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 $P f = \delta$ where the inhomogeneous term $\delta$ is a delta function (in appropriate boundary conditions).
The fundamental solution of a heat equation is called the heat kernel.
The study of heat kernel led to a new simpler proof of the index theorem by Atiyah, Bott and Patodi.
Let $E\to X$ be a smooth vector bundle over a Riemannian manifold $X$, $\Gamma(E)$ the space of the smooth sections of $E$ and $P:\Gamma(E)\to\Gamma(E)$ a positive self-adjoint elliptic differential operator. The heat operator symbolically denoted by $e^{-tP}:\Gamma(E)\to\Gamma(E)$ is an infinitely smoothening operator characterized by the property that
for all $u\in\Gamma(E)$. The heat kernel $K$ for $P$ is then the kernel of an integral operator? representing the heat operator:
$K_t(x,y):E_y\to E_x$ is a linear map for all $x,y$ and $t$. Of course, one needs to justify this definition by the proof of the existence.
The Schrödinger 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 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
Discussion in the context of renormalization in quantum field theory is around section 6.5 of
See also
H. Blaine Lawson, Jr. , Marie-Louise Michelson, Spin geometry, Princeton Univ. Press 1989.
Wikipedia, Heat kernel