A Flow Equation Approach Striving Towards an Energy-Separating Hamiltonian Unitary Equivalent to the Dirac Hamiltonian with Coupling to Electromagnetic Fields

Abstract

The Dirac Hamiltonian H (D) for relativistic charged fermions minimally coupled to (possibly time-dependent) electromagnetic fields is transformed with a purpose-built flow equation method, so that the result of that transformation is unitary equivalent to H (D) and granted to strive towards a limiting value H (N W ) commuting with the Dirac β-matrix. Upon expansion of H (N W ) to order v 2 c 2 the nonrelativistic Hamiltonian H (SP ) of Schrödinger-Pauli quantum mechanics emerges as the leading order term adding to the rest energy mc 2 . All the relativistic corrections to H (SP ) are explicitly taken into account in the guise of a Magnus type series expansion, the series coefficients generated to order v 2 c 2 n for n ≥ 2 comprising partial sums of iterated commutators only. In the special case of static fields the equivalence of the flow equation method with the well known energyseparating unitary transformation of Eriksen is established on the basis of an exact solution of a reverse flow equation transforming the β-matrix into the energy-sign operator associated with H (D) .That way the identityunambiguously. In contrast to H (D) it's unitary equivalent H (N W ) generates the motion of electrons and positrons in the presence of weak external fields now as entirely separated wave packets carrying mass m, charge ±|e| and spin ± 2 respectively, yet those wave packets being by construction bare of any "Zitterbewegung", akin to classical particles moving along individual trajectories under the influence of the Lorentz force. Upon expansion of H (N W ) to order v 2 c 2 n for n = 1, 2, 3, ... our results agree with results obtained recently by Silenko with a correction scheme developed for the original step-by-step FW-transformation method, the latter long-since known for not generating unambiguously a unitary equivalent Hamiltonian being energy-separating.