  We classify all possible allowed constitutive relations of relativistic
fluids in a statistical mechanical limit using the Schwinger-Keldysh effective
action for hydrodynamics. We find that microscopic unitarity enforces genuinely
new constraints on the allowed transport coefficients that are invisible in the
classical hydrodynamic description; they are not implied by the second law or
the Onsager relations. We term these conditions Schwinger-Keldysh positivity
and provide explicit examples of the various allowed terms.
