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

$$F_{\mu\nu}^{ik}\;F^{\mu\nu\,ki}$$

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

$$F_{\mu\nu}^{ik}\;\to\;U^\dagger_{im}\,F_{\mu\nu}^{mn}\,U_{nk}$$

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]

I'm asking because I read that the Pauli Villars regulator is not gauge covariant

What I remember is that it violates unitarity.

 

[Haydo]

I know this thread is two years dead, but it's the best I can find. While reading an article by Machleidt, Entem / Physics Report 503 (2011) 1-75, I came across an assertion that the QCD Lagrangian contained a gauge-covariant derivative within it. So I'm strongly inclined to think it's not a typo either, but like Majon, I don't know what the implications of that is within low-energy QCD. Note that this gauge covariance was discussed with respect to Chiral symmetry with the purpose of explaining the support for EFT. Any help is appreciated.

It might be worth adding that the cause of your prior confusion might possibly be from the fact the the QCD Lagrangian is invariant under relevant transformations WHEN RESTRICTED to the up and down quark. Or that might be something else entirely.

 

[tom.stoer]

I think I don't fully understand what you are asking.

The Lagrangian is always gauge invariant; this does by no means depend on the flavors or chiral symmetry

 

[Haydo]

See, that's what I was lead to believe, that the Lagrangian was always gauge invariant, but the claim here is that the QCD Lagrangian (sorry for the poor formatting) is

L_QCD=$\overline{q}$(i$\gamma$^μD_μ-M)q-(Gluon field strength tensor)

where D_μ is a gauge-covariant derivative:
D_μ=$\partial$_μ-ig($\frac{λ}{2}$)A_μ,a

So my assumption was that if L_QCD contains a gauge-covariant derivative, then L_QCD is itself gauge covariant. Am I just interpreting something incorrectly? I definitely don't understand gauge invariance or covariance well enough to say either way.

 

[The_Duck]

So my assumption was that if L_QCD contains a gauge-covariant derivative, then L_QCD is itself gauge covariant. Am I just interpreting something incorrectly?

Yes, you are incorrect. The QCD Lagrangian is gauge-invariant. Gauge invariant things can be (and usually are) built from gauge-covariant parts.

I definitely don't understand gauge invariance or covariance well enough to say either way.

Check out any QFT textbook's discussion of nonabelian gauge theory.

 

[Haelfix]

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?

There are some regulators that do not preserve manifest gauge invariance, but Pauli-Villars does. Perhaps you are remembering something else?

 

[Lapidus]

Covariant gauges are (manifestly) Lorentz-invariant gauges. Non-covariant are not.

Covariant gauges: Feynman-'t Hooft gauge, Lorentz gauge, unitary gauge.

Non-covariant gauges: lightcone gauge, Coulomb gauge, radial gauge.

 

[tom.stoer]

Look at

$$\bar{q}(i\gamma^\mu D_\mu)q$$

The quarks transform in the fundamental rep. of SU(3), i.e.

$$q \to q^\prime = U\,q$$

$$\bar{q} \to \bar{q}^\prime = \bar{q}\,U^\dagger$$

Gluons and the covariant derivative transform in the adjoint rep.

$$D_\mu \to D^\prime_\mu = U \, D_\mu \, U^\dagger$$

So each individual term transforms non-trivially, whereas the complete lagrangian does b/c the U's cancel