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I am reading Schwartz's QFT textbook. In Eq. (10.104) he writes: $$ \left[i\partial_\mu-eA_\mu,i\partial_\nu-eA_\nu\right]~=~-e i[\partial_\mu A_{\nu}-\partial_\nu A_{\mu}]~=~ -e i F_{\mu \nu}. \tag{10.104} $$ I was wondering what has happened to the $-ie(A_\mu\partial_\nu-A_\nu\partial_\mu)$ term?

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There is no such term. OP's question seems to be caused by ambiguities in what is meant by the derivative symbol, as explained in my Phys.SE answer here:

  1. Does the first-order differential operator act on everything (written or not written) to the right of it? This is the case for the left-hand side of eq. (10.104). The commutator turns out to be a differential operator of zero order, i.e. an left multiplication operator, which multiplies from left with a function (the field strength). In turn, we identify a left multiplication operator with its function.

  2. Or does the first-order differential operator only act on the next object to the right of it? This is the case for the middle expression of eq. (10.104). Presumably the square bracket (that is not a commutator) is supposed to indicate that the differential operator doesn't act further.

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