Spin vs. Helicity: Transformation under Poincare Group

In summary, the conversation discusses the difference between spin and helicity and the transformation of momentum-spin and momentum-helicity states under certain elements of the Poincare group. It is mentioned that helicity is Lorentz invariant and can be rotated using SO(3) matrices.
  • #1
paweld
255
0
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 [tex]\psi=|{p_1}^\mu=(\sqrt{p^2+m^2},0,0,p); s=1,s_{z}=1 \rangle [/tex]
(these informations determine the state of this particle)
(2) Photon with momentum-helicity state [tex]\phi=|p_2^\mu=(p,0,0,p); h=1\rangle [/tex]

I'm interested in how above states transform under some elements of Poinacre group.
I. space rotation through angle [tex]\alpha=\pi/2 [/tex] around z axis.
(1) [tex]\psi' = \exp(i \pi/2 ) |p_1^\mu=(\sqrt{p^2+m^2},0,0,p); s=1,s_{z}=1 \rangle [/tex]
(2) [tex] \phi' = \exp(i \pi/2 )|p_2^\mu=(p,0,0,p); h=1\rangle [/tex]
II. space rotation through angle [tex]\alpha=\pi/2 [/tex] around x axis:
(1) [tex]\psi' =1/2 |p_1^\mu=(\sqrt{p^2+m^2},0,-p,0); s=1,s_{z}=1 \rangle +[/tex]
[tex]+ i/\sqrt{2}|p_1^\mu=(\sqrt{p^2+m^2},0,-p,0); s=1,s_{z}=0 \rangle +[/tex]
[tex]-1/2|p_1^\mu=(\sqrt{p^2+m^2},0,-p,0); s=1,s_{z}=-1 \rangle [/tex]
(2) [tex]\phi' = |p_2^\mu=(p,0,-p,0), h=1\rangle [/tex]
 
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  • #2
Helicity is Lorentz invariant. So you just rotate three spin and momentum components with the appropriate SO(3) matrix.
 
Last edited:

1. What is the difference between spin and helicity?

Spin and helicity are both properties of particles, but they refer to different aspects. Spin is a measure of the intrinsic angular momentum of a particle, while helicity is a measure of the projection of the particle's spin onto its direction of motion.

2. How are spin and helicity related to the Poincare group?

The Poincare group is a mathematical framework that describes the symmetries of spacetime. Both spin and helicity are conserved quantities under the transformations of the Poincare group.

3. Can a particle have different spin and helicity values?

Yes, a particle can have different values for spin and helicity. For example, a photon has a spin of 1 but a helicity of either +1 or -1, depending on its direction of motion.

4. How do spin and helicity affect a particle's behavior?

Spin and helicity can affect a particle's behavior in different ways. For example, the spin of an electron can determine its magnetic moment, while the helicity of a neutrino can affect its interactions with other particles.

5. Can spin and helicity be changed?

Under certain conditions, spin and helicity can be changed. For example, a particle's spin can be flipped through interactions with other particles, while a particle's helicity can be changed through interactions with a magnetic field.

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