  We discuss the reactions $\pi p \to \pi p$ and $\pi p \to \pi p \gamma$ from
a general quantum field theory (QFT) point of view. We consider the pion-proton
elastic scattering both off shell and on shell. The on-shell amplitudes for
$\pi^{\pm} p \to \pi^{\pm} p$ scattering are described by two invariant
amplitudes, while the off-shell amplitudes contain eight invariant amplitudes.
We study the photon emission amplitudes in the soft-photon limit where the
photon energy $\omega \to 0$. The expansion of the $\pi^{\pm} p \to \pi^{\pm} p
\gamma$ amplitudes to the orders $\omega^{-1}$ and $\omega^{0}$ is derived.
These terms can be expressed by the on-shell invariant amplitudes and their
partial derivatives with respect to $s$ and $t$. The term of order
$\omega^{-1}$ is well known from the literature. Our term of order $\omega^{0}$
is new. The formulas given for the amplitudes in the limit $\omega \to 0$ are
valid for both real and virtual photons. We also discuss the behavior of the
corresponding cross-sections.
