Thoughts on late-time physics in Quantum gravity
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In general, solving something like the Hamiltonian constraint (called the Wheeler-DeWitt equation, which we will abbreviate to “WDW”) in the framework of canonical quantum gravity is hard. Usually, one picks up an additional ansatz that simplifies the situation, such as minisuperspace and so on, although I am not aware of such cases. In a paper called “Hilbert Space of Quantum Gravity in de Sitter Space”, it was shown that the solutions to the WDW equation (while a WDW state also corresponds to the momentum constraint, usually it is more preferrable to consider the explicit study of the Hamiltonian constraint and an implicit study of the momentum constraint, due to certain reasons I will expand on at the end of this post) in de Sitter take up a nice form, which was found by an ansatz, where one starts by saying that at late-time slices (found by setting a conformal factor $\Omega$ such that $\log\int\sqrt{|g|}d^{D}x\to\infty$ corresponds to the factor $\Omega\to+\infty$), the wavefunctional takes up the form [\Psi=e^{i\mathcal{F}}\;,] where $\mathcal{F}$ is a functional that will not be explicitly worked with at the present level. Read more