A Field Theory Framework of Incompressible Fluid Dynamics

This study develops an effective theoretical framework that couples two vector fields: the velocity field $\mathbf{u}$ and an auxiliary vorticity field $\boldsymbol{\xi}$. Together, these fields form a larger conserved dynamical system. Within this framework, the incompressible Navier-Stokes (NS) equation and a complementary vorticity equation with negative viscosity are derived. By introducing the concept of light-cone vorticity $\boldsymbol{\eta}_\pm = \mathbf{w} \pm \boldsymbol{\xi}$, the paper constructs a unified framework for coupled dynamics. Furthermore, it explores the mechanism of spontaneous symmetry breaking from $SU(2)$ gauge theory to $U(1) \times U(1)$, which leads to the emergence of the coupled vector field theory in the non-relativistic limit. This approach uncovers a connection between fluid dynamics and fundamental gauge theories, suggesting that the NS equations describe a subsystem where dissipation results from energy transfer between the velocity and auxiliary fields. The study concludes by linking the complete dynamical framework to the Abrikosov-Nielsen-Olesen-Zumino (ANOZ) theory, a non-Abelian generalization of Bardeen-Cooper-Schrieffer (BCS) theory, offering new insights into fluid dynamics and quantum fluid theory.