Helmholtz Path Integrals

摘要

The multidimensional, scalar Helmholtz equation of mathematical physics is addressed. Rather than pursuing traditional approaches for the representation and computation of the fundamental solution, path integral representations, originating in quantum physics, are considered. Constructions focusing on the global, two-way nature of the Helmholtz equation, such as the Feynman/Fradkin, Feynman/Garrod, and Feynman/DeWitt-Morette representations, are reviewed, in addition to the complementary phase space constructions based on the exact, well-posed, one-way reformulation of the Helmholtz equation. Exact, Feynman/Kac, stochastic representations are also briefly addressed. These complementary path integral approaches provide an effective means of highlighting the underlying physics in the solution representation, and, subsequently, exploiting this more transparent structure in natural computational algorithms.