# Helicity is invariant under boosts along $\hat{p}$?

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?

• 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. – Buzz Oct 31 '18 at 1:14

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."