Path integral solution of linear second order partial differential equations I: the general construction

Abstract

A path integral is presented that solves a general class of linear second order partial differential equations with Dirichlet/Neumann boundary conditions. Elementary kernels are constructed for both Dirichlet and Neumann boundary conditions. The general solution can be specialized to solve elliptic, parabolic, and hyperbolic partial differential equations with boundary conditions. This extends the well-known path integral solution of the Schrödinger/diffusion equation in unbounded space. The construction is based on a framework for functional integration introduced by Cartier/DeWitt-Morette.

Path integral solution of linear second order partial differential equations II: elliptic, parabolic, and hyperbolic cases

“…For reference purposes, the theorem of [1] is stated without proof. Relevant definitions and notation can be found in [1].…”

“…For reference purposes, the theorem of [1] is stated without proof. Relevant definitions and notation can be found in [1]. Theorem 2.1 Let M be a real(complex) m-dimensional (m ≥ 2) paracompact differentiable manifold with a linear connection, and let U be a bounded orientable open region in M with boundary ∂U.…”

“…Theorem 2.1 Let M be a real(complex) m-dimensional (m ≥ 2) paracompact differentiable manifold with a linear connection, and let U be a bounded orientable open region in M with boundary ∂U. 1 Let f and ϕ be elements of the space of sections or section distributions of the (r, s)-tensor bundle over M. Assume given the functional S(x(τ a ′ , z)) whose associated bilinear form Q satisfies Re(Q(x(τ a ′ , z)) > 0 for (x(τ a ′ , z)) = 0.…”

“…The second technique relies on the result of Appendix B to express the kernel of a bounded region in terms of the kernel on its covering space. The third technique is based on Subsection 3.2.1 in [1] and is akin to the method of images.…”

“…Material and notation presented in [1] will be assumed here. The reader requiring motivation to fully digest [1] may wish to begin with perhaps more familiar material in the examples presented here in Subsections 3.1 (the Poisson, diffusion/Schrödinger, and wave equations in unbounded space), 3.2.1 (the diffusion equation in the halfplane), and 3.3.1 (the Laplace/Poisson equations for a ball in R n ). As in [1] the issues of existence and uniqueness are not addressed; and functions, distributions, boundaries, etc.…”