On the Hilbert Space of Quantum Gravity from the Gravitational Path Integral

Explore the fundamental question of whether the gravitational path integral can define a valid Hilbert space for quantum gravity in this advanced physics lecture. Examine the positivity criteria that must be satisfied for both open and closed universe states to maintain positive norm, and discover how these seemingly restrictive conditions are remarkably fulfilled in several important cases. Learn about the construction of a well-defined Hilbert space that enables meaningful discussions of transition probabilities between different universe states. Investigate the no-boundary state as a detailed case study, focusing on how to properly compute transition probabilities while accounting for non-perturbative corrections in the gravitational path integral. Understand how these calculations yield results that significantly diverge from the traditional Hartle-Hawking approach, revealing new insights into the quantum nature of gravity and spacetime. The presentation addresses cutting-edge research in theoretical physics, connecting path integral formulations with the mathematical foundations of quantum gravity theory.