Transverse mode of a field

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Hello! I am reading some QFT and at a point I read that any vector field (here we are working with massive spin 1 particles) can be written as: $$A_\mu(x)=A^T_\mu(x)+\partial_\mu\pi(x)$$ with $$\partial_\mu A^T_\mu(x)=0$$ They don't talk about notation, but from the context I understand that ##A^T_\mu(x)## is the transverse component of ##A_\mu(x)##. Is ##\partial_\mu A^T_\mu(x)=0## the definition of the transverse component? And if so, why? Thank you!
 

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nrqed
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Hello! I am reading some QFT and at a point I read that any vector field (here we are working with massive spin 1 particles) can be written as: $$A_\mu(x)=A^T_\mu(x)+\partial_\mu\pi(x)$$ with $$\partial_\mu A^T_\mu(x)=0$$ They don't talk about notation, but from the context I understand that ##A^T_\mu(x)## is the transverse component of ##A_\mu(x)##. Is ##\partial_\mu A^T_\mu(x)=0## the definition of the transverse component? And if so, why? Thank you!
Yes, that's the definition. Note that you meant ##\partial^\mu A^T_\mu(x)=0##.
 
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Orodruin
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Note that you meant ##\partial^\mu A^T_\mu(x)=0##.
He might not be if he is reading Schwartz's QFT book. If I remember correctly, Schwartz starts by saying that he takes it for granted that students know that when summation inices appear one is covariant and the other contravariant and therefore puts all indices as subindices.
 

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