Consequences of Gauge Invariance for Electromagnetic Transitions. II

Some theorems developed by Sachs and Austern, regarding the electromagnetic properties of gauge-invariant Schrödinger systems, are generalized so that they may be applied to any gauge-invariant, bound system of particles which is describable by a normalizable state vector. In particular, the following theorems are established: (1) At low photon energy, purely electric radiative or absorptive transitions will depend only on the initial and final state vector of the matter system, and on the energy change (the Siegert theorem). (2) $f$-sum rules are obtained for electric multipole radiation of arbitrary order, and are expressed in terms of the electromagnetic interactions of the system. (3) If the static moments of the particle system are neglected, the cross section for gamma-ray scattering will depend on the fourth or higher power of the photon energy in the low energy limit. Theorem (3) shows that the energy-independent term in the transition amplitude computed by Sachs and Foldy for the scattering of gamma rays from nucleons, occurs because the nucleon state vector used in their work is not normalizable.