Conformal Transformations of the Wigner Function and Solutions of the Quantum Corrected Vlasov Equation

TL;DR

This paper investigates how the quantum kinetic equations, specifically the Wigner function and Vlasov equation, behave under conformal transformations in curved spacetime, revealing invariance properties and deriving quantum corrections.

Contribution

It establishes the conformal invariance of the quantum corrected Vlasov equation and derives transformation laws for the covariant Wigner function in curved spacetime.

Findings

Derived a transformation law for the covariant Wigner function under conformal changes.

Proved the conformal invariance of the quantum corrected Vlasov equation.

Identified quantum corrections to locally isotropic distributions and their stress-energy tensor structures.

Abstract

We study conformal properties of the quantum kinetic equations in curved spacetime. A transformation law for the covariant Wigner function under conformal transformations of a spacetime is derived by using the formalism of tangent bundles. The conformal invariance of the quantum corrected Vlasov equation is proven. This provides a basis for generating new solutions of the quantum kinetic equations in the presence of gravitational and other external fields. We use our method to find explicit quantum corrections to the class of locally isotropic distributions, to which equilibrium distributions belong. We show that the quantum corrected stress--energy tensor for such distributions has, in general, a non--equilibrium structure. Local thermal equilibrium is possible in quantum systems only if an underlying spacetime is conformally static (not stationary). Possible applications of our results are discussed.