Quantum Gravity

THE GEOMETRODYNAMICAL WAVE FUNCTION 151Gaussian functionals are used frequently in quantum field theory with externalfields. Examples are an external electric field in QED or an externalDe Sitter-space background in a gravitational context (Jackiw 1995). The functionalSchrödinger picture can also be formulated for fermions (see e.g. Kieferand Wipf 1994; Jackiw 1995; Barvinsky et al. 1999b). In the context of linearizedgravity, we have already encountered the Gaussian functional describingthe graviton ground state; see the end of Section 2.1. Many discussions of the geometrodynamicalwave functional take their inspiration from the above discussedproperties of the Schrödinger picture.5.3.4 Connection with path integralsWe have discussed in Section 2.2 the formulation of a quantum-gravitationalpath integral. In quantum mechanics, the path integral can be shown to satisfythe Schrödinger equation (Feynman and Hibbs 1965). It is, therefore, ofinterest to see if a similar property holds in quantum gravity, that is, if thequantum-gravitational path integral (2.71) obeys the quantum constraints (5.18)and (5.19). This is not straightforward since there are two major differences to ordinaryquantum theory: first, one has constraints instead of the usual Schrödingerequation. Second, the path integral (2.71) contains an integration over the wholefour-metric, that is, including ‘time’ (in the form of the lapse function). Sincethe ordinary path integral in quantum mechanics is a propagator, denoted by〈q ′′ ,T|q ′ , 0〉, the quantum-gravitational path integral corresponds to an expressionof the form ∫dT 〈q ′′ ,T|q ′ , 0〉 ≡G(q ′′ ,q ′ ; E)| E=0 ,where the ‘energy Green function’∫G(q ′′ ,q ′ ; E) =dT e iET 〈q ′′ ,T|q ′ , 0〉 (5.49)has been introduced. The quantum-gravitational path integral thus resembles anenergy Green function instead of a propagator, and due to the T -integration nocomposition law holds in the ordinary sense (Kiefer 1991, 2001a). All this is, ofcourse, true already for the models with reparametrization invariance discussedin Section 3.1. In general, an integration over T yields a divergence. One thereforehas to choose appropriate contours in a complex T -plane in order to get a sensibleresult.A formal derivation of the constraints from (2.71) is straightforward (Hartleand Hawking 1983). Taking a matter field φ into account, the path integral reads∫Z = DgDφ e iS[g,φ] , (5.50)where the integration over Dg includes an integration over the three-metric aswell as lapse function N and shift vector N a . From the demand that Z beindependent of N and N a at the three-dimensional boundaries, one gets