Quantum theory and the beginning of the Universe
Rewriting Hartle's and Hawking's no-boundary proposal as a path integral with Robin boundary conditions
Two AEI researchers have looked at how quantum theory relates to the idea that a finite Universe might have appeared out of nothing due to something akin of a quantum tunnelling effect. This idea was described in the “no-boundary proposal” by Hartle and Hawking. The authors show that this no-boundary proposal can be reformulated as describing a universe arising not from the complete absence of space and time, but rather from specific quantum fluctuations of spacetime. This is in agreement with Heisenberg’s uncertainty principle applied to the geometry of the universe.
Realising 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 et al.that the sum over all universes starting from 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 zero size. The Robin boundary conditions may be interpreted as an initial state with Euclidean momentum, with the quantum uncertainty shared between initial size and momentum.