Path integral for the loop representation of lattice gauge theories

Abstract

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.

The basis of the physical Hilbert space of lattice gauge theories

“…Very similar techniques have been used in the path integral approach [10], and to construct a q-deformed lattice gauge theory [11]. Our work generalizes the results obtained in [12] for the case of SU (2) lattice gauge theory in 2 + 1 dimensions, to arbitrary dimensions and compact gauge groups. Moreover, our procedure allows the explicit calculation of the matrix elements of the Hamiltonian.…”

“…Since when q is an n th -root of unity the number of the irreducible representation is finite, this could provide a natural way to obtain a finite dimensional model [9].Very similar techniques have been used in the path integral approach [10], and to construct a q-deformed lattice gauge theory [11]. Our work generalizes the results obtained in [12] for the case of SU (2) lattice gauge theory in 2 + 1 dimensions, to arbitrary dimensions and compact gauge groups. Moreover, our procedure allows the explicit calculation of the matrix elements of the Hamiltonian.One of the technical difficulties in manipulating the derived formulas is the proliferation of indexes, which turns out in quite cumbersome expressions.…”

“…x Therefore, the result reduces to the computation of the trace of the intertwing matrix. This can be done by using equations (12), (22) and (23). This result shows that the choice of an explicit basis is irrelevant, as expected.…”

Path integral loop representation of 2+1 lattice non-Abelian gauge theories

“…A path integral formulation of SU (2) gauge theories in terms of the worldsheets swept out by spin networks has been developed in [8] and [9]. The worldsheets are branched, coloured surfaces, known in the mathematical literature as spines [10].…”

“…The loop actions of the abelian gauge theories are written in terms of the surfaces swept by the time evolution of the loops. The explicit form of the loop actions for lattice non abelian gauge theories is known to be related with coloured surfaces [8], [9], known as spines in the mathematical lenguage [10]. They are related with the world sheet swept by the evolution in time of the spin networks.…”