Kiefer C. Quantum gravity

CANONICAL QUANTIZATION OF THE SCHWARZSCHILD BLACK HOLE 213
7.2.2 Quantization
Quantization then proceeds in the way discussed in Chapter 5 by acting with an
operator version of the constraints on wave functionals Ψ[R(r); τ,λ). Since (7.32)
leads to δΨ/δR = 0, one is left with a purely quantum mechanical wave function
ψ(τ,λ). One could call this a ‘quantum Birkhoff theorem’. The implementation
of the constraints (7.34) and (7.35) then yields
which can be readily solved to give
∂ψ
+ Mψ =0,
i ∂τ
(7.38)
∂ψ
+ qψ =0,
i ∂λ
(7.39)
ψ(τ,λ) =χ(M,q)e −i(Mτ+qλ)/ (7.40)
with an arbitrary function χ(M,q). Note that M and q are considered here as
being fixed. The reason for this is that up to now we have restricted attention to
one semiclassical component of the wave function only (eigenstates of mass and
charge).
If the hypersurface goes through the whole Kruskal diagram of the eternal
hole, only the boundary term at r →∞(and an analogous one for r →−∞)
contributes. Of particular interest in the black-hole case, however, is the case
where the surface originates at the ‘bifurcation surface’ (r → 0) of past and
future horizons. This makes sense since data on such a surface suffice to construct
the whole right Kruskal wedge, which is all that is accessible to an observer in
this region. Moreover, this mimicks the situation where a black hole is formed
by collapse, in which the regions III and IV of the Kruskal diagram are absent.
What are the boundary conditions to be adopted at r → 0? They are chosen
in such a way that the classical solutions have a non-degenerate horizon and
that the hypersurfaces start at r = 0 asymptotically to hypersurfaces of constant
Killing time (Louko and Whiting 1995). In particular,
N(r, t) =N 1 (t)r + O(r 3 ) , (7.41)
L(r, t) =L 0 (t)+O(r 2 ) , (7.42)
R(r, t) =R 0 (t)+R 2 (t)r 2 + O(r 4 ) . (7.43)
Variation leads, similarly to the situation at r →∞, to a boundary term at
r =0,
−N 1 R 0 (GL 0 ) −1 δR 0 .
If N 1 ≠ 0, this term must be subtracted (N 1 = 0 corresponds to the case of
extremal holes, |q| = √ GM, which is characterized by ∂N/∂r(r =0)=0).