© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:0706.0833

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

Fumio Hiroshima, József Lőrinczi

Published Jun 6, 2007• 4 citations

Abstract

A Feynman-Kac-type formula for a Lévy 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.

DOI: 10.48550/arxiv.0706.0833

Related papers

Functional Integral Representation of a Model in Quantum Electrodynamics

Fumio Hiroshima

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.

Reviews in Mathematical Physics•May 1, 1997•45 citations

Functional Integral Representation of a Model in QED(Analysis of Operators on Gaussian Space and Quantum Probability Theory)

Fumio Hiroshima

This article presents functional integral representations for the heat semigroups with the infinitesimal generators given by self-adjoint Hamiltonians describing an interaction of a non-relativistic charged particle and a quantized radiation field in the Coulomb gauge without the dipole approximation.Special attention is paid to definition of the "time-ordered Hilbert space-valued stochastic integrals associated with a family of isometries from a Hilbert space t,o another one" and semigroup techniques.Some inequalities are derived, which are infinite degree versions of those known for finite dimensional Schr\"odinger operators with classical vector potentials.

数理解析研究所講究録•Sep 1, 1995•0 citations

Imaginary-Time Path Integral for a Relativistic Spin-(1/2) Particle in a Magnetic Field

Gian Fabrizio De Angelis, A Rinaldi, Maurizio Serva

We construct a Feynman-Kac-Itô formula for positive-energy spin-(1/2) relativistic particles in a purely magnetic external field. We also introduce a new class of quantum Hamiltonians, the "relativistic Pauli operators", along with a path integral representation of their semi-groups. These Hamiltonians differ from the relativistic Schrödinger operators of Carmona, Masters and Simon only when a magnetic field is present.

Europhysics Letters (EPL)•Jan 15, 1991•12 citations

Functional Integral Representation of a Model in QED

Fumio Hiroshima

This article presents functional integral representations for the heat semigroups with the infinitesimal generators given by self-adjoint Hamiltonians describing an interaction of a. non-relativistic charged particle and a quantized radiation field in the Coulomb gauge without the dipole approximation. Special attention is paid to definition of the "time-ordered Hilbert space-valued stochastic integrals associated with a family of isometries from a Hilbert space to another one" and semigroup techniques. Some inequalities are derived, which are infinite degree versions of those known for finite dimensional Schrodinger operators with classical vector potentials.

Preprint Series of Department of Mathematics, Hokkaido University•Apr 1, 1995•5 citations

Relativistic quantum mechanics and path integral for Klein-Gordon equation

Gian Fabrizio De Angelis, Maurizio Serva

The subject of this paper is a path integral representations for the semigroup {exp(-t H_1)}{t≥0} generated by the quantum Hamiltonian H_1 of a relativistic spinless particle in an external electromagnetic field. The result is compared with the "Feynman-Kac" formula which holds for relativistic Schroedinger operators.

•Jan 1, 1995•0 citations

Imaginary-Time Path Integrals from Klein-Gordon Equation

Gian Fabrizio De Angelis, Maurizio Serva

The ordinary Feynman-Kac-Itô formula gives a path integral representation of the Schrödinger semi-groups. We discuss here an analogous probabilistic expression for the (positive energy) solutions of the imaginary-time Klein-Gordon equation in a static external electromagnetic field. When the external field is not purely magnetic, our result is different from the path integral associated to the semi-group t ∊ [0, + ∞ ) → exp[− t ( H − mc 2 )/ℏ] of a relativistic Schrödinger operator H = { c 2 [− i ℏ∇−( e/c ) A ] 2 + m 2 c 4 } 1/2 + V .

Europhysics Letters (EPL)•Mar 15, 1992•3 citations

A Feynman-Kac-type formula for a Lévy 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.

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.

This article presents functional integral representations for the heat semigroups with the infinitesimal generators given by self-adjoint Hamiltonians describing an interaction of a non-relativistic charged particle and a quantized radiation field in the Coulomb gauge without the dipole approximation.Special attention is paid to definition of the "time-ordered Hilbert space-valued stochastic integrals associated with a family of isometries from a Hilbert space t,o another one" and semigroup techniques.Some inequalities are derived, which are infinite degree versions of those known for finite dimensional Schr\"odinger operators with classical vector potentials.

This article presents functional integral representations for the heat semigroups with the infinitesimal generators given by self-adjoint Hamiltonians describing an interaction of a. non-relativistic charged particle and a quantized radiation field in the Coulomb gauge without the dipole approximation. Special attention is paid to definition of the "time-ordered Hilbert space-valued stochastic integrals associated with a family of isometries from a Hilbert space to another one" and semigroup techniques. Some inequalities are derived, which are infinite degree versions of those known for finite dimensional Schrodinger operators with classical vector potentials.

The ordinary Feynman-Kac-Itô formula gives a path integral representation of the Schrödinger semi-groups. We discuss here an analogous probabilistic expression for the (positive energy) solutions of the imaginary-time Klein-Gordon equation in a static external electromagnetic field. When the external field is not purely magnetic, our result is different from the path integral associated to the semi-group t ∊ [0, + ∞ ) → exp[− t ( H − mc 2 )/ℏ] of a relativistic Schrödinger operator H = { c 2 [− i ℏ∇−( e/c ) A ] 2 + m 2 c 4 } 1/2 + V .