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

Area of Science:

• Cosmology and Theoretical Physics
• Quantum Gravity
• Mathematical Physics

Background:

• The Hartle-Hawking no-boundary proposal offers a quantum mechanical description of the universe's origin.
• Previous attempts to formulate this proposal as a gravitational path integral faced challenges, particularly an unstable saddle point geometry.
• This instability was linked to summing over universes starting from zero size.

Purpose of the Study:

• To resolve the long-standing puzzle of realizing the Hartle-Hawking no-boundary proposal as a consistent gravitational path integral.
• To identify a method that overcomes the instability issue associated with zero-size initial universes.
• To explore the implications of alternative boundary conditions for quantum cosmology.

Main Methods:

• Investigated gravitational path integrals within the context of gravity featuring a positive cosmological constant.
• Introduced and analyzed a specific family of Robin boundary conditions.
• Examined the convergence properties and saddle point geometries of these path integrals.

Main Results:

• Demonstrated that specific Robin boundary conditions lead to manifestly convergent path integrals.
• Showed that these convergent path integrals are approximated by stable Hartle-Hawking saddle point geometries.
• Identified that the off-shell geometries under these conditions do not initiate from a zero size.

Conclusions:

• The use of Robin boundary conditions provides a consistent framework for the Hartle-Hawking no-boundary proposal in quantum gravity with a positive cosmological constant.
• This approach circumvents the previously identified instability by modifying the initial conditions of the universe.
• Robin boundary conditions can be interpreted as representing an initial state with Euclidean momentum, distributing quantum uncertainty between initial size and momentum.