Quantum Gravity

THE SEMICLASSICAL APPROXIMATION 165called a ‘Born–Oppenheimer’ type of approximation scheme. It is successfullyapplied in molecular physics, where the division is into the heavy nuclei (movingslowly) and the light electrons (following adiabatically the motion of the nuclei).Many molecular spectra can be explained in this way. In quantum gravity, the‘heavy’ part is often taken to be the full gravitational field (motivated by thelarge value of the Planck mass), while the ‘light’ part are all non-gravitationaldegrees of freedom (see e.g. Kiefer 1994). This has the formal advantage thatan expansion with respect to the Planck mass can be performed. It is, however,fully consistent to consider part of the gravitational degrees of freedom as fullyquantum and therefore include them in the ‘light’ part (see e.g. Halliwell andHawking 1985; Vilenkin 1989). Physically, these degrees of freedom are gravitonsand quantum density fluctuations. It depends on the actual situation one isinterested in whether this ‘light’ gravitational part has to be taken into accountor not.For full quantum geometrodynamics, this semiclassical expansion exists onlyon a formal level. Therefore, it will be appropriate to discuss quantum-mechanicalanalogies in this subsection. Albeit formal, the expansion scheme is of the utmostconceptual importance. As we shall see, it enables one to recover the usual timeas an approximate concept from ‘timeless’ quantum gravity (Section 5.4.2). Thisis the relevant approach for observers within the Universe (‘intrinsic viewpont’).Moreover, the scheme allows to go to higher orders and calculate quantumgravitationalcorrection terms to the functional Schrödinger equation, whichcould have observational significance (Section 5.4.3).Let us now consider in some detail a simple quantum-mechanical model (seee.g. Kiefer 1994; Bertoni et al. 1996; Briggs and Rost 2001). The total systemconsists of the ‘heavy part’ described by the variable Q, while the ‘light-part’variable is called q. 17 It is assumed that the full system is described by a stationarySchrödinger equation,HΨ(q, Q) =EΨ(q, Q) , (5.122)with the Hamilton operator to be of the formH = − 2 ∂ 2+ V (Q)+h(q, Q) , (5.123)2M ∂Q2 where h(q, Q) contains the pure q-part and the interaction between q and Q. Inthe case of the Wheeler–DeWitt equation, the total energy is zero, E =0.Onenow makes an expansion of the formΨ(q, Q) = ∑ nχ n (Q)ψ n (q, Q) (5.124)and assumes that 〈ψ n |ψ m 〉 = δ nm for each value of Q. The inner product hereis the ordinary scalar product with respect to q only (only this part will be17 For simplicity, the total system is considered to be two dimensional. The extension to moredimensions is straightforward.