Multivector Solutions to the Hyper-Holomorphic Massive Dirac Equation

TL;DR

This paper introduces a new set of multivector solutions to the massive Dirac equation, enabling a geometric path integral approach that models quantum particles through algebraic groupings.

Contribution

It constructs a Hilbert space of solutions and proposes a novel integral kernel application, generalizing the Feynman Path Integral with multivectors.

Findings

New multivector solutions form a Hilbert space

A geometric path integral formulation is developed

Quantum particle behavior is modeled through geometric groupings

Abstract

Attention is given to the interface of mathematics and physics, specifically noting that fundamental principles limit the usefulness of otherwise perfectly good mathematical general integral solutions. A new set of multivector solutions to the meta-monogenic (massive) Dirac equation is constructed which form a Hilbert space. A new integral solution is proposed which involves application of a kernel to the right side of the function, instead of to the left as usual. This allows for the introduction of a multivector generalization of the Feynman Path Integral formulation, which shows that particular ``geometric groupings'' of solutions evolve in the manner to which we ascribe the term ``quantum particle''. Further, it is shown that the role of usual $i$ is subplanted by the unit time basis vector, applied on the right side of the functions. Summary of talk, to appear in: Proceedings of the 17th Annual Lecture Series in the Mathematical Sciences, April 8-10, 1993, University of Arkansas, `Clifford Algebas in Analysis', J. Ryan, editor (CRC Press, expected 1994)