https://doi.org/10.1140/epjc/s10052-025-14205-x
Regular Article - Theoretical Physics
The Wigner formalism on black hole geometries
1
Departamento de Física Teórica, Facultad de Ciencias Físicas, Universidad Complutense de Madrid, Plaza de Ciencias 1, 28040, Madrid, Spain
2
Facultad de Ciencias Físicas, Institute of Particle and Cosmos Physics (IPARCOS), Universidad Complutense de Madrid, Plaza de Ciencias 1, 28040, Madrid, Spain
Received:
26
December
2024
Accepted:
17
April
2025
Published online:
2
May
2025
This work explores the intersection of quantum mechanics and curved spacetime by employing the Wigner formalism to investigate quantum systems in the vicinity of black holes. Specifically, we study the quantum dynamics of a probe particle bound to a Schwarzschild black hole using a phase-space representation of quantum mechanics. The analysis begins with a review of the covariant Wigner function in curved spacetime, highlighting its application to spherically symmetric, uncharged black holes. We then derive an effective potential from the Schwarzschild metric, which defines the Hamiltonian for the electron. Relativistic corrections are treated perturbatively to estimate energy levels and associated Wigner functions for the bound state. Additionally, we compare the results obtained through the Schrödinger equation with those derived directly using the symplectic formalism, demonstrating the consistency and versatility of the phase-space approach. The study sheds light on quantum behavior near black holes and suggests new avenues for combining quantum kinetic theory with relativistic gravitational settings.
© The Author(s) 2025
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