Galilean invariance of the pion-nucleon interaction

We examine the role of Galilean invariance in a nonrelativistic theory of pions and nucleons. We define a nonrelativistic pion-nucleon interaction $h$ as one which in perturbation theory gives the same matrix for a physical reaction as the limit of small nucleon velocities of the relativistic matrix. The latter matrices are always Galilean invariant, but this does not require that $h$ be Galilean invariant. If pion emission or absorption occurs from a nucleon moving in a potential, then the Galilean correction term can be shown to be ambiguously of order $\frac{v}{c}$ or of order $\frac{v^2}{c^2}$. We show that $h$ cannot always reproduce the nonrelativistic limit of a relativistic matrix. Finally, we suggest that pion production by nucleons on nuclei with excitation of giant dipole and quadrupole states may be particularly sensitive to the presence of a Galilean correction term in the production matrix.