Some Aspects of the Quantization of Theories with a Gauge Invariance

We discuss some problems that arise when one tries to quantize a theory that possesses gauge degrees of freedom. First, we identify the Gribov problem that is encountered when the Faddeev-Popov procedure of fixing the gauge is employed to define a perturbation expansion. We propose a modification of the procedure that takes this problem into account. We then apply this method to two-dimensional gauge theories where the exact answer is known. Second, we try to build chiral theories that are consistent in the presence of anomalies, without making use of additional degrees of freedom. We are able to solve the model exactly in two dimensions, arriving at a gauge-invariant theory. We discuss the four-dimensional case and also the application of this method to string theory. In the latter, we obtain a model that lives in arbitrary dimensions. However, we do not compute the spectrum of the model. Third, we investigate the possibility of compactifying the unwanted dimensions of superstrings on a group manifold. We give a complete list of conformally invariant models. We also discuss one-loop modular invariance. We consider both type-II and heterotic superstring theories. Fourth, we discuss quantization of string field theory. We start by presenting the lagrangian approach, to demonstrate the non-uniqueness of the measure in the path-integral. It is fixed by demanding unitarity, which manifests itself in the hamiltonian formulation, studied next.