4D spherically symmetric time-dependent quantum gravity amplitudes

In these short notes, we compute non-perturbatively the time-dependent quantum gravity amplitudes for a four-dimensional spherically symmetric space-time with space-like and time-like boundaries. We solve the 4D classical and quantum constraints in a novel way. We identify the classical solution of the constraints as a canonical transformation, where the integration constants are the new variables. We apply this canonical transformation to the path integral representation of the amplitudes we are interested in. We use both, the canonical and the path integral formalism. This procedure allows us to reduce the action to the actual degree of freedom of the theory. In the end, we get the time-dependent amplitudes from the path integral without performing an actual path integration. From these amplitudes, we show that for most of the boundary conditions time evolution in quantum gravity is non-unitary. There is however a special case where unitary evolution could be achieved.