Analytic solutions of the Wheeler-DeWitt equation in spherically symmetric geometry
Right from the early days when the canonical formalism of general relativity was formulated on the basis of the special lectures of Dirac and the Arnowitt-Deser-Misner (ADM) work, interest in further extension and consolidation of analytic solutions of various kinds has considerably increased. This, indeed, is a rich field of research still. After an introduction detailing different problems in solving the Wheeler-DeWitt (WD) equation concerning non-operator ordering, time, interpretation of wave function, realization of classical systems from quantum systems, regularization etc., the paper discusses a canonical formalism of the Einstein-Maxwell theory in a spherically symmetric spacetime with a cosmological constant term. In the third section a canonical quantisation in the Schrödinger picture is carried out leading to a set of constrained wave equations, momentum constraint equation, WD equation, mass and charge eigenstate equations. Relevant Hamiltonian and some commutation relations are covered. In the final section a summary is provided touching on comparison of the author's solutions with WKB (semi-classical) solution, Schwarzschild black hole and de Sitter universe case \((m=0)\). It is claimed that the present solutions are useful to investigate the early universe, inflation epoch and the cosmological quantum black hole. There is an extensive bibliography consisting of 34 valuable references.
Schwarzschild black hole and de Sitter universe
scientific article; zbMATH DE number 1495554