Quantum Gravity

THE PROBLEM OF TIME 143∫ ∏〈Ψ 1 |Ψ 2 〉 =ix(dΣ ab (x)Ψ ∗ 1[h ab ] ·G abcd−→δδh cd−←− )δG abcd Ψ 2 [h ab ]=〈Ψ 2 |Ψ 1 〉 ∗ . (5.22)δh cdHere, the integration is over a 5 ×∞ 3 -dimensional surface in the 6 ×∞ 3 -dimensional space Riem Σ, and dΣ ab denotes the corresponding surface element.In view of (5.21), the integration can be taken over the variables ˜h ab , referringto ‘constant time √ h=constant’. Of course, the lack of mathematical rigour isthe same as with (5.20).The inner product (5.22) has the advantage that it is invariant under deformationsof the 5 ×∞ 3 -dimensional surface. This expresses its ‘time independence’.However, this inner product is—like the usual inner product for theKlein–Gordon equation—not positive definite. In particular, one has 〈Ψ|Ψ〉 =0for real solutions of (5.18). Since the Wheeler–DeWitt equation is a real equation(unlike the Schrödinger equation), real solutions should possess some significance.For the standard Klein–Gordon equation in Minkowski space, one can makea separation between ‘positive’ frequencies and ‘negative’ frequencies. As long asone can stay within the one-particle picture, it is consistent to make a restrictionto the positive-frequency sector. For such solutions, the inner product is positive.On curved backgrounds, a separation into positive and negative frequencies canbe made if both the space–time metric and the potential are stationary, that is,if there is a time-like Killing field and if the potential is constant along its orbits.The Killing field can also be a conformal Killing field, but then the potentialmust obey certain scaling properties. Moreover, the potential must be positive. Ifthese conditions are violated, particles are produced and the one-particle picturebreaks down.Can such a separation into positive and negative frequencies be made forthe Wheeler–DeWitt equation? The clear answer is no (Kuchař 1992). Thereexists a conformal Killing field for the DeWitt metric, namely the three-metrich ab . The potential is, however, neither positive definite nor scales in the correctway. Therefore, no Klein–Gordon inner product can be constructed which ispositive definite for the generic case (although this might be achievable for specialmodels). For the standard Klein–Gordon equation, the failure of the one-particlepicture leads to ‘second quantization’ and quantum field theory. The Wheeler–DeWitt equation, however, corresponds already to a field-theoretic situation.It has, therefore, been suggested to proceed with a ‘third quantization’ and toturn the wave function Ψ[h] into an operator (see Kuchař 1992 for review andreferences). No final progress, however, has been achieved with such attempts.One might wonder whether the failure of the above attempts is an indicationof the absence of time at the most fundamental level. As will be discussed inSection 5.4, the usual concept of time emerges as an approximate notion on asemiclassical level. This is, in fact, all that is needed to have accordance with ex-