Reduced Phase Space Quantization and Dirac Observables

TL;DR

This paper discusses a new approach to reduced phase space quantization for constrained systems using Dirac observables, with applications to General Relativity and potential advantages over traditional methods.

Contribution

It proposes a practical framework for explicit reduced phase space quantization without computing gauge equivalence classes, extending Dittrich's partial observables approach.

Findings

Framework for implementing reduced phase space quantization.

Connection between time evolution and automorphisms of Dirac observables.

Application to quantizing Hamiltonian constraints in General Relativity.

Abstract

In her recent work, Dittrich generalized Rovelli's idea of partial observables to construct Dirac observables for constrained systems to the general case of an arbitrary first class constraint algebra with structure functions rather than structure constants. Here we use this framework and propose a new way for how to implement explicitly a reduced phase space quantization of a given system, at least in principle, without the need to compute the gauge equivalence classes. The degree of practicality of this programme depends on the choice of the partial observables involved. The (multi-fingered) time evolution was shown to correspond to an automorphism on the set of Dirac observables so generated and interesting representations of the latter will be those for which a suitable preferred subgroup is realized unitarily. We sketch how such a programme might look like for General Relativity. We also observe that the ideas by Dittrich can be used in order to generate constraints equivalent to those of the Hamiltonian constraints for General Relativity such that they are spatially diffeomorphism invariant. This has the important consequence that one can now quantize the new Hamiltonian constraints on the partially reduced Hilbert space of spatially diffeomorphism invariant states, just as for the recently proposed Master constraint programme.