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 is a solution of the PDE where the inhomogeneous term is a delta function? (in appropriate boundary conditions).
The fundamental solution of a heat equation is called the heat kernel.
Let be a smooth vector bundle over a Riemannian manifold , the space of the smooth sections of and a positive self-adjoint elliptic differential operator. The heat operator symbolically denoted by is an infinitely smoothening operator characterized by the property that
for all . The heat kernel for is then the kernel of an integral operator? representing the heat operator:
is a linear map for all and . Of course, one needs to justify this definition by the proof of the existence.
The Schrödinger equation without potential term is similar to the heat equation? (there is an additional ); 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 measure 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.
A standard textbook account is
For the relation to the index theorem see also
H. Blaine Lawson, Jr. , Marie-Louise Michelson, Spin geometry, Princeton Univ. Press 1989.
Wikipedia, Heat kernel