From Angular Momentum to Helicity

Angular momentum in ordinary quantum mechanics is usually labeled by a magnitude $j$ and a projection $m$ along a chosen axis. For relativistic particles, especially massless particles, the natural axis is often the particle’s own momentum. The angular momentum projection along momentum is called helicity.

For a sharp momentum state, the dimensionless helicity operator is

$$\Lambda = \frac{\mathbf J\cdot\mathbf p} {\hbar|\mathbf p|}.$$

Here $\mathbf J$ is total angular momentum and $\mathbf p\ne0$ is the momentum label. For a single particle the orbital contribution satisfies $(\mathbf x\times\mathbf p)\cdot\mathbf p=0$, leaving the spin projection along momentum. Helicity is the bridge from familiar magnetic quantum numbers to relativistic spin and polarization labels.

From a Fixed Axis to the Momentum Axis

In nonrelativistic angular momentum problems one often chooses a fixed laboratory axis and diagonalizes

$$J^2, \qquad J_z.$$

The resulting labels are

$$\lvert j,m\rangle.$$

This is natural for atoms in a central potential, spins in a magnetic field, and spectroscopy with a chosen quantization axis.

For a freely propagating particle, however, the momentum direction is physically distinguished. One can ask for the angular momentum projection along $\hat{\mathbf p}$ instead of along a fixed $\hat z$:

$$\mathbf J\cdot\hat{\mathbf p}.$$

The corresponding eigenvalue is the helicity label. In a frame where $\mathbf p$ points along $z$, helicity reduces to the familiar angular momentum projection along $z$.

Massive Particles

For a massive particle, there is a rest frame. The standard spin label is still the rest-frame little-group spin $s$, with projections

$$m_s=-s,-s+1,\ldots,s.$$

Helicity is also useful. For a one-particle state with momentum $\mathbf p$, a helicity basis uses labels

$$\lambda=-s,-s+1,\ldots,s,$$

where $\lambda$ is the spin projection along the direction of motion. For spin-$1/2$, this gives $\lambda=\pm1/2$.

The important caveat is that massive-particle helicity is not invariant under all Lorentz transformations. A boost can overtake the particle and reverse its momentum direction while leaving the spin direction continuously related. The helicity can then change sign.

This does not make helicity useless. It is extremely useful in scattering, high-energy limits, and decay kinematics. It only means that for massive particles helicity is a basis choice, not an invariant particle species label.

Massless Particles

Massless particles have no rest frame. The rest-frame spin projection story is unavailable. For physically realized finite-helicity massless particles, helicity becomes the spin-like invariant label.

A rotation can certainly reverse the momentum direction, but it rotates angular momentum with it and preserves helicity. Unlike the massive case, there is no rest frame to pass through by overtaking the particle. More precisely, the massless little group is locally $ISO(2)$; in a finite-helicity representation its translation subgroup acts trivially and its rotation subgroup supplies a phase labeled by helicity.

For massless particles, one writes one-particle states schematically as

$$\lvert p,\lambda\rangle,$$

where

$$p^2=0, \qquad \lambda \in \{\text{allowed helicities}\}.$$

For a photon, the physical helicities are

$$\lambda=+1, \qquad \lambda=-1.$$

For a graviton in linearized relativistic field theory, the corresponding physical helicities are $\lambda=\pm2$.

Why a Photon Has No Longitudinal Helicity State

A massive spin-$1$ particle has three spin projections:

$$\lambda=-1,0,+1.$$

A photon is massless and has only two physical polarizations. The missing $\lambda=0$ mode is not an experimental accident. It is tied to transversality and gauge redundancy in electromagnetism.

In a plane wave with momentum $\mathbf k$, the physical electric-field polarization is transverse:

$$\boldsymbol\epsilon\cdot\mathbf k=0.$$

The two transverse circular polarization states correspond to the two photon helicities. A longitudinal polarization would point along $\mathbf k$ and is not a physical photon state in the free theory.

This is one of the cleanest places where the word “spin” must be refined. Saying “the photon has spin one” does not mean it has three rest-frame spin projections like a massive spin-$1$ particle.

Circular and Linear Polarization

For a wave propagating along $+z$, one common convention uses transverse basis vectors

$$\boldsymbol\epsilon_\pm = \frac{1}{\sqrt2} (\hat{\mathbf x}\pm i\hat{\mathbf y}).$$

Depending on convention, $\boldsymbol\epsilon_+$ and $\boldsymbol\epsilon_-$ are assigned to right- and left-circular polarization or the reverse. The invariant statement is that the two circular polarization states are helicity eigenstates with opposite helicities.

Linear polarization is a superposition of the two helicities:

$$\hat{\mathbf x} = \frac{1}{\sqrt2} (\boldsymbol\epsilon_+ + \boldsymbol\epsilon_-),$$

and similarly for $\hat{\mathbf y}$ with a relative phase. A linearly polarized photon is therefore not a helicity eigenstate. A classical linearly polarized plane wave likewise contains both circular components.

The mode-language point is developed more generally in Mode Decompositions.

Relation to Selection Rules

In atomic and molecular transitions, polarization labels often appear as spherical components $q=0,\pm1$ of a vector operator. The angular momentum rule is

$$\Delta m=q.$$

For radiation propagating along the quantization axis, the two circular polarizations correspond to the two transverse $q=\pm1$ components. This is why circularly polarized light can carry angular momentum projection along its propagation direction.

There are two cautions.

First, the signs attached to $\sigma^+$ and $\sigma^-$ depend on propagation direction and convention. The safer label is the tensor component $q$ or the helicity $\lambda$ after the convention is fixed.

Second, dipole selection rules are about matrix elements in a chosen system and geometry. Helicity is a particle or mode label. The two meet in light-matter coupling, but they are not the same definition.

See Dipole Transitions and Selection Rules for the nonrelativistic angular momentum side.

Spin-Half Example

For a spin-$1/2$ particle with momentum along $+z$, the spin part of helicity reduces to

$$\frac{S_z}{\hbar} = \frac12\sigma_z.$$

Thus

$$\lvert\uparrow_z\rangle \quad \text{has} \quad \lambda=+\frac12,$$

and

$$\lvert\downarrow_z\rangle \quad \text{has} \quad \lambda=-\frac12.$$

If the same spin quantization states are paired with momentum along $-z$, the helicity signs are reversed. This is the simplest way to see why helicity is not merely “spin up” or “spin down”; it is spin projection relative to the momentum direction.

Helicity and Chirality

Chirality is a Lorentz-representation label for Weyl spinors. Helicity is an angular momentum projection along momentum. The two are closely related for massless spin-$1/2$ particles in standard relativistic theories, but they are not the same definition.

For massive spinors, a mass term couples left- and right-chiral components, and helicity can change under boosts. The bridge from $SU(2)$ spinors to Weyl and Dirac spinors is developed in From SU(2) Spinors to Lorentz Spinors.

Parity

Parity reverses momentum:

$$\mathbf p \longmapsto -\mathbf p.$$

Angular momentum is an axial vector, so under parity

$$\mathbf J \longmapsto \mathbf J.$$

Therefore helicity changes sign:

$$\Lambda \longmapsto -\Lambda.$$

For a nonzero-helicity massless particle, parity invariance requires the corresponding opposite-helicity state and the appropriate transformation of interactions. The helicity-zero representation can map to itself. The presence of both helicities alone does not guarantee parity-invariant interactions. This is one reason helicity language becomes central in relativistic weak-interaction physics.

Bridge to QFT

In quantum field theory and relativistic scattering, amplitudes are often organized by external momenta and helicities:

$$\mathcal M (\lambda_1,\lambda_2,\ldots).$$

For massless particles this is more than a convenient basis; helicity is part of the particle’s representation data. For massive particles it is a useful basis tied to a chosen frame and momentum assignment.

The representation-theoretic context is summarized in From Spin to Relativistic Representations. The compact reference bridge for spinors is Spinors.

References

• A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957.
• M. Jacob and G. C. Wick, “On the general theory of collisions for particles with spin,” Annals of Physics 7, 404-428, 1959.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• M. Srednicki, Quantum Field Theory, Cambridge University Press, 2007.
• S. Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995.
• E. P. Wigner, “On unitary representations of the inhomogeneous Lorentz group,” Annals of Mathematics 40, 149-204, 1939.

Exercises

1. A spin-$1/2$ particle has momentum along $+z$. What are the helicities of $\lvert\uparrow_z\rangle$ and $\lvert\downarrow_z\rangle$?

Solution

For momentum along $+z$,

$$\Lambda = \frac{S_z}{\hbar} = \frac12\sigma_z.$$

Therefore $\lvert\uparrow_z\rangle$ has $\lambda=+1/2$, and $\lvert\downarrow_z\rangle$ has $\lambda=-1/2$.

2. The same spin states are paired with momentum along $-z$. What happens?

Solution

For momentum along $-z$,

$$\Lambda = -\frac{S_z}{\hbar}.$$

Thus $\lvert\uparrow_z\rangle$ now has $\lambda=-1/2$, while $\lvert\downarrow_z\rangle$ has $\lambda=+1/2$. Helicity is spin projection along momentum, not along a fixed laboratory axis.

3. Why does a free photon have no physical helicity-zero state?

Solution

A free photon is massless and has transverse physical polarizations. For a plane wave,

$$\boldsymbol\epsilon\cdot\mathbf k=0.$$

The two independent transverse circular polarizations have helicities $+1$ and $-1$. A helicity-zero longitudinal mode would be aligned with $\mathbf k$ and is removed from the physical spectrum by transversality and gauge redundancy.

4. What does parity do to helicity?

Solution

Parity sends $\mathbf p\mapsto-\mathbf p$ while leaving angular momentum $\mathbf J$ unchanged because $\mathbf J$ is an axial vector. Therefore

$$\frac{\mathbf J\cdot\mathbf p} {\hbar|\mathbf p|} \longmapsto - \frac{\mathbf J\cdot\mathbf p} {\hbar|\mathbf p|}.$$

Parity flips helicity.