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Usually massive particles can be described as eigenstates $|p,\lambda\rangle$ of the angular $J^3$ operator, while massless particles are eigenstates $|p (m=0),\lambda\rangle$ of the helicity operator $\Lambda=\frac{\vec{p}\cdot \vec{J}}{|\vec{p}|}$.

In order to understand the differences between massive particles versus massless particles, our lecturer suggested to try to describe the massive particles also as eigenstates of the helicity operator (in analogy to the massless case).

In this particular description for massive particles, there is no mixture of the $\lambda's$ under (Lorentz) rotations, and thus, helicity doesn't change under rotations. On the other hand, helicity does change under (Lorentz) boosts (in contrast to massless particles, where it remains the same).

However, I'm having a bit of trouble in understanding the physical meaning of the last statements. What does it imply that helicity remains constant for Lorentz rotations in both cases, but changes for Lorentz boosts only for massive particles? How does that make the two particles different?

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    $\begingroup$ Recall that Lorentz transforms take you between various reference frames what kind of properties do you want your eigenstates to have when you move from frame to frame? $\endgroup$ – Triatticus Sep 4 '18 at 20:17
  • $\begingroup$ I want my eigenstates to be invariant under Lorentz transformations, so the observables I can extract from them don't depend on the reference frame. From that reasoning, since helicity is just the projection of spin in the direction of momentum, and since massless particles move at the speed of light, that means that no matter the boost I apply, I cannot place myself going faster than the speed of light and change the helicity. In this case, then helicity and chirality must coincide. $\endgroup$ – Charlie Sep 4 '18 at 23:50
  • $\begingroup$ On the other hand, since massive particles can't travel at the speed of light, I could theoretically apply a certain boost in which I could "reverse" the helicity of my particle (by placing myself going faster than the particle). Therefore, the helicity doesn't coincide with the chirality of the particle, or in other words, it can change under Lorentz boosts. Is my reasoning correct? $\endgroup$ – Charlie Sep 4 '18 at 23:52

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