Question

What is the physical interpretation of the automorphism on bounded operators induced by an S matrix?

Answer 1

Some comments (that I may expand into an answer if I have time): 1) the S matrix is not always unitary, its unitarity is equivalent to the so-called asymptotic completeness and it is in general very difficult to prove; 2) it seems unlikely that the properties you enlisted are sufficient to fix the S-matrix in a unique fashion (different operators may have the same vacuum/ground state for example, also the identity operator satisfies all your other properties); 3) Ad(S) is considered in the language of C∗ algebras, that is utilized mostly in statistical mechanics