A Gauge Theory of Turbulence:

Abstract

We formalize the gauge structure of the Navier--Stokes equation for incompressible fluids, interpreting $\nabla \cdot \mathbf{v}=0$ as a gauge fixing analogous to the Coulomb gauge in electromagnetism. We construct the Martin--Siggia--Rose action and its full BRST symmetry, introducing BRST doublets for the Gribov parameter $\gamma$ and the monodromy phase $\theta$. Through a Higgs mechanism, $\gamma$ acquires a vacuum expectation value ${\gamma }_0$, generating a mass scale for the vorticity and the ghosts; we compute the one-loop effective potential and analyze vacuum stability. Intermittency --- measured by the exponents ${\zeta }_n$ --- is described by fluctuations of the Higgs field $\sigma$ around the condensate, via a Gribov-inspired log-Poisson hierarchy that satisfies exactly ${\zeta }_3=1$ (Kolmogorov's four-fifths law). Fits to DNS data favor filamentary vorticity structures, with $D_f\approx 1$ (one parameter) or $D_f\approx 2.1$, $\Delta \approx 0.44$ (two parameters). The one-loop anomalous dimension in $d=3$, ${\Delta }_{\sigma }\approx 0.50$, supports the identification $D_f=2{\Delta }_{\sigma }$. The formalism unifies classical hydrodynamics with gauge theory and spontaneous symmetry breaking, opening the study of turbulence and intermittency to quantum field theory methods.

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