Spinor-Helicity Variables

In four dimensions, a null four-momentum is equivalent to a rank-one $2\times2$ bispinor. Factoring that matrix into one undotted and one dotted spinor turns Lorentz products into antisymmetric brackets, makes helicity weights manifest, and represents physical vector polarizations without carrying unphysical components. The formalism is compact because it solves the on-shell condition before the amplitude calculation begins.

Required background. On-Shell States and Little-Group Scaling supplies the helicity weight assigned to each external leg. Weyl Fields and Chirality supplies the two-component spinor representations and index conventions.

Helpful background. Spinors, Conjugations, Bilinears, Chirality, and Fierz Identities develops the underlying spinor algebra.

Null momenta as rank-one bispinors

Use the mostly-minus metric and

$$\sigma^\mu=(\mathbf1,\boldsymbol\sigma), \qquad \bar\sigma^\mu=(\mathbf1,-\boldsymbol\sigma),$$

with $p_{\alpha\dot\alpha}=p_\mu\sigma^\mu_{\alpha\dot\alpha}$. Then

$$\det(p_{\alpha\dot\alpha})=p^2.$$

For $p^2=0$ the matrix has rank one, so over the complex numbers

$$\boxed{p_{\alpha\dot\alpha} =\lambda_\alpha\widetilde\lambda_{\dot\alpha}} \equiv |p\rangle[p|.$$

This factorization has one complex redundancy,

$$(\lambda,\widetilde\lambda)\sim (t\lambda,t^{-1}\widetilde\lambda),$$

which is the complexified helicity scaling of the massless little group; the translation-like part of $\operatorname{ISO}(2)$ acts trivially on the finite-helicity states considered here. For a real future-directed momentum, dotted and undotted spinors are related by Hermitian conjugation up to this phase. For complex momenta they are independent. For a real momentum with negative energy—as occurs after crossing to an all-incoming or all-outgoing convention—the reality relation includes a sign choice. It is safer to impose momentum conservation algebraically and take the desired physical boundary value only at the end Elvang and Huang 2014, §§ 2.2 and 2.6, pp. 9–11 and 27–30, PDF.

Schwartz derives the rank-one factorization in the site’s metric convention in Schwartz 2014, § 27.1.1, pp. 537–539.

Brackets and Lorentz invariants

With $\epsilon^{12}=-\epsilon_{12}=1$, define

$$\langle ij\rangle =\epsilon^{\alpha\beta}\lambda_{i\alpha}\lambda_{j\beta}, \qquad [ij] =\epsilon_{\dot\alpha\dot\beta} \widetilde\lambda_i^{\dot\alpha} \widetilde\lambda_j^{\dot\beta} =\epsilon^{\dot\alpha\dot\beta} \widetilde\lambda_{i\dot\beta} \widetilde\lambda_{j\dot\alpha}.$$

The spinors commute, while $\epsilon$ is antisymmetric, hence

$$\langle ij\rangle=-\langle ji\rangle, \qquad [ij]=-[ji], \qquad \langle ii\rangle=[ii]=0.$$

Our invariant dictionary is

$$\boxed{2p_i\!\cdot p_j=\langle ij\rangle[ji]}.$$

Thus for massless all-outgoing momenta,

$$s_{ij}=(p_i+p_j)^2=\langle ij\rangle[ji].$$

The order of the square bracket is part of the convention; silently replacing $[ji]$ by $[ij]$ flips a sign. Brackets have mass dimension one and little-group weights

$$\langle ij\rangle\mapsto t_it_j\langle ij\rangle, \qquad [ij]\mapsto t_i^{-1}t_j^{-1}[ij].$$

Because the spinor space is two-dimensional, any three undotted spinors are linearly dependent. The Schouten identity is

$$\langle ij\rangle\langle k\ell\rangle +\langle ik\rangle\langle\ell j\rangle +\langle i\ell\rangle\langle jk\rangle=0,$$

with an identical square-bracket version. Momentum conservation becomes

$$\sum_i |i\rangle[i|=0, \qquad \sum_i\langle r i\rangle[i s]=0$$

for arbitrary reference spinors $r,s$. These identities, not numerical component fitting, are the standard way to prove equality of two compact amplitude formulas Schwartz 2014, § 27.1.1, pp. 538–539.

Polarization vectors

Let $q$ be a null reference momentum not collinear with $p$. A convention compatible with the invariant identity above is

$$\varepsilon_{+\,\alpha\dot\alpha}(p;q) =\sqrt2\, \frac{q_\alpha\widetilde\lambda_{\dot\alpha}} {\langle q\lambda\rangle}, \qquad \varepsilon_{-\,\alpha\dot\alpha}(p;q) =\sqrt2\, \frac{\lambda_\alpha\widetilde q_{\dot\alpha}} {[\lambda q]}.$$

They obey

$$p\!\cdot\varepsilon_\pm=0, \qquad \varepsilon_\pm^2=0, \qquad \varepsilon_+\!\cdot\varepsilon_-=-1.$$

Their little-group weights are $t^{-2}$ and $t^{+2}$, respectively. Changing $q$ produces a shift proportional to $p^\mu$; therefore any physical amplitude is $q$-independent precisely when the corresponding Ward identity holds. The normalization, reference-vector shift, and useful contractions are derived in Schwartz 2014, § 27.1.2, pp. 539–541.

Spinor-helicity dictionary

This table is the convention-fixed reference used by the massless pages in this chapter. Its cells use a linear text form so the definitions remain meaningful in table navigation; the equations immediately above provide the fully typeset versions.

Spinor-helicity dictionary in the QFT.org mostly-minus convention

Quantity	Definition in this volume	Weight on leg i	Independent check
Null momentum	pᵢ = |i⟩[i| as a bispinor	zero	det pᵢ = pᵢ² = 0
Lorentz product	2 pᵢ·pⱼ = ⟨ij⟩[ji]	zero	evaluate one real back-to-back configuration
Massless Mandelstam invariant	sᵢⱼ = ⟨ij⟩[ji]	zero	compare with (pᵢ+pⱼ)²
Angle bracket	⟨ij⟩ = ε λᵢ λⱼ	tᵢ	antisymmetry and mass dimension one
Square bracket	[ij] with the raised-dotted-index order defined above	tᵢ⁻¹	antisymmetry and mass dimension one
Positive-helicity vector	ε₊(i;q)	tᵢ⁻²	transversality, norm, and reference-q independence of the complete amplitude
Negative-helicity vector	ε₋(i;q)	tᵢ²	transversality, norm, and reference-q independence of the complete amplitude
Helicity-hᵢ amplitude	Aₙ(…,iʰⁱ,…)	tᵢ⁻²ʰⁱ	apply the homogeneity operator
Massive momentum interface	p = λI λ̃I with the SU(2) index contracted	SU(2) covariance on I	det p = m²; continue to the massive-variable page

The table does not fix phases of individual spinors. Those phases are redundant; invariant products, residues, and complete amplitudes are the transferable objects.

A component check

Take

$$p^\mu=(E,0,0,E), \qquad q^\mu=(E,0,0,-E),$$

so that $2p\!\cdot q=4E^2$. With the index and bracket conventions above, one consistent factorization is

$$\lambda_p=\widetilde\lambda_p= \begin{pmatrix}0\\-\sqrt{2E}\end{pmatrix}, \qquad \lambda_q=\widetilde\lambda_q= \begin{pmatrix}\sqrt{2E}\\0\end{pmatrix}.$$

It gives $\langle pq\rangle=2E$ and $[qp]=2E$, hence

$$\langle pq\rangle[qp]=4E^2=2p\!\cdot q.$$

A different little-group phase changes the two brackets oppositely and leaves their product fixed. This single back-to-back check catches the most common metric and bracket-order sign error before a longer calculation begins.

Common pitfalls

Imposing Lorentzian conjugation during a complex deformation. BCFW shifts make $\lambda$ and $\widetilde\lambda$ independent. Restore the appropriate real boundary condition only after the analytic calculation.

Forgetting the square-bracket order in $2p_i\cdot p_j$. With the definitions above the invariant is $\langle ij\rangle[ji]$. Check it once on real back-to-back momenta.

Choosing $q\parallel p$ in a polarization. The denominator then vanishes. Choose a noncollinear reference and verify that the complete amplitude is independent of it.

Treating Schouten as the only identity. Momentum conservation, on-shell conditions, and dimension-specific Gram relations can also relate expressions. State which relations were used.

Exercises

Multiply $\sum_i|i\rangle[i|=0$ on the left by $\langle r|$ and on the right by $|s]$.

Solution

Linearity gives

$$0=\langle r|\left(\sum_i|i\rangle[i|\right)|s] =\sum_i\langle ri\rangle[is].$$

The bracket order follows from matrix multiplication; reversing $[is]$ would introduce a minus sign.

Where to continue

• Three-Point Amplitudes uses bracket weights and complex kinematics to classify local seeds.
• Complex Momenta and Factorization turns bracket deformations into physical factorization poles.
• Massive On-Shell Variables and Little-Group Covariance supplies the $SU(2)$ extension.

References

• Elvang, Henriette, and Yu-tin Huang. Scattering Amplitudes in Gauge Theory and Gravity. Cambridge: Cambridge University Press, 2015. Open prepublication version. Open PDF.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI.