The scattering amplitudes should be unchanged if we make a field redefinition. Suppose, for example, we have
L=−21∂μφ∂μφ−21m2φ2,(10.15)
and we make the field redefinition
φ→φ+λφ2.(10.16)
Work out the lagrangian in terms of the redefined field, and the corresponding Feynman rules. Compute (at tree level) the φφ→φφ scattering amplitude. You should get zero, because this is a free-field theory in disguise. (At the loop level, we also have to take into account the transformation of the functional measure Dφ; see section 85.)