The world sheet formulation is introduced for lattice gauge theories with dynamical fermions. The partition function of lattice compact QED with staggered fermions is expressed as a sum over surfaces with a border on self-avoiding fermionic paths. The surfaces correspond to the world sheets of loop-like pure electric flux excitations and meson-like configurations ͑open electric flux tubes carrying matter fields at their ends͒. The proposed formulation does not have the problem of the additional doubling of the fermion species due to the discretization of time. The gauge non-redundancy and the geometric transparency are two appealing features of this description. From the computational point of view, the partition function involves fewer degrees of freedom than the Kogut-Susskind formulation and offers an alternative and more economic framework to perform numerical computations with dynamical fermions. ͓S0556-2821͑98͒00106-4͔

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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.

It is showed how the Hamiltonian lattice loop representation can be cast straightforwardly in the Lagrangian formalism. The procedure is general and here we present the simplest case: pure compact QED. This connection has been shaded by the non canonical character of the algebra of the fundamental loop operators. The loops represent tubes of electric ux and can be considered the dual objects to the NielsenOlesen strings supported by the Higgs broken phase. The lattice loop classical action corresponding to the Villain form is proportional to the quadratic area of the loop world sheets and thus it is similar to the Nambu string action. This loop action is used in a Monte Carlo simulation and its appealing features are discussed.

We present the extension of the Lagrangian loop gauge invariant representation in such a way to include matter fields. The partition function of lattice compact U (1)-Higgs model is expressed as a sum over closed as much as open surfaces. These surfaces correspond to world sheets of loop-like pure electric flux excitations and open electric flux tubes carrying matter fields at their ends. This representation is connected by a duality transformation with the topological representation of the partition function (in terms of world sheets of Nielsen-Olesen strings both closed and open connecting pairs of magnetic monopoles). We have simulated numerically the loop action equivalent to the Villain form of the action and mapped out the β-γ phase diagram of this model. By virtue of the gauge invariance of this description the equilibrium configurations seems to be reached faster than with the ordinary gauge-variant descriptions.