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| {{About|helicity in particle physics||Helicity (disambiguation)}}
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| In [[particle physics]], '''helicity''' is the projection of the [[Spin (physics)|spin]] <math>\scriptstyle\vec S</math> onto the direction of momentum, <math>\scriptstyle \hat p</math>:
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| :<math>h = \vec J\cdot\hat p = \vec L\cdot\hat p + \vec S\cdot \hat p = \vec S\cdot \hat p,\qquad \hat p = \frac{\vec p}{\left|\vec p\right|}</math>
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| as the projection of orbital angular momentum along the linear momentum is zero, <math>\scriptstyle\vec L\cdot\hat p\,=\, 0</math>. Because the eigenvalues of spin with respect to an axis have discrete values, the eigenvalues of helicity are also discrete. For a particle of spin ''S'', the eigenvalues of helicity are ''S'', {{nowrap|''S'' − 1}}, ..., −''S''. The measured helicity of a spin ''S'' particle will range from −''S'' to +''S''.
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| In {{nowrap|3 + 1}} dimensions, the [[little group]] for a [[massless particle]] is the [[Double covering group|double cover]] of [[Euclidean group|SE(2)]]. This has [[unitary representation]]s which are invariant under the SE(2) "translations" and transform as e<sup>i''hθ''</sup> under a SE(2) rotation by ''θ''. This is the helicity ''h'' representation. There is also another unitary representation which transforms non-trivially under the SE(2) translations. This is the ''continuous spin'' representation.
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| In {{nowrap|''d'' + 1}} dimensions, the little group is the double cover of SE({{nowrap|''d'' − 1}}) (the case where {{nowrap|''d'' ≤ 2}} is more complicated because of [[anyon]]s, etc.). As before, there are unitary representations which don't transform under the SE({{nowrap|''d'' − 1}}) "translations" (the "standard" representations) and "continuous spin" representations.
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| For massless [[spin-1/2|spin-{{frac|1|2}} particles]], helicity is equivalent to the [[chirality (physics)|chirality operator]] multiplied by <math>\scriptstyle \frac{1}{2}\hbar</math>.
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| ==See also==
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| * [[Wigner's classification]]
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| * [[Chirality (physics)|Chirality]]
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| * [[Pauli–Lubanski pseudovector]]
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| {{DEFAULTSORT:Helicity (Particle Physics)}}
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| [[Category:Particle physics]]
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| [[Category:Quantum field theory]]
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| {{particle-stub}}
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