Quantum Gravity

234 QUANTIZATION OF BLACK HOLESandB(R, Γ) = 0 . (7.135)The first equation (7.133) is the Hamilton–Jacobi equation. The second equationpresents an additional restriction on solutions of (7.133). The last equation(7.135) tells us that working on the lattice is only possible if the factor orderingdoes not contribute to the potential term. If we find solutions to all three equations,we can do all other calculations on the lattice, since these solutions havea well-defined continuum limit and satisfy the momentum constraint.At this point we have to make a few comments on the regularization procedurewe are using. It was already noted that the lattice regularization does notsolve the factor-ordering problem. The lattice regularization just represents anad hoc regularization in which the divergent terms have to cancel each other.Put differently, it is equivalent to a type of regularization that was employed inDeWitt (1967a) (which means setting δ(0) = 0) with an additional constrainton the solutions.The signature in the kinetic part of the Hamiltonian constraint (7.118) canchange from elliptic (outside the horizon) to hyperbolic (inside the horizon).This thus occurs for the kinetic term of the Wheeler–DeWitt equation (7.121),too. As discussed in Brotz and Kiefer (1997), we can say that the part insidethe horizon is always classically allowed, whereas this is not necessarily the casefor the outside part. The usual initial-value problem appropriate for hyperbolicequations can thus only be applied for the region corresponding to the black-holeinterior.The two equations (7.133) and (7.134) have to be satisfied in order to geta diffeomorphism-invariant solution to the Wheeler–DeWitt equation. (We nowset G =1=.) Looking for particular solutions of the separating form W =α(τ)+β(R), we recognize immediately that this system of equations is consistentonly for special factor orderings. Tackling the problem from the opposite pointof view, one can ask for which factor orderings we do get a separating solution.We findA(R, F )=− F ()12R 2 1+1 − a 2 . (7.136)FThis leads to∫W (τ, Γ, R,a)=const. ± aτ ±dR√1 − a2 FF. (7.137)It turns out that these functions are also solutions to the Hamilton–Jacobi equation.From there one gets the identification 2E =1/a 2 − 1. Since classicallyE ≥−1/2, it follows that a should be real. The integral appearing in (7.137)can be evaluated exactly. One gets