No-Boundary Proposal as a Path Integral with Robin Boundary Conditions

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.

Figure 1

The minisuperspace path integral contains four saddle points (orange dots) in the plane of the complex lapse $N$. Steepest descent or ascent lines of the magnitude of the action are drawn as black lines, with the arrows indicating descent. Asymptotic regions of convergence are shown in green, and divergent ones are red. For the path integral with Dirichlet boundary conditions, the defining contour of a positive real lapse (orange line) can be deformed to the (orange dashed) thimble flowing through saddle point 1 only. With Robin boundary conditions, saddle points 1 and 2 and the singularity of the action at $N=0$ move. Shown here is the direction of motion for negative imaginary $\beta$.

Figure 2

For sufficiently large negative imaginary $\beta$, the unstable saddle point 1 moves below the Hartle-Hawking saddle point 4. The thimble emanating from the singularity of the action at $N^{\star }$ now passes through only this stable saddle point, and the problem with instabilities is avoided. In white, we indicate the locus of geometries that contain a singularity and which we require the thimble to avoid.

Figure 3

On the left, we have the smooth saddle point geometry of Hartle-Hawking type. By contrast, with Robin boundary conditions a typical off-shell geometry will not start at a zero size (middle). Some off-shell geometries contain a recollapse to a zero size, and it is these geometries that we would like to avoid summing over (right).