On consistent gauge fixing conditions in polymerized gravitational systems

On the fate of quantum black holes

“…The classical gauge-fixing conditions for the dust-time and areal gauges in LTB space-times are known, and it would be interesting to determine their form as operators in LQC; in particular, it will be necessary to express the components of the Ashtekar-Barbero connection in terms of holonomies. Although deriving the specific form of such gauge-fixing conditions for LQC lies beyond the scope of this paper, it can be shown that since the operator for N x is not related to the classical one by a simple 'polymerization' where each b n term is replaced by bn , similarly the LQC gauge-fixing conditions cannot be obtained simply by taking the classical expressions and performing a direct polymerization [105]. This is not surprising for two reasons: first, the LQC shift vector is not given by the direct polymerization of the classical expression, and second, the non-gauge-fixed LQC scalar and diffeomorphism constraints for the LTB space-times are not related to the classical expressions through a direct polymerization, since this would result in a quantum constraint algebra which does not close and hence would give an inconsistent theory.…”