On the consistency of the constraint algebra in spin network quantum gravity

TL;DR

The paper investigates the consistency of Thiemann's Euclidean gravity Hamiltonian constraints by examining the quantum realization of the right-hand side ${\cal O}(N,M)$ and the constraint algebra. It demonstrates that, on the enlarged state space ${\cal T}'_*$, the standard quantization of ${\cal O}(N,M)$ vanishes, effectively rendering ${\hat q}^{ab}$ as the zero operator on many states, though regularization choices can yield nonzero ${\hat q}^{ab}$ with the commutator remaining vanishing or becoming anomalous. Through schematic computations on specific spin-network states, it shows that typical regularizations fail to reproduce the classical hypersurface-deformation algebra because the Hamiltonians do not move the original vertex, eliminating the necessary derivative terms. Overall, the work highlights a tension between achieving an anomaly-free quantum constraint algebra and faithfully reproducing the classical algebra, suggesting that new ideas are needed to reconcile vertex movement or alternative regularizations with the classical structure.

Abstract

We point out several features of the quantum Hamiltonian constraints recently introduced by Thiemann for Euclidean gravity. In particular we discuss the issue of the constraint algebra and of the quantum realization of the object $q^{ab}V_b$, which is classically the Poisson Bracket of two Hamiltonians.

Figures (2)

• Figure 1: The type of spin network state on which we act with the commutator of two Hamiltonians. Our notation is that $j_1,j_1+1/2, J_1+1,J_2,J_3,1,1/2$ are spins that label edges while $q,v,w,x,y,z$ are vertices of the graph.
• Figure 2: The type of state that results from the action of two Hamiltonians on a state of the form shown in figure 1. As we see, the action of the Hamiltonians is to "eliminate" lines from the vertices of the graph.