Path integral for the loop representation of lattice gauge theories

We show how the Hamiltonian lattice loop representation can be cast straightforwardly in the path integral formalism. The procedure is general for any gauge theory. Here we present in detail the simplest case: pure compact QED. The lattice loop path integral approach allows us to knit together the power of statistical algorithms with the transparency of the gauge-invariant loop description. The results produced by numerical simulations with the loop classical action for different lattice models are discussed. We also analyze the lattice path integral in terms of loops for the non-Abelian theory.