A new conserved current for electromagnetic fields in the Kerr spacetime

For any classical linear Lagrangian field theory, the symplectic product provides a conserved current that is bilinear on the space of solutions. Given a linear mapping from the space of solutions into itself, a ``symmetry operator'', one can therefore generate quadratic conserved currents for any linear classical field theory. We apply this procedure to the case of electromagnetism on a Kerr background, showing that this procedure generates the conserved currents given in Andersson, B\"ackdahl, and Blue, as well as a new conserved current. These currents reduce to the sum of the Carter constants of the photons in the geometric optics limit, and generalize the current for scalar fields discovered by Carter.