Analytic methods from finite to infinite volumes

Abstract

We shall review the analytic methods that have been developed over the past few years to understand the volume dependence of field theories on a torus, which is the geometry relevant for lattice gauge theories. First we start in small to intermediate volumes, below one cubic fermi, and discuss the results for the low-lying spectrum of pure SU(3) gauge theory. Then we assume a mass gap is formed in the infinite volume and show how this leads to a precise asymptotic expression for the volume dependence of the stable one-particle masses, due to polarization effects and for the two-particle masses, due to scattering processes in the finite volume. The latter is first illustrated in the two dimensional O(N) model, using the exact S-matrix. The generalization to four (or any) dimensions for the relation between the two-particle masses in a finite volume and the scattering phase shifts, based on a very recent analysis by Lüscher, will be discussed. In the near future this will make a careful study of resonances in a finite volume feasible. Finally we discuss how the Bethe Ansatz for the two dimensional O(N) model can be used to calculate the infinite volume mass gap, providing an important test for lattice Monte Carlo studies to extract this quantity.