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I read this paper on Lorentzian Quantum Cosmology. But I couldn't understand the term minisuperspace which is used to define the path integral $$\int\mathcal{D}N \mathcal{D}\pi\mathcal{D}a\mathcal{D}p\mathcal{D}c \mathcal{D}\bar{P} e^{\frac{i}{h}\int_0^1 dt(\dot{N}\pi+\dot{a}p+\dot{c}\bar{P}-NH)} $$ where $a,N$ are related to metric by $ds^2=-N(t)^2dt^2+a(t)^2d\Omega_3^2$, $c$ is fermionic ghost; $p,\pi,\bar{P}$ their corresponding conjugate momenta

Their whole analysis is done on this path integral which is told to be residing in the minisuperspace.

Why is the ghost field needed in minisuperspace though it's told in the paper it's needed because of diffeomorphism invariance of minisuperspace action but I couldn't get it. All I know about ghost fields is that they are needed when we want to define path integrals of gauge theory.

Is the minisuperspace related to supersymmetry of QFT where the spin particles are given their counterparts particle?

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In the full gravitational path integral, you would integrate over all components $g_{\mu\nu}$ of the metric. You would also need some way to account for diffeomorphism invariance.

The minisuperspace is an approximation which consists of only integrating over the lapse $N(t)$ and the scale factor $a(t)$, assuming both are functions of time only and not space as well. Including the lapse accounts for time reparameterization invariance.

The hope when you make this approximation, is that the parts of the metric you are ignoring (for instance, the off diagonal terms, and the dependence on spatial coordinates) do not make a large contribution to the path integral. It's not easy to justify this assumption, since no one knows how to do the full gravitational path integral, where you don't make some kind of approximation like the minisuperspace approximation, to check.

Finally, minisuperspace has nothing whatsoever to do with supersymmetry. People just liked the word "super" in the 70s.

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    $\begingroup$ Very nice summary. "It's not easy to justify this assumption" – the converse can be justified by looking at different models of quantum cosmology and noticing how their predictions differ drastically. $\endgroup$ Nov 17, 2020 at 8:10
  • $\begingroup$ @Prof.Legolasov Indeed. $\endgroup$
    – Andrew
    Nov 17, 2020 at 8:13

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