3. Electronic Theses and Dissertations (ETDs) - All submissions
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Item Radial dynamics of the large N limit of multimatrix models(2016-01-22) Masuku, MthokozisiMatrix 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.Item Matrix polar coordinates(2011-03-28) Masuku, MthokozisiMatrix 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.