  We present an alternative Eulerian hydrodynamic model for the electromagnetic
field in which the discrete vector indices in Maxwell\s equations are replaced
by continuous angular freedoms, and develop the corresponding Lagrangian
picture in which the fluid particles have rotational and translational
freedoms. This enables us to extend to the electromagnetic field the exact
method of state construction proposed previously for spin 0 systems, in which
the time-dependent wavefunction is computed from a single-valued continuum of
deterministic trajectories where two spacetime points are linked by at most a
single orbit. The deduction of Maxwell\s equations from continuum mechanics is
achieved by generalizing the spin 0 theory to a general Riemannian manifold
from which the electromagnetic construction is extracted as a special case. In
particular, the flat-space Maxwell equations are represented as a curved-space
Schr\"odinger equation for a massive system. The Lorentz covariance of the
Eulerian field theory is obtained from the non-covariant Lagrangian-coordinate
model as a kind of collective effect. The method makes manifest the
electromagnetic analogue of the quantum potential that is tacit in Maxwell\s
equations. This implies a novel definition of the \classical limit\ of
Maxwell\s equations that differs from geometrical optics. It is shown that
Maxwell\s equations may be obtained by canonical quantization of the classical
model. Using the classical trajectories a novel expression is derived for the
propagator of the electromagnetic field in the Eulerian picture. The trajectory
and propagator methods of solution are illustrated for the case of a light
wave.
