I am trying to understand to which realistic objects one can apply the definition of single-particle states as irreducible representations of the Poincare group. Let me start with a couple of motivational questions.

  1. In $\phi^4$ theory, would we call the states of definite momentum and energy single-particle states?

  2. What are the single-particle states in a theory like QCD, with multiple fields and various charges? Are those what we intuitively think of as "dressed quarks"/"dressed gluons" or do "bound states" count as well?

My intuitive understanding is that one should call a single-particle state the state with the maximum number of charges being defined. For example, in QED, I would say that the single-particle states are the states of definite energy, momentum, and electric charge. How does the presence of multiple fields and charges affect the definition?


I am not asking "what do we call a particle in QCD?", but rather "what is the physical meaning of states defined in Weinberg (2.5.1)?" and "how does this definition work for theories with multiple fields"?


1 Answer 1


An invariant way to understand single particle states and bound states are to look for poles and resonances in S-matrix elements. In fact you can get this information from the propagator (two-point function), using the Kallen-Lehmann spectral representation, which is defined in an interacting theory. Single particle states appear as poles in the propagator, and unstable bound states appear as resonances.

In $\phi^4$ theory, you can associate the energy and momentum eigenstates with poles in the propagator. This is less scary than it might sound; the propagator is $(k^2+m^2)^{-1}$, which is zero when the field obeys $k^2+m^2=0$, which is the Klein-Gordon equation in momentum space. (Note that in the interacting theory, this $m$ is not the same as the bare mass $m_0$ multiplying $\phi^2$ in the Lagrangian, and additionally there is a wavefunction renormalization condition that should make the residue of this pole equal to one.)

In QCD, the stable particles at low energies will be color-neutral combinations like baryons and hadrons, and "glue balls." The "Yang-Mills mass gap" Millennium prize problem is precisely to show that the spectrum of Yang-Mills theory does not include free massless gluons, even though you would expect it to perturbatively. Quarks and gluons are always off shell. But if we are interested in processes high energies above the $\Lambda_{\rm QCD}$, where QCD becomes weakly coupled due to asymptotic freedom, we can meaningfully talk about free quarks and gluons scattering.

  • $\begingroup$ When you say "stable particles", is this the same as your definition of "single-particle states" or "bound states"? $\endgroup$
    – mavzolej
    Aug 12, 2021 at 14:54
  • $\begingroup$ Please see the update to my question. $\endgroup$
    – mavzolej
    Aug 12, 2021 at 15:41
  • $\begingroup$ @mavzolej Stable particles are states that don't decay. If you think of a proton as a bound state of quarks and gluons, then the proton would appear as a pole but is still a bound state. $\endgroup$
    – Andrew
    Aug 12, 2021 at 16:10
  • $\begingroup$ The physical meaning of the free states is that they are particle states that propagate at asymptotically large distances and correspond to particles we see in a detector in a scattering experiment. You can map them to poles in the propagator/two point function. The latter is invariant under redefinitions of fields and avoids all the mess about what fields are fundamental or not when talking about QCD. $\endgroup$
    – Andrew
    Aug 12, 2021 at 16:11
  • $\begingroup$ Around (2.5.1) Weinberg emphasizes that this definition is valid not only for free theories. My question is how to interpret "single-particle state" according to this definition in the interacting theories. $\endgroup$
    – mavzolej
    Aug 12, 2021 at 16:12

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