The LSZ theorem for calculating the S-matrix

\[\begin{eqnarray} \langle f ¦ S ¦ i \rangle & = & \left [ i \int d^4x_1 e^{-ip_1x_1}(\square + m^2 ) \right ] \dotsc \left [ i \int d^4x_n e^{-ip_nx_n}(\square + m^2 ) \right ] \\ & & \times \langle \Omega¦ T(\phi(x_1) \dotsc \phi(x_n)) ¦ \Omega \rangle \end{eqnarray} \]

is as I understand it only valid for Klein-Gordon fields as can for example be seen by the appearance of the operator $(\square + m^2)$.

**Is there a generalization of this theorem which allows to calculate the S-matrix for scattering events of particles of any (including half-integer) spin?**

Looking at the derivation of the LSZ formula in chapter 6.1 of this book, it seems that the following things should be considered to derive an LSZ formula for fermions for example:

- The initial $¦ i\rangle$ and final $¦f\rangle$ asymptotic state should be constracted by using fermionic creation and annihilaton operators
- The Fourier decomposition of scalar fields should be replaced by the decomposition of spinorial fields
- The dispersion relation $\omega_p = \sqrt{\vec{p}^2 + m^2}$ would still be valid (?)
- In the imporant relation $i\int d^4 x e^{ipx}(\square + m^2 ) \phi(x) = \sqrt{2\omega_p}[a_p(\infty)] -a_p[-\infty]$ the the operator $\square + m^2$ should additionally be replaced by $i\gamma^{\mu}\partial_{\mu} -m$
- When time-ordering the fermionic creation and annihilation operators (or spinorial fields) one would need to carefully keep track of sign changes

Just from these naive considerations I would expect an LSZ formula for fermions to look like something as

\(\begin{eqnarray} \langle f ¦ S ¦ i \rangle & = & \left [ i \int d^4x_1 e^{-ip_1x_1}(i\gamma^{\mu}\partial_{\mu} -m) \right ] \dotsc \left [ i \int d^4x_n e^{-ip_nx_n}(i\gamma^{\mu}\partial_{\mu} -m ) \right ] \\ & & \times \langle \Omega¦ T(\psi(x_1) \dotsc \psi(x_n)) ¦ \Omega \rangle \end{eqnarray} \)