The question about Lorentz invariance of the helicity quantum number for the massless particles

I need to show that helicity is Lorentz invariant (under the proper Lorentz transformation) for the massless particles. I heard about most frequently used argument which contains an idea of impossibility to "outrun" the massless particle, so the sign of helicity is Lorentz invariant. But how about the absolute value of the helicity (when we don't divide this operator on the spin operator norm)?

I want to ask about the method of proof of Lorentz invariance of helicity value $h$, which is determined for the massless particles as $$W_{\mu} = hp_{\mu}, \qquad (1)$$ or in the spinor language $$W_{c \dot {c}}\psi_{a_{1}...a_{n}\dot {b}_{1}...\dot {b}_{k}} = hp_{c \dot {c}}\psi_{a_{1}...a_{n}\dot {b}_{1}...\dot {b}_{k}}, \qquad (2)$$ where $h = \frac{n - k}{2}$ and $\psi_{a_{1}...a_{n}\dot {b}_{1}...\dot {b}_{n}}$ has only one independent component.

Do expressions $(1), (2)$ give us the authomatical proof of the invariance of the helicity value? For example, the left part of $(2)$ transforms under spinor representation of the Lorentz group as the product of $n + 1$ undotted spinors and $k + 1$ dotted spinors, so the right side must transforms by the same way, so it means that $h$ is Lorentz scalar? Analogical thinking may be passed for $(1)$.

• Are you sure about those definitions? You can show that helicity is equal to chirality for massless particles, would that suffice? Commented Feb 19, 2014 at 19:32
• @innisfree : yes, I'm sure. "...You can show that helicity is equal to chirality for massless particles, would that suffice?..", - can I do it in general for an arbitrary spin particle? Commented Feb 19, 2014 at 19:41
• @AndrewMcAddams: What do you mean by "But how about the absolute value of the helicity"? It also doesn't change under boosts, just by the definition of helicity, $h={\bf J} \cdot {\mathbf{p }} / \left| {\mathbf{p}} \right|$, as boosts don't change the direction of ${\mathbf{p}}$ for massless particles. Commented Feb 20, 2014 at 9:36
• @JeffDror : what is a helicity? It is a scalar product of a 3-vectors. Yes, boosts don't change the direction of $\mathbf p$, but they may change the absolute value $h$ (sometimes one uses operator $h = \frac{(\mathbf J \cdot \mathbf p )}{|\mathbf J||\mathbf p |}$, so it formally may be only $\pm 1$. Commented Feb 20, 2014 at 16:52
• @AndrewMcAddams: If I understand correctly, what you then want to show is that the momentum of a massless particle can't change direction under a boost. If you take momentum to be $p^\mu=(E,0,0,E)$ and then boost it in the $z$ direction, you find that if $\gamma >\gamma \beta$ ($\gamma$ is the Lorentz factor, $\beta$ is the velocity), then $p^\mu$ won't change direction. Since $\beta<1$ this is always true. Commented Feb 20, 2014 at 17:27

$$[\frac{J\cdot P}{H},K_i]=i\bigg(\frac{\epsilon_{ijk}K_jP_k}{H}+J_i-\frac{P_i}{H^2}J\cdot P\bigg)\qquad(1)$$
So it does not follow from the Poincare algebra that $$\frac{J\cdot P}{H}$$ is boost invariant. Instead only for those states for which $$(\sigma P^{\mu}-W^{\mu})|\sigma\rangle=0$$ do we have that helicity is boost invariant. One can show this by using that this condition requires
$$\sigma\vec{P}=H\vec{J}-\vec{P}\times\vec{K}\qquad (2)$$
$$[\frac{J\cdot P}{H},K_i]|\sigma\rangle=0$$