New perspectives on the Kähler calculus and wave functions (Q1038786)

The authors aim to recast Kähler's calculus, introduced at the beginning of the sixties of the last century, in the language of Cartan's calculus of Clifford bundle-valued differential forms, to obtain a modern formulation of that geometric formulation. This leads to consider manifolds with torsion and to generalize the Dirac equation (D) to the so-called Kähler-Dirac equation (KD). Some applications in mathematical physics, of this point of view, are stressed too. The paper, after a detailed introduction, splits in five more sections. 2. Kähler's ansatz derivatives. 3. Of the connection and the interior derivative. 4. Schmeikal's model for law energy quarks. 5. Reducing through symmetries (KD) solutions to (D) solutions. 6. Dynamical consequence of Schmeikal's quark phenomenology. Remark. After the great development of differential geometry in the last decades, it appears that Kähler's calculus is full of cumbersome formulas (as noted in this paper). However, what really remains actual in that formulation is its pioneering vision to consider Cartan's calculus of differential forms a common language for relativity and quantum mechanics. Of course, it was necessary to add many further generalizations to arrive at the actual geometric architecture of noncommutative differential geometry, that unifies relativity and quantum field theory. But Kähler shows the right way. The Cartan-Kähler theory is a very important part in this beautiful mathematical building.