Weaving a classical metric with quantum threads

Because level spacing Δa_S goes to zero as a_S→∞, considerations of Bekenstein and Mukhanov [15] do not apply to large black holes in this approach and there is no reason to expect deviations from Hawking's semi-classical results.

“Because the 'level spacing' ∆ a S goes to zero as a S goes to infinity, the considerations of [15] do not apply to large black holes in our approach and there is no reason to expect deviations from Hawking's semi-classical results.”

Local quantities tend to be divergent or ill-defined in non-trivial generally covariant quantum field theories.

“Local quantities tend to be divergent or ill-defined in non-trivial generally covariant quantum field theories [3, 4, 11], a fact likely to be related to the non-renormalizability of the perturbative series.”

The volume operator construction corrects certain difficulties of a previous definition [4, 11].

“This construction corrects certain difficulties of a previous definition [4, 11].”

The volume operator is finite and diffeomorphism invariant.

“Furthermore, as a consequence, for reasons that have been discussed previously[3, 4, 11] the operator is finite and diffeomorphism invariant.”

Any spinfoam model admits a group field theory formulation, which is triangulation independent.

“It was shown in [16] that any spinfoam model [17], as the one we define here, admits a group field theory (GFT) [18] formulation, which is triangulation independent.”

Birefringence was also considered in the context of modifications of electromagnetism and nonsymmetric gravity [9], but in those cases the effect was not frequency dependent.

“Birefringence was also considered in the context of modifications of electromagnetism and also in nonsymmetric gravity [9]. In those cases the effect was not frequency dependent.”

The kinematics of loop quantum gravity is rather well understood, while its dynamics is not understood as cleanly.

“While the kinematics of loop quantum gravity (LQG) [1] is rather well understood [2, 3], its dynamics is not understood as cleanly.”

The ultraviolet continuum limit is trivial in the quantum theory.

“The ultraviolet continuum limit is trivial in the quantum theory (for a discussion of this point, see [3]).”

The new sign factor takes care of the sign complications discussed in reference [19].

“The new sign takes care immediately of the sign complications extensively discussed in reference [19].”

The new sign choice implements the sign factor for the number of loops recognized in [19] as key to transform the spinor relation into a local relation.

“This shows that the new sign choice implements in the formalism the sign factor for the number of loops that was recognized in [19] as the key to transform the spinor relation into a local relation.”

To keep track of the overall sign, one needs oriented trivalent graphs.

“To keep track of this overall sign, one needs oriented trivalent graphs, as in reference [43] where Penrose considered oriented spin networks and in [19].”