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Helicity (particle physics)

 

 
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Helicity (particle physics)
 
 
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In particle physics
Particle physics

Particle physics is a branch of physics that studies the elementary particle constituents of matter and radiation, and the interactions between them....
, helicity is the projection of the spin
Spin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
  onto the direction of momentum, :

Because the eigenvalues of spin with respect to an axis has discrete values, the eigenvalues of helicity are also discrete. For a particle of spin S, the eigenvalues of helicity are S, (S-1}, ..., -S. The measured helicity of a spin S particle will range from -S to + S. Note that helicity can equivalently be written with the total angular momentum operator , instead of , because the projection of orbital angular momentum along the linear momentum vanishes: .

In 3+1 dimensions, the little group for a massless particle is the double cover of SE(2)
Euclidean group

In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space. Its elements, the isometry associated with the Euclidean Metric , are called Euclidean moves....
.






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In particle physics
Particle physics

Particle physics is a branch of physics that studies the elementary particle constituents of matter and radiation, and the interactions between them....
, helicity is the projection of the spin
Spin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
  onto the direction of momentum, :

Because the eigenvalues of spin with respect to an axis has discrete values, the eigenvalues of helicity are also discrete. For a particle of spin S, the eigenvalues of helicity are S, (S-1}, ..., -S. The measured helicity of a spin S particle will range from -S to + S. Note that helicity can equivalently be written with the total angular momentum operator , instead of , because the projection of orbital angular momentum along the linear momentum vanishes: .

In 3+1 dimensions, the little group for a massless particle is the double cover of SE(2)
Euclidean group

In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space. Its elements, the isometry associated with the Euclidean Metric , are called Euclidean moves....
. This has unitary representation
Unitary representation

In mathematics, a unitary representation of a Group G is a linear representation p of G on a complex Hilbert space V such that p is a unitary operator for every g ? G....
s which are invariant under the SE(2) "translation"s and transform as eih? under a SE(2) rotation by ?. This is the helicity h representation. We also have another unitary representation which transforms nontrivially under the SE(2) translations. This is the continuous spin representation.

In d+1 dimensions, the little group is the double cover of SE(d-1) (the case where d≤2 is more complicated because of anyon
Anyon

In mathematics and physics, an anyon is a type of particle that occurs only in two-dimensional systems. It is a generalization of the fermion and boson concept....
s, etc). As before, we have unitary reps which don't transform under the SE(d-1) "translations" (the "standard" reps) and "continuous spin" reps.

For massless (or extremely light) spin-1/2 particles, helicity is equivalent to the operator of chirality
Chirality (physics)

A phenomenon is said to be chiral if it is not identical to its mirror image . The Spin of a particle may be used to define a handedness for that particle....
 multiplied by .

Etymology

Helicity derives from the Latin "helix", from Greek; akin to Greek eilyein to roll, wrap.

See also

  • Wigner's classification
    Wigner's classification

    In mathematics and theoretical physics, Wigner's classificationis a classification of the nonnegative energy Irreducible representations of the Poincar? group, which have sharp mass eigenvalues....