Translational Inertial Spin Effect with Photons

The Borgnis technique is used to solve the Maxwell equations under the following conditions: (a) Complex field strengths are used such that $\mathbf{H}=\pm i\mathbf{E}$, that is, circularly polarized waves of positive or negative helicity; (b) there is only harmonic time dependence through a factor $\exp \mathrm{}\left(\mathrm{iWt}\right)\mathrm{}$, that is, a pure energy state of the photons; (c) there is no $z$ dependence, that is, no $k_z$ component of the photons' momentum; (c) there is no restriction on the $x$, $y$ dependence of the solutions. It is then shown that the $S_z$ component of the Poynting vector obeys the formula

$\pm 2W{S}_z={\partial }_x{S}_y-{\partial }_y{S}_x$

and is in general nonzero. This fact, together with the postulated nullity of $k_z$, is the expression of the "translational inertial spin effect." An experiment using the limiting case of total reflection is proposed to test the effect. A discussion of gauge-dependent expressions of the effect, using potentials, is also given, in connection with de Broglie's formulas for the current- and spin-density 4-vectors of the photon waves.