# Relation between Borchers class and the LSZ formula on S-matrix equivalence

The comment in my book (cited by Diffycue) originated in my own puzzlement when I was a student and reading my own favourite QFT book: that by K. Nishijima, *Fields and Particles: Field Theory and Dispersion Relations*. In it he seems to use the LSZ technique show that any two fields $\phi_i$ and $\phi_2$ that have matrix elemements between the vacuum and the one particle states of the form
$$
\langle k\vert \phi_i(x)\vert 0\rangle = \sqrt Z_i e^{ikx}
$$
will have the same $S$-matrix. Since this is a property possesed by $\phi_{\rm in}(x)$ and $\phi_{\rm out}(x)$, which are free, he seemed to have proved that all theories were free. It took me ages and a very careful reading of his argument to realize that he made use of Lorentz transformation properties of time-ordered correlators such as
$$
\langle 0\vert T\{ \phi_1(x_1) \phi_2(x_2)\ldots \}\vert0\rangle
$$
that are messed up unless $\phi_1$ and $\phi_2$ commute outside the lightcone. Thus Nishijima was tacitly assuming the mutual locality of the various fields.

As a consequence $\phi$ and $\phi+\phi^3$ will have the same $S$ matrix for $\lambda \phi^4$ theories with their ${\mathbb Z}_2$ symmetry, but $\phi$ and $\phi^2$ will not because $\phi^2$ cannot couple the vacuum to the $\phi$ state because it has the wrong ${\mathbb Z}_2$ parity.