Reduced Phase Space Quantization and Dirac Observables
Thomas Thiemann
TL;DR
Dittrich's partial observables framework is extended to general first-class systems with structure functions, enabling a reduced phase space quantization that avoids explicit gauge orbit computations. By selecting clocks that render the Dirac observable algebra canonical, the paper outlines a path to unitary multi-fingered time evolution and sketches concrete applications to General Relativity coupled to matter, including using Higgs-derived clocks. It also contrasts reduced quantization with Dirac (constraint) quantization and explores how Master Constraint techniques can yield spatially diffeomorphism-invariant Hamiltonians on a suitable reduced space. Together, these ideas offer a principled, albeit technically challenging, route to constructing and quantizing Dirac observables in gravity and gauge theories, with potential implications for LQG and alternative quantization schemes.
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.