Electromagnetic Hamiltonian Expansion with neither Foldy-Wouthuysen Transformations nor Akhieser-Berestetski Method

Abstract:
This poster is the first complement to the talk by the present author "A unifying calculus for physics: a demonstration in relativistic quantum mechanics". The calculus in question is Kaehler's exterior-interior Calculus (KC) of differential forms. For physicists' benefit, we approach it by enriching the Clifford structure underlying Dirac's theory to turn the former into a Kaehler-Atiyah structure. The spinor is replaced with a "state's inhomogeneous differential form" (wave function for brevity), so that the Dirac theory evolves into the pseudo-Cartesian flat-spacetime version of Kaehler's. In this and the accompanying posters, we preserve the original flavor of the KC in computing old results in new ways (in this poster) and new results (in accompanying posters). Those computations are only browsed in the talk. In this poster, we reproduce the first (i.e. Pauli's) and next approximation to the electromagnetic (EM) Hamiltonian in terms of non-relativistic operators. After choosing the ideal that picks the electrons, we develop the Kaehler-Dirac (KD) equation with minimal EM coupling. Other than that and without choice of representation (an unnecessary concept), one writes the wave function with exponential factor where time multiplies the dominant rest mass in the exponent. Reduction of terms followed by simple inspection allows one to write a system of coupled equations for the odd and even parts of the wave function. This "mother system" parallels term by term the one in Dirac's theory obtained with a choice of representation leading directly to the Pauli equation, but no further (Bjorken & Drell, chapter I). In the KC calculus, however, the mother system is further expanded in a straightforward way, not requiring some clever new idea (Foldy-Wouthuysen, Akhieser-Berestetski).