The Cosmological Probability Density Function for Bianchi Class A Models in Quantum Supergravity

TL;DR

This paper derives a cosmological probability density function for Bianchi Class A models in quantum supergravity, connecting it to the wormhole state and interpreting it through a stochastic Fokker-Planck framework.

Contribution

It introduces a novel approach to quantum supergravity by deriving a Fokker-Planck type evolution equation for cosmological models, linking static solutions to wormhole states.

Findings

The evolution equation takes the form of a Fokker-Planck equation.

The static solution matches the square of the wormhole wave function.

A boundary term and initial conditions are crucial for the formulation.

Abstract

Nicolai's theorem suggests a simple stochastic interpetation for supersymmetric Euclidean quantum theories, without requiring any inner product to be defined on the space of states. In order to apply this idea to supergravity, we first reduce to a one-dimensional theory with local supersymmetry by the imposition of homogeneity conditions. We then make the supersymmetry rigid by imposing gauge conditions, and quantise to obtain the evolution equation for a time-dependent wave function. Owing to the inclusion of a certain boundary term in the classical action, and a careful treatment of the initial conditions, the evolution equation has the form of a Fokker-Planck equation. Of particular interest is the static solution, as this satisfies all the standard quantum constraints. This is naturally interpreted as a cosmological probability density function, and is found to coincide with the square of the magnitude of the conventional wave function for the wormhole state.