I Why do the extra terms cancel in the derivation of the EM field strength tensor?

AI Thread Summary
The discussion revolves around the derivation of the field strength tensor using covariant derivatives defined as Du = ∂u - iqAu. The field strength is expressed through the commutator [Du, Dv] = -iqFuv, leading to the expected result Fuv = ∂uAv - ∂vAu. The user initially encounters extra terms, iq(Av∂u - Au∂v), which they believe should cancel out. It is clarified that these terms arise from the operator nature of the commutator, where the differentiation and multiplication by A must be properly accounted for. Ultimately, the user confirms that the derivation aligns correctly after considering the operator's action.
dyn
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Hi
I am trying to follow the derivation in some notes I have for the field strength tensor using covariant derivatives defined by Du = ∂u - iqAu . The field strength is the defined by [ Du , Dv ] = -iqFuv
The given answer is Fuv = ∂uAv - ∂vAu .When I expand the commutator I get this answer but I have 2 terms left over which I presume should cancel but I don't understand why . I have the extra terms iq(Av∂u - Au∂v ) . Do these terms cancel and if so , why ?
Thanks
 
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You are forgetting that this commutator is an operator that should act on something and it should be seen as the operator commutator. The ##\partial A## terms represent first multiplying by A then differentiating. Therefore ##\partial A f = (\partial A) f + A \partial f##. Add the corresponding indices.
 
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Thanks for your reply. It all works out now.
 
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