Canonical Quantization of Gravitational Field

The six dynamical variables $g_{\mathrm{mn}}\left(\mathrm{x}\right)$ are expressed in terms of three "coordinatelike" variables $y^A\left(\mathrm{x}\right)$ and three invariant variables ${\gamma }^{\alpha }\left(\mathrm{k}\right)$. The longitudinal constraints ${R_n}^0+{T_n}^0=0\mathrm{}$ imply the vanishing of the momenta canonically conjugate to the $y^A\left(\mathrm{x}\right)$, and the only remaining constraint can be written invariantly in terms of the ${\gamma }^{\alpha }\left(\mathrm{k}\right)$, their conjugate momenta, and the matter variables. The quantization of this fourth constraint then leads to a Schwinger-Tomonaga equation $\frac{i\delta \Psi }{\delta \sigma }=H\left[\sigma \right]\Psi$, 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 $\frac{i\partial \Psi }{\partial T}=H\Psi$, where the dynamical variable $T$ (which is a known functional of the metric and the matter variables at $t=0\mathrm{}$) plays the role of an "invariant time."