Helicity (particle physics)

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In physics, helicity is the projection of the spin onto the direction of momentum. Mathematically, helicity is the sign of the projection of the spin vector onto the momentum vector: "left" is negative, "right" is positive. Thus, it can be represented with the operator p ^ ⋅ Σ ^ | p ^ | {\displaystyle {\frac {{\hat {\mathbf {p} }}\cdot {\hat {\mathbf {\Sigma } }}}{|{\hat {\mathbf {p} }}|}}} , where p ^ {\displaystyle {\hat {\mathbf {p} }}} is the momentum operator and Σ ^ {\displaystyle {\hat {\mathbf {\Sigma } }}} is the spin operator. The angular momentum J is the sum of an orbital angular momentum L and a spin S. The relationship between orbital angular momentum L, the position operator r and the linear momentum (orbit part) p is so L's component in the direction of p is zero. Thus, helicity is just the projection of the spin onto the direction of linear momentum. The helicity of a particle is positive ("right-handed") if the direction of its spin is the same as the direction of its motion and negative ("left-handed") if opposite. Helicity is conserved. That is, the helicity commutes with the Hamiltonian, and thus, in the absence of external forces, is time-invariant. It is also rotationally invariant, in that a rotation applied to the system leaves the helicity unchanged. Helicity, however, is not Lorentz invariant; under the action of a Lorentz boost, the helicity may change sign.