What does it mean if something Transforms Covariantly ? (Quantum Field Theory)

[Elwin.Martin]

What does it mean if something "Transforms Covariantly"? (Quantum Field Theory)

Referring to an isospinor, \psi which transforms as \psi(x)→ \psi'(x)=S(x) \psi(x) (S(x) being an n by n matrix)

I'm told that it is clear that ∂_{μ}\psi does not transform covariantly.

Now, correct me if I'm wrong, but it would appear that ∂_{μ}\psi ' can be found by the product rule to be S(∂_{μ}\psi) + (∂_{μ}S)\psi.

What is meant by that it doesn't transform covariantly?

I know what covariant vs. contravariant indices are, but I don't know what it means for something to transform covariantly. I understand that we *want* a covariant derivative, but I don't understand why =|

Any and all help would be great!

**from Ryder, in case anyone was wondering

 

[dextercioby]

I don't a copy of Ryder handy, however I can tell you that \partial_{\mu}\psi does not transform covariantly with respect S(x), because you have another term - (\partial_{\mu} S)\psi - which spoils covariance. If the gradient of psi had transformed covariantly, this extra term would not have been there. Covariant comes from co-variant, in other words, the derivative of the spinor should have transformed like the spinor, i.e. with an identical matrix.