  A complete and explicit classification of all independent local conservation
laws of Maxwell's equations in four dimensional Minkowski space is given.
Besides the elementary linear conservation laws, and the well-known quadratic
conservation laws associated to the conserved stress-energy and zilch tensors,
there are also chiral quadratic conservation laws which are associated to a new
conserved tensor. The chiral conservation laws possess odd parity under the
electric-magnetic duality transformation of Maxwell's equations, in contrast to
the even parity of the stress-energy and zilch conservation laws. The main
result of the classification establishes that every local conservation law of
Maxwell's equations is equivalent to a linear combination of the elementary
conservation laws, the stress-energy and zilch conservation laws, the chiral
conservation laws, and their higher order extensions obtained by replacing the
electromagnetic field tensor by its repeated Lie derivatives with respect to
the conformal Killing vectors on Minkowski space. The classification is based
on spinorial methods and provides a direct, unified characterization of the
conservation laws in terms of Killing spinors.
