Canonical Quantization of Gravitational Field

The six dynamical variables g₌₍ (x) are expressed in terms of three "coordinatelike" variables y^A (x) and three invariant variables ^ (k). The longitudinal constraints {R₍}^0+{T₍}^0=0 imply the vanishing of the momenta canonically conjugate to the y^A (x), and the only remaining constraint can be written invariantly in terms of the ^ (k), their conjugate momenta, and the matter variables. The quantization of this fourth constraint then leads to a Schwinger-Tomonaga equation i=H, whose solution yields a complete set of commuting observables for the gravitational field. For most applications, it is possible to select the initial hypersurface so as to get the much simpler equation i=H, where the dynamical variable T (which is a known functional of the metric and the matter variables at t=0) plays the role of an "invariant time. "