Gauge invariance of the helicity continuity equation

Abstract

The derivation of the helicity continuity equation in electromagnetic theory is performed without specifying a gauge. In contrast with previous proposals, the form of this equation is shown to be gauge invariant without invoking a Helmholtz decomposition. The helicity and its flow, the latter associated with the spin in quantized fields, involve two sets of a vector and a scalar potential, where each set can independently undergo a gauge transformation. There are alternative definitions of the helicity and flow densities that arise from different grouping of terms in the continuity differential equation. The various definitions acquire an unambiguous meaning, depending on the gauge and the physical context. The helicity density, defined as ϱAC2≔μεA⋅B−C⋅E and flow density JAC2≔μεE−∇ϕA×A+B−∇ϕC×C, include all the rotational content of the free fields regardless of the gauge. In free space, these quantities satisfy a gauge invariant conservation equation without gauge-fixing source terms. A further asset of the present formulation is that charge and current source terms can be readily incorporated. The helicity source terms are of the form μB⋅∫Jdt−μ∫Bdt⋅J. A helicity continuity equation in terms solely of transverse fields is derived in the Coulomb gauge.