Polydimensional Relativity, a Classical Generalization of the Automorphism Invariance Principle

TL;DR

This paper proposes a classical generalization of automorphism invariance using polydimensional transformations within Clifford algebra, aiming to extend symmetry principles to broader physical laws and improve the treatment of spinning particles in relativity.

Contribution

It introduces a polydimensional invariance principle applicable to classical physics, generalizing automorphism invariance and providing a new framework for relativistic spinning particles.

Findings

Polydimensional transformations preserve algebraic structure while reshuffling geometric components.

New treatment of Papapetrou equations for spinning particles in curved space.

Natural coupling between linear and spinning motion via changing geometric rank.

Abstract

The automorphism invariant theory of Crawford[J. Math. Phys. 35, 2701 (1994)] has show great promise, however its application is limited by the paradigm to the domain of spin space. Our conjecture is that there is a broader principle at work which applies even to classical physics. Specifically, the laws of physics should be invariant under polydimensional transformations which reshuffle the geometry (e.g. exchanges vectors for trivectors) but preserves the algebra. To complete the symmetry, it follows that the laws of physics must be themselves polydimensional, having scalar, vector, bivector etc. parts in one multivector equation. Clifford algebra is the natural language in which to formulate this principle, as vectors/tensors were for relativity. This allows for a new treatment of the relativistic spinning particle (the Papapetrou equations) which is problematic in standard theory. In curved space the rank of the geometry will change under parallel transport, yielding a new basis for Weyl's connection and a natural coupling between linear and spinning motion.