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I'm not sure if someone's already asked this before, but I was wondering, in field theory,

  1. when we say that a certain field is gauge invariant but not gauge covariant, what does this mean? In particular, in Wikipedia, the regulator of Pauli-Villars is said to be as such.

  2. Moreover, as a consequence of not being gauge covariant, the Wikipedia article says that this regulator can't be used in QCD. How to see the link between not being gauge covariant and QCD here? And, why can one use it in QED then?

Qmechanic
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Joman
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1 Answers1

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There are gauge conditions that are very useful but are not covariant. Working for example in Coulomb/Transverse gauge you can obtain gauge invariant results that are not manifestly covariant - this is called Gupta-Bleuler approach in QED. The BRS scheme is another example in non-Abelian gauge theories. In such situations Poincare symmetry is restored by restricting the space of available states appropriately.

About Wiki (often wrong) on Pauli Villars: there is no problem creating a gauge invariant and Lorentz covariant scheme, just lots of computations and theoretical complications as gauge invariance arises in combinations of several terms from different order of perturbation theory, a normal feature in any higher order theory including "normal" scalar particle.

I cannot imagine anyone doing PV in QCD.... that is like converting a VW into race car, will work if you must win a bet.....

JohannR
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