Spatial Geometry of Hamiltonian Gauge Theories

TL;DR

This paper reformulates Hamiltonians of $SU(2)$ and $SU(3)$ gauge theories in 3+1 dimensions using gauge-invariant spatial geometric variables, leading to local Hamiltonians, and extends the approach to $SU(2)$ in 2+1 dimensions.

Contribution

It introduces a novel geometric variable framework that makes the Hamiltonians local and simplifies their structure in non-Abelian gauge theories.

Findings

Hamiltonians expressed in geometric variables are local.

The approach applies to both 3+1 and 2+1 dimensional theories.

Provides new insights into the structure of gauge theories.

Abstract

The Hamiltonians of $SU(2)$ and $SU(3)$ gauge theories in 3+1 dimensions can be expressed in terms of gauge invariant spatial geometric variables, i.e., metrics, connections and curvature tensors which are simple local functions of the non-Abelian electric field. The transformed Hamiltonians are local. New results from the same procedure applied to the $SU(2)$ gauge theory in 2+1 dimensions are also given.