Mixmaster revisited: wormhole solutions to the Bianchi IX Wheeler–DeWitt equation using the Euclidean-signature semi-classical method

Abstract

A modified semi-classical method is used to construct both ground and excited state solutions to the canonically quantized vacuum Bianchi IX (Mixmaster) cosmological models. Employing a modified form of the semi-classical Ansatz we solve the relevant Wheeler–DeWitt equation asymptotically by integrating a set of linear transport equations along the flow of a suitably chosen solution to the corresponding Euclidean-signature Hamilton–Jacobi equation. For the Moncrief–Ryan (or ‘wormhole’) Hamilton–Jacobi solution, we compute the ground state quantum correction term associated with operator ordering ambiguities and show how higher order correction terms can be computed. We also determine the explicit, leading order forms of a family of excited states and show how to compute their quantum corrections as smooth, globally defined functions on the Bianchi IX minisuperspace. These excited state solutions are peaked away from the minisuperspace origin and are labeled by a pair of positive integers that can be plausibly interpreted as graviton excitation numbers for the two independent anisotropy degrees of freedom. The Euclidean-signature semi-classical method used here is applicable to more general models, representing a significant progress in the Wheeler–DeWitt approach to quantum gravity.