Quantum gravitational collapse of a scalar field and the wave function of black hole decay

We develop a quantum description of spherically symmetric gravitational collapse of a massless scalar field. The canonical quantization procedure is applied to a minisuperspace model in which the four-metric and scalar field variables are restricted to a self-similar functional form. Our main purpose is to propose a mechanism of black hole decay in terms of a time evolution of the wave function defined on the minisuperspace. For the self-similar classical dynamics there exists a one-parameter family of solutions for the Einstein-scalar equations, and the Hamiltonian turns out to play the role of the critical parameter which separates the supercritical solution corresponding to a black hole formation from the subcritical one corresponding to a wave reflection to infinity. We derive a set of the eigenfunctions of the quantum Hamiltonian operator, which is interpreted to be the quantum version of the supercritical and subcritical solutions. One of the metric components is also treated as a quantum operator which does not commute with the Hamiltonian operator. The expansion of the eigenfunction of the metric operator in terms of the Hamiltonian eigenfunctions motivates us to introduce the superposition principle and the time-dependent Schrödinger equation which lead to the uncertainty of the Hamiltonian eigenvalue. Then we study the time evolution of the initial black hole state, which is chosen to be one of the supercritical Hamiltonian eigenstates. We arrive at the conclusion that the time evolution toward black hole decay is essentially due to the breakdown of orthogonality of the Hamiltonian eigenfunctions: The quantum black hole behaves like a wave packet written by a superposition of the various subcritical states. The localized structure of the black hole wave packet begins to spread as each subcritical state evolves with time according to the Schrödinger equation. We can estimate the decay time and find the final behavior of the wave function describing an outgoing flux of scalar waves observable at future null infinity.