Spin vs. Helicity: Transformation under Poincare Group

[paweld]

Can somenone check if my reasoning is correct. I would like to have deeper
insight into the difference between spin and helicity.

Let's consider two particles:
(1) Particle with mass m in momentum-spin state $$\psi=|{p_1}^\mu=(\sqrt{p^2+m^2},0,0,p); s=1,s_{z}=1 \rangle$$
(these informations determine the state of this particle)
(2) Photon with momentum-helicity state $$\phi=|p_2^\mu=(p,0,0,p); h=1\rangle$$

I'm interested in how above states transform under some elements of Poinacre group.
I. space rotation through angle $$\alpha=\pi/2$$ around z axis.
(1) $$\psi' = \exp(i \pi/2 ) |p_1^\mu=(\sqrt{p^2+m^2},0,0,p); s=1,s_{z}=1 \rangle$$
(2) $$\phi' = \exp(i \pi/2 )|p_2^\mu=(p,0,0,p); h=1\rangle$$
II. space rotation through angle $$\alpha=\pi/2$$ around x axis:
(1) $$\psi' =1/2 |p_1^\mu=(\sqrt{p^2+m^2},0,-p,0); s=1,s_{z}=1 \rangle +$$
$$+ i/\sqrt{2}|p_1^\mu=(\sqrt{p^2+m^2},0,-p,0); s=1,s_{z}=0 \rangle +$$
$$-1/2|p_1^\mu=(\sqrt{p^2+m^2},0,-p,0); s=1,s_{z}=-1 \rangle$$
(2) $$\phi' = |p_2^\mu=(p,0,-p,0), h=1\rangle$$

 

[arkajad]

Helicity is Lorentz invariant. So you just rotate three spin and momentum components with the appropriate SO(3) matrix.