# What picture is the $S$-matrix defined in?

I am looking into the definition of the $S$-matrix, and have found two different cases. Firstly I have seen it derived that (see here, link to Google books p110): $$S=U_S(\infty,-\infty)$$ But more commonly that (see here, link to Google books p169): $$S=U_I(\infty,-\infty)$$ (Where subscript $S$ represents the Schrödinger picture and $I$ the interaction). Are these two definitions identical? If so can this be shown and if not why two different definitions? Either way the latter seems to be preferred, why?

## Further Information/My guess:

This may come down to how we define $S$ from what I can tell it is define such that if your particle is in the Schrödinger state $|\psi_S(t)\rangle$ and you want to find the amplitude that at $t=\infty$ it is in the state $|\phi_S(t)\rangle$ then $S$ is defined such that: $$\langle\psi_S(\infty)|S|\psi_S(-\infty)\rangle=\langle\psi_S(\infty)|\psi_S(\infty)\rangle$$ which would indicate my former rather then my latter definition. Thus different authors may be defining it in different ways?

• Well, the S-matrix corresponds to matrix elements, and you are free to use any picture to compute those, expecting, of course, to find the same answer; so, typically, the Schroedinger, Heisenberg, or even interaction picture. There are advantages and disadvantages to each, but the WP article details them all. – Cosmas Zachos Sep 19 '16 at 16:05