Diffraction in the Semiclassical Approximation to Feynman's Path Integral Representation of the Green Function

TL;DR

This paper develops a semiclassical approximation for the Green function of a massless particle near an obstacle, revealing how classical paths and diffraction effects can be analyzed through non-holonomic constraints and analytic continuation.

Contribution

It introduces a novel semiclassical method for approximating the Green function in shadow regions, accounting for creeping and whispering gallery paths with non-holonomic constraints.

Findings

Derived asymptotic expressions for Green function in shadow regions.

Linked creeping and whispering gallery diffraction via analytic continuation.

Provided solutions for extremal rays with constant curvature.

Abstract

We derive the semiclassical approximation to Feynman's path integral representation of the energy Green function of a massless particle in the shadow region of an ideal obstacle in a medium. The wavelength of the particle is assumed to be comparable to or smaller than any relevant length of the problem. Classical paths with extremal length partially creep along the obstacle and their fluctuations are subject to non-holonomic constraints. If the medium is a vacuum, the asymptotic contribution from a single classical path of overall length L to the energy Green function at energy E is that of a non-relativistic particle of mass E/c^2 moving in the two-dimensional space orthogonal to the classical path for a time \tau=L/c. Dirichlet boundary conditions at the surface of the obstacle constrain the motion of the particle to the exterior half-space and result in an effective time-dependent but spatially constant force that is inversely proportional to the radius of curvature of the classical path. We relate the diffractive, classically forbidden motion in the "creeping" case to the classically allowed motion in the "whispering gallery" case by analytic continuation in the curvature of the classical path. The non-holonomic constraint implies that the surface of the obstacle becomes a zero-dimensional caustic of the particle's motion. We solve this problem for extremal rays with piecewise constant curvature and provide uniform asymptotic expressions that are approximately valid in the penumbra as well as in the deep shadow of a sphere.