From phase space to multivector matrix models

TL;DR

This paper introduces a novel framework combining twistor-space, phase space, and Clifford algebras to construct and quantize Lorentz-covariant multivector geometries, linking non-relativistic mechanics with relativistic space-time.

Contribution

It develops a new approach to model fuzzy geometries using multivectors and matrix models derived from Yang-Mills interactions, with implications for quantum geometry and cosmology.

Findings

Constructed Lorentz-covariant multivectors parametrized by phase space polynomials

Introduced Lorentz-covariant non-commutativity via Groenewold-Moyal *-product

Proposed multivector matrix models as large-N limits of Yang-Mills theories

Abstract

Combining elements of twistor-space, phase space and Clifford algebras, we propose a framework for the construction and quantization of certain (quadric) varieties described by Lorentz-covariant multivector coordiantes. The correspondent multivectors can be parametrized by second order polynomials in the phase space. Thus the multivectors play a double role, as covariant objects in $D=2,3,4 \texttt{ Mod } 8$ space-time dimensions, and as mechanical observables of a non-relativistic system in $2^{[D/2]-1}$ euclidean dimensions. The latter attribute permits a dual interpretation of concepts of non-relativistic mechanics as applying to relativistic space-time geometry. Introducing the Groenewold-Moyal *-product and Wigner distributions in phase space induces Lorentz-covariant non-commutativity and it provides the spectra of geometrical observables. We propose also new (multivector) matrix models, interpreted as descending from the interaction term of a Yang-Mills theory with minimally coupled massive fermions, in the large-$N$ limit, which serves as a physical model containing the constructed multivector (fuzzy) geometries. We also include a section on speculative aspects on a possible cosmological effect and the origin of space-time entropy.