Mixmaster Ground State

Basic set-up

Mixmaster universe:

$\theta$, $\psi$, and $\varphi$ are Euler angles parametrizing our spatial manifold which is topologically $S^{3}$.

Use the noncoordinate basis vectors fields

(1)
\begin{align} e_{1} = \cos \psi \frac{\partial }{\partial \theta} - \sin \psi \left( \cot \theta \frac{\partial}{\partial \psi} - \frac{1}{\sin \theta} \frac{\partial}{\partial \varphi} \right) \\ e_{2} = \sin \psi \frac{\partial }{\partial \theta} + \cos \psi \left( \cot \theta \frac{\partial}{\partial \psi} - \frac{1}{\sin \theta} \frac{\partial}{\partial \varphi} \right) \\ e_{3} = \frac{\partial }{\partial \psi} \end{align}

The commutation coefficients of these basis vectors, defined by the formula $\left[ e_{ \alpha } , e_{ \beta } \right] = c_{ \alpha \beta }^{ \ \ \ \gamma} e_{ \gamma }$, are

(2)
\begin{align} c_{ \alpha \beta }^{ \ \ \ \gamma}= - \varepsilon_{ \alpha \beta \gamma}. \end{align}

The dual 1-forms corresponding to these basis vectors are

(3)
\begin{align} \sigma_{1} = \cos \psi d \theta + \sin \psi \sin \theta d \varphi \\ \sigma_{2} = \sin \psi d \theta - \cos \psi \sin \theta d \varphi \\ \sigma_{3} = d \psi + \cos \theta d \varphi. \end{align}

In terms of these 1-forms, the mixmaster metric is given by

(4)
\begin{align} ds^{2} = -N^{2} dt^{2} + \left( 6 \pi \right) ^{-1} e^{2\alpha \left( t \right) } e^{2 \beta_{ij} \left( t \right) } \sigma^{i} \sigma^{j} \end{align}

where $\alpha \left( t \right)$ is a scalar and $\beta _{ij} \left( t \right)$ is a $3 \times 3$ diagonal matrix given by

(5)
\begin{align} \beta_{ij} \left( t \right) = diag \left[ \beta_{+} \left( t \right) + \sqrt{3} \beta_{-} \left( t \right) , \beta_{+} \left( t \right) - \sqrt{3} \beta_{-} \left( t \right) , -2 \beta_{+} \left( t \right) \right]. \end{align}

Thus $\sqrt{^{(3)}g} = e^{3 \alpha}$. $N$ is the lapse function from the ADM splitting. The action can be written

(6)
\begin{align} I = \int p_{+} d \beta_{+} + p_{-} d \beta_{-} + p_{\alpha} d \alpha - \left( \frac{3 \pi}{2} \right) ^{1/2} N \mathcal{H} dt \end{align}

where

(7)
\begin{align} \mathcal{H} & = & \frac{1}{2} e^{-3 \alpha} \left( - p_{ \alpha}^{2} + p_{+}^{2} + p_{-}^{2} - \frac{2}{3} e^{6 \alpha} R \right) \\ \ \ \ & = & \frac{1}{2} e^{-3 \alpha} \left( - p_{ \alpha}^{2} + p_{+}^{2} + p_{-}^{2} + e^{4 \alpha} V \right) \end{align}

In the first line $R = ^{(3)}R = \frac{1}{2} e^{-2 \alpha} tr \left( 2e^{-2 \beta} - e^{4 \beta} \right)$ is the scalar curvature of a spatial slice.

(8)
\begin{align} V & = & V \left( \beta \right) = \frac{1}{3} Tr \left( - 2e^{- 2 \beta} + e^{4 \beta} \right) \\ \ \ & = & \frac{1}{3}e^{-8 \beta_{+}} - \frac{4}{3}e^{-2 \beta_{+}} \cosh \left( 2 \sqrt{3} \beta_{-} \right) + \frac{2}{3} e^{4 \beta_{+}} \left( \cosh \left( 4 \sqrt{3} \beta_{-} \right) - 1 \right) \end{align}

(Note that this is $V_{Moncrief} = V_{MTW} - 1$.)

Coordinate condition: $N = 1.$

The 3-volume of a spatial slice is given by

(9)
\begin{align} ^{ \left( 3 \right) } V \left( t \right) = e^{ 3 \alpha }. \end{align}

Curvature tensor

Einstein curvature tensor:

(10)
\begin{align} G_{\alpha \beta} = R_{ \alpha \beta} - \frac{1}{2} g_{\alpha \beta} R \end{align}
(11)
\begin{align} R_{\alpha \beta} = R^{\mu}_{\alpha \mu \beta} = \Gamma^{\mu}_{\alpha \beta , \mu} - \Gamma^{\mu}_{\alpha \mu , \beta} + \Gamma^{\mu}_{\delta \mu} \Gamma^{\delta}_{\alpha \beta} - \Gamma^{\mu}_{\delta \beta} \Gamma^{\delta}_{\alpha \mu} - \Gamma^{\mu}_{\alpha \delta} c^{\delta}_{\mu \beta} \end{align}
(12)
\begin{align} R = R^{\alpha}_{\ \alpha} = g^{\alpha \beta} R_{\beta \alpha} \end{align}

Moncrief-Ryan ground state

For quantized mixmaster in nonlinear normal ordering, Moncrief-Ryan ground state is

(13)
\begin{align} \Omega \left( \alpha, \beta_{\pm} \right) = \mathcal{N} e^{-S \left( \alpha, \beta_{\pm} \right) } \end{align}

where

(14)
\begin{align} S \left( \alpha, \beta_{\pm} \right) & = \frac{1}{6} e^{2 \alpha} \left[ e^{-4 \beta_{+}} + 2e^{2 \beta_{+}} \cosh \left( 2 \sqrt{3} \beta_{-} \right) \right] \\ \ \ \ & = \frac{1}{6} e^{2 \alpha} \left[ e^{-4 \beta_{+}} + e^{2 \beta_{+}} \left( e^{2 \sqrt{3} \beta_{-}} + e^{-2 \sqrt{3} \beta_{-}} \right) \right] . \end{align}

The animation below shows the (unnormalized) $\alpha = \text{constant}$ slice of the mixmaster wavefunction $e^{-S \left( \alpha, \beta_{\pm} \right)}$, over the $\beta_{+} \beta_{-}$ plane. The animation runs over values of $\alpha$ from $-2$ to $2$.

movie.gif

The normalization constant $\mathcal{N} = \mathcal{N} \left( \alpha \right)$ is such that

(15)
\begin{align} \int_{- \infty} ^{\infty} \int_{- \infty} ^{\infty} \Omega^{2} \left( \alpha , \beta_{ \pm} \right) d \beta_{+} d \beta_{-} = 1. \end{align}

The following plot shows the natural logarithm of $\mathcal{N} = \mathcal{N} \left( \alpha \right)$.

normalization.jpg

Observables

Scalar curvature of 3d spatial slice

From MTW p 808, the scalar curvature of a spacelike slice of the Mixmaster universe is

(16)
\begin{align} ^{ \left( 3 \right) } R & = & \frac{1}{2} e^{-2 \alpha} tr \left( 2e^{-2 \beta} - e^{4 \beta} \right) \\ \ \ \ & = & \frac{1}{2}e^{-2 \alpha} \left( 2 \left( e^{-2 \left( \beta_{+} + \sqrt{3} \beta_{-} \right) } + e^{-2 \left( \beta_{+} - \sqrt{3} \beta_{-} \right) } + e^{4 \beta_{+} } \right) \right \\ \ \ & \ \ & \ \ \ \ \ \ \ \ \ \left - \left( e^{4 \left( \beta_{+} + \sqrt{3} \beta_{-} \right)} + e^{4 \left( \beta_{+} - \sqrt{3} \beta_{-} \right)} +e^{-8 \beta_{+} } \right) \right) \end{align}

Since the curvature is constant over a given spatial slice, we can find the total curvature $\int_{ \Sigma } \left ^{ \left( 3 \right) } R \right$ by multiplying $^{ \left( 3 \right) } R$ by the 3-volume $e^{3 \alpha}$ to obtain

(17)
\begin{align} R_{tot} & = & \int_{ \Sigma } ^{ \left( 3 \right) } R \\ \ \ \ & = & \frac{1}{2}e^{ \alpha} \left( 2 \left( e^{-2 \left( \beta_{+} + \sqrt{3} \beta_{-} \right) } + e^{-2 \left( \beta_{+} - \sqrt{3} \beta_{-} \right) } + e^{4 \beta_{+} } \right) - \left( e^{4 \left( \beta_{+} + \sqrt{3} \beta_{-} \right)} + e^{4 \left( \beta_{+} - \sqrt{3} \beta_{-} \right)} +e^{-8 \beta_{+} } \right) \right) \end{align}

Expectation value of total scalar 3-curvature, for given $\alpha$:

(18)
\begin{align} \left< R_{tot} \right> = \int_{- \infty} ^{ \infty } \int_{- \infty} ^{ \infty } R_{tot} \Omega^{2} \left( \alpha, \beta_{\pm} \right) d \beta_{+} d \beta_{-} \end{align}

Below is a plot of the expected value of the quantum observable corresponding to total scalar curvature, as a function of $\alpha$:

curv_exp.jpg

Dispersion:

(19)
\begin{align} D_{R_{tot}} = \left< R_{tot} ^{2} \right> - \left< R_{tot} \right> ^{2} \end{align}

where

(20)
\begin{align} \left< R_{tot} ^{2} \right> = \int_{- \infty} ^{ \infty } \int_{- \infty} ^{ \infty } R_{tot}^{2} \Omega^{2} \left( \alpha, \beta_{\pm} \right) d \beta_{+} d \beta_{-}. \end{align}

Below is a logarithmic plot of dispersion of the quantum observable corresponding to total scalar curvature, as a function of $\alpha$:

curv_disp.jpg
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