Long-range potentials and the electromagnetic polarizabilities

The long-range spin and velocity independent forces of electromagnetic origin which act between any two systems are studied for those cases in which no forces of this type exist to order e². It is shown that they are uniquely determined by the charge, magnetic moment, and polarizabilities of both systems, not only to the dominant order r^?n, but also to the next one r^?(n+1). These potentials provide the link between Compton scattering polarizabilities (response to real photons) and classically defined polarizabilities (response to static electromagnetic field). The two definitions are shown to be equivalent for neutral spinless systems; the problems arising for a neutral particle with magnetic moment are studied in detail. The r^?(n+1) terms have no classical counterpart, since they are due to the relativistic quantum propagation of the system which carries charge or magnetic moment. The results are of general validity with analyticity, crossing, unitarity, and gauge invariance as only inputs. The most general conclusion is that the polarizabilities represent electromagnetic properties of a system at order e², as the charge and magnetic moment do at order e. Thus they give the strength of the response to electric and magnetic fields, independently of the specific characteristics of the electromagnetic agent.