Lorentz Transformation Vector Form Formula

elibj123
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What is the vector form formula (not the martrice form) for lorentz transformation
from system S' to S for a particle with 4-vectors (\vec{r'},ct') and (\vec{p'},E'/c) where S moves with an arbitary speed \vec{v} in relation to S'?

thanks
 
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Well I saw that, I guess I'm doing something wrong in my proof.

I'm trying to prove phase invariance, yet some expressions don't sum up very nice.
 
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