Loop Equations and bootstrap methods in the lattice

Abstract

Pure gauge theories can be formulated in terms of Wilson Loops correlators by means of the loop equation. In the large-N limit this equation closes in the expectation value of single loops. In particular, using the lattice as a regulator, it becomes a well defined equation for a discrete set of loops. In this paper we study different numerical approaches to solving this equation. Previous ideas gave good results in the strong coupling region. Here we propose an alternative method based on the observation that certain matrices $\hat{\rho}$ of Wilson loop expectation values are positive definite. They also have unit trace ($\hat{\rho}\succeq 0, \mbox{tr} \hat{\rho}=1$), in fact they can be defined as density matrices in the space of open loops after tracing over color indices and can be used to define an entropy associated with the loss of information due to such trace $S_{WL}=-\mbox{tr}[ \hat{\rho}\ln \hat{\rho}]$. The condition that such matrices are positive definite allows us to study the weak coupling region which is relevant for the continuum limit. In the exactly solvable case of two dimensions this approach gives very good results by considering just a few loops. In four dimensions it gives good results in the weak coupling region and therefore is complementary to the strong coupling expansion. We compare the results with standard Monte Carlo simulations.