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