics, its significance goes way beyond giving a mathematical model for heat distribution. The name cannot be changed — it’s too late for that — but the impression some people may have that an occurrence of a certain kernel called the heat kernel means that one is necessarily doing physics is a false impression. Maybe one is and may be one isn’t. Also one may be using the language of physics and heat distribution simply as a convenient backdrop for some theorems giving bounds or asymptotic estimates for the heat kernel, as we do below (“diffusion”). For a recent example, cf. Norris [Nor 97], with a notable bibliography. The issue of Acta where this paper appears actually contains two articles on the heat kernel (the other is [BiJ 97]), constituting 3/5th of the issue, which is one way of being ubiquitous. Less anecdotically, there is a universal gadget which is a dominant factor practically everywhere in mathematics, also in physics, and has very simple and powerful properties. We have no a priori explanation (psychological, philosophical, mathematical) for the phenomenon of the existence of such a universal gadget.

Date: 2001
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DOI: 10.1007/978-3-642-56478-9_34

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