Joseph Hongchul Bae

A Euclidean-signature semi-classical method is used to construct both ground and excited state solutions to the canonically quantized 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 terms associated with operator-ordering ambiguities. We next determine the explicit, leading-order forms of a discrete spectrum of excited states and show how to compute their quantum corrections as smooth, globally-defined functions on the Bianchi IX minisuperspace. These excited states 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.

To help with the interpretation of these solution states, we perturb about the Lorentz-signature Taub solutions using the Jacobi method of second variation. Through a series of canonical transformations, we decouple the final linearized Hamiltonian into a gauge-invariant and gauge-dependent part, further splitting the gauge-invariant part into two Hamiltonians that are of time-dependent harmonic oscillator form. For the perturbation Q+’ = p_+ a’ + p_a b_+’ that stays entirely within Taub models, we solve for the quantum discrete state and squeezed state solutions explicitly, using the method of invariants of Lewis & Riesenfeld.

While our analysis is currently limited to spatially homogeneous cosmological models, we anticipate that the techniques under development should eventually be much more generally applicable and thus represent a significant advance in the Wheeler-DeWitt approach to quantum gravity.