point out, for reasons not well understood, for $\theta ^{\mu\nu}\neq 0$, the LSZ S-matrix differs from the interaction representation S-matrix and leads to different cross sections.

The LSZ Formalism for Twisted Quantum Theories
==============================================

In the next two subsections below, we review scattering theory, including the LSZ formalism for standard (untwisted fields). We then generalise the discussion to the twisted case.

Formal scattering theory
------------------------

In standard scattering theory, the Hamiltonian $H$ is split into a “free” Hamiltonian $H_0$ and an “interaction” piece $H_I$, $$H = H_0 + H_I,$$ and $H_0$ is used to define the states in the infinite past and future. Then the states at $t=0$ which in the infinite past (future) become states evolving by $H_0$ are the in (out) states: $$\begin{aligned}
e^{-i H T_-}|\psi; {\rm in}\rangle &\stackrel{T_- \rightarrow -\infty}
\longrightarrow& e^{-i H_0 T_-} |\psi; {\rm F}\rangle, \quad {\rm F}
\equiv {\rm free} \\
e^{-i H T_+}|\psi; {\rm out}\rangle &\stackrel{T_+ \rightarrow +\infty}
\longrightarrow& e^{-i H_0 T_+} |\psi; {\rm F}\rangle.\end{aligned}$$

Hence $$\begin{aligned}
|\psi; {\rm in}\rangle &=& \Omega_+ |\psi;{\rm F}\rangle, \\
|\psi; {\rm out}\rangle &=& \Omega_- |\psi;{\rm F}\rangle, \\
\Omega_\pm &\equiv& e^{i H T_\mp} e^{-i H_0 T_\mp}, \quad {\rm as} \quad
T_\mp \rightarrow \mp \infty, \\
&=& {\rm M\phi 