Spectral Analysis of the Dirac Polaron

TL;DR

This paper investigates the spectral properties of a Dirac particle coupled with a quantized radiation field, establishing conditions under which the lowest energy level is an eigenvalue and analyzing related physical properties.

Contribution

The study proves that the lowest energy of the Dirac polaron system is an eigenvalue under specific conditions, advancing understanding of its spectral structure.

Findings

Lowest energy is an eigenvalue under infrared regularization.

Eigenvalue condition: E(p,m) < E(p,0).

Discussion of polarization vectors and angular momenta.

Abstract

A system of a Dirac particle interacting with the radiation field is considered. The Hamiltonian of the system is defined by $H = \alpha\cdot(\hat\mathbf{p}-q\mathbf{A}(\hat\mathbf{x}))+m\beta + H_f$ where $q\in\mathbb{R}$ is a coupling constant, $\mathbf{A}(\hat\mathbf{x})$ denotes the quantized vector potential and $H_f$ denotes the free photon Hamiltonian. Since the total momentum is conserved, $H$ is decomposed with respect to the total momentum with fiber Hamiltonian $H(\mathbf{p}), (\mathbf{p}\in\mathbb{R}^3)$. Since the self-adjoint operator $H(\mathbf{p})$ is bounded from below, one can define the lowest energy $E(\mathbf{p},m):=\inf\sigma(H(\mathbf{p}))$. We prove that $E(\mathbf{p},m)$ is an eigenvalue of $H(\mathbf{p})$ under the following conditions: (i) infrared regularization and (ii) $E(\mathbf{p},m)<E(\mathbf{p},0)$. We also discuss the polarization vectors and the angular momenta.