Different algebras for one reality

TL;DR

This paper revises the formalism of geometric algebra used in physics, explores isomorphic algebras with different spacetime signatures, and shows their equivalence in formulating physical theories and equations.

Contribution

It demonstrates the equivalence of multiple isomorphic algebras in physics and connects different formulations, including Clifford, G(2,3), G(4,1), and Q x Q x C.

Findings

Equivalence of various algebraic formulations of physics.

Conversion methods among isomorphic algebras.

Relation between monogenic functions and nilpotent entities.

Abstract

The most familiar formalism for the description of geometry applicable to physics comprises operations among 4-component vectors and complex real numbers; few people realize that this formalism has indeed 32 degrees of freedom and can thus be called 32-dimensional. We will revise this formalism and we will briefly show that it is best accommodated in the Clifford or geometric algebra G(1,3) x C, the algebra of 4-dimensional spacetime over the complex field. We will then explore other algebras isomorphic to that one, namely G(2,3), G(4,1) and Q x Q x C, all of which have been used in the past by PIRT participants to formulate their respective approaches to physics. G(2,3)is the algebra of 3-space with two time dimensions, which John Carroll used implicitely in his formulation of electromagnetism in 3 + 3 spacetime, G(4,1) was and it still is used by myself in a tentative to unify the formulation of physics and Q x Q x C is the choice of Peter Rowlands for his nilpotent formulation of quantum mechanics. We will show how the equations can be converted among isomorphic algebras and we also examine how the monogenic functions that I use are equivalent in many ways to Peter Rowlands nilpotent entities.