Functional integrals and canonical quantum gravity

The functional-integral sum-over-histories formulation of quantum gravity in the canonical Arnowitt-Deser-Misner formalism is examined. Reduced phase-space quantization (RPSQ) is contrasted with Dirac quantization (DQ). While it does not appear that RPSQ is even defined for gravity, there do exist minisuperspace models in which different identifications of time correspond to inequivalent RSPQ’s none of which appears to correspond to DQ. While the Batalin-Fradkin-Vilkovisky Becchi-Rouet-Stora-Tyutin-invariant functional integral provides a representation for the propagators of DQ, its (assumed) independence of the choice of gauge function has been used (incorrectly) to demonstrate that the canonical limit corresponds to a functional integral in RPSQ. There is clearly something amiss. To identify the source of the inconsistency we use a minisuperspace model. In this way, we can demonstrate explicitly that the apparent coincidence is due, in part, to a mistaken counting of factors appearing in the respective measures. We also reexamine the Batalin-Vilkovisky theorem, demonstrating how it might break down when the two gauges compared are not infinitesimally related.