Diffraction in the semiclassical approximation to Feynman's path integral representation of the Green function

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 s ¼ 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.

Semiclassical estimates of electromagnetic Casimir self-energies of spherical and cylindrical metallic shells

“…In disjoint systems, the Casimir interaction at small separations usually is dominated by fluctuations about periodic orbits [26]. If there are no stationary classical periodic orbits, the leading interaction typically is due to periodic orbits of extremal-rather than stationary-length [44]. These correspond to diffractive effects that lead to relatively weak but sometimes rather interesting interactions.…”

“…For the cylindrical-and sphericalshells we will be considering, extremal periodic rays creep about the exterior of the cavity a number of times. They lead to rather small diffractive corrections [38,44,58] that will be ignored. The stationary points of the action in Eq.…”

Geometrical theory of diffracted rays, orbiting and complex rays

“…Moreover, in theoretical physics, GTD is closely related to the Feynman integral approach. It is worth mentioning the recent paper by Schaden and Spruch [6], where a link between diffraction in the semiclassical approximation and Feynman's path integral representation of the Green function can be found. These authors note that the evaluation of the Casimir effects needs the consideration of diffraction, at least semiclassically.…”

WKB propagators in position and momentum space for a linear potential with a ‘ceiling’ boundary

“…(Of course, it could at least equally well be a constant electrostatic force. The problem also arises in the study of diffraction by a smooth convex obstacle [8].) The extension to dimension 3 with a flat ceiling is immediate, since the propagator factors and the WKB approximation for the transverse propagator is exact.…”