# Does there exist a generalization of the LSZ theorem for scattering particles of any (including half-integer) spin?

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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

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

edited Apr 19, 2015

Yes there does, LSZ is one of the corner stones of any modern teaching of QFT, so you should expect it to have enough generality. The proof given in Weinberg's textbook covers arbitrary spin, but somewhat obscures the connection with asymptotic states, and given the bullet points you presented, you are probably more comfortable with the proof given in these slides, fermion case starts from page 29.

Yes, of course there exist generalisations of LSZ reduction to other spins, see e.g.  for Dirac fields. edit: @JiaYiyang Oops, I was writing at the same time.

As far as the usual QM can be cast into the occupation number representation, there should be some analogue to LSZ formula for scattering in QM. What may be the difference between QM and QFT in this respect?

The question was asked for arbitrary spin.

What happens in the spin 1 case? Slide 46 of Romao's slides pointed to by Yiyang gives the answer for photons, massless U(1) spin 1 particles. What in the massive case? For nonabelian gauge theories?

What happens for asymptotic composite states? These are good asymptotic states, so there should be S-matrix elelemts. In particular, hadrons in QCD can have quite high spin content. So there should be a formula for massive particles of arbitrary spin.

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