If Λ is a (inhomogeneous) proper orthochronous Lorentz transformation, then Wigner's theorem guarantees the existence of a unitary operator U(Λ) acting either on HiorHf. A theory is said to be Lorentz invariant if the same U(Λ) acts on HiandHf. Using the unitarity of U(Λ), Sβα = ⟨i, β|f, α⟩ = ⟨i, β|U(Λ)†U(Λ)|f, α⟩. The right-hand side can be expanded using knowledge about how the non-interacting states transform to obtain an expression, and that expression is to be taken as a definition of what it means for the S-matrix to be Lorentz invariant. See Weinberg (2002), equation 3.3.1 gives an explicit form. Weinberg, S. (2002), The Quantum Theory of Fields, vol I, Cambridge University Press, ISBN0-521-55001-7
Here the postulate of asymptotic completeness is employed. The in and out states span the same Hilbert space, which is assumed to agree with the Hilbert space of the interacting theory. This is not a trivial postulate. If particles can be permanently combined into bound states, the structure of the Hilbert space changes. See Greiner & Reinhardt 1996, section 9.2. Greiner, W.; Reinhardt, J. (1996), Field Quantization, Springer Publishing, ISBN3-540-59179-6
