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I have field as an element of a Lie algebra as $\Phi = \phi^at^a$ and I want to calculate the commutator $$[D_{\mu}, \Phi],$$ with $$D_{\mu} = \partial_{\mu} + igA^a_{\mu}t^a,$$ the covariant derivative, where ${t^a}$ are the generators of the group. When the professor did the calculation explicitly, he did $$[\partial_{\mu}, \Phi] = \partial_{\mu}\Phi - \Phi\partial_{\mu} = (\partial_{\mu}\Phi) + \Phi\partial_{\mu} - \Phi\partial_{\mu}.$$ I don't understand this last step.

Any guidance is much appreciated!

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    $\begingroup$ Remember what the commutator acts on and then use the product rule. $\endgroup$
    – joseph h
    Jun 18 at 21:45
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    $\begingroup$ If you like this question you may also enjoy reading this related Phys.SE post. $\endgroup$
    – Qmechanic
    Jun 18 at 22:18

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It may help to evaluate the commutator on a function, i.e.

$$[\partial_\mu,\Phi]f=\partial_\mu(\Phi f)-\Phi\partial_\mu f=(\partial_\mu \Phi)f+\Phi \partial_\mu f-\Phi \partial_\mu f.$$

The last step is just applying the product rule for differentiation. The RHS can also be written as $(\partial_\mu \Phi+\Phi \partial_\mu-\Phi \partial_\mu)f$, so $[\partial_\mu,\Phi]=\partial_\mu \Phi+\Phi \partial_\mu-\Phi \partial_\mu$.

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  • $\begingroup$ Thanks so much! $\endgroup$
    – zequi
    Jun 18 at 22:04

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