Quantum transitions among de Sitter and Minkowski spacetimes through bubble nucleation are revisited using the Hamiltonian formalism. We interpret tunnelling probabilities as relative probabilities: the ratio of the squared wave functionals P=false|normalΨscriptNfalse|2false|normalΨscriptBfalse|2, with ΨscriptB,scriptN solutions of the Wheeler‐DeWitt equation corresponding to the spacetimes scriptN and scriptB, gives the probability of nucleating the state scriptN relative to the probability of having the state scriptB. We find that the transition amplitude from de Sitter to de Sitter for both up‐ and down‐tunnelling agrees with the original result based on Euclidean instanton methods. Expanding on the work of Fischler, Morgan and Polchinski we find that the Minkowski to de Sitter transition is possible as in the original Euclidean approach of Farhi, Guth and Guven. We further generalise existing calculations by computing the wave function away from the turning points for the classical motion of the wall in de Sitter to de Sitter transitions. We address several challenges for the viability of the Minkowski to de Sitter transition, including consistency with detailed balance and AdS/CFT. This sets this transition on firmer grounds but opens further questions. Our arguments also validate the Coleman‐De Luccia formulae in the presence of gravity since it has no issues involving negative eigenmodes and other ambiguities of the Euclidean approach. We briefly discuss the implications of our results for early universe cosmology and the string landscape.
We calculate amplitudes for 2D vacuum transitions by means of the Euclidean methods of Coleman-De Luccia (CDL) and Brown-Teitelboim (BT), as well as the Hamiltonian formalism of Fischler, Morgan and Polchinski (FMP). The resulting similarities and differences in between the three approaches are compared with their respective 4D realisations. For CDL, the total bounce can be expressed as the product of relative entropies, whereas, for the case of BT and FMP, the transition rate can be written as the difference of two generalised entropies. By means of holographic arguments, we show that the Euclidean methods, as well as the Lorentzian cases without non-extremal black holes, provide examples of an AdS2/CFT1 ⊂ AdS3/CFT2 correspondence. Such embedding is not possible in the presence of islands for which the setup corresponds to AdS2/CFT1 ⊄ AdS3/CFT2. We find that whenever an island is present, up-tunnelling is possible.
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