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I just got confused about the meaning of confinement in QFT.

The naive definition is that in QCD one cannot observe isolated quarks and gluons. This is a trivial statement because in any gauge theory you can only observe gauge singlets, very much by definition: observables are always gauge invariant. This naive definition of confinement is vacuous and has no deep meaning. It is a tautology.

A similar definition, concerning the potential energy of quarks, suffers the same drawback, because again quarks are not observable, they are not gauge invariant and hence the potential itself is not something with invariant meaning. One can write down gauge theories of quark which, in a dual frame, have no quarks at all (e.g., Seiberg duality with $N_f=N_c+1$).

(It is also not uncommon to find people using the word "confinement" when they actually mean "asymptotic freedom". These are independent concepts. There are asymptotically-free theories that confine, and asymptotically-free theories that do not confine.)

So, what exactly do we mean when we say a given theory is (or is not) confining? I've seen a formal definition concerning the spontaneous breaking of a one-form symmetry, i.e., about whether unscreened Wilson loops have area law or perimeter law, but I don't really get the physics of this definition. What is the physical interpretation of confinement? How can it be measured in practice?

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    $\begingroup$ Considering that the confinement and Higgs phases are analytically connected to each other (in many models), I'm not sure confinement has any perfectly clean definition. If you like, if nobody else steps in, then I could post an "answer" consisting of excerpts from chapter 3 ("What is confinement?") in Greensite (2011), An Introduction to the Confinement Problem, where the author bemoans the deficiencies in various attempts to define what confinement means and then chooses the definition that he considers to be the least deficient. $\endgroup$ Sep 6, 2021 at 19:30
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    $\begingroup$ The naive definition of confinement of particles is not vacuous: it was speculated about before a gauge QFT was agreed to describe it. Indeed, the gauge invariant field operators can overlap with physical phenomena involved in the relevant physical spectra. You might focus your misgivings on QGP... $\endgroup$ Sep 6, 2021 at 20:42
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    $\begingroup$ @CosmasZachos I agree about the historical significance of my the definition. Indeed, before the advent of gauge theories, claiming that partons cannot be isolated is a very non-trivial statement. But once we agree that these partons are charged under a gauge symmetry, the claim becomes vacuous (we replace a non-trivial claim for another non-trivial claim, the second one superseding the first one). My question is not about history though. If, today, someone says that a given theory "confines for a given range of paramenters, and deconfines for another range", what do they mean, exactly? $\endgroup$ Sep 7, 2021 at 6:17
  • $\begingroup$ What do they mean by electroweakobservables? $\endgroup$ Sep 7, 2021 at 7:53
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    $\begingroup$ @Einj ? Are you confusing me with the OP? What paragraph or question are you talking about?? Do you understand the dispositive difference between QED and the zero flavor triality condition, which was shoehorned into confinement relatively late in the game? $\endgroup$ Sep 13, 2021 at 13:22

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This is a trivial statement because in any gauge theory you can only observe gauge singlets, very much by definition: observables are always gauge invariant.

While this sounds like it should be true, I think it doesn't actually imply what you think it implies. In the following I'll argue that "we can't observe an isolated gluon" is, in fact, a perfectly fine operational definition of what we mean by "confinement".

In order to see this, let us first think about how one builds the asymptotic Fock spaces in QFT where our usual "free particles" live: We "turn off" the interaction between the fields of our theory, and build a Fock space from the mode expansions of the free fields. Now, a non-Abelian gauge theory is not, in fact, a free theory, even in the absence of additional fields: The Yang-Mills Lagrangian density $$ L_\text{YM}[A] = -\frac{1}{4}F^{a\mu\nu}F^a_{\mu\nu}$$ contains three- and four-gluon terms with a "self-coupling" constant $g$ since we have $$ F^a_{\mu\nu} = \partial_\mu A^a_\nu - \partial_\nu A_\mu^a + gf^{abc}A^b_\mu A^c_\nu$$ and so this is not by any stretch of the imagination a non-interacting theory. The question of the correct construction of the asymptotic Fock space for such theories is therefore not straightforward. The original reference to this is probably the three-paper series "Manifestly Covariant Canonical Formulation of Yang-Mills Field Theories" by Kugo and Ojima (I, II, III), where they construct the BRST charge for the interacting theory and then "transfer" it to the space of asymptotic non-interacting fields in order to determine the correct space of physical states.

In any case, if we believe the computations of Kugo and Ojima then it turns out the asymptotic Fock space we "should" associate to such a theory contains indeed "colored states" created by the various components of $A$, just with the non-transversal parts eliminated like in the Abelian case. I think the intuition here is that the "non-interacting" clause effectively means that that $A^a$ cannot act on $A^b$ anymore, not even by gauge transformations, which of course runs counter to the "one-level-more-naive" intuition that $A^a$ and $A^b$ are related by gauge transformations in the interacting theory and hence cannot create gauge-invariant states.

Note that this is not a contradiction to the statement "color is not a gauge-invariant observable", since it is not obvious at all that there is a straightforward relation between these asymptotic states and "true" interacting states on which a potential color observable would act.

Therefore, we can again talk about confinement in the "naive" sense: Confinement is the phenomenon that Kugo und Ojima's procedure does not yield the correct asymptotic space for non-Abelian gauge theories that have a confining phase: We do not observe free gluons or quarks, but we do observe "free" hadrons (in the confining phase).

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