The third book in the series deals with the mathematics of quantum mechanics that forms part of the natural evolution of the calculus, if we accept in the first place that the calculus is about differential forms. It is to the exterior calculus of differential forms what Clifford algebra is to exterior algebra. A word about this algebra: it should be called Euclidean algebra, because it is the algebra built upon the interior (or dot, or scalar) product and exterior products (the latter supersedes the vector product) of vectors.
This calculus was created or discovered by Erich Kähler, of Kähler manifolds fame in 1960-62. It constitutes a formidable piece of mathematics, as it extends, for instance, the theory of harmonic functions to the ring of differential firms. For a physicist, iit permits one to do quantum physics without gamma matrices. One simply endows differential forms with a richer structure. The Kähler equation supersedes the Dirac equation. As an example of its power, the obtaining of the hydrogen atom’s fine structure is just an exercise following the solution of the equation of (strict) harmonic differentials. And yet, in doing so, one unleashes only a very minimal part of the power that the Kähler equation has. With that perspective, it is impossible to repress seeing quarks emerge in high energy scattering experiments if one were to use the Kähler equation for its study. Very important: his quantities have three series of indices, two of them are for covariant tensors with different behaviors under differentiation, which speaks of the fact that to define differential forms as antisymmetric covariant tensors is a very poor road to the Kähler calculus. Of course, one can avoid this debacle of indices by writing the bases throughout, which is what Cartan does, and would have done for the Kähler calculus if he had been alive.
So, in the third book we plan to first do Clifford algebra. Then the Kähler calculus, but in a more Cartanian style. We shall also extend it to connections other than the Levi-Civita connection. We shalll also prepare the ground for computing tin quantum electrodynamics with this tool, and possibly avoid the divergence problems.
Jose Vargas and Doug Torr will make every effort to respond to questions on their papers and the subject of Cartan-Kähler Unification.