# Feynman-Kac formulas for Dirichlet-Pauli-Fierz operators with singular coefficients

## TL;DR

This paper develops Feynman-Kac formulas for Dirichlet realizations of Pauli-Fierz operators, accommodating singular coefficients and boundary singularities, thereby advancing mathematical tools for quantum systems with complex boundary conditions.

## Contribution

It introduces Feynman-Kac formulas for Dirichlet-Pauli-Fierz operators with singular coefficients, extending applicability to systems with boundary singularities and minimal regularity assumptions.

## Key findings

- Derived Feynman-Kac formulas for Dirichlet-Pauli-Fierz operators.
- Handled singular coefficients and boundary singularities in quantum systems.
- Maintained familiar formulas under minimal regularity assumptions.

## Abstract

We derive Feynman-Kac formulas for Dirichlet realizations of Pauli-Fierz operators generating the dynamics of nonrelativistic quantum mechanical matter particles, which are minimally coupled to both classical and quantized radiation fields and confined to an arbitrary open subset of the Euclidean space. Thanks to a suitable interpretation of the involved Stratonovich integrals, we are able to retain familiar formulas for the Feynman-Kac integrands merely assuming local square-integrability of the classical vector potential and the coupling function in the quantized vector potential. Allowing for fairly general coupling functions becomes relevant when the matter-radiation system is confined to cavities with inward pointing boundary singularities.