Quantum theory of the Bianchi II model

We describe the quantum evolution of the vacuum Bianchi II universe in terms of the transition amplitude between two asymptotic quantum Kasner-like states. For large values of the momentum variable, the classical and quantum calculations give similar results. The difference occurs for small values of this variable due to the Heisenberg uncertainty principle. Our results can be used, to some extent, as a building block of the quantum evolution of the vacuum Bianchi IX universe.

Figure 1

Plot of the phase shift $\delta ({\pi }_{+})$ of Eq. (52) for $n_1=1$ (left) and for $n_1=4$ (right). Changing the value of $n_1$ introduces a linear additive term.

Figure 2

Plots of ${\beta }_{+}^{(C)}$ (red curve) and ${\beta }_{+}^{(Q)}$ (blue curves) for $n_1=1$ (top) and for $n_1=4$ (bottom). Changing the value of $n_1$ introduces the same shift on ${\beta }_{+}^{(C)}$ and ${\beta }_{+}^{(Q)}$. The left plots are without any correction; on the right, ${\beta }_{+}^{(C)}$ has been shifted with $\delta {\beta }_{\infty }=\ln 2/e$.

Figure 3

Plots of ${\tilde{\beta }}_{+}^{(C)}$ (red curve) and ${\beta }_{+}^{(Q)}$ (blue curve) for $n_1=1$ (left) and for $n_1=4$ (right) and small values of ${\pi }_{+}^{\infty }$.