Matrix polar coordinates

Abstract

Matrix models feature prominently when studying string theory. In this project we extend well known single matrix model results to two matrix models. The two matrix model is represented using polar coordinates and then used to compute the kinetic piece of the quantum mechanical Hamiltonian operator of two, space indexed, hermitian matrices with a radially invariant potential. As an extension of these matrix polar coordinates, we determine the form(s) of the Laplacian(s) that act on invariant states. The radially dependent Hamiltonian operator is shown to be equivalent to a system of non interacting (2+1) dimensional fermions. Further on, we consider the integral of the two matrix model in polar coordinates to show the standard solution which emulates the Wigner distribution in the free case, when g2Y M = 0. Also in the large N limit we find that the polar coordinate matrix model, when solved using perturbation theory, agrees with the well known result of perturbation theory up to order λ, where λ is the ’t Hooft coupling constant.