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Looking at the Wheeler-deWitt equation. My attempt to describe it in words is this:

"For each 3d-manifold given by metric tensor field $\gamma$, associate a complex number $\Psi$. The Wheeler-de-Witt equation says that the rate of change of the phase $\Psi$ compared to a small change in the manifold is proportional to the total curvature, $R$, of the manifold (plus a constant $\Lambda$)."

But then I guess it depends what one means by a "small change". If one can put a "small change in the metric" into more pedestrian terms. And I guess this is given by the super-metric $G$.

From this one would conclude the Universe "favours" smoother spaces as the wave function of highly curved spaces would tend to cancel as it would change a lot under a small perturbation.

Is this a good way to describe the equation in words? If not what is a better way?

\begin{eqnarray} \left[ -G_{ijkl} \frac{\delta^2}{\delta \gamma_{ij} \delta \gamma_{kl}} - ^{(3)}R(\gamma) \gamma^{\frac{1}{2}} + 2 \Lambda \gamma^{\frac{1}{2}} \right] \Psi[\gamma_{ij}] &=& 0\\ \frac{1}{2} \gamma^{-\frac{1}{2}} \left( \gamma_{ik} \gamma_{jl} + \gamma_{il} \gamma_{jk} - \gamma_{ij} \gamma_{kl} \right)&=& G_{ijkl} \end{eqnarray}

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  • $\begingroup$ Please use MathJax, not scans of math. $\endgroup$ – G. Smith Nov 13 '19 at 21:44
  • $\begingroup$ the rate of change of the phase $\Psi$ Did you mean to write “phase of”? How do you conclude that the phase of $\Psi$ is being extracted? $\endgroup$ – G. Smith Nov 13 '19 at 21:52
  • $\begingroup$ Well it's more or less a wave equation in the form $y''+a^2y=0$ which has a solution $y=e^{iax}$. So $a$ would correspond the the frequency of the wave or how fast the phase changes. In fact $a$ would be the square root of the curvature in this case. $\endgroup$ – zooby Nov 14 '19 at 5:22

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