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What makes helicity an important quantity in quantum field theory? I know that one can classify particles by mass and spin. For particles without mass one uses helicity (correct me if this is wrong). Is this the only thing that makes helicity important?

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Helicity is the quantity by which we characterize massless particles. The root of this statement is based on the fact that mathematically the free particles are defined as the irreducible representations of the Poincare group (which just reflects the fact that our world is (locally) Poincare-symmetric). Within this definition, massive particles are characterized by the mass and the spin, but for massless particles both of these quantities (squared) vanish. In the latter case, however, the spin 4-operator is parallel to the 4-momentum, and the proportionality coefficient is essentially the helicity. It is the only quantity by which we characterize massless representations (so your statement in the question is true).

Precisely, if we choose the given helicity of the massless particle, we automatically obtain the equations of motion for the corresponding field representing the particle. And as you know, the free theories in the QFT are the key for making the calculations based on perturbation theory, with its propagators and so on...

That's why helicity is so important.

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