Functional Integral Representation of a Model in Quantum Electrodynamics

Abstract

This paper presents functional integral representations for heat semigroups with infinitesimal generators given by self-adjoint Hamiltonians (Pauli–Fierz Hamiltonians) describing an interaction of a non-relativistic charged particle and a quantized radiation field in the Coulomb gauge without the dipole approximation. By the functional integral representations, some inequalities are derived, which are infinite degree versions of those known for finite dimensional Schrödinger operators with classical vector potentials.

On the Feynman Path Integral for Nonrelativistic Quantum Electrodynamics

“…We begin with the Lagrangian function of the corresponding classical mechanics, differently from the models in [12,14,16,32], and construct rigorously the Feynman path integral. Usually in physics, the Feynman path integral for nonrelativistic QED is only heuristically defined.…”

“…e.g. (9)(10)(11)(12)(13)(14)(15)(16)(17) in [11], (A-7) in [29], (13.10) in [32] and (7.38) in [33]) when electrodynamics is quantized from the classical mechanics. It is a reason for introducing the constraints that a momentum canonically conjugate to the scalar potential is absent.…”

“…If we take into the constraints ∇ · E = 4πρ as in (9)(10)(11)(12)(13)(14)(15)(16)(17) in [11] and (7.38) in [33], we have …”

“…[10,11,29,33]). A functional integral representation for the above nonrelativistic QED model with imaginary time was also obtained by Hiroshima [16] by means of the probabilistic method. We can see from Theorem 3.1 in the present paper that the Hamiltonian operators in [12,14,16,32] are formally like (3.10) in the present paper.…”

Gibbs measures on Brownian currents

“…A substantial difficulty to solve in this case is the proper definition of the double integrals in the first place, which will be done below. For the specific model applications see [24] in which we perform an ultraviolet renormalization of Nelson's Hamiltonian by using path measures whose densities are of the above form, and [26,28] for the Pauli-Fierz model. The similar point charge limit in the latter model is an open problem.…”