Quantum Gravity

226 QUANTIZATION OF BLACK HOLESsecond term contains the gauge variables. Both classical solutions are containedin the single constraintp u p v =0, (7.82)that is, one has p v =0forη =1andp u =0forη = −1. Observe that thePoisson algebra of the chosen set of observables p u and u for η =+1aswellas p v and v for η = −1 is gauge-invariant in spite of the fact that it has beenobtained by a calculation based on a gauge choice (the double-null coordinatesU and V ). Therefore, the quantum theory will also be gauge-invariant. A crucialpoint is that the new phase space has non-trivial boundaries,−u + vp u ≤ 0 , p v ≤ 0 , > 0 . (7.83)2The boundary defined by the last inequality is due to the classical singularity.The system has now been brought into a form that can be taken as the startingpoint for quantization.7.4.3 QuantizationThe task is to quantize the physical degrees of freedom defined by the action∫S phys = dτ (p u ˙u + p v ˙v − np u p v ) ; (7.84)cf. (7.81). The appropriate method is group quantization; see e.g. Isham (1984).This method is suited in particular to implement conditions such as (7.83). Itis based on the choice of a set of Dirac observables forming a Lie algebra. Thisalgebra generates a group of transformations respecting all boundaries which insuresthat information about such boundaries are implemented in the quantumtheory. The method automatically leads to self-adjoint operators for the observables.One obtains in particular a self-adjoint Hamiltonian and, consequently, aunitary dynamics.The application of this method to the null-dust shell was presented in detailin Hájíček (2001, 2003). A complete system of Dirac observables is given by p u ,p v ,aswellasD u ≡ up u and D v ≡ vp v . Thus, they commute with the constraintp u p v . The only non-vanishing Poisson brackets are{D u ,p u } = p u , {D v ,p v } = p v . (7.85)The Hilbert space is constructed from complex functions ψ u (p) andψ v (p), wherep ∈ [0, ∞). The scalar product is defined by(ψ u ,φ u ):=∫ ∞0dpp ψ∗ u(p)φ u (p) (7.86)(similarly for ψ v (p)). To handle the inequalities (7.83), it is useful to perform thefollowing canonical transformation,