  It is explicitly shown, for optical processes arbitrarily comprising two-,
three- or four-photon interactions, that the sum over all matter states of any
optical susceptibility is exactly zero. The result remains true even in
frequency regions where damping is prominent. Using a quantum electrodynamical
framework to render the photonic nature of the fundamental interactions, the
result emerges in the form of a traceless operator in Hilbert space. The
generality of the sum rule and its significance as a thermodynamic limit are
discussed, and the applicability to real systems is assessed.
