Spatial geometry of non-abelian gauge theory in 2 + 1 dimensions Original Research Article

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 Ψphys[Eai] = exp(iΩ[E]/g)F[Gij], where the phase factor is a simple local functional required to satisfy the Gauss law constraint, and Gij is a dynamical metric tensor which is bilinear in Eak. The Hamiltonian acting on F[Gij] is local, but the energy density is infinite for degenerate configurations where detG(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[Gij] = Fc[Gij] R[Gij], and the product Ψc [E] - exp (iΩ [E] /g) Fc [ Gif ] 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.