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For general gauge theories, the total Lagrangian density is given as $$L=-\frac{1}{4}F^2+L_M(\psi, D\psi)$$ where $L_M(\psi, D\psi)$ is the matter field with the ordinary derivative replaced by the covariant derivative $D$. Here $\psi$ is a generic matter field other than the gauge field.

Then, for calculating the equation of motion for the matter field $\psi$, I am confused whether I have to calculate with $\partial_\mu \phi$ or $D_\mu \psi$. That is, which one is correct?:

\begin{equation} \frac{\partial L}{\partial\psi}-\partial_\mu \frac{\partial L}{\partial(\partial_\mu\psi)}=0,\tag{1} \end{equation}

\begin{equation} \frac{\partial L}{\partial\psi}-D_\mu \frac{\partial L}{\partial(D_\mu\psi)}=0.\tag{2} \end{equation}

This kind of stuff have always confused me...so I desperately feel a need to clarify.

Qmechanic
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Keith
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  • Would't the fact that you'd be applying the same action principle to the whole Lagrangian action (all three pieces, gauge, coupling and matter) entail that the same derivative is used in the EOM? – DanielC Oct 04 '20 at 07:43
  • Could you clarify more? What do you mean by "the same derivative"? – Keith Oct 04 '20 at 07:44

1 Answers1

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  1. The principle of stationary action always implies the EL equations (1) with partial derivatives, so (1) is a safe bet.

  2. By imposing further conditions on the theory, the EL equations (2) with covariant derivatives may hold as well, cf. this related Phys.SE post.

Qmechanic
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