Phase Space structure on Clifford Algebras

TL;DR

This work addresses the problem of extending Kähler structure to Clifford algebras and extracting a phase-space dynamics for multilinear variables. It uses the Hodge relation $g(A,B)\mathcal{I}=A\wedge\star B$ to relate inner and outer products and defines dual subspaces $\mathcal{X}_k,\mathcal{P}_k$ with $P_k=X_k^{-1} I$ to form a Clifford-phase-space framework. The framework reveals parity-dependent dynamics: the pairs $(X_k,P_k)$ are either commuting (bosonic) or anticommuting (Grassmann) according to $k(n-k)$, and a Clifford-valued Poisson bracket $\{F,G\}=(F_{x_k}G_{p_k}+(-1)^{k(n-k)}F_{p_k}G_{x_k})I^{\dagger}$ is established, enabling Hamiltonian dynamics on $Cl(n)$. The results suggest a natural higher-dimensional generalization of Kähler geometry and a rich phase-space structure with potential physical applications, with planned extensions to curvature, non-Euclidean metrics, and connections to noncommutative geometries.

Abstract

I argue that the Hodge structure on a Euclidean Clifford algebra $Cl(n)$ provides a way to generalise Kähler 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.