Quantization of Reparameterization Invariant Models.

The technique of BRST quantization is applied to a selection of models based on classically constrained Hamiltonian systems. Unlike the case of Quantum Chromodynamics, where the local symmetry is an internal one, the underlying symmetry group of reparameterization invariant systems gives rise to constraints, that only act as generators of the gauge group away from the initial and final surfaces. Consequently the standard techniques, developed for Quantum Chromodynamics, need to be generalized when studying such systems as the Bosonic string. By first studying a constrained Hamiltonian system possessing a local symmetry corresponding to any Lie group, it is possible to establish precisely what roles the various ghost fields should play when using BRST quantizing the system. It is found that when the anti-commuting ghosts are integrated out of the path integral, representing the matrix elements of the theory, one naturally obtains a group invariant measure, and a set of physical state projection operators which automatically project out the non-physical degrees of freedom. This interpretation is then applied to the case of the free-Bosonic string, however, due to the non-compactness of the conformal group in two dimensions, it is not possible to completely evaluate the BRST invariant matrix elements between the initial and final string states. Instead, these matrix elements are studied in the 'unitary limit' where it is shown that they correspond to the results normally obtained in the light-cone gauge. Due to the rather elaborate constraint algebra of the covariant superstring of Green and Schwarz, a full understanding of the quantized system is presently lacking. By first considering the much simpler example of a free superparticle, it is shown how one can construct a new set of classical variables, relative to which the system is seen to be in canonical form.