Hamiltonian Flow Equations for a Dirac Particle in an External Potential

Abstract

We derive and solve the Hamiltonian flow equations for a Dirac particle in an external static potential. The method shows a general procedure for the set up of continuous unitary transformations to reduce the Hamiltonian to a quasidiagonal form. Supported by DFG Grant 436 RUS 113/205/1 on Russian German CooperationThe well known Foldy-Wouthuysen (FW) transformation [1] is a unitary transformation that separates big and small components in the Dirac equation for a particle interacting with an external electromagnetic field. If the external field is stationary then the FW transformation reduces to a unitary transformation of the Hamiltonian H of the Dirac particle H ′ = e iS (⃗α(⃗p − e ⃗ A(⃗x)) + βm + eA0(⃗x))e −iS (1) where S can be expanded in a sequence of hermitean operators in powers of 1/m. Recently a novel method to diagonalize Hamiltonians via continuous unitary transformations has been proposed [2], where the Hamiltonian H and the S are considered to be functions of a continuous variable, the flow parameter l. This new method has been applied to several topics like the n − orbital model [2], superconductivity and the electron phonon interaction [3, 4]. We want to apply it here to the Dirac particle in an external field in order to explore its possibilities with a well known problem. Thereby we can gain experience in solving problems like the elimination of negative energy states in near light cone QCD [5], where the solution has yet to be discovered. For infinitesimal steps exp(η(l)dl) a continuous unitary transformation of the Hamiltonian H leads to the so-called Hamiltonian flow equation dH(l) dl = [η(l), H(l)] (2) with initial condition H(0) = H. The transformation matrix η(l) is antihermitian and related to S as follows η(l) = ( d dl eiS(l))e −iS(l). The parameter l has dimension of energy −2 and characterizes the amount of diagonalization which has been achieved. In the energy representation the matrix elements of the Hamiltonian H will be grouped in a band of width