# 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