nLab
heat kernel

Heat kernel as a fundamental solution

One of the simplest linear partial differential equations of parabolic type is the heat (conductivity) equation. Recall that a fundamental solution of a linear partial differential operator $P$ is a solution of the PDE $P f = δ$ where the inhomogeneous term $δ$ is a delta function? (in appropriate boundary conditions).

The fundamental solution of a heat equation is called the heat kernel.

Role in index theory

The study of heat kernel led to a new simpler proof of the index theorem? by Atiyah?, Bott and Patodi.

Heat kernel for operators over Riemannian manifolds

Let $E \to X$ be a smooth vector bundle over a Riemannian manifold $X$ , $Γ (E)$ the space of the smooth sections of $E$ and $P : Γ (E) \to Γ (E)$ a positive self-adjoint elliptic differential operator. The heat operator symbolically denoted by $e^{- tP} : Γ (E) \to Γ (E)$ is an infinitely smoothening operator characterized by the property that

\frac{d}{dt} (e^{- tP} u) = - {Pe}^{- tP} u

for all $u \in Γ (E)$ . The heat kernel $K$ for $P$ is then the kernel of an integral operator? representing the heat operator:

(e^{- tP} u) (x) = \int_{X} K_{t} (x, y) u (y) dy

$K_{t} (x, y) : E_{y} \to E_{x}$ is a linear map for all $x, y$ and $t$ . Of course, one needs to justify this definition by the proof of the existence.

Heat kernel and path integrals

The Schroedinger equation? without potential term is similar to the heat equation (there is an additional $\sqrt{- 1}$ ); hence its fundamental solution is similar. The heat equation on the other hand can describe diffusion. Therefore also the similarity in the path integral description: the Wiener integral describes diffusion using Brownian motion, similarly the Feynman path integral (for a finite-dimensional system) describes quantum mechanics; many points in the standard calculations are parallel.

References

wikipedia
H. Blaine Lawson, Jr. and Marie-Louise Michelson, Spin geometry, Princeton Univ. Press 1989.