Non-perturbative Quantum Propagators in Bounded Spaces

TL;DR

This paper introduces a novel method for calculating quantum propagators with complex boundary conditions using a generalized hit function and Padé approximants, providing analytical solutions in certain dimensions and applications to Casimir energy computations.

Contribution

It develops a new approach to compute quantum propagators in bounded spaces with Dirichlet conditions, extending previous methods with analytical formulas and recursive relations.

Findings

Analytical expressions for hit functions in 1D and 3D

Recursion relations between hit functions in different dimensions

Application to propagation with conducting plates and conjecture of a general formula

Abstract

We outline a new approach to calculating the quantum mechanical propagator in the presence of geometrically non-trivial Dirichlet boundary conditions based upon a generalisation of an integral transform of the propagator studied in previous work (the so-called ``hit function''), and a convergent sequence of Pad\'e approximants. In this paper the generalised hit function is defined as a many-point propagator and we describe its relation to the sum over trajectories in the Feynman path integral. We then show how it can be used to calculate the Feynman propagator. We calculate analytically all such hit functions in $D=1$ and $D=3$ dimensions, giving recursion relations between them in the same or different dimensions and apply the results to the simple cases of propagation in the presence of perfectly conducting planar and spherical plates. We use these results to conjecture a general analytical formula for the propagator when Dirichlet boundary conditions are present in a given geometry, also explaining how it can be extended for application for more general, non-localised potentials. Our work has resonance with previous results obtained by Grosche in the study of path integrals in the presence of delta potentials. We indicate the eventual application in a relativistic context to determining Casimir energies using this technique.