The Kähler-Dirac Equation with Non-Scalar-Valued Differential Form

Abstract

The original, matrix-free Kähler-Dirac (KD) equation of 1960 has an unpleasant feature when the input differential form is not scalar-valued. Possibly to deal with that feature, Kähler proposed in 1962 a highly cumbersome alternative equation involving matrices. It coincides with the one from 1960 when the input differential form is scalar-valued.In the 5th Clifford Conference, we proposed our own alternative, where the tensor (1960) and matrix (1962) products of the valuedness factors were replaced with the Clifford product. We also suggested a sort of Kaluza-Klein (KK) space canonically determined by a subset of the fundamental invariants that define the spacetime manifold. In this paper we show that our KK space seems to accommodate the physics well for appropriate choice of the torsion. We further show that one can choose an input differential form highly appropriate for when the aforementioned torsion is assumed to be the output differential form of a geometrized KD equation. This leads to equations of structure for the torsion that have a more quantum-mechanical and less moving frame theory flavor than the standard ones.

New Perspectives on the Kähler Calculus and Wave Functions

“…in auxiliary bundles). In the accompanying paper [6], we show how classical differential geometry may help in this regard. It would be risky to leave such option unexplored, even from a mathematical point of view: indeed it is not yet clear from Kähler's work what his KD equation should be when the input differential form is tensor-valued.…”

The Foundations of Quantum Mechanics and the Evolution of the Cartan-Kähler Calculus

“…Without explanation, Kähler produced in 1962 an equation for tensor-valued input a. It differs from (12), even though the latter is well defined also in the general case (for a detailed discussion see [23]). Unaware of his 1962 equation, that unwelcome feature was addressed [24] in the alternative way that we proceed to summarize.…”

“…He showed that, if u is a solution of the Kähler equation with EM coupling, ηū is a solution of its conjugate equation. Hence, if u and v are both solutions of the direct equation, we have, in view of (23), that u, ηv 1 is conserved.…”

“…In order to take care of the motion of the particles as distinct from the motion of origins of the frames, one may develop a most natural Kaluza-Klein type of theory suggested by the Finsler bundle structure and where the fifth dimension is used to represent precisely the trajectory of particles. It is reassuring that the form of the Lorentz force emerges also in the autoparallels of that Kaluza-Klein space [23], like it does in Finsler geometry [36].…”