SPATIAL GEOMETRY OF THE ELECTRIC-FIELD REPRESENTATION OF NON-ABELIAN GAUGE-THEORIES

A unitary transformation Psi [E] = exp (i Omega[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 omega(i)(a) equivalent to -delta Omega[E]/delta E(ai) transforms as a (composite) connection. The geometric information in omega(i)(a) is transferred to a gauge invariant spatial connection Gamma(jk)(i) 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.