Description
Matrix models are the simplest examples of quantum gauge theories, the type of quantum field theories which describe the electromagnetic, weak and strong nuclear forces: they are quantum gauge theories in zero spacetime dimensions. The basic variables are matrices, for instance an N by N hermitian matrix. The entries of these matrices can take arbitrary values due to quantum fluctuations, but certain values are more likely than others, and any two values which are related by conjugation by a unitary matrix (a "gauge transformation") are equally likely to occur and are considered to be equivalent. So the quantities of interest in a matrix model are given by integrals over spaces of matrices, with an integration measure which respects the symmetry under gauge transformations.
Physical observables in quantum gauge theories in more spacetime dimensions are also given by integrals, but these are hard-to-compute and mathematically ill-defined integrals over infinite-dimensional spaces of field configurations. In contrast, the integrals of interest in matrix models are well-defined ordinary finite-dimensional integrals of the type that you are familar with from high school, with the only complication that there are multiple integration variables (the entries of the matrix). As the size of the matrix increases, you might be excused to think that the integrals get harder and harder, but it turns out instead that for many matrix models these integrals can be computed exactly, for matrices of arbitrary size. The solution can be obtained by a variety of methods, all of which cleverly bypass brute force integration.
Despite being very simple to formulate, matrix models have found many uses in theoretical physics (nuclear physics, gauge theory, condensed matter physics, quantum gravity) and mathematics (combinatorics, number theory, algebraic geometry). A lot can be understood by studying integrals over matrices! The aim of this project will be to investigate some of the methods of solution of matrix models (large N expansion, orthogonal polynomials, loop equations) and explore some of their physical and mathematical applications. There is vast literature on the subject and the applications are many, so the project can be approached at different levels and there is ample scope for specialization.
Prerequisites
2H Mathematical Physics
3H Quantum Mechanics.
No prior knowledge of quantum field theory is needed, but familiarity with basic linear algebra will be assumed.
Resources
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Reading material:
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