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In this source (In Introduction, p. 1) we find the claim that the helicity operator $h=\vec{S}\cdot\hat{p}$ is invariant under rotations and boosts.

I agree that is is clearly invariant under rotations, but if I boost $p$ in such a way, that it changes direction, the the helicity does flip sign as well.

Where am I wrong?

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  • $\begingroup$ They seems to be sloppy about this. However, the helicity formalism is useful mostly when the daughter products of a reaction are ultrarelativistic, so the misstatement is of little practical consequence. $\endgroup$ – Buzz Oct 31 '18 at 1:14
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For reference, here is the complete sentence from page 1 in the paper:

The helicity formalism is well suited to relativistic problems bcause the helicity operator $h=\vec S\cdot\hat p$ is invariant under both rotations and boosts along $\hat p$.

The context suggests that the hat on $\hat p$ is meant to indicate a unit vector. Even if we ignore any statements about helicity, this is already in conflict with any Lorentz boost that transforms the momentum to zero: the zero vector cannot be a unit vector.

This problem goes away if the particle is massless because then its momentum is never zero, and a simple Lorentz boost along $\hat p$ cannot change the sign of $\hat p$. However, I did not see anything in the context that assumes a massless particle. Maybe the author is assuming that "helicity" implies "massless."

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