The Dirac Equation and the Normalization of its Solutions in a Closed Friedmann-Robertson-Walker Universe

TL;DR

This paper formulates the Dirac equation in a closed Friedmann-Robertson-Walker universe, explicitly solves the spatial part, and analyzes the normalization of solutions to define the fermionic projector with global and local normalization.

Contribution

It introduces a method to normalize Dirac solutions in a closed universe and constructs the fermionic projector with specified normalization conditions.

Findings

Explicit solutions for Dirac equation in closed universe

Analysis of space-time normalization integral

Definition of fermionic projector with global and local normalization

Abstract

We set up the Dirac equation in a Friedmann-Robertson-Walker geometry and separate the spatial and time variables. In the case of a closed universe, the spatial dependence is solved explicitly, giving rise to a discrete set of solutions. We compute the probability integral and analyze a space-time normalization integral. This analysis allows us to introduce the fermionic projector in a closed Friedmann-Robertson-Walker geometry and to specify its global normalization as well as its local form.