Conditions required to make gauge-invariant the electromagnetic spin and orbital angular momenta and helicity densities in isotropic and anisotropic media

The spin and orbital angular momenta densities (sn and ln) and helicity density (hn) of the electro- magnetic (EM) fields are defined using an electric vector potential C in analogy with the magnetic vector potential A, where C is introduced to observe the duality invariance and electric-magnetic democracy. However, their definitions for sn, ln and hn have an essential drawback (lacking for the gauge invariance) that they vary widely depending on the choice of two kinds of A and C. To solve this problem, it is necessary to make sn, ln and hn gauge-invariant. The gauge invariance for sn, ln and hn in isotropic and anisotropic media can be established by an appropriate combination of gauge functions and scalar potentials. As a result, it is proved in a distinct way from the second quantization procedure that the gauge invariance for sn and ln depending on the phase difference between A and C leads to the quantization of electromagnetic spin. Moreover, the gauge-invariant hn gives a reasonable explanation for the Faraday rotation caused in an anisotropic medium.