# Helicity

The helicity ( ancient Greek ἕλιξ helix , German , the Spiral ' ) is in the particle component of the spins of a particle which in the direction of its pulse , d. H. in the direction of movement. ${\ displaystyle h \! \,}$

## Definitions

The helicity is defined as

${\ displaystyle h = {\ vec {S}} \ cdot {\ hat {p}}}$,

where denotes the vector of the spin and the pulse direction. ${\ displaystyle {\ vec {S}}}$${\ displaystyle {\ hat {p}} = {\ vec {p}} / | {\ vec {p}} |}$

• For a mass particle with total spin S , the helicity 2 S + 1 can assume different eigenvalues (cf. multiplicity ):
• for integer S : - S , - S +1,…, 0,…, S −1, S
• for half-integer S : - S , - S +1,…, −1/2, +1/2,…, S −1, S
• For a massless particle that always moves with the speed of light, only the two values ​​- S and + S are possible; the helicity in this case coincides with the chirality except for a factor S ; for an almost massless particle (movement with almost the speed of light) this applies approximately.

Sometimes the helicity is also defined as the component of the total angular momentum in the direction of the momentum : ${\ displaystyle {\ vec {J}}}$

${\ displaystyle h \, = \, {\ vec {J}} \ cdot {\ hat {p}}}$.

The two definitions are equivalent, because the orbital angular momentum , which links spin and total angular momentum, is always perpendicular to the momentum vector and therefore cannot contribute to the scalar product ( ). ${\ displaystyle {\ vec {L}}}$${\ displaystyle {\ vec {L}} \ cdot {\ hat {p}} = 0}$

For a massless particle helicity is the proportionality constant between the four-momentum of the particle and its Pauli Lubanski pseudo vector , . ${\ displaystyle W ^ {\ mu} = hp ^ {\ mu}}$

## Clear description

L: left-handed helix,
R: right-handed helix

Helicity clearly defines the direction of rotation or the handedness of a particle. If you look at the term in the sense of classical mechanics , positive helicity means that the axis of rotation of the particle is inclined “forwards”, ie in the direction of movement. The direction of the axis of rotation is determined in such a way that the particle rotates in the direction of the fingers of the right hand when the thumb of the same hand points in the direction of the axis of rotation. If you consider the path of a point on the surface of such a classical particle, it would run through a “right-handed helix ”, as you know it from the thread of a normal screw . Particles with positive helicity are therefore referred to as right-handed , those with negative helicity as left-handed .

Spin direction
mostly inclined ...
Helicity Helix
(in fig.)
applies under weak interaction
for ...
in the
direction of impulse / movement
positive right handed
(R)
massless antiparticles
against the direction of impulse /
movement
negative left handed
(L)
massless particles
1. s. u. Helicity and quantum theory

It should be noted, however, that these are analogy considerations for the purpose of illustration, which do not fully reflect the true quantum mechanical nature of the particles.

## Helicity and Theory of Relativity

Within the framework of the theory of relativity , helicity is only clearly determined for massless particles (which always move at the speed of light ). For all mass-afflicted particles, on the other hand, a reference system can always be selected that “overtakes” the particle, thereby reversing the direction of its momentum and thus its helicity.

## Helicity and quantum field theory

Since helicity is not Lorentz-invariant , it can only be used with restrictions in quantum field theory . However, for massless particles, helicity and chirality are equivalent to one another and for massless antiparticles they are opposite. Therefore, the Lorentz-invariant quantity of chirality is used in quantum field theory : only particles with left-handed chirality are subject to the charged currents of the weak interaction (exchange of W bosons ). For helicity this means that only (massless) particles with negative helicity and antiparticles with positive helicity can interact weakly charged.

For a long time no mass could be proven experimentally for neutrinos . Because they interact only weakly with matter , it was assumed that there are only left-handed neutrinos and right-handed antineutrinos. From the discovery of neutrino oscillations it can be deduced that neutrinos have a non-vanishing mass. According to current physical understanding, it follows that there must also be right-handed neutrinos and left-handed antineutrinos. Another consequence of a non-zero mass is that neutrinos do not move quite at the speed of light .

## Individual evidence

1. V. Devanathan: The helicity Formalism . In: Angular Momentum Techniques in Quantum Mechanics . Springer, 1999, ISBN 978-0-7923-5866-4 , chap. 13 , doi : 10.1007 / 0-306-47123-X_13 ( springer.com [PDF; accessed January 15, 2018]).