Loop States in Lattice Gauge Theories

Abstract

We solve the Gauss law as well as the corresponding Mandelstam constraints of (d+1) dimensional SU(2) lattice gauge theory in terms of harmonic oscillator prepotentials. This enables us to explicitly construct a complete orthonormal and manifestly gauge invariant basis in the physical Hilbert space. Further, we show that this gauge invariant description represents networks of unoriented loops carrying certain non-negative abelian fluxes created by the harmonic oscillator prepotentials. The loop network is characterized by $3(d-1)$ gauge invariant integers at every lattice site which is the number of physical degrees of freedom. Time evolution involves local fluctuations of these loops. The loop Hamiltonian is derived. The generalization to SU(N) gauge group is discussed.