Asao Arai
Heisenberg Operators of a Dirac Particle Interacting with the Quantum Radiation Field
(209K, LaTex 2e)
ABSTRACT. We consider a quantum system of a Dirac particle interacting with the quantum
radiation field, where the Dirac particle is in a $4\times 4$-Hermitian matrix-valued
potential $V$.
Under the assumption that the total Hamiltonian $H_V$ is
essentially self-adjoint (we denote its closure by ${\bar H}_V$),
we investigate properties of the Heisenberg operator
$x_j(t):=e^{it{\bar H}_V}x_je^{-it{\bar H}_V}$ ($j=1,2,3$)
of the $j$-th position operator of the Dirac particle at time $t\in\R$ and
its strong derivative $dx_j(t)/dt$ (the $j$-th velocity operator),
where $x_j$ is the multiplication operator
by the $j$-th coordinate variable $x_j$ (the $j$-th position operator at time $t=0$).
We prove that $D(x_j)$, the domain of the position operator $x_j$, is invariant under the action of
the unitary operator $e^{-it{\bar H}_V}$ for all $t\in \R$ and establish a mathematically rigorous formula
for $x_j(t)$. Moreover, we derive asymptotic expansions of Heisenberg operators
in the coupling constant $q\in \R$ (the electric charge of the Dirac particle).