Time decay of maxwell—klein—gordon equations in 4—dimensional minkowski space

. In this paper I derive a gauge invariant decay estimate of the solutions of massive Maxwell—Klein—Gordon fields equations in the 4—dimensional Minkowski space, provided that the initial energy of the system is bounded. This estimate implies that the Klein—Gordon field decays to zero in the local L2 norm. I also show that the local energy decays. The proof is based on gauge invariant energy identities.