Functional Integral Representation of the Pauli-Fierz Model with Spin 1/2

TL;DR

This paper develops a functional integral representation for the Pauli-Fierz Hamiltonian with spin 1/2, enabling analysis of quantum electrodynamics systems with spin via scalar kernels and deriving energy inequalities.

Contribution

It introduces a Feynman-Kac-type formula for the Pauli-Fierz model with spin 1/2, including a scalar kernel representation despite the presence of spin, and provides energy comparison inequalities.

Findings

Derived a Feynman-Kac-type formula for the Pauli-Fierz Hamiltonian with spin 1/2.

Constructed a scalar kernel functional integral representation for the model.

Established energy comparison inequalities using the integral representations.

Abstract

A Feynman-Kac-type formula for a L\'evy and an infinite dimensional Gaussian random process associated with a quantized radiation field is derived. In particular, a functional integral representation of $e^{-t\PF}$ generated by the Pauli-Fierz Hamiltonian with spin $\han$ in non-relativistic quantum electrodynamics is constructed. When no external potential is applied $\PF$ turns translation invariant and it is decomposed as a direct integral $\PF = \int_\BR^\oplus \PF(P) dP$. The functional integral representation of $e^{-t\PF(P)}$ is also given. Although all these Hamiltonians include spin, nevertheless the kernels obtained for the path measures are scalar rather than matrix expressions. As an application of the functional integral representations energy comparison inequalities are derived.