Quantum Gravity

170 QUANTUM GEOMETRODYNAMICSan expansion of the Dirac equation must be employed). A more general case isthe Klein–Gordon equation coupled to gravity and the electromagnetic field. Thisleads to additional relativistic correction terms (Lämmerzahl 1995).A major difference of the Klein–Gordon example to the first example is theindefinite structure of the kinetic term (d’Alembertian instead of Laplacian).Therefore, on the full level, the conserved inner product is the Klein–Gordonone (cf. Section 5.2.2). In order c 0 of the approximation, one obtains from thisinner product the standard Schrödinger inner product as an approximation. Thenext order yields corrections to the Schrödinger inner product proportional toc −2 . Does this mean that unitarity is violated at this order? Not necessarily. Inthe case of the Klein–Gordon equation in external gravitational and electromagneticfields, one can make a (t-dependent!) redefinition of wave functions andHamiltonian to arrive at a conserved Schrödinger inner product with respect towhich the Hamiltonian is Hermitian (Lämmerzahl 1995).5.4.2 Derivation of the Schrödinger equationSimilar to the discussion of the examples in the last subsection one can performa semiclassical (‘Born–Oppenheimer’) approximation for the Wheeler–DeWittequation and the momentum constraints. In this way one can recover approximatelythe limit of ordinary quantum field theory in an external gravitationalbackground. This is done in the Schrödinger picture, so this limit emerges throughthe functional Schrödinger equation, not the quantum-mechanical Schrödingerequation as in the last subsection. In the following, we shall mainly follow, withelaborations, the presentation in Barvinsky and Kiefer (1998); see also Kiefer(1994) and references therein.The starting point is the Wheeler–DeWitt equation (5.18) and the momentumconstraint (5.19). Taking into account non-gravitational degrees of freedom, theseequations can be written in the following form:{− 1 δ 22m 2 G abcd − 2m 2 PPδh ab δh cd{− 2 i h abD c√h (3) R + Ĥm ⊥δ+δh Ĥm abc}|Ψ[h ab ]〉 =0, (5.150)}|Ψ[h ab ]〉 =0. (5.151)Here, m 2 P =(32πG)−1 , =1,Λ=0,andĤm ⊥ and Ĥm a denote the contributionsfrom non-gravitational fields. To be concrete, we think about the presence of ascalar field. The notation |Ψ[h ab ]〉 means: Ψ is a wave functional with respect tothe three-metric h ab and a state in the standard Hilbert space referring to thescalar field (bra- and ket-notation).The situation is now formally similar to the example discussed in the previoussubsection. One of the main differences is the presence of the momentum constraints(5.151), which have no analogue in the quantum-mechanical example.Comparing (5.150) with (5.123), one notes the following formal correspondencebetween the terms: