Perfect magnetohydrodynamics as a field theory

TL;DR

This paper formulates perfect magnetohydrodynamics (MHD) as a covariant field theory using a complex scalar field and electromagnetism, revealing new conserved quantities and quantization conditions related to circulation and helicity.

Contribution

It introduces a covariant action for MHD based on a scalar field, deriving classical MHD equations, conserved quantities, and quantization conditions, which are novel in the field theory context.

Findings

Derivation of MHD equations from a covariant scalar field action

Identification of a quantized circulation related to the scalar field's single-valuedness

Discovery of new conserved helicity and Bernoulli-like theorems in MHD

Abstract

We propose the generally covariant action for the theory of a self-coupled complex scalar field and electromagnetism which by virtue of constraints is equivalent, in the regime of long wavelengths, to perfect magnetohydrodynamics (MHD). We recover from it the Euler equation with Lorentz force, and the thermodynamic relations for a prefect fluid. The equation of state of the latter is related to the scalar field's self potential. We introduce 1+3 notation to elucidate the relation between MHD and field variables. In our approach the requirement that the scalar field be single valued leads to the quantization of a certain circulation in steps of $\hbar$; this feature leads, in the classical limit, to the conservation of that circulation. The circulation is identical to that in Oron's generalization of Kelvin's circulation theorem to perfect MHD; we here characterize the new conserved helicity associated with it. We also demonstrate the existence for MHD of two Bernoulli-like theorems for each spacetime symmetry of the flow and geometry; one of these is pertinent to suitably defined potential flow. We exhibit the conserved quantities explicitly in the case that two symmetries are simultaneously present, and give examples. Also in this case we exhibit a new conserved MHD circulation distinct from Oron's, and provide an example.