In quantum field theory with a mass gap, why do states in the asymptotic future/past turn out to have a Fock space structure? For a free quantum field theory, that's trivial, but why is that the case for interacting theories? In fact, the more one thinks about it, the less clear it becomes. If the quanta of the "fundamental" field is unstable, it doesn't show up in the asymptotic Fock space. If the quanta is confined, it also doesn't show up. If there is a stable bound state, it does show up. If there is a stable solitonic particle, it also shows up.

I am very aware of the LSZ formalism, but that presupposes the existence of an asymptotic Fock space structure as a starting point. Besides, it doesn't handle stable solitons.


Asymptotically, particles and bound states in a QFT with mass gap behave as noninteracting particles, as the interaction decays exponentially because of the mass gap. Hence they are described by free fields. The physical S-matrix is between all these asymptotic states, and only these. Thus on the level of the asymptotic states we have particle democracy - elementary, composite and nonlocal (solitonic) particles appear on the same footing.

In particular, soliton states are asymptotically also describes by free fields, though LSZ is not directly applicable. (Textbooks conventionally just treat the case where there are neither bound states nor solitons. This includes Weinberg's QFT treatise; however, he at least acknowledges the problem in somewhat cryptic remarks on p.110 of his Vol. 1.)

If there is no mass gap (for example in QED), the situation is significantly more complicated, because then all asymptotic states describe so-called infraparticles, and asymptotic states are not Fock states but Fock states dressed with coherent states made from the massless particles. Working instead with Fock states (as in traditional QFT books) produces infrared divergences which must then be eliminated by ad hoc techniques. If coherent states are used, these infrared divergences do not appear.

On the other hand, unstable particles are visible asympotically only through their decay products, hence are not represented by Fock states. However, in approximate theories neglecting interactions that cause instability, one may regard the unstable particles as asymptotic particles.

Similarly, confined particles are represented asymptotically only through the bound states in which they appear, so they are also not represented by Fock states.

  • $\begingroup$ I am bit confused. How does this situation compare with QED ? Photons are massless but we do nevertheless calculate the S-matrix elements, whose corresponding states are nevertheless treated as Fock states. $\endgroup$ – user91411 Feb 26 at 16:26
  • $\begingroup$ @user91411: In QED, the masslessness of the photons gives rise to infrared divergences when computing the S-matrix by the traditional formulas. Only suitably integrated cross sections are finite. Closer analysis shows that this is related to dressing by soft photon clouds. $\endgroup$ – Arnold Neumaier Feb 26 at 19:31

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