Consequences of Guage Invariance for Radiative Transitions

Consideration is given to any system of particles whose behavior under the influence of an external electromagnetic field can be described by a guage invariant Schroedinger equation. Detailed restrictions on the form of the hamiltonian which are imposed by the condition of guage invariance are derived. These provide a simple means to the solution of many problems of the interaction of a system with the electromagnetic field. In particular the following consequences are established: (1) In multipole expansions for single photon processes the electric multipole operators have the usual form but the form of the magnetic multipole operators may depend in a detailed way on the interactions between particles and electromagnetic field. (2) The $f$-sum rule can be expressed in closed form in terms of the interactions. (3) A generalization of the $f$-sum rule to all electric multipole orders is given. (4) The cross section for scattering of a low energy photon can be expressed in terms of the electrostatic polarizability quite independently of the interactions. Applications of these methods to problems in nuclear physics are given in an accompanying paper.