A Unifying Calculus for Physics: A Demonstration in Relativistic Quantum Mechanics
In 1960-62, Erich Kaehler produced his largely ignored extension of the calculus of differential forms, i.e. of the language of differential geometry and general relativity, so as to be adequate also for quantum mechanics. Most significant in that work is his "Kaehler-Dirac equation" (KDE), which he used to reproduce the fine structure of the hydrogen atom and to show that positrons emerge with the same sign of the energy as electrons. It is shown that with the very basics of the exterior calculus and algebra of gamma matrices, one easily immerses oneself in a Kaehler Calculus (KC) with Cartesian coordinates. Since the Lorentz group is implicit in the KC through the metric, knowledge of relativistic physics is not even necessary to do relativistic quantum mechanics. We make the point of superiority of the KC through as follows:
(a) the electromagnetic Hamiltonian is developed without Foldy-Wouthuysen transformations, both for electrons and positrons and to a standard degree of approximation beyond Pauli's,
(b) it is shown that for stationary fields the expansion can be stopped exactly at any point, i.e. without residual,
(c) the emergence of negative energy solutions in the Dirac theory is shown to be, from a KC perspective, a spurious effect arising from using spinors instead of inhomogeneous differential forms,
(d) spinors are shown to arise from ideals through the use of symmetries in the "wave differential form".
It is clear from the nature of the arguments in (a)-(d) that the same computational straightforwardness and simplifications can be expected for quantum electrodynamics based on that very simple KC. As for its more sophisticated versions, suffice to say that all of the above only involves scalar-valued input and output. New Hamiltonians can be generated in principle from more sophisticated inputs, by leaving alone on the left hand side of the KDE equation the Lie operator pertaining to time translations. In this regard, it should be noticed that the co-derivative is connected with metric-compatible affine connections in the KC, not with the metric itself in general (the usual co-derivative is obtained for Levi-Civita connection). Hence, non-linear generalizations of the KDE, and thus of the Dirac equation, emerge by two different mechanisms. First, a quantity of affine nature (specifically the torsion) is made to satisfy a KDE. Second, the "input" differential form is the potential for the torsion, and also is to be solved for.