Pure Yang-Mills, noncommutative Chern-Simons and noncommutative quantum mechanics: A Hamiltonian approach

This thesis addresses three topics: calculation of the invariant measure for the pure Yang-Mills configuration space in (3 + 1) dimensions, Hamiltonian analysis of the pure Chern-Simons theory on the noncommutative plane and noncommutative quantum mechanics in the presence of singular potentials.; In Chapter 1 we consider a gauge-invariant Hamiltonian analysis for Yang-Mills theories in three spatial dimensions. The gauge potentials are parameterized in terms of a matrix variable which facilitates the elimination of the gauge degrees of freedom. We develop an approximate calculation of the volume element on the gauge-invariant configuration space. We also make a rough estimate of the ratio of 0++ glueball mass and the square root of string tension by comparison with (2 + 1)-dimensional Yang-Mills theory.; In Chapter 2 the Hamiltonian analysis of the pure Chern-Simons theory on the noncommutative plane is performed. We use the techniques of geometric quantization to show that the classical reduced phase space of the theory has nontrivial topology and that quantization of the symplectic structure on this space is possible only if the Chern-Simons coefficient is quantized. Also we show that the physical Hilbert space of the theory is one-dimensional and give an explicit expression for the vacuum wavefunction. This vacuum state is found to be related to the noncommutative Wess-Zumino-Witten action.; And finally in Chapter 3 we address the question of two-dimensional quantum mechanics in the presence of delta-function potentials which is known to be plagued by UV divergences resulting from the singular nature of the potentials in question. The two particularly interesting examples of this kind are non-relativistic spin zero particles in delta-function potential and Dirac particles in Aharonov-Bohm magnetic background. We show that by extending the corresponding Schrodinger and Dirac equations onto the flat noncommutative space a well-defined quantum theory can be obtained. Complete analytic solution is found in both cases. In the limit of vanishing noncommutativity we recover the standard expressions corresponding to certain self-adjoint extensions of the Hamiltonians in question.