Phase Space structure on Clifford Algebras

TL;DR

This paper introduces a phase space structure on Euclidean Clifford algebras using Hodge structures, enabling Hamiltonian dynamics on multilinear variables and revealing new commuting and anticommuting behaviors.

Contribution

It generalizes Kähler structures to higher dimensions within Clifford algebras, establishing a novel phase space framework and dynamics for multilinear variables.

Findings

Phase space structure on Clifford algebras established.

Hamiltonian dynamics on multilinear variables constructed.

Discovery of commuting and anticommuting behaviors in subspace dynamics.

Abstract

I argue that the Hodge structure on a Euclidean Clifford algebra $Cl(n)$ provides a way to generalise K\"ahler structure to higher dimensions, in the sense that the paired variables are now associated with $k-$ and $(n-k)-$ dimensional subspaces rather than with vectors. This puts a phase space structure on Clifford algebras, and so allows us to construct a Hamiltonian dynamics on these multilinear variables. This construction shows that alternating pairs of subspaces obey commuting and anticommuting dynamics, hinting that this construction is indeed a natural one, with interesting new behaviour.