Levy Differential Operators and Gauge Invariant Equations for Dirac and Higgs Fields

TL;DR

This paper explores Levy differential operators in infinite-dimensional spaces and their connection to gauge-invariant equations like Yang-Mills, revealing new formulations for fundamental quantum field equations.

Contribution

It introduces new infinite-dimensional Levy differential operators and establishes their equivalence and relation to Yang-Mills-Higgs and Yang-Mills-Dirac equations.

Findings

Levy differential operators relate to solutions of Yang-Mills equations.

Infinite-dimensional equations with Levy operators are equivalent to quantum chromodynamics equations.

Parallel transport on curve spaces solves gauge-invariant differential systems.

Abstract

We study the Levy infinite-dimensional differential operators (differential operators defined by the analogy with the Levy Laplacian) and their relationship to the Yang-Mills equations. We consider the parallel transport on the space of curves as an infinite-dimensional analogue of chiral fields and show that it is a solution to the system of differential equations if and only if the associated curvature is a solution to the Yang-Mills equations. This system is an analogue of the equation of motion of chiral fields and contains the Levy divergence. The systems of infinite-dimensional equations containing Levy differential operators, that are equivalent to the Yang-Mills-Higgs equations and the Yang-Mills-Dirac equations (the equations of quantum chromodinamics), are obtained. The equivalence of two ways to define the Levy differential operators is shown.