On phase-space integrals with Heaviside functions

Phase Functions and Path Integrals

This note is an introduction to our forthcoming paper [17]. There we show how to construct the metaplectic representation using Feynman path integrals. We were led to this by our attempts to understand Atiyah’s explanation of topological quantum field theory in [2]. Like Feynman’s original approach in [9] (see also [10]) an action integral plays the role of a phase function. Unlike Feynman, we ...

Wave functions on phase space

Theories of Torres-Vega and Fredrick [5], Harriman [6], and of Ban [11], in which phase space points (p,q) are used as configurational variables to formulate quantum mechanics are considered from the standpoint of a class of quantization schemes associating phase space functions with operators. The connection between these schemes and the theories given in [5] to [11] is made by means of augmen...

Feynman path integrals on phase space and the metaplectic representation

High order numerical methods to two dimensional Heaviside function integrals

In this paper we design and analyze a class of high order numerical methods to two dimensional Heaviside function integrals. Inspired by our high order numerical methods to two dimensional delta function integrals [19], the methods comprise approximating the mesh cell restrictions of the Heaviside function integral. In each mesh cell the two dimensional Heaviside function integral can be rewrit...

Finite difference methods for approximating Heaviside functions

We present a finite difference method for discretizing a Heaviside function H(u(~x)), where u is a level set function u : Rn 7→ R that is positive on a bounded region Ω ⊂ R. There are two variants of our algorithm, both of which are adapted from finite difference methods that we proposed for discretizing delta functions in [13–15]. We consider our approximate Heaviside functions as they are use...