Path Integrals on Manifolds and their asymptotic Expansions

Path integrals, i.e. integrals where the integration domain is an infinite-dimensional space of paths, were introduced into physics by Richard Feynman and since then have been an important tool in thermodynamics, quantum mechanics and quantum field theory. Mathematically however, they are notoriously ill-defined, and there are several different approaches to turn path integrals into a rigorous mathematical concept.

We give an introduction to the method which is probably closest to Feynman's original idea, called time-slicing or finite-dimensional approximation. Here, the time interval is cut into small pieces, and the infinite-dimensional space of all paths in a Riemannian manifold $M$ is replaced by the finite-dimensional space of piecewise geodesic paths, subordinate to the time partition. The path integral is then defined as the limit (if it exists!) when the mesh of the partition tends to zero. We show how on a compact Riemannian manifolds $M$, the heat kernel of a Laplace type operator can be represented as a path integral in this sense.

Because one of the most important features of path integrals are their (formal) short-time asymptotic expansions, we are especially interested in time uniformity of the approximation. We present some new results in this direction and argue how the asymptotic expansions of path integrals connect with infinite-dimensional geometry.