Matrix polar coordinates
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Date
2011-03-28
Authors
Masuku, Mthokozisi
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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.