Is the Sign in the Covariant Derivative Important for Local Gauge Invariance?

[Ravendark]

Homework Statement

Consider the fermionic part of the QCD Lagrangian: $$\mathcal{L} = \bar\psi (\mathrm{i} {\not{\!\partial}} - m) \psi \; ,$$ where I used a matrix notation to supress all the colour indices (i.e., $\psi$ is understood to be a three-component vector in colour space whilst each component is a Dirac four spinor).

To achieve local gauge invariance, we introduce a covariant derivative $D_\mu$ containing a gauge field with a transformation property that ensures local gauge invariance of $\mathcal{L}$.

My question: Does the sign in the covariant derivative really matters? Or is it more like a convention and every author choose its own? I wrote down my thoughts in section 3.

Homework Equations

Covariant derivative: $D_\mu = \partial_\mu \pm A_\mu$.

The Attempt at a Solution

$\mathcal{L}$ is not invariant under local gauge transformations $U = U(x) \in \mathrm{SU}(3)$ of the fields since $$\partial_\mu \psi \to \partial_\mu \psi' = \partial_\mu U \psi = U \partial_\mu \psi + (\partial_\mu U) \psi \; .$$ Now we introduce a modified derivative $D_\mu$ and demand that it transforms like the fields, i.e., $$D_\mu \psi \stackrel{!}{\to} U D_\mu \psi \; .$$This would lead to a locally gauge invariant Lagrangian.

Now the sign of the covariant derivative comes into play:

• The choice $D_\mu = \partial_\mu + A_\mu$ implies that the gauge field has to transform as $A_\mu \to U A_\mu U^{-1} - (\partial_\mu U)U^{-1}$ to achieve local gauge invariance.
• The choice $D_\mu = \partial_\mu - A_\mu$ implies that the gauge field has to transform as $A_\mu \to U A_\mu U^{-1} + (\partial_\mu U)U^{-1}$ to achieve local gauge invariance.

Thus, from my point of view the sign in the covariant derivative does not really matters since we simply have to demand a slighty different transformation behaviour of the gauge field. Is this correct?

 

[strangerep]

That's my understanding -- one simply has to maintain consistent conventions for all the fields.

 

[Henriamaa]

Anything that keeps your Lagrangian invariant under a local SU(3) transformations should work. If you think about calculating the amplitude for a process the distinction between the two signs disappears also both describe the same principal bundle.