The theta parameter in loop quantum gravity: effects on quantum geometry and black hole entropy

TL;DR

This work shows that in loop quantum gravity a real θ parameter, arising from large SU(2) gauge transformations, yields inequivalent quantum theories and renders kinematical area and volume ill-defined for θ ≠ 0, while leaving gauge-invariant observables unaffected at the level of leading semiclassical behavior. By incorporating a θ-dependent Chern-Simons extension, the authors demonstrate that the isolated horizon boundary conditions still give a discrete horizon area spectrum, and hence a finite entropy that scales with the macroscopic area with a leading term independent of θ. The results emphasize that physical content resides in Dirac observables, with BH entropy providing a nontrivial test of universality in θ-augmented LQG. Overall, the θ parameter acts as a parity-violating quantization ambiguity that does not spoil the qualitative BH entropy result but reshapes the underlying kinematical geometry and its domain.

Abstract

The precise analog of the theta-quantization ambiguity of Yang-Mills theory exists for the real SU(2) connection formulation of general relativity. As in the former case theta labels representations of large gauge transformations, which are super-selection sectors in loop quantum gravity. We show that unless theta=0, the (kinematical) geometric operators such as area and volume are not well defined on spin network states. More precisely the intersection of their domain with the dense set Cyl in the kinematical Hilbert space H of loop quantum gravity is empty. The absence of a well defined notion of area operator acting on spin network states seems at first in conflict with the expected finite black hole entropy. However, we show that the black hole (isolated) horizon area--which in contrast to kinematical area is a (Dirac) physical observable--is indeed well defined, and quantized so that the black hole entropy is proportional to the area. The effect of theta is negligible in the semiclassical limit where proportionality to area holds.