Dirac Bracket Transformations in Phase Space

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.

From this subgroup, we can construct the factor group. All members of the original group have generators, but the generators of the invariant subgroup are zero. There is, then, a one-to-one correspondence between the nonvanishing generators and the members of the factor group. The Dirac bracket is uniquely defined for all dynamical variables that can serve as generators; excluded are variables that have no invariant significance (such as the divergence of the electromagnetic vector potential). If it is possible to identify all these permissible generators, then the theory can be reformulated in terms of these and will be free of constraints. It is proposed to adopt the set of generators and the Dirac brackets between them as the point of departure for the formulation of covariant quantum theories.