Solutions to the Dirac equation in a universe with a spacetime-anti-spacetime boundary

Abstract

Just as particles have corresponding antiparticles, spacetime has an analogue known as anti-spacetime. The sign of the determinant of the vielbein—the "square root" of the metric—distinguishes between spacetime (positive determinant) and anti-spacetime (negative determinant). While bosonic fields and classical particles couple only to the metric and cannot differentiate between the two, fermions couple to the vielbeins. This suggests that fermions could potentially be used to detect a spacetime-anti-spacetime boundary. In this work, we investigate the solutions to the Dirac equation within an anti-spacetime bubble in a spacetime universe, and vice versa. We find that the physical solutions correspond to fermions either becoming localized at the boundary or treating it as an infinite potential, depending on key parameters such as the fermion's angular momentum, energy, and the steepness of the anti-spacetime to spacetime transition at the boundary.