Spinor helicity formalism

Conventions

We set up the conventions of spinor helicity formalism in a signature-independent way.

It is noteworthy that while the choice of convention for physical observables is purely a "gauge", it propagates along computations. From the perspective of programming, this is actually a nonlocal effect that should be encapsulated within different modules. Therefore, we align our spinor conventions to the particle physics literature, ensuring they remain independent of the metric signature.

The signature is denoted as ¹

\[\begin{equation} \label{eq: signature} \signature= \begin{cases} +1 \, , \quad (-,+,+,+)\, ,\\ -1 \, , \quad (+,-,-,-)\, .\\ \end{cases} \end{equation}\]
The convention of spinor contraction follows from Wess&Bagger

\[\begin{align} \tag{NW-SE} &\psi\chi=\psi^{a}\chi_{a} \, , \\ \tag{NE-SW} &\psib\chib=\psib_{\dota}\chib^{\dota} \, . \end{align}\]
A generic momentum is denoted as \(P\), massive as \(p\) and massless as \(q\).

Comparison

In Dreiner et al.'s review ², the authors provide both signature versions and a detailed discussion on convention changing.

\((-,+,+,+)\)

In Elvang&Huang's book ³, the signature is \(\signature=1\), propagates into momentum bispinors, see \eqref{eq: momentum bispinor} vs.

\[\begin{equation} \tag{2.7 [^2]} \label{eq: test} P_{a\dota}\eqq P_{\mu} \sigma^{\mu}_{a\dota} \, , \end{equation}\]

then ceases at helicity spinors, see \eqref{eq: helicity spinor} vs.

\[\begin{equation} \tag{2.16 [^2]} q_{a\dota}=-\ketS{q}_{a}\braA{q}_{\dota} \, , \quad q^{\dota a}=-\ketA{q}^{\dota}\braS{q}^{a} \, . \end{equation}\]
- In Srednicki's book ⁴, the conventions are almost the same as ³, except for incoming/outgoing. For this, we follow ³ as discussed later.

\((+,-,-,-)\)

In Taylor's lecture notes ⁵, the signature is \(\signature=-1\). The type of helicity spinors is opposite to ours, i.e. angle spinor is undotted, see \eqref{eq: pairing A} vs.

\[\begin{equation} \tag{4.31 [^4]} \braketA{q_{1}q_{2}}\eqq \braA{q_{1}}^{a} \ketA{q_{2}}_{a} \, . \end{equation}\]

The type of polarization vectors is also opposite to ours, see \eqref{eq: polarization component} vs.

\[\begin{equation} \tag{5.7 [^4]} \epsilon_{\pm}=\frac{1}{\sqrt{2}}(0,1,\pm i,0) \, . \end{equation}\]

The net effect is that the polarization vectors match ours up to \(\pm\sqrt{2}\), see \eqref{eq: polarization} vs.

\[\begin{equation} \tag{5.8 [^4]} \epsilon_{+}^{\mu}(q,q') = \frac{1}{\sqrt{2}} \frac{ \braketAS{q'}{\sigma^{\mu}}{q} }{ \braketA{q' q} } \, , \quad \epsilon_{-}^{\mu}(q,q') = -\frac{1}{\sqrt{2}} \frac{ \braketSA{q'}{\sigmab^{\mu}}{q} }{ \braketS{q' q} } \, , \end{equation}\]

for the reference momentum \(q'\).
- In Schwartz's book ⁶, the conventions are almost the same as ⁵. The contraction of spinor indices is opposite to Wess&Bagger's convention, see Section 10.6.2.
- In Cheung's lecture notes ⁷, the conventions are almost the same as ⁵. The relation between Mandelstam variables and helicity spinors omits a sign, possibly due to the convention of spinor inner products.
- In Witten's paper ⁸,

Helicity spinor

Helicity spinors are essentially bosonic Weyl spinors that satisfy the massless Weyl equation.

Square spinor

The square ket \(\ketS{q}_{a}\), as a left-handed (bosonic) Weyl spinor associated to the complex massless momentum \(q\in\CC^{4}\), lies in the fundamental representation \((\half,0)\) of \(\slgroup(2,\CC)\), and the square bra \(\braS{q}^{a}\) lies in its dual \((\half,0)^{\dual}\):

\[\begin{equation} \ketS{q}_{a} \to M^{b}_{a} \ketS{q}_{b} \, , \quad \braS{q}^{a} \to (M^{-\tp})^{a}_{b} \braS{q}^{b} \, , \end{equation}\]

then the square braket is the natural pairing

\[\begin{equation} \label{eq: pairing S} \braketS{q_{1}q_{2}}\eqq \braS{q_{1}}^{a} \ketS{q_{2}}_{a} \, . \end{equation}\]

Due to the intertwiner and its inverse

\[\begin{equation} \epsilon^{ab}=-\epsilon_{ab}= \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \, , \quad \epsilon^{ab}\epsilon_{bc}=\delta^{a}_{b} \, , \end{equation}\]

the two representations are isomorphic by

\[\begin{equation} \label{eq: raising lowering S} \braS{q}^{a}=\epsilon^{ab}\ketS{q}_{b} \, , \quad \ketS{q}_{a}=\epsilon_{ab}\braS{q}^{b} \, . \end{equation}\]

Equivalently, \(\epsilon\) is an invariant bilinear form on \((\half,0)\) or on its dual, and then \(\braketS{q_{1}q_{2}}\) is antisymmetric.

Angle spinor

Repeating the previous discussion onto the complex conjugate representation \((0,\half)\) and its dual \((0,\half)^{\dual}\), the angle bra and ket satisfy

\[\begin{equation} \braA{q}_{\dota} \to (M^{*})_{\dota}^{\dotb}\braA{q}_{\dotb} \, , \quad \ketA{q}^{\dota} \to (M^{-\dagger})^{\dota}_{\dotb}\ketA{q}^{\dotb} \, , \end{equation}\]

and the pairing is

\[\begin{equation} \label{eq: pairing A} \braketA{q_{1}q_{2}} \eqq \braA{q_{1}}_{\dota}\ketA{q_{2}}^{\dota} \, . \end{equation}\]

The normalization of the intertwiner is chosen to commute with complex conjugation:

\[\begin{equation} \epsilon^{\dota \dotb}=-\epsilon_{\dota \dotb}= \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \, , \quad \epsilon^{\dota \dotb}\epsilon_{\dotb \dotc}=\delta^{\dota}_{\dotb} \, , \end{equation}\]

then the indices are lowered/raised as

\[\begin{equation} \label{eq: raising lowering A} \ketA{q}^{\dota}=\epsilon^{\dota \dotb}\braA{q}_{\dotb} \, , \quad \braA{q}_{\dota}=\epsilon_{\dota \dotb}\ketA{q}^{\dotb} \, . \end{equation}\]

Momentum bispinor

The Pauli matrices \(\vec\sigma\) intertwine the action of \(\sugroup(2)\) on \(1\in \half\ox\half\) and the action of \(\sogroup(3,\RR)\) on the vector representation through the group morphism \(\sugroup(2)\to \sogroup(3,\RR)\), given by

\[\begin{equation} \sigma^{1}= \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \, , \quad \sigma^{2}= \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \, , \quad \sigma^{3}= \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \, . \end{equation}\]

Similarly, the four-component Pauli matrices \(\sigma\) intertwine the action of \(\slgroup(2,\CC)\) on \((\half,\half)\) and the action of \(\sogroup(3,1,\RR)\) on the complex vector representation \(V\simeq \CC^{4}\) through the group morphism, given by

\[\begin{equation} \label{eq: Pauli matrix} \sigma^{\mu}_{a \dota}=(\id,\vec\sigma) \, , \quad \sigmab^{\mu,\dota a}\eqq\epsilon^{ab}\epsilon^{\dota\dotb}\sigma^{\mu}_{b \dotb}=(\id,-\vec\sigma) \, . \end{equation}\]

Then the metric \(g_{\mu\nu}\), i.e. the invariant bilinear form on \(V\) is intertwined by \(\sigma\) to the one \(\epsilon_{ab}\epsilon_{\dota\dotb}\) on \((\half,\half)\), given by

\[\begin{equation} \label{eq: Pauli matrix and epsilon tensor} g_{\mu\nu}\sigma^{\mu}_{a\dota}\sigma^{\nu}_{b\dotb} = - 2 \signature \epsilon_{ab}\epsilon_{\dota\dotb} \, , \quad \epsilon^{ab}\epsilon^{\dota\dotb}\sigma^{\mu}_{a\dota}\sigma^{\nu}_{b\dotb} = - 2 \signature g^{\mu\nu} \, . \end{equation}\]

With this isomorphism, a vector \(P^{\mu}\) can be rewritten into bispinors

\[\begin{equation} \label{eq: momentum bispinor} \begin{aligned} & P_{a\dota} \eqq - \signature P_{\mu}\sigma^{\mu}_{a\dota} = \begin{pmatrix} P^0-P^3 & -P^1+i P^2 \\ -P^1-i P^2 & P^0+P^3 \\ \end{pmatrix} \, , \\ & P^{\dota a} \eqq - \signature P_{\mu}\sigmab^{\mu,\dota a} = \begin{pmatrix} P^0+P^3 & P^1-i P^2 \\ P^1+i P^2 & P^0-P^3 \\ \end{pmatrix} \, , \end{aligned} \end{equation}\]

and \(\det P = - \signature P^{2}\).

Particularly, when the momentum is massless \(q^{2}=0\), the bispinors factorize into two helicity spinors

\[\begin{equation} \label{eq: helicity spinor} q_{a\dota}=\ketS{q}_{a}\braA{q}_{\dota} \, , \quad q^{\dota a}=\ketA{q}^{\dota}\braS{q}^{a} \, , \end{equation}\]

and then the square and angle spinors satisfy the massless Weyl equations

\[\begin{equation} \label{eq: Weyl equation} q^{\dota a}\ketS{q}_{a}=0 \, , \quad \braA{q}_{\dota} q^{\dota a}=0 \, , \quad q_{a\dota}\ketA{q}^{\dota}=0 \, , \quad \braS{q}^{a} q_{a\dota}=0 \, . \end{equation}\]

Real momentum

Let's count the degrees of freedom. For a complex massless momentum \(q\), the square and angle spinors associated to \(q\) are independent. The left side of \eqref{eq: helicity spinor} is of \(\dim_{\CC}=3\), while the right side is of \(\dim_{\CC}=4\). The reason for this discrepancy is that this factorization is only determined up to a rescaling

\[\begin{equation} \ketS{q} \to \lambda^{-1} \ketS{q} \, , \quad \braA{q} \to \lambda \braA{q} \, , \end{equation}\]

which turns out to be exactly the action of the complexified little group.

While for real momentum, the square and angle spinors are related by complex conjugate. Then the left side of \eqref{eq: helicity spinor} is of \(\dim_{\RR}=3\), the right side is of \(\dim_{\RR}=4\), and the little group scaling is a phase factor in \(\sogroup(2,\RR)\). We also need to distinguish incoming \((-)\) for negative energy \(E<0\) and outgoing \((+)\) for positive energy \(E>0\).

We adopt the following conventions:

\[\begin{align} \label{eq: incoming} \ketS{-q}=\ketS{q} \, , \quad \braA{-q}=-\braA{q} \, , \end{align}\]

and

\[\begin{align} \label{eq: conjugate} \ketS{q}^{*}=\inout{q}\braA{q} \, , \quad \braA{q}^{*}=\inout{q} \ketS{q} \, , \end{align}\]

hence

\[\begin{equation} \label{eq: conjugate braket} \braketS{q_{1}q_{2}}^{*} =-\inout{q_{1}}\inout{q_{2}}\braketA{q_{1}q_{2}} \, . \end{equation}\]

Here the sign notation is

\[\begin{equation} \inout{q}=\cases{ +1 \quad \text{for outgoing} \\ -1 \quad \text{for incoming} } \, . \end{equation}\]

\eqref{eq: incoming}, \eqref{eq: conjugate}

Since the intertwiner \(\epsilon\) commutes with complex conjugate, we only need to consider \(\ketS{q}\) and \(\braA{q}\),

\[\begin{equation} \ketS{q}^{*}=\alpha(q)\braA{q} \, , \quad \braA{q}^{*}=\alpha^{-*}(q) \ketS{q} \, , \end{equation}\]

and from \eqref{eq: helicity spinor} we have

\[\begin{equation} \alpha(q)=\alpha^{*}(q) \in \RR \, . \end{equation}\]

Continuing \(q\) to \(-q\), there is only one minus sign in \eqref{eq: helicity spinor}, so we need to absorb it either in both of \(\ketS{q}\) and \(\braA{q}\) or in one of them. Starting from the ansatz

\[\begin{equation} \ketS{-q}=\beta(q) \ketS{q} \, , \quad \braA{-q}=-\beta^{-1}(q) \braA{q} \, , \end{equation}\]

and then by the compatibility of complex conjugate, we have

\[\begin{align} \ketS{-q}^{*} &=\beta^{*}(q) \ketS{q}^{*} =\beta^{*}(q) \alpha(q) \braA{q} \\ &=\alpha(-q)\braA{-q}=-\alpha(-q)\beta^{-1}(q)\braA{q} \, , \end{align}\]

which implies

\[\begin{equation} \beta(q)\beta^{*}(q)=-\alpha(-q)\alpha^{-1}(q)>0 \, . \end{equation}\]

Furthermore, to make the square/angle brakets balanced

\[\begin{equation} \braketS{q_{1}q_{2}}^{*} =-\alpha(q_{1})\alpha(q_{2})\braketA{q_{1}q_{2}} \, , \end{equation}\]

a natural choice of \(\alpha(q)\) would be

\[\begin{equation} \alpha(q)=\inout{q} \, . \end{equation}\]

Then for \(\abs{\beta(q)}=1\) we choose

\[\begin{equation} \beta(q)=1 \, . \end{equation}\]

This is the convention in ³. While ⁴ chooses \(\beta(q)=i\), then one should be careful that \(\ketS{-(-q)}=-\ketS{q}\), i.e., \(\ketS{q}\) is double-valued in the complex plane of \(E\).

Polarization

Lacia-TimeStamp-2025-05-10-04:25:04

\[\begin{equation} \label{eq: polarization} \epsilon^{\mu}_{+}(q,q')=\frac{\braketSA{q}{\sigma^{\mu} \lor \gamma^{\mu}}{q'}}{\braketA{qq'}} \, , \quad \epsilon^{\mu}_{-,1}(q,q')=\frac{\braketAS{q}{\sigmab^{\mu} \lor \gamma^{\mu}}{q'}}{\braketS{qq'}} \, , \end{equation}\]

for the reference momentum \(q'\).

Useful properties

Lacia-TimeStamp-2025-05-10-04:25:12

For massless momenta, \(q\) will be omitted, e.g.

\[\begin{equation} \braketS{12}\eqq \braketS{q_{1}q_{2}} \, . \end{equation}\]

The contracted indices between Pauli/Dirac matrices and momentum spinors will be omitted, e.g.

\[\begin{equation} \braketSA{1}{P}{2} \eqq \braS{1}^{a}P_{a \dota}\ketA{2}^{\dota} = -\signature P_{\mu}\braS{1}^{a}\sigma^{\mu}_{a\dota}\ketA{2}^{\dota} \, . \end{equation}\]

Property

Mandelstam variable:

\[\begin{equation} \label{eq: momentum squared} \braketS{12}\braketA{12} = 2\signature q_{1} \cdot q_{2} =\signature (q_{1}+q_{2})^{2} \, . \end{equation}\]

Bispinor contraction:

\[\begin{equation} \label{eq: bispinor contraction} \begin{aligned} &\braketSA{1}{P \lor \sigma^{\mu} \lor \gamma^{\mu}}{2}=\braketAS{2}{P \lor \sigmab^{\mu} \lor \gamma^{\mu}}{1} \, , \\ &\braketAS{1}{2}{3}=\braketA{12}\braketS{23} \, , \\ &\braketSA{1}{2}{3}=\braketS{12}\braketA{23} \, . \end{aligned} \end{equation}\]

Fierz identity:

\[\begin{equation} \label{eq Fierz identity} \braA{1}\sigma^{\mu},\gamma^{\mu}\ketS{2} \braA{3}\sigma_{\mu},\gamma_{\mu}\ketS{4} = 2\signature\braketA{13}\braketS{24} \, . \end{equation}\]

Little group scaling

Under the little group scaling, the weights are

Object	Weight
square bra/ket	\(1\)
angle bra/ket	\(-1\)
wavefunction with helicity \(J\)	\(2J\)
amplitude with helicities \(J_i\)	\(2J_i\)

Component

The massless momentum can be parametrized by the energy \(E\) and angular coordinates \((\theta,\phi)\) as

\[\begin{equation} q=E(1,\sin \theta \cos \phi ,\sin \theta \sin \phi ,\cos \theta ) \, . \end{equation}\]

In the celestial CFT literature, we also use the following parametrization:

\[\begin{equation} q=\omega( 1+z \bar{z},z+\bar{z},-i (z-\bar{z}),1-z \bar{z} ) \, . \end{equation}\]

The two are related by the stereographic projection

\[\begin{equation} \omega= E \cos^2\frac{\theta}{2} \, , \quad z= e^{i \phi} \tan \frac{\theta}{2} \, , \end{equation}\]

with the north pole mapped to the origin.

The helicity spinors can be parametrized as

\[\begin{align} \label{eq: bra ket parametrization} & \braS{q}=\sqrt{2\omega}(1,\bar{z}) \, , \\ & \ketS{q}=\sqrt{2\omega}(-\bar{z},1) \, , \\ & \braA{q}=\inout{q}\sqrt{2\omega}(-z,1) \, , \\ & \ketA{q}=\inout{q}\sqrt{2\omega}(1,z) \, . \end{align}\]

For massless spin-1 particles, the polarization vectors are \(\epsilon_{\pm}=\pp_{\pm}\qhat\) where \(\pp_{+}=\pp_{z}\) and \(\pp_{-}=\pp_{\zb}\), and can be parametrized as

\[\begin{equation} \label{eq: polarization component} \begin{aligned} &\epsilon_{+}=(\zb,1,-i,-\zb) \, , \\ &\epsilon_{-}=(z,1,i,-z) \, . \end{aligned} \end{equation}\]

Verification test

Directory: ~/TestSource/SpinorHelicity/

Test	Equation(s)
`bispinor-contraction.wlt`	\(\eqref{eq: bispinor contraction}\)
`bispinor.wlt`	\(\eqref{eq: momentum bispinor} \eqref{eq: helicity spinor}\)
`epsilon-tensor.wlt`	\(\eqref{eq: raising lowering S} \eqref{eq: raising lowering A}\)
`Fierz-identity.wlt`	\(\eqref{eq Fierz identity}\)
`incoming-outgoing.wlt`	\(\eqref{eq: incoming}\)
`momentum-squared.wlt`	\(\eqref{eq: momentum squared}\)
`Pauli-Dirac-matrix.wlt`	\(\eqref{eq: Pauli matrix} \eqref{eq: Pauli matrix and epsilon tensor}\)
`spinor-component-2-vs-4.wlt`
`spinor-component.wlt`	\(\eqref{eq: bra ket parametrization}\)
`spinor-conjugate.wlt`	\(\eqref{eq: conjugate} \eqref{eq: conjugate braket}\)
`spinor-pairing.wlt`	\(\eqref{eq: pairing S} \eqref{eq: pairing A}\)
`Weyl-equation.wlt`	\(\eqref{eq: Weyl equation}\)

When altering the signature, we need to flip the signs of \(g_{\mu\nu}\), \(g^{\mu\nu}\) and the derived objects, e.g. momentum and Pauli matrix with lower index \(P_{\mu}\), \(\sigma_{\mu}\), \(\sigmab_{\mu}\), and covariant derivative with upper index \(\pp^{\mu}\), \(D^{\mu}\). ↩
Herbi K. Dreiner, Howard E. Haber, and Stephen P. Martin. Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry. Phys. Rept., 494:1–196, 2010. arXiv:0812.1594, doi:10.1016/j.physrep.2010.05.002. ↩
H. Elvang and Y. Huang. Scattering Amplitudes in Gauge Theory and Gravity. Cambridge University Press, 2015. ISBN 9781316195130. URL: https://books.google.com/books?id=1AG7BwAAQBAJ. ↩↩↩↩
Mark Srednicki. Quantum Field Theory. Cambridge University Press, 2007. ISBN 9781139462761. URL: https://books.google.com/books?id=5OepxIG42B4C. ↩↩
Tomasz R. Taylor. A Course in Amplitudes. Phys. Rept., 691:1–37, 2017. arXiv:1703.05670, doi:10.1016/j.physrep.2017.05.002. ↩↩↩
M.D. Schwartz. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014. ISBN 9781107034730. URL: https://books.google.com/books?id=HbdEAgAAQBAJ. ↩
Clifford Cheung. TASI lectures on scattering amplitudes. pages 571–623, 2018. arXiv:1708.03872, doi:10.1142/9789813233348_0008. ↩
Edward Witten. Perturbative gauge theory as a string theory in twistor space. Commun. Math. Phys., 252:189–258, 2004. arXiv:hep-th/0312171, doi:10.1007/s00220-004-1187-3. ↩