I have question about four-derivative on spinors.
$$ \bar{\psi}\partial_\mu $$
Does the derivative act on the spinor psi bar?
$$ (\bar{\psi}\partial_\mu)\psi $$
$$ \bar{\psi}\partial_\mu\psi $$
Is it different from the results of these two?
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$\begingroup$ When acting on the left, the differentiation operator is written $\overleftarrow{\partial}$. $\endgroup$– Jeanbaptiste RouxCommented Jan 10, 2022 at 14:03
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$\begingroup$ So for the two cases below, the operator acts on psi? Is it possible that the arrow is skipped? $\endgroup$– lalalaCommented Jan 10, 2022 at 14:08
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1$\begingroup$ If the first quantity comes from the adjoint dirac equation it is to be understood as $\partial \bar{\psi}$. Sometimes authors don't write the over arrow unfortunately and you will have to use context to know where the derivative is acting. If your asking about deriving the four current by multiplying the adjoint dirac formula by $\psi$, then the derivative still acts to the left. $\endgroup$– TriatticusCommented Jan 10, 2022 at 18:43
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