The Geometrical Basis of the Nonlinear Gauge

TL;DR

This paper explores the geometric structure of Yang-Mills theory's configuration space, introducing a non-linear gauge that applies in the non-perturbative regime but does not globally foliate the space.

Contribution

It develops a geometric framework for Yang-Mills theory, proposing a non-linear gauge condition that extends the orthogonal gauge concept to non-Abelian cases.

Findings

Coulomb-like surfaces foliate the configuration space in Abelian theory.

Non-linear gauge modifies orthogonality condition for non-Abelian theory.

Foliation by the non-linear gauge is limited to the non-perturbative regime.

Abstract

We consider Yang-Mills theory in Euclidean space-time $(R^4)$ and construct its configuration space. The orbits are first shown to form a congruence set. Then we discuss the orthogonal gauge condition in Abelian theory and show that Coulomb-like surfaces foliate the entire configuration space. In the non-Abelian case, where these exists no global orthogonal gauge, we derive the non-linear gauge proposed previously by the author by modifying the orthogonality condition. However, unlike the Abelian case, the entire configuration space cannot be foliated by submanifolds defined by the non-linear gauge. The foliation is only limited to the non-perturbative regime of Yang-Mills theory.