Spatial Geometry of the Electric Field Representation of Non-Abelian Gauge Theories

TL;DR

This paper reveals a geometric interpretation of non-abelian gauge theories in the electric field representation, transforming the Gauss law constraint into a geometric framework involving connections, metrics, and torsion, with explicit results for SU(2) and SU(3).

Contribution

It introduces a novel geometric formulation of non-abelian gauge theories using electric field variables, connecting gauge invariance with spatial geometry and simplifying the Hamiltonian.

Findings

Spatial geometry for SU(2) is Riemannian.

For SU(3), geometry includes both conventional and unconventional torsion.

The transformed Hamiltonian is local and expressed in gauge-invariant geometric variables.

Abstract

A unitary transformation $\Ps [E]=\exp (i\O [E]/g) F[E]$ is used to simplify the Gauss law constraint of non-abelian gauge theories in the electric field representation. This leads to an unexpected geometrization because $\o^a_i\equiv -\d\O [E]/\d E^{ai}$ transforms as a (composite) connection. The geometric information in $\o^a_i$ is transferred to a gauge invariant spatial connection $\G^i_{jk}$ and torsion by a suitable choice of basis vectors for the adjoint representation which are constructed from the electric field $E^{ai}$. A metric is also constructed from $E^{ai}$. For gauge group $SU(2)$, the spatial geometry is the standard Riemannian geometry of a 3-manifold, and for $SU(3)$ it is a metric preserving geometry with both conventional and unconventional torsion. The transformed Hamiltonian is local. For a broad class of physical states, it can be expressed entirely in terms of spatial geometric, gauge invariant variables.