# Hydrodynamic construction of the electromagnetic field

Peter Holland

## Abstract

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 Schrödinger 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.

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## Footnotes

• The uniqueness of this identification needs careful discussion. For examination of some of the issues involved in an analogous problem for the Dirac field, see Holland (2003) and Holland & Philippidis (2003).

• The preservation of the identity of each fluid element (labelled by its initial position) is of fundamental importance. In the electromagnetic application it apparently constitutes an answer to the objection of Lorentz (2003) to the meaningfulness of energy flow lines, which is based upon a claimed loss of identity of individual energy elements when combining with others.

• For further discussion on the curved-space propagator, see Kleinert (2004).

• Received May 9, 2005.
• Accepted June 8, 2005.
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