Helicity and Chirality

Helicity measures spin along momentum; chirality labels the two Weyl components of a Dirac spinor. They agree in a specific sense for positive-energy massless modes, but a massive helicity eigenstate has both chiralities. Helicity is conserved by the free Dirac Hamiltonian yet can change under a change of inertial frame for a massive particle. Chirality is preserved by connected Lorentz transformations but generally fails to commute with a massive Hamiltonian. These statements concern different operations.

Required background. Free Dirac Spinors fixes the modes; Dirac Spinors fixes chirality and mass coupling; Lorentz Transformations fixes the frame comparison.

Two operators with different definitions

Use natural units, $\mathbf p\ne0$, and $\boldsymbol\Sigma=\operatorname{diag}(\boldsymbol\sigma,\boldsymbol\sigma)$. Define dimensionless helicity by

$$\widehat h=\frac{\boldsymbol\Sigma\cdot\mathbf p}{2|\mathbf p|}.$$

Its eigenvalues for spin one half are $\pm1/2$. Some sources call $2\widehat h$ the helicity operator; the factor two must be translated when comparing formulas.

Chirality has eigenvalues $\pm1$ under $\gamma^5$ and projectors $P_{L,R}=(1\mp\gamma^5)/2$. It is a component-space operator with no momentum direction in its definition. At $\mathbf p=0$, helicity has no preferred axis and is undefined; chirality remains defined.

Free helicity is conserved

For $H=\boldsymbol\alpha\cdot\mathbf p+\beta m$, the Pauli algebra gives

$$[H,\boldsymbol\Sigma\cdot\mathbf p]=0,\qquad [H,|\mathbf p|]=0.$$

Thus $[H,\widehat h]=0$ away from zero momentum. The antisymmetry of the Pauli commutator contracts with the symmetric product $p_ip_j$ and vanishes; the mass matrix commutes with $\boldsymbol\Sigma$.

This does not say that all spin components are separately conserved, or that helicity survives a general external field. It is a property of the free Hamiltonian.

Chirality instead obeys

$$[H,\gamma^5]=2m\beta\gamma^5,\qquad \frac{d\gamma^5}{dt}=2im\beta\gamma^5$$

in the Heisenberg equation, with all operators at the same time in the second expression. The kinetic term commutes with $\gamma^5$, while the mass term anticommutes with it. For $m=0$, both helicity and chirality commute with $H$.

Chiral weights of a massive helicity eigenstate

Choose a Pauli spinor with $(\boldsymbol\sigma\cdot\widehat{\mathbf p})\chi_\lambda =\lambda\chi_\lambda$, $\lambda=\pm1$, and form the positive-energy spinor $u_\lambda(p)$ of norm $2E$. In the Dirac basis, $\gamma^5$ exchanges the upper and lower blocks. Direct multiplication gives

$$\frac{u_\lambda^\dagger\gamma^5u_\lambda} {u_\lambda^\dagger u_\lambda} =\lambda\,\frac{|\mathbf p|}{E}.$$

Consequently

$$f_L=\frac12\left(1-\lambda\frac{|\mathbf p|}{E}\right), \qquad f_R=\frac12\left(1+\lambda\frac{|\mathbf p|}{E}\right).$$

These are normalized projector weights in this frame, not extra particle species. At low speed they are nearly equal; at high speed one dominates. For positive helicity, the suppressed left weight is

$$f_L=\frac{m^2}{2E(E+|\mathbf p|)} =\frac{m^2}{4E^2}+O(m^4/E^4).$$

A left-chiral projection therefore has a small but nonzero norm on a massive positive-helicity mode. Its amplitude suppression is of order $m/E$; the squared weight is of order $m^2/E^2$. A cross section additionally depends on the interaction and the other external states.

For example, at $|\mathbf p|/E=3/5$, a positive-helicity particle has $f_R=4/5$ and $f_L=1/5$. It is neither a pure right-chiral spinor nor a negative-energy admixture.

Operator nonconservation does not force every state to oscillate

A stationary positive-energy spinor keeps its chiral weights constant because its entire state acquires one common phase. The same holds for the integrated chirality expectation of any freely evolving packet confined to one energy sector. In momentum space $H$ acts as $+E_{\mathbf p}$ or $-E_{\mathbf p}$ times the identity within that sector; the phases cancel in its diagonal expectation.

More explicitly, the anticommutator $\{H,\gamma^5\}=2\boldsymbol\Sigma\cdot\mathbf p$ gives

$$P_\pm\gamma^5P_\pm =\pm\frac{\boldsymbol\Sigma\cdot\mathbf p}{E_{\mathbf p}}P_\pm.$$

Here $P_\pm$ are energy projectors, not chiral projectors. Time-dependent integrated chirality in the free theory comes from coherence between opposite energy sectors. Local chiral densities of a packet can still redistribute in space. The rest-spinor oscillation on Dirac Spinors is an explicit example with both energy signs, not a universal oscillation of every massive electron.

A boost can reverse massive helicity

Consider a particle moving along $+z$ with momentum $p>0$ and a spin eigenstate along $z$. A passive boost with speed $v$ along the same axis gives

$$p'_z=\gamma_v(p-vE).$$

If $p/E<v<1$, the observer overtakes the particle and $p'_z<0$. A collinear boost does not rotate the canonical spin axis, so the spin projection along $z$ keeps its sign while the momentum direction reverses. The helicity flips. At the intermediate rest frame it is undefined.

This is compatible with free helicity conservation: conservation describes time evolution in a fixed frame, whereas the boost compares different frames. For a massless particle no inertial observer can overtake it. Its finite-helicity label is invariant under proper orthochronous Lorentz transformations.

The chiral subspaces are each invariant under the connected Lorentz cover, but normalized chiral weights need not be frame invariant. Boosts are not unitary in the local component norm and reweight the two nonzero blocks. “Chirality is Lorentz invariant” means that a pure chirality transforms into the same chirality, not that every massive mode’s $f_L$ is the same in every frame.

The massless limit and antiparticle labels

For a positive-energy mode at fixed nonzero momentum, $m\to0$ gives $\gamma^5u_\lambda=\lambda u_\lambda$. Its chirality eigenvalue is twice its helicity. The limit is not taken through $\mathbf p=0$, where helicity has no definition.

For a negative-frequency coefficient $v(p)e^{+ip\cdot x}$, the future label $p$ differs from its differential four-momentum $-p$. Moreover, after quantization the coefficient multiplies an antiparticle creation operator, whose state labels must be read from the field transformation law. One must not infer the created state’s spin merely from the eigenvalue of a Pauli column inside $v$.

With the usual free-field interpretation, a massless left-chiral field annihilates helicity $-1/2$ particles and creates helicity $+1/2$ antiparticles. A right-chiral field has the opposite assignment. Thus “left-chiral field” and “only negative-helicity excitations” are different claims.

Exercises

1. At $|\mathbf p|=m$, find the left and right fractions of a positive-helicity, positive-energy spinor.

Solution

$E=\sqrt2m$ gives $f_L=(1-1/\sqrt2)/2\approx0.1464$ and $f_R\approx0.8536$. Its negative-energy projector weight is zero despite its nonzero left-chiral fraction.

2. A particle has $p/E=3/5$ along $+z$. Apply a collinear passive boost with $v=4/5$. Determine the new velocity and the helicity change for spin along $+z$.

Solution

$v'=(3/5-4/5)/(1-12/25)=-5/13$. The spin axis is unchanged but the momentum reverses, so helicity changes from $+1/2$ to $-1/2$. The chiral weights also change: the new negative-helicity mode has $f_L=(1+5/13)/2=9/13$ and $f_R=4/13$.

3. Can a nonzero $[H,\gamma^5]$ establish oscillating chirality in every free positive-energy wave packet?

Solution

No. The commutator is an operator statement on the full Dirac space. Its expectation vanishes for a state restricted to one free energy sector, whose integrated chiral weights are constant. Intersector coherence supplies the free oscillatory contribution.

References

• H. K. Dreiner, H. E. Haber, and S. P. Martin, “Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry,” Physics Reports 494, 1–196, 2010, doi:10.1016/j.physrep.2010.05.002; corrected arXiv:0812.1594v6, 2022 — helicity modes and chiral components.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014, doi:10.1017/9781139540940 — helicity, chirality, and antiparticle states.
• B. Thaller, The Dirac Equation, Springer, 1992 — free spectral sectors and Dirac observables.