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Your conclusion that I think a pseudovector is actually a function of vectors equipped with preallocated arguments and every transformation on a vector has a counterpart on a pseudovector like $\tilde P $. is, I think, essentially correct; the reason parity is confusing is that we tend to drop the tilde. More precisely, whenever parity considerations ...


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This all links back to the fact that any time you speak about a transformation, you require all the relevant physics to transform along. E.g. velocity would not transform the way it transforms without requiring the trajectory $x^i(t)$ it is tangent to to transform also. In the sense of the notation you use we have $v \equiv <x^i(t),\frac{d}{dt}>$ and ...



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