Heisenberg operators of a Dirac particle interacting with the quantum radiation field

We consider a quantum system of a Dirac particle interacting with the quantum radiation field, where the Dirac particle is in a 4 × 4 -Hermitian matrix-valued potential V. Under the assumption that the total Hamiltonian H V is essentially self-adjoint (we denote its closure by H ¯ V ), we investigate properties of the Heisenberg operator x j ( t ) : = e i t H ¯ V x j e − i t H ¯ V ( j = 1 , 2 , 3 ) of the j-th position operator of the Dirac particle at time t ∈ R and its strong derivative d x j ( t ) / d t (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 − i t H ¯ V for all t ∈ R and establish a mathematically rigorous formula for x j ( t ) . Moreover, we derive asymptotic expansions of Heisenberg operators in the coupling constant q ∈ R (the electric charge of the Dirac particle).