On Finite Matrix Bi-Dimensional Formulation of $D=4n+2$ Classical Field Models

TL;DR

This paper develops a bi-dimensional finite matrix calculus and action principle to analyze scalar and spinorial fields in dimensions of the form 4n+2, revealing a matrix holomorphic property for massless fields.

Contribution

It introduces a novel bi-dimensional matrix calculus and action principle tailored for D=4n+2 dimensions, incorporating a Dirac-algebra-modified Leibniz rule.

Findings

Established a bi-dimensional finite matrix calculus framework.

Derived matrix holomorphic features for massless scalar and spinorial fields.

Verified the necessity of a modified Leibniz rule for the action principle.

Abstract

We introduce a basis for a bi-dimensional finite matrix calculus and a bi-dimensional finite matrix action principle. As an application, we analyze scalar and spinorial fields in $D=4n+2$ in this approach. We verify that to establish a bi-dimensional matrix action principle we have to define a Dirac-algebra-modified Lebniz rule. From the bi-dimensional equations of motion, we obtain a matrix holomorphic feature for massless matrix scalar and spinorial fields.