Relativistic invariance of the two-photon transition amplitude of a composite system

Using a new approach to the problem of the interaction of a composite system with a quantized radiation field, an expression for the two-photon amplitude of a composite system is derived in its center-of-mass frame, correct to order $v^2$/$c^2$. In the expression for the amplitude there are terms which depend on the recoil of the composite system due to the emission of the photons. These terms which are of order $v^2$/$c^2$ compared to the nonrelativistic terms were neglected in the previous relativistic treatments of the problem. We also show that the amplitude is relativistically invariant to order $v^2$/$c^2$, when viewed from another Lorentz frame. We have used the relativistic center-of-mass and internal variables of Krajcik and Foldy [R. A. Krajcik and L. L. Foldy, Phys. Rev. D 10, 1777 (1974)] throughout the treatment.