2.12 The constraint algebra

This line of research was continued by Renteln [179], who proved that for a particular factor-ordering (all momenta to the left), the subalgebra of the discretized diffeomorphism constraints (smeared by lapse functions N),
∑ ∑ 𝒱latt[N, n) := N latt(n,ˆa )Tr(U ⊓⊔ τi)pi(n,ˆb), (9 ) n n ab
closes in the limit as the lattice spacing is taken to zero. This calculation was later extended to a variety of different symmetrizations for the lattice operator and to an arbitrary factor-ordering of the form α Tr (ˆUτ )pˆ+ (1 − α )pˆTr (ˆUτ), with 0 ≤ α ≤ 1 [156]. Again, one does not find any quantum anomalies. It would be highly desirable to extend this result to commutators involving also the discretized Hamiltonian constraint and to find the explicit functional form of the anomalies, if there were any.
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