  We derive positivity bounds for scattering amplitudes of particles with
arbitrary spin using unitarity, analyticity and crossing symmetry. The bounds
imply the positivity of certain low-energy coefficients of the effective action
that controls the dynamics of the light degrees of freedom. We show that
low-energy amplitudes strictly softer than $O(p^4)$ do not admit unitary
ultraviolet completions unless the theory is free. This enforces a bound on the
energy growth of scattering amplitudes in the region of validity of the
effective theory. We discuss explicit examples including the Goldstino from
spontaneous supersymmetry breaking, and the theory of a spin-1/2 fermion with a
shift symmetry.
