Optical helicity and Hertz vectors

Abstract: We study the conserved quantity associated with the dual symmetry of the Maxwell equations, called the optical helicity, by means of transverse Hertz vectors. In the presence of charges, its evolution yields the integral of $\bm{E}\cdot\bm{B}$ which is the anomaly term for chiral fermions. We also discuss the helicity change in condensed matter systems where topological magnetoelectric effect emerges. An alternative expression of the optical helicity is also found. Lastly, a dual symmetric Hertz Lagrangian is constructed and its conserved charge is derived.