LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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Null momenta admit unique spinor-helicity classes

Statement

For every $p \in C^{4}$ , $\det κ (p) = h (p, p)$ . Thus for every nonzero $p$ with $h (p, p) = 0$ , $κ (p)$ has a spinor factorization. If $κ (p)_{a \dot{a}} = λ_{a} {\tilde{λ}}_{\dot{a}} = λ_{a}^{'} {\tilde{λ}}_{\dot{a}}^{'}$ , then there is a unique $t \in C^{\times}$ such that $λ^{'} = t λ$ and ${\tilde{λ}}^{'} = t^{- 1} \tilde{λ}$ . Hence the spinor factorizations of each nonzero null momentum form exactly one equivalence class under the stated rescaling.

Statement references

Complex null momenta and helicity weights

Elvang and Huang, Scattering Amplitudes, §2.2

Marzolla, The four-dimensional on-shell three-point amplitude in spinor-helicity formalism and BCFW recursion relations, §3

Britto, Cachazo, Feng, and Witten, Direct Proof of Tree-Level Recursion Relation in Yang-Mills Theory, §2

Facts & Assumptions

Given: $p \neq 0$ , $h (p, p) = 0$ , and the coordinate-to-matrix map $κ$ of the spinor-helicity definition.

[F1]

Complex null momenta and helicity weights: $κ (p) = (\begin{array}{cc} p^{0} + p^{3} & p^{1} - i p^{2} \\ p^{1} + i p^{2} & p^{0} - p^{3} \end{array})$ .

[F2]

Complex null momenta and helicity weights: $h (p, q) = p^{0} q^{0} - p^{1} q^{1} - p^{2} q^{2} - p^{3} q^{3}$ .

[F3]

Complex null momenta and helicity weights: a spinor factorization is a pair of nonzero two-spinors with $M_{a \dot{a}} = λ_{a} {\tilde{λ}}_{\dot{a}}$ .

[F4]

Complex null momenta and helicity weights: the explicit inverse formula for $κ$ recovers all four coordinates from its matrix entries.

Proof

1.1givenF1F2F4algebra

Direct expansion gives $\det κ (p) = (p^{0})^{2} - (p^{1})^{2} - (p^{2})^{2} - (p^{3})^{2} = h (p, p) = 0$ . The explicit inverse formula in [F4] shows $κ (p) \neq 0$ because $p \neq 0$ . A nonzero $2 \times 2$ matrix with determinant zero has rank one.

2.1givenF1F3step 1.1choose

Choose a nonzero column $v$ of $κ (p)$ . Rank one means each of its two columns is a scalar multiple $a_{\dot{a}} v$ ; setting $λ = v$ and ${\tilde{λ}}_{\dot{a}} = a_{\dot{a}}$ gives the factorization required by [F3]. Both factors are nonzero because their product matrix is nonzero.

3.1givenstep 2.1algebra∎

Any second factorization has the same one-dimensional column space, so $λ^{'} = t λ$ for a unique nonzero scalar $t$ . Equality of the outer products then gives ${\tilde{λ}}^{'} = t^{- 1} \tilde{λ}$ by comparing a nonzero component of $λ$ . This proves uniqueness of the class.

Depends on

Complex null momenta and helicity weights