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I would like some help clarifying what Zinn Justin is saying in his book "Quantum Field Theory and Critical Phenomena" p.805 on detecting confinement of gauge theories.

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In particular, I understand Elitzur's theorem just states that there is no such thing as gauge symmetry breaking (since gauge is just a labeling redundancy of the theory). However I don't understand why I cannot find a local order parameter to detect confinement in gauge theories. I understand local observables cannot be charged (since charged operators don't commute with gauge transformations)

Could someone help understand why I cannot construct some local order parameter to detect gauge field confinement? (with equations would be nice).

I think one difficulty is I don't have a clear definition of what qualifies as a local order parameter. To prove the statement one would need to define what is a local order parameter.

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The reason is that non-abelian gauge theories are asymptotically free. This means that at short distance, one cannot distinguish between the confined and deconfined phase because the gauge interaction turns off. Therefore it is hard to imagine what kind of local observable could measure the confinement phase transition.

A local order parameter just means any physical observable one can use to detect a phase transition. It is a somewhat general definition, but phase transitions come in a large variety so it is hard to make a more precise definition.

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  • $\begingroup$ Then why does every copy of the QCD phsae diagram say that a high temperature (which makes the theory look more and more free) takes you out of the confining phase? $\endgroup$ Nov 17, 2021 at 21:52
  • $\begingroup$ I assume in that case $\beta \approx {1 \over \hbar}$ in euclidean field theory. High temperature means $\hbar$ is large which means quantum fluctuations matter, in which case the coupling is large. Is that what you are referring to? $\endgroup$ Nov 21, 2021 at 7:53
  • $\begingroup$ I'm just saying the gauge interaction is what causes confinement. So if it turns off you can surely say the matter becomes deconfined. It sounds like you have the right line of argument though. $\endgroup$ Nov 22, 2021 at 15:12
  • $\begingroup$ I think your comment does raise a good point: can we ever have local order param for phase transitions in theories that are asymptotically free? (if at short distance, the theory looks like a free theory, how can you "detect" locally anything non-trivial) $\endgroup$ Nov 22, 2021 at 23:57

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