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

Abstract

New non trivial solutions to the Lorentzian-signature symmetry reduced Bianchi VIII Wheeler DeWitt equation for Hartle Hawking ordering parameters $ \pm 2\sqrt{33} $ are obtained using an Euclidean-signature semi classical method. Using the aforementioned method we define 'excited' states for the Bianchi VIII wave function and construct leading order 'excited' states which appear to be a hybrid of scattering and bound states. Also six new solutions for the $\Lambda \ne 0$ Bianchi VIII Euclidean-signature Hamilton Jacobi equation are presented which can be used to construct semi-classical states corresponding to the Lorentzian signature symmetry reduced Wheeler DeWitt equation. In addition, new solutions to the Euclidean-signature Hamilton Jacobi equation for the Bianchi VIII models are found when $\Lambda \ne 0$ and an aligned primordial magnetic field is present. Furthermore, we find eight new complex solutions to the Euclidean-signature Hamilton Jacobi equation for the case when matter is not present, which can be used to construct additional semi-classical states. We also study the leading order wave functions which result from our aforementioned solutions when they are restricted to the $\beta+$ axis. Prior to presenting these results we explain the Euclidean-signature semi classical method and discuss how to solve the resultant equations this method generates when it is applied to the Lorentzian-signature symmetry reduced Bianchi VIII Wheeler DeWitt equation. The Euclidean-signature semi-classical method used here is applicable to other field theories as well. The work presented throughout this paper further shows the power Eulclidean-signature semi classical methods possess for solving problems in quantum cosmology and solving Lorentzian signature problems in general.