QSD III : Quantum Constraint Algebra and Physical Scalar Product in Quantum General Relativity
Thomas Thiemann
TL;DR
This work advances non-perturbative canonical quantum gravity by resolving key obstacles in implementing the Wheeler-DeWitt constraint in a diffeomorphism-invariant continuum setting. It shows that diffeomorphism superselection sectors are not physically distinct once the Hamiltonian constraint is included, and it derives a unique, diffeomorphism-invariant inner product via an adapted group-averaging procedure. The dual constraint algebra is shown to close anomaly-free on the diffeomorphism-invariant space, and a natural physical inner product is proposed for the space of solutions to all constraints, enabling a systematic construction of Dirac observables. The introduction of Theta moduli and the Theta-Superselection Assumption offers a path to separability, while the two-oscillator toy model illustrates how non-self-adjoint constraints can be treated within this framework. Overall, the paper provides a coherent mechanism to define physical states and observables in quantum general relativity, with clear implications for the interpretation, separability, and quantization of constrained gravity theories.
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.