The Hawking-Hartle No Boundary Proposal in Causal Set Quantum Gravity

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The Hartle-Hawking (HH) no-boundary proposal provides a Euclidean path integral prescription for a measure on the space of all possible initial conditions. Apart from saddle point and minisuper-space calculations, it is hard to obtain results using the unregulated path integral. A promising choice of spacetime regularisation comes from the causal set (CST) approach to quantum gravity. Using analytic results as well as Markov Chain Monte Carlo and numerical integration methods we obtain the HH wave function in a theory of non-perturbative 2d CST. We find that the wave function is sharply peaked with the peak geometry changing discretely with "temperature". In the low temperature regime the peak corresponds to causal sets which have no continuum counterpart but exhibit physically interesting behaviour. They show a rapid spatial expansion with respect to the discrete proper time as well as a high degree of spatial homogeneity due to extensive overlap of the causal past. While our results are limited to 2 dimensions they provide a concrete example of how quantum gravity could explain the initial conditions for our observable universe.