A random field that is empirically equivalent to the quantized electromagnetic field is constructed. A mapping between the creation and annihilation operator algebras of a random field and of the quantized electromagnetic field provides a functor between the algebras and the Hilbert spaces generated by the vacuum states over those algebras. The functor inevitably does not extend to a functorial relationship between the local algebras generated by the random field and by the quantized electromagnetic field, but the empirical content provided by the vacuum state restores an empirical equivalence through the Hilbert spaces. The isomorphism from one creation and annihilation algebra to the other is not translation invariant because it depends on mapping positive frequency modes of one helicity to equivalent negative frequency modes, but the two theories taken independently are presented in equally well-defined and manifestly Lorentz and translation covariant ways.
We propose a scheme for the generation of an ion vortex beam and we study the properties of its electromagnetic field on a plane transverse to its propagation direction. The ion vortex is created by the diffraction of an ionic beam by a light mask with a spiral-like transverse intensity pattern. The diffraction is a result of a short time optical dipole interaction between the light mask and the ion which results to a phase imprint on the ionic wave function. It is shown that as the order of the vortex states that are generated by the various diffraction gets higher the electromagnetic field components have reduced magnitudes. The properties of these fields are studied against the various parameters of the problem.