Quantization of Simple Parametrized Systems
Abstract
I study the canonical formulation and quantization of some simple parametrized systems using Dirac's formalism and the BecchiRouetStoraTyutin (BRST) extended phase space method. These systems include the parametrized particle and minisuperspace. Using Dirac's formalism I first analyze for each case the construction of the classical reduced phase space. There are two separate features of these systems that may make this construction difficult: (a) Because of the boundary conditions used, the actions are not gauge invariant at the boundaries. (b) The constraints may have a disconnected solution space. The relativistic particle and minisuperspace have such complicated constraints, while the nonrelativistic particle displays only the first feature. I first show that a change of gauge fixing is equivalent to a canonical transformation in the reduced phase space, thus resolving the problems associated with the first feature above. Then I consider the quantization of these systems using several approaches: Dirac's method, DiracFock quantization, and the BRST formalism. In the cases of the relativistic particle and minisuperspace I consider first the quantization of one branch of the constraint at the time and then discuss the backgrounds in which it is possible to quantize simultaneously both branches. I motivate and define the inner product, and obtain, for example, the KleinGordon inner product for the relativistic case. Then I show how to construct phase space path integral representations for amplitudes in these approachesthe BatalinFradkinVilkovisky (BFV) and the Faddeev path integrals from which one can then derive the path integrals in coordinate spacethe FaddeevPopov path integral and the geometric path integral. In particular I establish the connection between the Hilbert space representation and the range of the lapse in the path integrals. I also examine the class of paths that contribute in the path integrals and how they affect spacetime covariance, concluding that it is consistent to take paths that move forward in time only when there is no electric field. The key elements in this analysis are the spacelike paths and the behavior of the action under the nontrivial ( Z_2) element of the reparametrization group.
 Publication:

Ph.D. Thesis
 Pub Date:
 January 1995
 Bibcode:
 1995PhDT........40R
 Keywords:

 DIRAC'S FORMALISM;
 BRST FORMALISM;
 PATH INTEGRALS;
 PHASE SPACE;
 Physics: Elementary Particles and High Energy; Mathematics