A Wavefunction Description for a Localized Quantum Particle in Curved Spacetimes

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 Schrödinger equation associated with a symmetric Hamiltonian. When compared to the standard Schrödinger 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.

26 pages, 1 figure. RevTeX 4.1. V4. Added a more detailed explanation in Appendix C and fixed minor typos

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• Finite Pulse-Time Effects in Long-Baseline Quantum Clock Interferometry
• Unified laboratory-frame analysis of atomic gravitational-wave sensors
• General Relativistic Center-of-Mass Coordinates for Composite Quantum Particles