Dirac Bracket Transformations in Phase Space

Abstract

The purpose of this paper is twofold. One is to analyze the group-theoretical significance of the Dirac bracket and to examine, in particular, the apparent ambiguities in the presence of both first-class and second-class constraints. The other is to prepare the ground for the utilization of the Dirac bracket for the quantization of generally covariant theories. It is shown that the Dirac bracket represents the commutator of infinitesimal transformations in phase space which are not canonical but form a group, in that they are the only transformations that preserve the form of all the constraints of a theory as well as the canonical form of the equations of motion. This group of transformations possesses an invariant subgroup: those transformations that correspond to coordinate transformations, gauge transformations and the like.