The Wigner formalism on black hole geometries

TL;DR

This paper applies the covariant Wigner formalism to analyze quantum dynamics of particles near Schwarzschild black holes, bridging quantum mechanics and curved spacetime.

Contribution

It introduces a phase-space approach to study quantum bound states in black hole geometries, including relativistic corrections and comparison with Schrödinger solutions.

Findings

Derived effective potential from Schwarzschild metric for bound states.

Estimated energy levels and Wigner functions for particles near black holes.

Demonstrated consistency between phase-space and Schrödinger equation results.

Abstract

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 Schrodinger 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.