QSD III : Quantum Constraint Algebra and Physical Scalar Product in Quantum General Relativity

TL;DR

This paper advances non-perturbative quantum gravity by clarifying the constraint algebra, fixing the inner product for diffeomorphism-invariant states, and proposing a natural physical scalar product for quantum general relativity.

Contribution

It introduces a new physical scalar product and clarifies the constraint algebra in canonical quantum gravity, addressing issues with the Wheeler-DeWitt operator.

Findings

Constraint algebra is non-anomalous and faithfully implemented.

Inner product for diffeomorphism-invariant states is fixed via group averaging.

A natural physical scalar product for quantum gravity is proposed.

Abstract

This paper deals with several technical issues of non-perturbative four-dimensional Lorentzian canonical quantum gravity in the continuum that arose in connection with the recently constructed Wheeler-DeWitt quantum constraint operator. 1) The Wheeler-DeWitt constraint mixes the previously discussed diffeomorphism superselection sectors which thus become spurious, 2) Thus, the inner product for diffeomorphism invariant states can be fixed by requiring that diffeomorphism group averaging is a partial isometry, 3) The established non-anomalous constraint algebra is clarified by computing commutators of duals of constraint operators, 4) The full classical constraint algebra is faithfully implemented on the diffeomorphism invariant Hilbert space in an appropriate sense, 5) The Hilbert space of diffeomorphism invariant states can be made separable if a natural new superselection principle is satisfied, 6) We propose a natural physical scalar product for quantum general relativity by extending the group average approach to the case of non-self-adjoint constraint operators like the Wheeler-DeWitt constraint and 7) Equipped with this inner product, the construction of physical observables is straightforward.