Tuesday, 3 December 2019

quantum field theory - Wilson loops and gauge invariant operators (Part 2)


These questions are sort of a continuation of this previous question.





  • I would like to know of the proof/reference to the fact that in a pure gauge theory Wilson loops are all the possible gauge invariant operators (... that apparently even the local operators can be gotten from the not-so-obviously-well-defined infinitesimal loop limit of them!..)




  • If the pure gauge theory moves into a confining phase then shouldn't there be more observables than just the Wilson loops... like "glue balls" etc? Or are they also somehow captured by Wilson loops?




  • If matter is coupled to the gauge theory are the so called ``chiral primary" operators, $Tr[\Phi_{i1}\Phi_{i2}\cdots\Phi_{im}]$ a separate class of observables than either the baryons or mesons (for those field which occur in the fundmanetal and the anti-fundamental of the gauge group) or Wilson loops?..is there a complete classification of all observables in the confined phase?





{..like as I was saying last time..isn't the above classification the same thing as what in Geometric Invariant Theory is well studied as asking for all G-invariant polynomials in a polynomial ring (..often mapped to $\mathbb{C}^n$ for some $n$..) for some group $G$?..)




  • But can there be gauge invariant (and hence colour singlet?) operators which are exponential in the matter fields?




  • In the context of having new colour singlet( or equivalently gauge invariant?) observables in the confining phase I would like to ask the following - if one is working on a compact space-(time?) then does the Gauss's law (equation of motion for $A_0$) somehow enforce the colour-singlet/confinement condition on the matter observables?..hence possibly unlike on flat space-time, here even at zero gauge coupling one still has to keep track of the confinement constraint?






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