Stochastic mechanics of Abelian lattice theories

Abstract

A stochastic equation for lattice theories is written, which describes a Markov process on a space lattice evolving in (stochastic) time. At the cost of requiring the construction of the drift function this reduces one dimension in numerical simulations, as compared to Monte Carlo methods. The drift can be obtained either from the asymptotic solution of an auxiliary equation or from a ground state ansatz. It is shown that for Abelian theories a drift can be constructed from ground state ansatze which are exact eigenstates of Hamiltonians with the same continuum limit as the Kogut-Susskind Hamiltonian. Lattice observables may be obtained from stochastic time correlations. In addition, a new method is obtained to measure the lowest excited state (mass gap) from the exit times of the stochastic process from a bounded region. In some cases the mass gap may be obtained at weak coupling from the theory of small random perturbations of dynamical systems.