Abstract:
Matrix models, and their associated integrals, are encoded with a rich structure,
especially when studied in the large N limit. In our project we study the dynamics
of a Gaussian ensemble of m complex matrices or 2m hermitian matrices for d = 0
and d = 1 systems.
We rst investigate the two hermitian matrix model parameterized in \matrix
valued polar coordinates", and study the integral and the quantum mechanics of
this system. In the Hamiltonian picture, the full Laplacian is derived, and in the
process, the radial part of the Jacobian is identi ed. Loop variables which depend
only on the eigenvalues of the radial matrix turn out to form a closed subsector
of the theory. Using collective eld theory methods and a density description,
this Jacobian is independently veri ed.
For potentials that depend only on the eigenvalues of the radial matrix, the
system is shown to be equivalent to a system of non-interacting (2+1)-dimensional
\radial fermions" in a harmonic potential.
The matrix integral of the single complex matrix system, (d = 0 system), is
studied in the large N semi-classical approximation. The solutions of the stationary
condition are investigated on the complex plane, and the eigenvalue density
function is obtained for both the single and symmetrically extended intervals of
the complex plane.
The single complex matrix model is then generalized to a Gaussian ensemble
of m complex matrices or 2m hermitian matrices. Similarly, for this generalized
ensemble of matrices, we study both the integral of the system and the Hamiltonian
of the system.
A closed sector of the system is again identi ed consisting of loop variables
that only depend on the eigenvalues of a matrix that has a natural interpretation
as that of a radial matrix. This closed subsector possess an enhanced U(N)m+1
symmetry. Using the Schwinger-Dyson equations which close on this radial sector
we derive the Jacobian of the change of variables to this radial sector.
The integral of the system of m complex matrices is evaluated in the large
N semi-classical approximation in a density description, where we observe the
emergence of a new logarithmic term when m 2. The solutions of the stationary
condition of the system are investigated on the complex plane, and the eigenvalue
density functions for m 2 are obtained in the large N limit.
The \fermionic description" of the Gaussian ensemble of m complex matrices
in radially invariant potentials is developed resulting in a sum of non-interacting
Hamiltonians in (2m + 1)-dimensions with an induced singular term, that acts
on radially anti-symmetric wavefunctions.
In the last chapter of our work, the Hamiltonian of the system of m complex
matrices is formulated in the collective eld theory formalism. In this density description
we will study the large N background and obtain the eigenvalue density
function.
Description:
A thesis submitted to the Faculty of Science,
University of the Witwatersrand, Johannesburg,
in ful lment of the requirements for the degree of
Doctor of Philosophy.
Johannesburg, 2014