Helicity (particle physics)

# Helicity (particle physics)

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In particle physics
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

, helicity is the projection of the spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

onto the direction of momentum, :

as the projection of orbital angular momentum along the linear momentum is zero, . 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, , ..., −S. The measured helicity of a spin S particle will range from −S to +S.

In 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...

. This has unitary representation
Unitary representation
In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π is a unitary operator for every g ∈ G...

s which are invariant under the SE(2) "translations" and transform as ei 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.

In dimensions, the little group is the double cover of SE (the case where is more complicated because of anyon
Anyon
In 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.-From theory to reality:...

s, etc.). As before, there are unitary representations which don't transform under the SE "translations" (the "standard" representations) and "continuous spin" representations.

For massless spin-

In particle physics
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

, helicity is the projection of the spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

onto the direction of momentum, :

as the projection of orbital angular momentum along the linear momentum is zero, . 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.

In {{nowrap|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...

. This has unitary representation
Unitary representation
In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π is a unitary operator for every g ∈ G...

s which are invariant under the SE(2) "translations" and transform as ei 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.

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
Anyon
In 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.-From theory to reality:...

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.

For massless spin-

In particle physics
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

, helicity is the projection of the spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

onto the direction of momentum, :

as the projection of orbital angular momentum along the linear momentum is zero, . 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.

In {{nowrap|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...

. This has unitary representation
Unitary representation
In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π is a unitary operator for every g ∈ G...

s which are invariant under the SE(2) "translations" and transform as ei 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.

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
Anyon
In 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.-From theory to reality:...

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.

For massless spin-{{frac, helicity is equivalent to the chirality operator
Chirality (physics)
A chiral phenomenon is one that is not identical to its mirror image . The spin of a particle may be used to define a handedness for that particle. A symmetry transformation between the two is called parity...

multiplied by .