A Wavefunction Description for a Localized Quantum Particle in Curved Spacetimes

TL;DR

This paper develops a formalism that describes a localized quantum particle in curved spacetimes using a complex wavefunction, incorporating effects of acceleration and curvature into a modified Schrödinger equation.

Contribution

It generalizes Dirac's formalism to a wavefunction approach in curved spacetimes, including non-relativistic approximations and curvature corrections.

Findings

Derived a wavefunction-based quantum description in curved spacetime.

Obtained a modified Schrödinger equation with acceleration and curvature corrections.

Provided a formalism for analyzing localized quantum particles in gravitational fields.

Abstract

We reduce Dirac's spinor formalism for a spin 1/2 particle to a complex wavefunction description in curved spacetimes. We consider a localized fermionic particle in curved spacetimes and perform an expansion in terms of the acceleration and curvature around the center of mass of the system, generalizing the results of [Phys. Rev. D 22, 1922]. Under a non-relativistic approximation, one obtains a quantum description in a Hilbert space of complex wavefunctions defined in the rest space of the system. The wavefunction of the particle then evolves according to a modified Schr\"odinger equation associated with a symmetric Hamiltonian. When compared to the standard Schr\"odinger equation for a wavefunction, we obtain corrections in terms of the acceleration of the system's center of mass and curvature of spacetime along its trajectory. In summary, we provide a formalism for the use of a complex wavefunction to describe a localized quantum particle in curved spacetimes.