Helicity

Prerequisites: Spin, Special Relativity. Full path (20 pages)

Why is it interesting?
Intuitive
Concrete
Abstract

Why is it interesting?

Helicity is an important experimental observable, which describes the projection of the spin onto the direction of the momentum. Moreover, there is a close connection to the more fundamental concept chirality, because for massless particles the chirality and helicity are the same (for antiparticles they are opposite: a left-chiral field annihilates a left-handed particle and creates a right-handed antiparticle).

Important Related Concepts:

Intuitive

A particle with spin behaves a bit like a small spinning top, and a moving particle also has a direction of motion. Helicity compares the two: it is the component of the spin along the direction in which the particle moves. If the spin points along the motion, the particle rotates like a right-handed screw being driven forward, and we call it right-handed (positive helicity). If the spin points against the motion, it is left-handed (negative helicity).

For a massive particle helicity depends on the observer. An observer who runs faster than the particle and overtakes it sees the particle moving backwards, while the spin still points the same way. For this observer the helicity has the opposite sign. The faster the particle, the harder it is to overtake, so for fast particles helicity is almost a fixed property.

A massless particle moves at the speed of light and no observer can overtake it. Its helicity is the same for every observer, so for massless particles helicity is a basic label of the particle itself. A photon has helicity plus one or minus one, which correspond to the two circular polarizations of light. There is no helicity zero photon, which is why light has only two polarization states although the photon has spin one.

In experiments helicity is useful because it is easy to define for a particle flying through a detector, and the weak interaction treats left-handed and right-handed particles differently.

Concrete

Whether spin, helicity or chirality is important depends on the physical question you are interested in. For free massless spinors, the spin eigenstates are also helicity eigenstates and chirality eigenstates. In other words, the Hamiltonian for the massless Dirac equation commutes with the operators for chirality, γ5, helicity, S⃗·p⃗, and the spin operators, S⃗. The QED interaction $\bar\psi \not{A}\psi = \bar\psi_L\not{A}\psi_L + \bar\psi_R\not{A}\psi_R$ is non-chiral, that is, it preserves chirality. Helicity, on the other hand, is not necessarily preserved by QED: if a left-handed spinor has its direction reversed by an electric field, its helicity flips. When particles are massless […]

In the massive case, it is also possible to take the non-relativistic limit. Then it is often better to talk about spin, the vector. Projecting on the direction of motion does not make so much sense when the particle is nearly at rest, or in a gas, say, when its direction of motion is constantly changing. The QED interactions do not preserve spin, however; only a strong magnetic field can flip an electron’s spin. So, as long as magnetic fields are weak, spin is a good quantum number. That is why spin is used in quantum mechanics.

[In] QED, we hardly ever talk about chirality. The word is basically reserved for chiral theories, which are theories that are not symmetric under L ↔ R, such as the theory of the weak interactions. We talk very often about helicity. In the high-energy limit, helicity is often used interchangeably with chirality. As a slight abuse of terminology, we say ψL and ψR are helicity eigenstates. In the non-relativistic limit, we use helicity for photons and spin (the vector) for spinors. Helicity eigenstates for photons are circularly polarized light.

Quantum Field Theory and Standard Model by M. Schwartz

Abstract

For a particle with momentum $\vec p \neq 0$ the helicity operator is

$h = \frac{\vec J \cdot \vec p}{|\vec p|} = \frac{\vec S\cdot \vec p}{|\vec p|} ,$

where the orbital part drops out because $\vec L \cdot \vec p = (\vec x\times\vec p)\cdot\vec p = 0$ . It commutes with the free Hamiltonian and with rotations. A boost that does not reverse the direction of $\vec p$ , for example one along $\vec p$ with speed below that of the particle, leaves the helicity unchanged, but a general boost does not.

The invariant meaning comes from Wigner's classification of the unitary irreducible representations of the Poincaré group, see Wigner's little groups. Use the metric signature $(+,-,-,-)$ and the Pauli and Lubanski vector

$W^\mu = \tfrac{1}{2}\,\epsilon^{\mu\nu\rho\sigma} J_{\nu\rho} P_\sigma ,$

with the sign of $\epsilon$ chosen such that $W^0 = \vec J\cdot \vec P$ . It satisfies $W^\mu P_\mu = 0$ and commutes with all $P^\mu$ , and $W_\mu W^\mu$ is a Casimir.

Massive case. In the rest frame $W^\mu = (0, m\vec S)$ , so $W_\mu W^\mu = -m^2 s(s+1)$ . The little group is $SU(2)$ and the states form a $2s+1$ dimensional multiplet. One can choose the basis $|p,\lambda\rangle$ obtained by preparing a state with $S_z = \lambda$ at rest, boosting it along $z$ and then rotating $z$ into $\hat p$ . These are helicity eigenstates with $\lambda = -s,\ldots,s$ . A boost that reverses the direction of $\vec p$ maps $\lambda$ to $-\lambda$ , so helicity is not a Lorentz invariant label.

Massless case. For $P^\mu = (E,0,0,E)$ the little group is $ISO(2)$ , generated by the rotation $J_3$ and two commuting null rotations. Representations with finitely many polarization states have the null rotations acting trivially. Then $W_\mu W^\mu = 0$ and, together with $W\cdot P = 0$ , this forces

$W^\mu = \lambda P^\mu ,$

with $\lambda$ the eigenvalue of $J_3$ , which is the helicity. Since $W^\mu$ and $P^\mu$ are both four-vectors, $\lambda$ is Lorentz invariant. Each irreducible representation carries a single value of $\lambda$ , and invariance under a rotation by $4\pi$ requires $\lambda \in \tfrac{1}{2}\mathbb Z$ . Parity and CPT map $\lambda \to -\lambda$ , so a local field theory that is CPT invariant contains $+\lambda$ and $-\lambda$ together: the photon has $\lambda = \pm 1$ , the graviton $\pm 2$ . The representations in which the null rotations act non-trivially are the continuous spin representations, which have infinitely many polarization states and are not known to describe observed particles.

For a massless Weyl field, chirality and helicity are tied together: a left-handed Weyl field annihilates particles of helicity $-\tfrac12$ and creates antiparticles of helicity $+\tfrac12$ . For massive fermions the mass term mixes the chiralities, helicity is frame dependent, and the two notions agree only up to corrections of order $m/E$ .

Helicity

Why is it interesting?

Intuitive

Concrete

Abstract

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