# Quantum cosmology of Bianchi VIII, IX LRS geometries

## TL;DR

This paper revisits Bianchi VIII and IX cosmological models, deriving classical solutions using integrals of motion and exploring quantum solutions via the Wheeler DeWitt equation, with implications for understanding collapsed configurations.

## Contribution

It introduces a novel method using quadratic integrals of motion and homothetic vectors to analyze classical and quantum Bianchi models.

## Key findings

- Classical solutions derived using new integrals of motion.
- Quantum solutions obtained from Wheeler DeWitt equation with two operators.
- A probabilistic interpretation that excludes collapsed configurations.

## Abstract

In the present work we revisit the axisymmetric Bianchi VIII and IX models. At the classical level we reproduce the known analytic solution, in a novel way making use of two quadratic integrals of motion, the constraint equation, as well as a linear non-local integral of motion. These quantities correspond to two second rank Killing tensors and a homothetic vector field existing on the relevant configuration space. On the corresponding phase space the two quadratic charges commute with the Hamiltonian constraint but not among themselves. Thus, after turning these charges into operators we obtain two different solutions to the Wheeler DeWitt equation utilizing each of the quadratic operators. The homothetic vector is then used, as a natural guide line, to define a normalizable conditional probability which assigns zero to the classically collapsed configurations.