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

In a QFT, the S-matrix $S$ is a unitary operator, that fixes the vacuum and commutes with the unitary operators implementing the action of the Poincare group on an appropriate Hilbert space $H$.

What is the physical interpretation of the automorphism on the space of bounded operators $B(H)$ that is induced by conjugation with $S$, i.e. $$\mathrm{Ad}(S) : B(H)\to B(H), O\mapsto SOS^\dagger$$ ? It seems to me that since $S$ fixes the vacuum that one could recover it from $\mathrm{Ad}(S)$ as the unique unitary operator that fixes the vacuum and implements $\mathrm{Ad}(S)$. If so, what is it called and what are some decent references? Also, $\mathrm{Ad}(S)$ is a unital automorphism of $B(H)$ that is Poincare covariant and preserves the vacuum state (the vector state of the vacuum). Thus it seems that $\mathrm{Ad}(S)$ is a Poincare covariant quantum channel from $B(H)$ to itself.

Is it standard to consider $\mathrm{Ad}(S)$ in physics? I imagine one might think of this as a limiting automorphism of evolution automorphisms from the distant past to the distant future, but would like confirmation from an honest physicist.

• 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 – yuggib Apr 13 '15 at 8:26
• @yuggib: Pardon the glibness of my question. I am interested in asymptotically complete theories. Regarding (2), I especially owe an apology as I did not check the claim sufficiently well...it may indeed be false. Although the vacuum may be cyclic and separating for local algebras in the Haag-Kastler picture in "nice" situations, this is probably not true for the ambient B(H)...part of my question might be whether I can hope for this ever to be true in an interesting situation. Finally, are you saying Ad(S) is used in stat mech. for an S-matrix or operators in general? Thanks for the comment! – Jon Bannon Apr 13 '15 at 13:28
• Also, I am really only interested in the factorizable case...not so much in non integrable models.. – Jon Bannon Apr 13 '15 at 13:49
• link.springer.com/chapter/10.1007/978-3-642-97306-2_12#page-1 has a comment in the top paragraph that seems related. – Arnold Neumaier Apr 13 '15 at 15:24
• scholar.google.at/… gives a number of possibly relevant references. – Arnold Neumaier Apr 13 '15 at 15:24