ar X iv : g r - qc / 9 41 10 03 v 1 1 N ov 1 99 4 COLO - HEP - 341 hep - th / 9411003 October 1994 Lessons of Quantum

Based on the analysis of two dimensional dilaton gravity we argue that the semiclassical equations of black hole formation and evaporation should not be interpreted in terms of expectation values of operators in the exact quantum theory, but rather as WKB trajectories. Thus at the semiclassical level it does not seem possible to formulate a notion of quantum mechanical information loss. The development of two-dimensional dilaton gravity models (CGHS) [1-5] raises the possibility of understanding some conceptual issues in quantum gravity. In particular it may be hoped that the question of information loss in the evaporation of black holes [6] might be resolved within this simple model. Another issue that may be addressed is the problem of time in a quantum theory of gravity. Actually, as discussed in [7], in order to resolve the first issue it is necessary to have some understanding of the second. In the literature on quantum gravity the semiclassical approximation is often confused with the Born-Oppenheimer approximation. As shown in [8] the latter approximation (which for 4D gravity consists of expanding in powers of M −1 P), when applied to the Wheeler-DeWitt (WDW) equation, leads to a Schrödinger equation for the matter wave function in a background geometry that is a solution of the vacuum Einstein equation. However, in much of the work on black hole evaporation that attempts to include the effect of backreaction, it is assumed that the gravitational field may be treated classically according to the equation G µν = 8πG < T µν >, where the expectation value is to be taken in a quantum state of matter that satisfies the Schrödinger equation. Although many authors have attempted to obtain this from the WDW equation, there is no real derivation of this approximation. In fact the source of the problem in our opinion is the appearance of an expectation value, i.e. a quantity that violates the superposition principle. ‡ Clearly, starting from a linear equation like the WDW equation, it is unlikely that there can be a satisfactory derivation of a non-linear " approximation ". § How then are we to understand the equations that have been discussed in ‡ The well known argument for information loss assumes that there is a notion of time evolution of quantum states defined on a series of space-like surfaces in an evaporating black hole space time that is a solution of the semiclassical equation. …