**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).