Derivation and local well-posedness of a relativistic quantum hydrodynamic system on the Heisenberg group

TL;DR

This paper derives a relativistic quantum hydrodynamic system on the Heisenberg group, addressing analytical challenges like vacuum singularities, and proves local existence and uniqueness of solutions.

Contribution

It introduces a new geometric formulation of RQHD on the Heisenberg group and establishes a framework for analyzing singular limits on nilpotent Lie groups.

Findings

Established uniform energy estimates for the extended system.

Proved local-in-time existence and uniqueness of classical solutions.

Reformulated the system to handle vacuum singularities.

Abstract

We derive and analyze a relativistic quantum hydrodynamic (RQHD) system on the Heisenberg group. Starting from the Klein--Gordon--Poisson system, we apply the Madelung transformation to obtain a fluid-type model in which the relativistic and quantum parameters are explicitly separated. The Heisenberg-group structure gives rise to an additional geometric term in the momentum equation, reflecting the underlying noncommutative structure. A central analytical difficulty is the possible appearance of vacuum, where the phase function and the quantum potential become singular. To address this issue, we reformulate the RQHD system as an extended hyperbolic--elliptic system with auxiliary variables. For this extended system, we establish uniform higher-order energy estimates on $\mathbb H^1$ by combining the Banach algebra property of sub-elliptic Sobolev spaces with noncommutative Fourier analysis. We then prove that the extended system is equivalent to the original RQHD system at the level of classical solutions. As a consequence, we obtain the local-in-time existence and uniqueness of non-vacuum classical solutions to the RQHD system on $\mathbb H^1$. The result also provides a framework for the study of related singular limits, including the semiclassical and non-relativistic limits on nilpotent Lie groups.