New Solutions To The Bianchi IX Wheeler DeWitt Equation And Leading Order Solutions For $\Lambda$ $\ne$ 0 And A Primordial Magnetic Field

TL;DR

This paper presents new closed-form solutions to the Bianchi IX Wheeler DeWitt equation using a semi-classical Euclidean approach, exploring effects of a cosmological constant and primordial magnetic fields on quantum cosmological models.

Contribution

It introduces novel solutions for Bianchi IX quantum cosmology with cosmological constant and magnetic fields, advancing semi-classical methods in Lorentzian quantum gravity.

Findings

New solutions for Bianchi IX Wheeler DeWitt equation with cosmological constant.

First order quantum corrections to Bianchi IX wave functions.

Cosmological constant influences anisotropy in models.

Abstract

New closed form solutions to the Lorentzian signature symmetry reduced Bianchi IX Wheeler DeWitt equation are obtained using a Euclidean-signature semi classical method for Hartle Hawking ordering parameters $ \pm 2\sqrt{33} $ for the wormhole Hamilton-Jacobi solution. Moving past the wormhole case we also compute first order quantum corrections to Bianchi IX wave functions restricted to the $\beta_+$ axis for the 'no boundary' solution and one of the 'arm' solutions. Furthermore, six solutions to the Bianchi IX Euclidean-signature Hamiltonian Jacobi equation for the case when a cosmological constant is present have been found and are used to construct a semi-classical which respects the symmetry of the Bianchi IX potential. Also two additional solutions were found when both a cosmological constant and an aligned primordial magnetic field are present, which are also used to construct a leading order wave function. Furthermore we construct leading order 'excited' states which are restricted to the $\beta_+$ axis for the cases when a cosmological constant and a primordial magnetic field are present. To conclude, we discuss the interesting features of our solutions, and point out how the cosmological constant appears to act as a driver of anisotropy in our Bianchi IX models. The Euclidean-signature semi-classical method used here is applicable to certain field theories as well. Because this semi classical method was able to find new solutions to a heavily studied problem the results in this paper represent significant progress in the Wheeler-DeWitt approach to quantum cosmology and in the application of Euclidean-signature semi classical methods to Lorentzian signature problems in general.