Wikipedia states that:

Although the LSZ reduction formula cannot handle bound states, massless particles and topological solitons

yet S-Matrix Elements, represented by Feynman-diagrams for processes with -in and outgoing photons are part of every book on QFT (I'm encountering these at the moment, bear with me, I didn't yet get accustomed to them).

Is Wikipedia wrong, and one can also state the LSZ-Reduction formula for photons? If so, can somebody provide a source to a derivation? (Schwartz only contains a derivation for scalars and fermions).

In case Wikipedia is wrong, why would somebody state something like the citation from wikipedia? Are there subleties?

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    $\begingroup$ (Also, see Srednicki, chapter 56 for the derivation) $\endgroup$ Jul 2, 2021 at 13:15
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    $\begingroup$ The Wikipedia statement is probably referring to IR divergences $\endgroup$ Jul 3, 2021 at 14:38

1 Answer 1


Wikipedia is correct. The derivation of Lehmann, Symanzik and Zimmermann is invalid if the theory contains long-range forces (massless gauge bosons). As pointed out by Nihar in the comments this is known as the infrared problem of QFT. It is (as far as I know) a still unsolved problem in the branch of mathematical physics that tries to construct a mathematically rigorous formulation of QFT. The best rigorous and at the same time (semi-)pedagogical treatment that I know of this topic can be found in O. Steinmann, "Perturbative Quantum Electrodynamics and Axiomatic Field Theory", https://link.springer.com/book/10.1007/978-3-662-04297-7, Chapter 6.3.

However, particle physics has a long and proud tradition of cutting corners on mathematical rigour and still being able to make correct predictions. Let's suspend our concerns for a moment and just apply to LSZ formulat to, say, QED. We then start calculating some scattering amplitudes and we eventually find that our amplitudes are divergent in some kinematic regions. These divergences are a sign that we've made a mistake: we've used the LSZ formula in a situation where it is not valid. We don't know how to fix the mistake (there's no version of the LSZ formula which holds for gauge theories), but we can deal with the divergences in a way that's very similar to the way we deal with ultraviolet (UV) divergences: we regularise them and cancel them out.

In QED the regularisation is rather straightforward. We can simply add a photon mass $\lambda$ to the theory. This means that for $\lambda > 0$ the LSZ formula actually applies. The IR divergences manifest as terms which diverge when $\lambda$ goes to zero.

To cancel the divergences you have to calculate sufficiently 'inclusive' cross sections. For example, one situation where you encounter an IR divergence is when you emit a very 'soft' (i.e. low energy) photon. Real detectors need a minimum amount of energy before they can register the photon, so if your detectors sees no extra photon that could mean there was no extra photon or there was one but it was too soft to register. So you have to combine your cross sections for processes with a soft photon in the final state with the cross sections for processes with no soft photon in the final state since you can't distinguish the two. The IR divergences cancel in the sum.

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    $\begingroup$ Hey Martin, I discovered (with the topic of "bremsstrahlung" that the divergence for small photon momentum happens in the denominator of the photon propagator. Does that mean that actually, The problem doesn't start at the LSZ Formula, but instead already before with the calculation of n point correlation functions? I could write down a diagram with incomming and outgoing fermions, and still encounter a divergence in a propagator in the middle, right? $\endgroup$ Jul 4, 2021 at 15:02
  • $\begingroup$ The way I see it is that the IR divergences are a symptom of the fact that we've applied the LSZ formula in a situation where it is not valid. So, if the assumptions of the LSZ theorem were met there shouldn't be any divergences. I've changed the second paragraph a bit to make this point clear. $\endgroup$ Jul 4, 2021 at 16:14
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    $\begingroup$ Can you name the exact requirements (in short) that have not been met? $\endgroup$ Jul 4, 2021 at 19:28
  • $\begingroup$ @Quantumwhisp, the assumption that fails is what LSZ call the 'asymptotic condition' that relates the bare fields to the asymptotic ones. I have added a reference to a good (but quite mathematical) textbook on the subject. What it all boils down to, though, is that the derivation of LSZ is invalid when massless gauge bosons are in the game. $\endgroup$ Jul 12, 2021 at 14:04

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