Gauge invariance Vs. Gauge covariance

majon

I know what gauge invariance is, but I'm not sure what gauge covariance is. Is it that a given field has a gauge covariant derivative?

And under which circumstances do we get a field that is gauge invariant but not gauge covariant? And I would appreciate an example (other than the one below).

Finally, what is the link between gauge covariance and QCD? I'm asking because I read that the Pauli Villars regulator is not gauge covariant (which I don't understand what it means) hence can't be used in QCD.

Could someone clear the confusion fog in my head, please?

Bill_K

Perhaps this is a Wikipedia misprint? Maybe the article meant to say, it's gauge invariant but it's not unitary.

tom.stoer

Invariance means that you have an expression that doesn't change under gauge transformation; a well known example in SU(n) gauge theories is the gauge-field part of the Lagrangian which looks like

[tex]F_{\mu\nu}^{ik}\;F^{\mu\nu\,ki}[/tex]

where i,k are the su(n) indices and which is invariant due to the trace w.r.t. i,k.

Covariance means that an expression is not gauge invariant but has a well-defined transformation w.r.t. to the gauge group; a well known example in SU(n) gauge theories is the field strength F which lives in the adjoint rep. and which transforms as

[tex]F_{\mu\nu}^{ik}\;\to\;U^\dagger_{im}\,F_{\mu\nu}^{mn}\,U_{nk}[/tex]

where U represents the gauge trf.

Bill_K

That's what they mean all right. But how do those definitions apply to the remark about the Pauli-Villars regulator?

tom.stoer

I am not sure; perhaps these papers plus references may help

http://arxiv.org/abs/hep-ph/9507443
http://arxiv.org/abs/hep-th/9502025
http://arxiv.org/abs/hep-lat/0001024

majon

Bill_K, I don't think it's a mistake, I think they put it there to indicate something I don't understand. Thanks you for your comments

tom.stoer, many thanks for stating the difference between the two expressions, and for mentioning the papers. I shall read them and see what can get out of them

Bill_K

Thanks, majon. I stand by my remark. It doesn't make any sense the way it's stated. Good luck.

tom.stoer

What I remember is that it violates unitarity.