Abstract
Realizing the no-boundary proposal of Hartle and Hawking as a consistent gravitational path integral has been a long-standing puzzle. In particular, it was demonstrated by Feldbrugge, Lehners, and Turok that the sum over all universes starting from a zero size results in an unstable saddle point geometry. Here we show that, in the context of gravity with a positive cosmological constant, path integrals with a specific family of Robin boundary conditions overcome this problem. These path integrals are manifestly convergent and are approximated by stable Hartle-Hawking saddle point geometries. The price to pay is that the off-shell geometries do not start at a zero size. The Robin boundary conditions may be interpreted as an initial state with Euclidean momentum, with the quantum uncertainty shared between the initial size and momentum.
Highlights
If the quantum theory is universal, and there currently is no reason to think otherwise, the Universe should be describable by a quantum state just like any other system
In this Letter, we will combine several of the ideas mentioned above, using Robin boundary conditions in order to impose a condition on a linear combination of the initial size and momentum of the Universe
For a family of such conditions, we find that the path integral can be approximated solely by the stable no-boundary saddle points, avoiding instabilities and representing a consistent definition of the no-boundary proposal
Summary
H2qÞ : The integration domain for the lapse is N ∈ ð0þ; ∞Þ, ensuring that the geometries in the sum have a Lorentzian signature This makes the path integral a propagator in the sense that it solves the inhomogeneous Wheeler-DeWitt equation H G1⁄2q1; 0 1⁄4 −iδðq1Þ, where His the quantum Hamiltonian. That the integral along the positive real N line is equivalent to the integral along the steepest descent path (“thimble”) running through this saddle point alone; see Fig. 1. There is, no convergent contour which can be deformed into a steepest descent path running through solely these two saddle points [6] In this sense, the wave function (3) is not the saddle point approximation of the no-boundary wave function for any Lorentzian path integral with these boundary conditions. Let us consider augmenting the action with a Robin boundary term at the initial surface: Stot
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