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.