Exterior Differential Systems for Field Theories

TL;DR

This paper formulates exterior differential systems for classical gauge theories, analyzing their well-posedness and revealing the classical necessity of ghost fields akin to those in quantum gauge theories.

Contribution

It introduces a classical exterior differential systems framework for gauge theories, showing the natural emergence of ghost fields for well-posedness, paralleling quantum theory.

Findings

EDS formulation for Maxwell and Yang-Mills fields with Dirac spinors

Well-posedness depends on inclusion of ghost fields in non-Abelian case

Classical analysis anticipates ghost fields used in quantum gauge theories

Abstract

Exterior Differential Systems (EDS) and Cartan forms, set in the state space of field variables taken together with four space-time variables, are formulated for classical gauge theories of Maxwell and SU(2) Yang-Mills fields minimally coupled to Dirac spinor multiplets. Cartan character tables are calculated, showing whether the EDS, and so the Euler-Lagrange partial differential equations, is well-posed. The first theory, with 22 dimensional state space (10 Maxwell field and potential components and 8 components of a Dirac field), anticipates QED. In the second, non-Abelian, case (30 Yang-Mills field components and 16 Dirac), only if three additional "ghost" fields are included (15 more scalar variables) is a well-posed EDS found. This classical formulation anticipates the need for introduction of Fadeev-Popov ghost fields in the quantum standard model.