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Sign in covariant derivative

  1. Aug 19, 2015 #1
    1. The problem statement, all variables and given/known data
    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.

    2. Relevant equations
    Covariant derivative: ##D_\mu = \partial_\mu \pm A_\mu##.


    3. 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?
     
  2. jcsd
  3. Aug 21, 2015 #2

    strangerep

    User Avatar
    Science Advisor

    That's my understanding -- one simply has to maintain consistent conventions for all the fields.
     
  4. Aug 23, 2015 #3
    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.
     
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