Loop Action for Lattice U(1) Gauge Theory

It is showed that the very recently introduced Lagrangian loop formulation of the lattice Maxwell theory is equivalent to the Villain form in 2+1 dimensions. A transparent description of the classical loop action is given in pure geometrical terms for the 2 + 1 and 3 + 1 dimensional cases. The loop formalism was introduced some time ago as a general analytical Hamiltonian approach based on the properties of the group of loops [1]. It provides a common framework for quantum gauge theories – it works for several models, as Maxwell theory [2], Chern-Simons theory[3], etc – and quantum gravity [4]. Furthermore, it works for Yang-Mills theories on a lattice [5] and recently has been extended in such a way to take account of dynamical fermions [6]. Very recently [7] a loop action for the U(1) gauge theory has been built which lattice version leads to the Kogut-Susskind Hamiltonian. A Lagrangian formulation in terms of loops is interesting for multiple reasons. First, it offers the possibility of knit together the intrinsic advantages of the nonlocal loop description (the non-redundancy of gauge degrees of freedom and its geometrical transparency) and the computational power of numerical simulations. This provides a a useful complement to the analytical loop studies. The classical action may be also relevant to perform semiclassical approximations. In this paper we show that the proposed action for the D=2+1 dimensional case leads after a duality transformation to the discrete Gaussian model form. It is known that the same happens for the Villain form. Thus, for D=2+1, the minimal description provided by the loops actually corresponds to that of Villain form instead of Wilson cosinus form. For D=3+1 the loop action and Villain form give a similar description but the connection between them is more subtle. According to reference [7] the continuum Euclidean loop action for the abelian theory in D = d+ 1 space-time dimensions is given by S = g 2 ∫ dtdx{−Ẋ i C(x) 1 ∆ Ẋ i C(x) +X i C(x)X i C(x)} (1) whereX i C(x) is the loop current ∮ C dy δ(x−y) and ∆ is the three-dimensional Laplace operator. The time derivative of X i C(x) can be written as limdt→0 1 dt ∮ Ct+dtC̄t dyδ(x−y), where C̄t denotes the loop Ct traversed in the opposite direction. In order to formulate (1) on a lattice we represent the continuum surface spanned by the loop C as a set of spatial loops Ct at different times t = 0, 1, . . . , T . We also replace the derivatives by finite difference operators and get