Derivation of the Wheeler-DeWitt equation from a path integral for minisuperspace models

Abstract

We explore the relationship between path-integral and Dirac quantization for a simple class of reparametrization-invariant theories. The main object is to study minisuperspace models in quantum cosmology---models for quantum gravity in which one restricts attention to a finite number of degrees of freedom. Our starting point for the construction of the (Lorentzian) path integral is the very general and powerful method introduced by Batalin, Fradkin, and Vilkovisky. Particular attention is paid to the measure in the large, i.e., to the range of integration of the Lagrange multiplier. We show how to derive the Wheeler-DeWitt equation from our path-integral expression. The relationship between the choice of measure in the path integral and the operator ordering in the Wheeler-DeWitt equation is thus determined. The operator-ordering ambiguity in the Wheeler-DeWitt equation is completely fixed by demanding invariance under field redefinitions of both the three-metric and the lapse function. Our results are applied to two simple examples: the nonrelativistic point particle in parametrized form and the relativistic point particle. We also consider a simple minisuperspace example and discuss a difficulty that arises: namely, the problem of incorporating the fact that det${h}_{\mathrm{ij}}$>0 into the quantization procedure.