Contents

# Contents

## Idea

### 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 $P f = \delta$ 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.

### 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 $X$, $\Gamma(E)$ 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

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

for all $u\in\Gamma(E)$. 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 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