# Spectral Analysis of the Dirac Polaron

### Itaru Sasaki

Shinshu University, Matsumoto, Japan

## Abstract

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

## Cite this article

Itaru Sasaki, Spectral Analysis of the Dirac Polaron. Publ. Res. Inst. Math. Sci. 50 (2014), no. 2, pp. 307–339

DOI 10.4171/PRIMS/135