Monday, January 18, 2016

differential geometry - Inconsistency between $d_A = d + A wedge$ and $d_A = d(..) + [A,..]$?


I am confused by something basic stated in this https://physics.stackexchange.com/a/429947/42982 by @ACuriousMind and some fact I knew of. Here $d_A$ is covariant derivative.




  1. $d_A A=F$ --- @ACuriousMind says "The field strength is the covariant derivative of the gauge field."




  2. The Bianchi identity is $d_A F=0.$








  • In the 1st case, we need to define



$$ d_A = d + A \wedge \tag{1} $$



So $$ d_A A= (d + A \wedge) A= d A + A \wedge A $$




  • In the 2nd case, we need to define



$$ d_A = d(..) + [A,..] \tag{2} $$



So we get a correct Bianchi identity which easily can be checked to be true $$ d_A F= d F+ [A,F]= d (dA+AA)+[A,dA+AA]=0 $$


However, eq (1) and (2) look different.


e.g. if we use eq(2) for "The field strength is the covariant derivative of the gauge field.", we get a wrong result


$$ d_A A = dA + [A,A] = dA \neq F !!!! $$


e.g. if we use eq(1) for "Bianchi identity", we get the wrong result we get $$ d_A F= d F+ A \wedge F\neq 0 $$




my puzzle: How to resolve def (1) and (2)?


Could it be that for the $p$-form $$ d_A \omega = d \omega + \dots, $$ where $ \dots$ depends on the $p$ of the $p$-form? How precisely?





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