Calculus of Variation and Path-Integrals with Non-Linear Generalized Functions

TL;DR

This paper develops a rigorous framework using non-linear generalized functions to analyze calculus of variation and path integrals, addressing boundary effects and modeling physical systems like harmonic oscillators and gravitational fields.

Contribution

It introduces a novel approach to calculus of variation and path integrals with non-linear generalized functions, overcoming limitations of distribution-based methods.

Findings

Boundary cost functions relate adjoint and state variables in harmonic oscillator and scalar field cases.

Path integral construction for optimal control actions is achieved within the generalized functions framework.

Discretization effects on the continuum limit are analyzed.

Abstract

The calculus of variation and the construction of path integrals is revisited within the framework of non-linear generalized functions. This allows us to make a rigorous analysis of the variation of an action that takes into account the boundary effects, even when the approach with distributions has pathological defects. A specific analysis is provided for optimal control actions, and we show how such kinds of actions can be used to model physical systems. Several examples are studied: the harmonic oscillator, the scalar field, and the gravitational field. For the first two cases, we demonstrate how the boundary cost function can be used to assimilate the optimal control adjoint state to the state of the system, hence recovering standard actions of the literature. For the gravitational field, we argue that a similar mechanism is not possible. Finally, we construct the path integral for the optimal control action within the framework of generalized functions. The effect of the discretization grid on the continuum limit is also discussed.