Gauge procedure with gauge fields of various ranks

TL;DR

This paper explores the possibility of gauging phase symmetries using gauge fields of various ranks, extending traditional gauge theory, and discusses the properties and challenges of such theories, including their nonrenormalizability.

Contribution

It introduces a method to gauge phase symmetries with higher rank fields starting from matter coupling, deriving gauge-invariant field strengths, and analyzing their similarities to general relativity.

Findings

Higher rank gauge theories can be constructed from matter couplings.

Many resulting Lagrangians are nonrenormalizable, akin to general relativity.

Some theories involve derivatives of gauge fields similar to Christoffel symbols.

Abstract

The standard procedure for making a global phase symmetry local involves the introduction of a rank 1, vector field in the definition of the covariant derivative. Here it is shown that it is possible to gauge a phase symmetry using fields of various ranks. In contrast to other formulations of higher rank gauge fields we begin with the coupling of the gauge field to some matter field, and then derive the gauge invariant, field strength tensor. Some of these gauge theories are similar to general relativity in that their covariant derivatives involve derivatives of the rank n gauge field rather than just the gauge field. For general relativity the covariant derivative involves the Christoffel symbols which are written in terms of derivatives of the metric tensor. Many (but not all) of the Lagrangians that we find for these higher rank gauge theories lead to nonrenormalizable quantum theories which is also similar to general relativity.