Riemannian Gauge Theory and Charge Quantization

TL;DR

This paper introduces a novel gauge theory framework inspired by Riemannian geometry, deriving gauge fields from embeddings rather than fundamental objects, leading to a universal charge coupling and a basis-independent action.

Contribution

It proposes a new gauge theory construction based on geometric embeddings, resulting in a universal matter coupling and an invariance group GL(n,R) or GL(n,C).

Findings

Recover traditional SO(n) or U(n) gauge theories from the new framework.

All matter fields couple with the same charge, regardless of their fiber type.

The theory suggests a correction to Einstein's gravity proportional to the curvature tensor squared.

Abstract

In a traditional gauge theory, the matter fields \phi^a and the gauge fields A^c_\mu are fundamental objects of the theory. The traditional gauge field is similar to the connection coefficient in the Riemannian geometry covariant derivative, and the field-strength tensor is similar to the curvature tensor. In contrast, the connection in Riemannian geometry is derived from the metric or an embedding space. Guided by the physical principal of increasing symmetry among the four forces, we propose a different construction. Instead of defining the transformation properties of a fundamental gauge field, we derive the gauge theory from an embedding of a gauge fiber F=R^n or F=C^n into a trivial, embedding vector bundle F=R^N or F=C^N where N>n. Our new action is symmetric between the gauge theory and the Riemannian geometry. By expressing gauge-covariant fields in terms of the orthonormal gauge basis vectors, we recover a traditional, SO(n) or U(n) gauge theory. In contrast, the new theory has all matter fields on a particular fiber couple with the same coupling constant. Even the matter fields on a C^1 fiber, which have a U(1) symmetry group, couple with the same charge of +/- q. The physical origin of this unique coupling constant is a generalization of the general relativity equivalence principle. Because our action is independent of the choice of basis, its natural invariance group is GL(n,R) or GL(n,C). Last, the new action also requires a small correction to the general-relativity action proportional to the square of the curvature tensor.