Quantum Gravity

THE PROBLEM OF TIME 141The main problem with this approach towards the issue of time in quantumgravity is perhaps its closeness to a classical space–time picture. From variousequations, such as (5.14), one gets the illusion that a space–time exists evenin the quantum theory, although this cannot be the case, see (5.3). One can,therefore, conclude that attempting to identify time before quantization doesnot solve the problem of time in the general case, although it might help inspecial cases (Section 7.4).5.2.2 Time after quantizationUsing the commutation rules (5.3) and their formal implementation (5.4) and(5.5) directly, one arrives at wave functionals Ψ[h ab (x)] defined on Riem Σ, thespace of all three-metrics. This is the central kinematical quantity. The ‘dynamics’must be implemented through the quantization of the constraints (4.69)and (4.70)—this is all that remains in the quantum theory. One then gets thefollowing equations for the wave functional:(Ĥ g ⊥ Ψ ≡ −16πG 2 δ 2G abcd −δh ab δh cd√ )h16πG ( (3) R − 2Λ) Ψ=0, (5.18)ĤaΨ g δΨ≡−2D b h ac =0. (5.19)i δh bcEquation (5.18) is called the Wheeler–DeWitt equation 6 in honour of the workby DeWitt (1967a) and Wheeler (1968). In fact, these are again infinitely manyequations. The constraints (5.19) are called the quantum diffeomorphism (ormomentum) constraints. Occasionally, both (5.18) and (5.19) are referred to asWheeler–DeWitt equations. In the presence of non-gravitational fields, theseequations are augmented by the corresponding terms.There are many problems associated with these equations. An obvious problemis the ‘factor-ordering problem’: the precise form of the kinetic term isopen—there could be additional terms proportional to containing at mostfirst derivatives in the metric. Since second functional derivatives at the samespace point usually lead to undefined expressions such as δ(0), a regularization(and perhaps renormalization) scheme has to be employed. Connected with thisis the potential presence of anomalies. The general discussion of these problemsis continued in Section 5.3. Here we shall address again the problem of time andthe related Hilbert-space problem. Since (5.18) does not have the structure ofa local Schrödinger equation (5.14), the choice of Hilbert space is not clear apriori.The first option for an appropriate Hilbert space is related to the use of aSchrödinger-type inner product, that is, the standard quantum-mechanical innerproduct as generalized to quantum field theory,∫〈Ψ 1 |Ψ 2 〉 = Dµ[h] Ψ ∗ 1 [h]Ψ 2[h] , (5.20)Riem Σ6 In earlier years the name ‘Einstein–Schrödinger equation’ was used.