Methods and applications of noncommutative geometry

This thesis is devoted to developing techniques for studying noncommutative physics. It can be divided into 3 parts.; In the first we introduce the generalized coherent state approach, which is the main tool in our investigation of the commutative limit of the different noncommutative models. After applying it to some known cases like the noncommutative plane and fuzzy sphere we consider novel noncommutative space, whose commutative limit is the punctured plane. These noncommutative spaces admit a differential structure. An important feature is that the spaces under consideration are the deformation of a commutative space with a boundary (though in this case boundary is just one point).; In the second we provide a detailed example of the construction of the Dirac operator, which is a very important ingredient of noncommutative geometry. We consider the case of the q-deformed sphere, which naturally appears in a specific limit in superstring theory. This space has two interesting limits. One of them is the well known fuzzy sphere. For this case we recover Watamura's Dirac operator. The second limit is the commutative one. In this case we have the Dirac operator for the deformed sphere.; The final part is devoted to the investigation of noncommutative Chern-Simons theory (NCCS) on spaces with a boundary. We address the problem of defining NCCS theory on a noncommutative space with a “boundary.” We show that the definition of the Moyal star should be modified in this case. We further propose some finite dimensional matrix model as a candidate for NCCS on the disk. Using the generalized coherent state approach we show how to recover a disk in the commutative limit at the level of the integration measure. Further we consider NCCS on the noncommutative punctured plane. Using the canonical approach we construct the algebra of the observables. The main result is that this algebra is some deformation of the classical w _∞ algebra. This is a very important result in connection to quantum Hall effect.; We believe that the methods developed in this dissertation will be very useful in recovering commutative limits of different NCCS theories.