  The conserved magnetic flux of U(1) electrodynamics coupled to matter in four
dimensions is associated with a generalized global symmetry. We study the
realization of such a symmetry at finite temperature and develop the
hydrodynamic theory describing fluctuations of a conserved 2-form current
around thermal equilibrium. This can be thought of as a systematic derivation
of relativistic magnetohydrodynamics, constrained only by symmetries and
effective field theory. We construct the entropy current and show that at first
order in derivatives, there are seven dissipative transport coefficients. We
present a universal definition of resistivity in a theory of dynamical
electromagnetism and derive a direct Kubo formula for the resistivity in terms
of correlation functions of the electric field operator. We also study
fluctuations and collective modes, deriving novel expressions for the
dissipative widths of magnetosonic and Alfven modes. Finally, we demonstrate
that a non-trivial truncation of the theory can be performed at low
temperatures compared to the magnetic field: this theory has an emergent
Lorentz invariance along magnetic field lines, and hydrodynamic fluctuations
are now parametrized by a fluid tensor rather than a fluid velocity.
Throughout, no assumption is made of weak electromagnetic coupling. Thus, our
theory may have phenomenological relevance for dense electromagnetic plasmas.
