Porting conventional tools to quantum geometrodynamics

As the prevailing theory of gravitation, the general theory of relativity successfully describes classical gravitation, but has yet to be consistently quantised, despite the efforts of generations of physicists in over a hundred years. One of the first attempts to quantise general relativity directly is the Wheeler–DeWitt approach. It begins with the Hamiltonian formulation of this theory by Arnowitt, Deser and Misner, and applies the quantisation scheme of Dirac, designed for constrained systems, including the Dirac spinors and the Maxwell theory, among others. This approach, also known as quantum geometrodynamics, is successful in the semi-classical method of Wentzel–Kramers–Brillouin (WKB) and Born–Oppenheimer, and has been applied to quantum models of universes and black holes. Unfortunately, because of the constrained nature of general relativity (from another perspective, its diffeomorphism invariance), its quantised version à la Dirac lacks many properties that are crucial in conventional quantum theory. Particularly, the scalar product of quantum states is difficult to define, rendering the non-existence of a Hilbert space, and of the analysis of self-adjoint operators. Moreover, the semi-classical approach described above only works for wave functions in the WKB form, which contain the classical Hamilton’s principal function as a phase factor in the leading-order approximation. For wave-packets, which naturally arise in many realistic systems, even their corresponding semi-classical trajectories cannot be calculated; in conventional quantum mechanics, in contrast, one can refer to the Ehrenfest theorem if the wave-packet is sharp. In this dissertation, we try to address these problems of the Wheeler–DeWitt approach by porting conventional tools in physics and mathematics to this context. We study a two-dimensional minisuperspace model, related to physical cosmological models, to illustrate our arguments. Under the WKB approximation, we show that a narrow Gaussian wave-packet has “maxima” on the semi-classical trajectory, which is given by the stationary phase principle, that also governs the WKB approach. In other words, these two semi-classical approaches are consistent in the semi-classical trajectory. By considering additional conditions, an effective Hilbert space emerges from our minisuperspace model, and the Hamiltonian, responsible for the energy spectrum, can have non-trivial self-adjoint extensions. We study its self-adjoint domains in detail and argue that these mathematical properties could lead to physical effects. In order to maintain consistency of our new tools for both quantum gravitation and conventional quantum theory, we construct a framework of stationary wave-packets, that make sense for both the minisuperspace Wheeler–DeWitt equation and the stationary Schrödinger equation. In doing so, we also argue for the suitable choice of amplitudes when constructing wave-packets. The framework is then tested by the model of two-dimensional hydrogen atom. Finally, we discuss approaches to find the semi-classical trajectories from arbitrary wave-packets, which are methods for ridge-detection. We discuss different mathematical descriptions of ridge-lines, which were historically developed for Riemannian geometry with Euclidean metric signature. Then we try to generalise these descriptions to pseudo-Riemannian geometry with Lorentzian metric signature, which is the usual case of minisuperspaces. In the end, we give proposals of prospective physical applications