Current Renormalization in Finite Volume

For finite volume field theories with discrete translational invariance, conserved currents can be additively renormalized by infrared effects. We demonstrate this for pions using chiral perturbation theory coupled to electromagnetism in a periodic box. Gauge invariant single particle effective theories are constructed to explain these results. In such theories, current renormalization arises from operators involving the zero mode of the gauge field. No contradictions with Ward identities, or low-energy theorems are encountered.