$\delta'$-Function Perturbations and Neumann Boundary-Conditions by Path Integration

TL;DR

This paper extends the path integral formalism to include $ abla$-function perturbations and Neumann boundary conditions, connecting relativistic point interactions with boundary conditions in quantum mechanics.

Contribution

It introduces a novel method to incorporate $ abla$-function perturbations and boundary conditions into path integrals, linking relativistic interactions with boundary phenomena.

Findings

Path integral representation for Dirac particles with point interactions.

Non-relativistic limit yields $ abla$-function perturbations and boundary conditions.

Infinite strength perturbations lead to Dirichlet and Neumann boundary conditions.

Abstract

$\delta'$-function perturbations and Neumann boundary conditions are incorporated into the path integral formalism. The starting point is the consideration of the path integral representation for the one dimensional Dirac particle together with a relativistic point interaction. The non-relativistic limit yields either a usual $\delta$-function or a $\delta'$-function perturbation; making their strengths infinitely repulsive one obtains Dirichlet, respectively Neumann boundary conditions in the path integral.