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Spinor helicity formalism

Conventions

本文旨在梳理旋量螺旋度方法 (spinor helicity formalism) 中的约定,主要遵循 1 2

  • 尽管对于物理观测量而言,约定的选择是一种“规范”,但它会沿着计算过程传播。从编程视角看,这是一种非局域效应,应当被封装在独立的模块中。
  • 任意动量记为 \(P\),有质量的记为 \(p\),无质量的记为 \(q\)。无质量动量的参数化为

    \[\begin{equation*} q=\omega( 1+z \bar{z},z+\bar{z},-i (z-\bar{z}),1-z \bar{z} ) \, , \textInMath{for} \omega>0,z\in\CC \, . \end{equation*}\]

Metric

度规号差记为

\[\begin{equation} \tag{Signature} \signature= \begin{cases} +1 \, , \quad (-,+,+,+)\, ,\\ -1 \, , \quad (+,-,-,-)\, .\\ \end{cases} \end{equation}\]

若要变更度规号差,需要翻转 \(g_{\mu\nu}, g^{\mu\nu}\) 以及相关物理量的符号,例如带下指标的动量 \(P_{\mu}\) 与带上指标的导数 \(\pp^{\mu}\)

采用度规 \((-,+,+,+)\) 的文献示例如下:

  • Wess & Bagger 1 Supersymmetry and Supergravity

  • Srednicki 3 Quantum Field Theory

  • Elvang & Huang 4 Scattering Amplitudes in Gauge Theory and Gravity

采用度规 \((+,-,-,-)\) 的文献示例如下:

  • SAGEX 2 The SAGEX Review on Scattering Amplitudes Chapter 1

  • Schwartz 5 Quantum Field Theory and the Standard Model

  • Badger et al. 6 Scattering Amplitudes in Quantum Field Theory

  • Dixon 7 A brief introduction to modern amplitude methods

  • Taylor 8 A Course in Amplitudes

  • Cheung 9 TASI lectures on scattering amplitudes

Pauli matrices

Pauli 矩阵的定义是约定差异的主要来源之一,有如下两种:

\[\begin{equation*} \sigma_{\mu}\sim(\id,\sigma^{i}) \, , \textInMath{vs.} \sigma^{\mu}\sim(\id,\sigma^{i}) \, . \end{equation*}\]

不同文献的选择示例如下:

  • Wess & Bagger 1 采用了前者,见 1 Appendix A & B。与之适配的旋量缩并为

    \[\begin{equation*} \psi\chi=\psi^{a}\chi_{a} \, , \quad \psib\chib=\psib_{\dota}\chib^{\dota} \, . \end{equation*}\]
  • SAGEX 2 与 Dixon 7 采用了前者。尖旋量 \(\ketA{q}_{a}\) 的指标为不带点的,旋量缩并与 Wess & Bagger 相同,见 2 Appendix A 与 7 Equation 3.6。

  • Srednicki 3 与 Elvang & Huang 4 采用了后者。尖旋量 \(\braA{q}_{\dota}\) 的指标为带点的,旋量缩并与 Wess & Bagger 相同,见 3 Section 35 与 4 Appendix A。

  • Schwartz 5 采用了后者。尖旋量 \(\ketA{q}_{a}\) 的指标为不带点的,旋量缩并与 Wess & Bagger 相反,见 5 Section 10.6.2。

尽管存在这些差异,通常会保证旋量内积 \(\braketA{q_{1}q_{2}}\sim z_{1}-z_{2}\) 是全纯的。

此外,在 twistor 理论的相关文献中,Pauli 矩阵往往带有归一化因子 \(\frac{1}{\sqrt{2}}\)

Spinor inner product

旋量内积是另一类正负号差异的来源:其一是等变映射 \(\varepsilon^{ab}, \varepsilon^{\dota\dotb}\) 的分量的选取,其二是旋量内积与自然配对的相对符号 \(\braketA{q_{1}q_{2}}= \pm\braA{q_{1}}^{a}\ketA{q_{2}}_{a}\)

需注意的差异是 Mandelstam 变量与旋量内积的关系

\[\begin{equation*} s_{12}=\pm \braketA{12}\braketS{12} \, . \end{equation*}\]

在粒子物理领域的相关文献中,多数为负号。

Incoming/Outgoing

出入态的转换对应于能量的解析研拓 \(\omega\to -\omega\)。通常有两种约定:例如文献 2 3 6 把负号分配到两种旋量中,

\[\begin{equation*} \ketA{-q}=i\ketA{q} \, , \quad \braS{-q}=i\braS{q} \, , \end{equation*}\]

此时有 \(\ketA{-(-q)}=-\ketA{q}\)。而例如文献 4 把负号吸收到尖旋量中,

\[\begin{align*} \ketA{-q}=-\ketA{q} \, , \quad \braS{-q}=\braS{q} \, , \end{align*}\]

此时有 \(\ketA{-(-q)}=\ketA{q}\)

Polarization

Lacia-TimeStamp-2025-06-24-07:07:04
  • 极化矢量的归一化为

    \[\begin{equation*} \epsilon_{+}\cdot\epsilon_{-}=2\signature\polar^{2} \, . \end{equation*}\]
  • 极化矢量的类型

Basic ingredients

Helicity spinor

螺旋度旋量 (helicity spinor) 是一类特殊的玻色型 Weyl 旋量,它解开了无质量 Weyl 方程的约束。

Angle spinor

与复化的无质量动量 \(q\in\CC^{4}\) 相关联的旋量称为尖右矢 (angle ket) \(\ketA{q}_{a}\),它处于 \(\slgroup(2,\CC)\) 的基本表示 \((\half,0)\) 中。而处于对偶表示 \((\half,0)^{\dual}\) 中的称为尖左矢 (angle bra) \(\braA{q}^{a}\)。两者的变换关系为

\[\begin{equation*} \ketA{q}_{a} \to M^{b}_{a} \ketA{q}_{b} \, , \quad \braA{q}^{a} \to (M^{-\tp})^{a}_{b} \braA{q}^{b} \, . \end{equation*}\]

其自然配对称为尖括号 (angle braket),

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

两者实际上是同构的,

\[\begin{equation} \label{eq: raising lowering A} \braA{q}^{a}=\varepsilon^{ab}\ketA{q}_{b} \, , \quad \ketA{q}_{a}=\varepsilon_{ab}\braA{q}^{b} \, , \end{equation}\]

其中 \(\varepsilon\) 为可逆等变映射,分量可取为

\[\begin{equation} \label{eq: varepsilon tensor A} \varepsilon^{ab}=-\varepsilon_{ab}= \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \, , \quad \varepsilon^{ab}\varepsilon_{bc}=\delta^{a}_{b} \, . \end{equation}\]

等价的,\(\varepsilon\) 诱导了 \((\half,0)\)\((\half,0)^{\dual}\) 上的反对称不变内积。

Square spinor

将前述讨论应用于复共轭表示 \((0,\half)\) 及其对偶表示 \((0,\half)^{\dual}\),方左矢 (square bra) 和方右矢 (square ket) 的变换关系为

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

两者的自然配对记为方括号 (square braket),

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

采用与复共轭交换的等变映射,

\[\begin{equation} \label{eq: varepsilon tensor S} \varepsilon^{\dota \dotb}=-\varepsilon_{\dota \dotb}= \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \, , \quad \varepsilon^{\dota \dotb}\varepsilon_{\dotb \dotc}=\delta^{\dota}_{\dotb} \, , \end{equation}\]

可得指标的升降关系:

\[\begin{equation} \label{eq: raising lowering S} \ketS{q}^{\dota}=\varepsilon^{\dota \dotb}\braS{q}_{\dotb} \, , \quad \braS{q}_{\dota}=\varepsilon_{\dota \dotb}\ketS{q}^{\dotb} \, . \end{equation}\]

Momentum bispinor

在群同态 \(\sugroup(2)\to \sogroup(3,\RR)\) 下,Pauli 矩阵 \(\vec\sigma\) 是从 \(\sugroup(2)\) 的表示 \(1\in \half\ox\half\)\(\sogroup(3,\RR)\) 的矢量表示的可逆等变映射。通常选为

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

类似的,在群同态 \(\slgroup(2,\CC) \to \sogroup(3,1,\RR)\) 下,四分量 Pauli 矩阵 \(\sigma\) 是从 \(\slgroup(2,\CC)\) 的表示 \((\half,\half)\)\(\sogroup(3,1,\RR)\) 的复矢量表示 \(V\simeq \CC^{4}\) 的可逆等变映射,可选为

\[\begin{equation} \label{eq: Pauli matrix} \sigma_{\mu,a \dota}=(\id,\vec\sigma) \, , \quad \sigmab_{\mu}^{\dota a} = (\id,-\vec\sigma) \, . \end{equation}\]

在此同构下,\(V\) 上的不变内积 \(g_{\mu\nu}\) 对应为 \((\half,\half)\) 上的 \(\epsilon_{ab}\epsilon_{\dota\dotb}\),有

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

矢量 \(P^{\mu}\) 可以被重写为双旋量 (bispinor)

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

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

对于无质量动量 \(q^{2}=0\),该双旋量因子化为两个螺旋度旋量

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

它们分别满足无质量 Weyl 方程

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

Real momentum

对于复的无质量动量 \(q\),尖旋量和方旋量是独立的。方程 \eqref{eq: helicity spinor} 的左边维数为 \(\dim_{\CC}=3\),而右边维数为 \(\dim_{\CC}=4\),不匹配的原因是因子化可以相差一个标度变换

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

这正是复化的小群变换,通常称为小群标度变换 (little group rescaling)。

对于实动量,尖旋量和方旋量互为复共轭。此时方程 \eqref{eq: helicity spinor} 的左边维数为 \(\dim_{\RR}=3\),右边维数为 \(\dim_{\RR}=4\)。 小群标度变换为 \(\sogroup(2,\RR)\) 的相位变换。 此外,我们还需要区分 \(q^{0}<0\) 的入态和 \(q^{0}>0\) 的出态 \((+)\),用 \(\inout{q}\) 表示出入态的符号

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

对于出入态和复共轭,我们采用如下约定:

\[\begin{align} \label{eq: incoming} \ketA{-q}=-\ketA{q} \, , \quad \braS{-q}=\braS{q} \, , \end{align}\]
\[\begin{align} \label{eq: conjugate} \ketA{q}^{*}=\inout{q} \braS{q} \, , \quad \braS{q}^{*}=\inout{q}\ketA{q} \, . \end{align}\]

因此对于旋量内积有

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

Since the intertwiner \(\varepsilon\) commutes with complex conjugate, we only need to consider \(\ketA{q}\) and \(\braS{q}\),

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

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 \(\ketA{q}\) and \(\braS{q}\) or in one of them. Starting from the ansatz

\[\begin{equation*} \ketA{-q}=\beta(q) \ketA{q} \, , \quad \braS{-q}=-\beta^{-1}(q) \braS{q} \, , \end{equation*}\]

and then by the compatibility of complex conjugate, we have

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

which implies

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

Furthermore, to make the angle/square brakets balanced

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

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

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

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

\[\begin{equation*} \beta(q)=\pm1 \textInMath{or} i \, . \end{equation*}\]

Polarization

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

Useful properties

对于无质量动量,\(q\) 通常会被省略,例如

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

Pauli/Dirac 矩阵与动量旋量的缩并指标也会被省略,例如

\[\begin{equation*} \braketAS{1}{P}{2} \eqq \braA{1}^{a}P_{a \dota}\ketS{2}^{\dota} = P^{\mu} \braA{1}^{a}\sigma_{\mu,a\dota}\ketS{2}^{\dota} \, . \end{equation*}\]

Property

Mandelstam 变量:

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

双旋量缩并:

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

Fierz 恒等式:

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

Little group scaling

在小群标度变换下,各对象的权为

Object Weight
angle bra/ket \(-1\)
square bra/ket \(1\)
wavefunction with helicity \(J\) \(2J\)
amplitude with helicities \(J_i\) \(2J_i\)
Lacia-TimeStamp-2025-06-24-10:11:13

Component

无质量动量 \(q\) 可以用能量 \(E\) 和角坐标 \((\theta,\phi)\) 参数化为

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

在天穹 CFT 领域的相关文献中,更常用的是

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

两者的关系由球极投影给出:

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

其中北极被映射到原点。

螺旋度旋量为

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

相应的,旋量内积为

\[\begin{equation} \label{eq: braket parametrization} \braketA{12}=2\inout{1}\inout{2}\sqrt{\omega_{1}\omega_{2}}z_{1,2} \, , \quad \braketS{12}=-2\sqrt{\omega_{1}\omega_{2}}\bar{z}_{1,2} \, . \end{equation}\]
Lacia-TimeStamp-2025-06-24-11:13:12

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 A} \eqref{eq: varepsilon tensor A} \eqref{eq: varepsilon tensor S} \eqref{eq: raising lowering S}
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} \eqref{eq: braket parametrization}
spinor-conjugate.wlt \eqref{eq: conjugate} \eqref{eq: conjugate braket}
spinor-pairing.wlt \eqref{eq: pairing A} \eqref{eq: pairing S}
Weyl-equation.wlt \eqref{eq: Weyl equation}


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  2. Andreas Brandhuber, Jan Plefka, and Gabriele Travaglini. The SAGEX Review on Scattering Amplitudes Chapter 1: Modern Fundamentals of Amplitudes. J. Phys. A, 55(44):443002, 2022. arXiv:2203.13012, doi:10.1088/1751-8121/ac8254

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  6. S. Badger, J. Henn, J.C. Plefka, and S. Zoia. Scattering Amplitudes in Quantum Field Theory. Springer International Publishing, 2023. ISBN 9783031469879. URL: https://books.google.com/books?id=-7frEAAAQBAJ

  7. Lance J. Dixon. A brief introduction to modern amplitude methods. In Theoretical Advanced Study Institute in Elementary Particle Physics: Particle Physics: The Higgs Boson and Beyond, 31–67. 2014. arXiv:1310.5353, doi:10.5170/CERN-2014-008.31

  8. Tomasz R. Taylor. A Course in Amplitudes. Phys. Rept., 691:1–37, 2017. arXiv:1703.05670, doi:10.1016/j.physrep.2017.05.002

  9. Clifford Cheung. TASI lectures on scattering amplitudes. pages 571–623, 2018. arXiv:1708.03872, doi:10.1142/9789813233348_0008