Derivation and application of extended parabolic wave theories. II. Path integral representations

The n-dimensional reduced scalar Helmholtz equation for a transversely inhomogeneous medium is naturally related to parabolic propagation models through (1) the n-dimensional extended parabolic (Weyl pseudodifferential) equation and (2) an imbedding in an (n+1)-dimensional parabolic (Schrödinger) equation. The first relationship provides the basis for the parabolic-based Hamiltonian phase space path integral representation of the half-space propagator. The second relationship provides the basis for the elliptic-based path integral representations associated with Feynman and Fradkin, Feynman and Garrod, and Feynman and DeWitt-Morette. Exact and approximate path integral constructions are derived for the homogeneous and transversely inhomogeneous cases corresponding to both narrow- and wide-angle extended parabolic wave theories. The path integrals allow for a global perspective of the transition from elliptic to parabolic wave theory in addition to providing a unifying framework for dynamical approximations, resolution of the square root operator, and the concept of an underlying stochastic process.