Spatial Geometry of Non-Abelian Gauge Theory in \(2 + 1\) Dimensions

Abstract

The Hamiltonian dynamics of \(2 + 1\) dimensional Yang-Mills theory with gauge group SU(2) is reformulated in gauge invariant, geometric variables, as in earlier work on the \(3 + 1\) dimensional case. Physical states in electric field representation have the product form \(Ψ_{\mathrm{phys}} [E^{a i}] = \exp ( i Ω[ E ] / g ) F [G_{ij}]\), where the phase factor is a simple local functional required to satisfy the Gauss law constraint, and \(G_{ij}\) is a dynamical metric tensor which is bilinear in \(E^{a k}\). The Hamiltonian acting on \(F [ G_{ij} ]\) is local, but the energy density is infinite for degenerate configurations where \(\det G (x)\) vanishes at points in space, so wave functionals must be specially constrained to avoid infinite total energy. Study of this situation leads to the further factorization \(F [G_{ij} ] = F_c [ G_{ij} ] \mathcal R [ G_{ij} ]\), and the product \(Ψ_c [E] \equiv \exp (i Ω[ E ] / g ) F_c [G_{ij}]\) is shown to be the wave functional of a topological field theory. Further information from topological field theory may illuminate the question of the behavior of physical gauge theory wave functionals for degenerate fields.