Efficient Computation of Functional Determinants

Feynman Path Integrals offer a more intuitive alternative to the Schrodinger approach in Quantum Mechanics. Green's functions for the Schrodinger operator are computed as the limit of sums of exponents of the classical action along all possible trajectories connecting two configurations on a lattice, as the lattice spacing tends to zero. Path integrals naturally reveal the classical trajectories as the limit of "trajectory beams" that contribute the most to the conditional probability. This naturally leads to asymptotics that use only some of the trajectories - e.g., those "sufficiently close" to the classic trajectories - in the integral evaluation.

However, with the exception of a few special cases of simple Hamiltonians, Feynman path integrals are notorious for their computational complexity, even for reduced "beams" of trajectories. Computing path integrals using the classic trajectories only, on the other hand, results in wrong Green's function amplitudes (although may still be useful for simple qualitative analysis of quantum phenomena as in the double-slit experiment) Such simplified Green's functions can be corrected by applying a normalizing "pre-factor" that can be shown to be the regularized operator determinant for the equation in variations. Therefore, our ability to compute functional determinants is key to successful application of path integrals to semi-classical approximation.

See the referenced paper for a discussion of numerical techniques for computing functional determinants.