Quantum Hydrodynamics in External Magnetic Fields: From Nonrelativistic to Relativistic Regimes

We explore nonrelativistic quantum electrodynamics in a background magnetic field, maintaining gauge invariance through a vector potential. The Lagrangian incorporates electron self-interactions and electromagnetic field interactions. The path integral formalism elucidates the effective action, focusing on the photon propagator and screening effects. Analyzing density and current components reveals Laurent expansions in the hydrodynamic limit. Equations of motion in real space are derived, including a discussion on Poiseuille-like flow and the Navier-Stokes equation. Nonrelativistic and relativistic hydrodynamics are discussed, with linearized equations presented. The study concludes with insights into the nonrelativistic limit, providing a comprehensive framework for quantum electrodynamics analysis.