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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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1 Answers1

4

  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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