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

TL;DR

This paper develops a flow equation method to transform the Dirac Hamiltonian into an energy-separating form, explicitly including relativistic corrections and establishing equivalence with known transformations in static fields.

Contribution

It introduces a novel flow equation approach to derive an energy-separating Hamiltonian for relativistic fermions, unifying and extending previous methods with explicit relativistic corrections.

Findings

Derived a unitary equivalent Hamiltonian approaching an energy-separating form.

Explicitly included relativistic corrections via a Magnus series expansion.

Established equivalence with Eriksen's transformation in static fields.

Abstract

The Dirac Hamiltonian $H^{\left(D\right)}$ 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 $H^{\left(D\right)}$ and granted to strive towards a limiting value $H^{\left(NW\right)}$ commuting with the Dirac $\beta$-matrix. Upon expansion of $H^{\left(NW\right)}$ to order $\frac{v^2}{c^2}$ the nonrelativistic Hamiltonian $H^{\left(SP\right)}$ of Schr\"odinger-Pauli quantum mechanics emerges as the leading order term adding to the rest energy $mc^2$. All the relativistic corrections to $H^{\left(SP\right)}$ are explicitly taken into account in the guise of a Magnus type series expansion, the series coefficients generated to order $\left(\frac{v^{2}}{c^{2}}\right)^{n}$ for $n\geq2$ 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 energy-separating unitary transformation of Eriksen is established on the basis of an exact solution of a reverse flow equation transforming the $\beta$-matrix into the energy-sign operator associated with $H^{\left(D\right)}$. That way the identity $H^{\left(NW\right)}=\beta\sqrt{H^{\left(NW\right)}H^{\left(NW\right)}}$ is established implying $H^{\left(NW\right)}$ being determined unambiguously.