Green's Bracket Algebras and Their Quantization

The Peierls algebra is equivalent to, though defined differently than, the Poisson algebra for a system without constraints or invariances. When invariances are present, the Peierls bracket is defined only on invariants, but we consider here the "Green's algebra" extensions to variant quantities. Because many such extensions can be defined, we find a general context for studying algebras in classical systems with invariances. The structure of each extension resembles the Poisson algebra used in Dirac's quantization prescription for systems with invariances and can be used in a similar manner to construct a quantum theory. As with Dirac's algebra, there may be equations of motion that are not respected by the Green's algebra. We refer to such equations as "constraints." The number of constraints depends on the extension. For any system, there is some extension that has exactly one constraint for each invariance. However, none have more constraints and usually some have fewer. The set of constraints that do exist is always first class. The complication of second class constraints is avoided. We then examine special properties of certain Green' s algebras. Dirac's Poisson algebra and gauge fixed algebras are found to be special cases of the general method. We also consider for which Green's algebras we can avoid the concept of a physical Hilbert space as a subspace of a larger space. We then provide some results concerning the equivalence of quantum theories built from different such extensions for some time reparameterization invariant systems. In a certain sense, Dirac quantization of systems with time reparameterization invariance is found not to be equivalent to reduction quantization. We also arrive at an unusual proposal for the quantization of some time reparameterization invariant systems.