Asao Arai
 Non-relativistic Limit 
of a Dirac Polaron 
in Relativistic Quantum Electrodynamics
(21K, Latex2.09)

ABSTRACT.  A quantum system of a Dirac particle interacting with the quantum radiation field 
is considered in the case where no external potentials exist. Then 
the total momentum of the system is conserved and the total Hamiltonian 
is unitarily equivalent to 
the direct integral $\int_{{\bf R}^3}^\oplus\overline{H({\bf p})}d{\bf p}$ 
of a family of self-adjoint operators 
$\overline{H({\bf p})}$ acting in the Hilbert space $\oplus^4{\cal F}_{\rm rad}$, 
where ${\cal F}_{\rm rad}$ is the Hilbert space of 
the quantum radiation field. The fibre operator $\overline{H({\bf p})}$ is 
called the Hamiltonian of the Dirac polaron with total momentum ${\bf p} 
\in {\bf R}^3$. The main result of this paper is concerned with 
the non-relativistic (scaling) limit of 
$\overline{H({\bf p})}$. It is proven that the non-relativistic limit 
of $\overline{H({\bf p})}$ yields 
a self-adjoint extension of 
a Hamiltonian of a polaron with spin $1/2$ 
in non-relativistic quantum electrodynamics.