Spinor Helicity Formalism

• Spinor helicity formalism is a technique that rewrites relativistic wave equations in terms of two-component spinors, enabling a clear treatment of helicity for massless particles.
• It simplifies the computation of scattering amplitudes by replacing cumbersome Dirac trace methods with efficient spinor bracket techniques, notably in gauge theory and QED.
• The method offers direct physical insights in ultrarelativistic regimes, decoupling equations into definite helicity eigenstates and streamlining amplitude calculations.

The spinor helicity formalism rewrites relativistic wave equations and amplitudes directly in terms of two-component spinors, offering a powerful framework for the calculation of scattering amplitudes, especially for massless and ultrarelativistic particles. Originating in the paper of Lorentz and Weyl representations, this formalism underpins much of the modern approach to high-energy scattering and forms the technical basis for helicity amplitude techniques prevalent in gauge theory, QED, and quantum field theory calculations.

1. Foundations: From the Klein–Gordon Equation to Weyl Spinors

The spinor helicity formalism traces its roots to the factorization of the Klein–Gordon (KG) operator. The KG equation for a scalar field $\phi$,

$(-\Box + m^2)\phi = 0,$

is factorized by introducing matrices $\boldsymbol{\alpha}$ such that the operator admits a square root in terms of first-order differential operators: $$-\frac{\partial^2}{\partial t^2} + \nabla^2 = [i \partial_t + (\boldsymbol{\alpha} \cdot \nabla)][i \partial_t + (\boldsymbol{\beta} \cdot \nabla)],$$ with $\boldsymbol{\beta} = -\boldsymbol{\alpha}$ and $(\boldsymbol{\alpha})^2 = -1$. Taking $\boldsymbol{\alpha} \equiv i\boldsymbol{\sigma}$ (with $\boldsymbol{\sigma}$ the Pauli matrices), and imposing the anti-commutation relation $\{\sigma_i, \sigma_j\} = 2\delta_{ij}$, leads directly to the two-component Weyl equations. These take the coupled form: $$i\partial_t\,\phi = i \boldsymbol{\sigma} \cdot \nabla\,\phi + m\chi, \qquad i\partial_t\,\chi = -i\boldsymbol{\sigma} \cdot \nabla\,\chi + m\phi.$$ In the massless limit ($m=0$), the equations decouple, and the eigenvalue of $\boldsymbol{\sigma}\cdot\mathbf{p}$ specifies the helicity—left- or right-handed.

2. Incorporation of Electromagnetic Interactions

To describe electromagnetically interacting systems, the principle of minimal coupling is employed: $\partial_\mu \to D_\mu = \partial_\mu - ie A_\mu$. For massless Weyl spinors, the kinetic equations become

$$-i \sigma^\mu D_\mu\,\phi^\dagger = 0, \qquad -i \bar{\sigma}^\mu D_\mu\,\chi = 0,$$

where $\sigma^\mu = (\mathbb{I}, \boldsymbol{\sigma})$ and $$\bar{\sigma}^\mu = (-\mathbb{I}, \boldsymbol{\sigma})$$. Applying a further $\bar{\sigma}^\nu D_\nu$ operator, and using the Clifford algebra,

$$\bar{\sigma}^\nu \sigma^\mu = g^{\nu\mu} - 2i\bar{\sigma}^{\nu\mu},$$

leads to the second-order equation: $$(D_\mu D^\mu - e\,\bar{\sigma}^{\nu\mu} F_{\nu\mu})\,\phi^\dagger = 0.$$ Upon decomposing $F_{\nu\mu}$ into electric and magnetic components, the extra term $e\bar{\sigma}^{\nu\mu} F_{\nu\mu}$ yields $$-\boldsymbol{\sigma}\cdot(\mathbf{B} - i\mathbf{E})$$, leading to magnetic and (complex) electric moment couplings: $$(D_\mu D^\mu + e\,\boldsymbol{\sigma}\cdot(\mathbf{B} - i\mathbf{E}))\,\phi^\dagger = 0.$$ The term $e\,\boldsymbol{\sigma}\cdot\mathbf{B}$ encodes the magnetic dipole interaction of the spinor particle.

3. Helicity Methods in Scattering Amplitudes: The Example of Compton Scattering

The spinor helicity formalism enables a streamlined computation of amplitudes such as those for Compton scattering ($e^-\gamma \to e^-\gamma$). Massless spinors are introduced as $|p]$ and $|p\rangle$, with fundamental spinor brackets $[pk] \equiv \phi^a \kappa_a$, $$\langle pk\rangle \equiv \phi_{\dot{a}} \kappa^{\dot{a}}$$, and the key relation: $$\langle k p \rangle [p k] = -2\, k \cdot p = -(k + p)^2.$$ Physical spinors are re-expressed in two-component form, e.g., $u_-(p) = (|p], 0)^T$, $u_+(p) = (0, |p\rangle)^T$. Photon polarization vectors use a reference lightlike momentum $q$: $$\epsilon_+^\mu(k) = -\frac{\langle q|\gamma^\mu|k]}{\sqrt{2}\langle qk\rangle}, \quad \epsilon_-^\mu(k) = -\frac{[q|\gamma^\mu|k\rangle}{\sqrt{2}[qk]}.$$ Amplitudes are then constructed in spinor brackets; for example, for the $\{+,-,+,-\}$ helicity configuration: $$\mathcal{M}_{+ - + -} = 2e^2 \frac{\langle 2\, 4\rangle^2}{\langle 1\, 3\rangle \langle 2\, 3\rangle},$$ with the indices labeling the external lines. The cross section, after helicity averaging and in terms of Mandelstam variables, matches the standard QED result: $$\langle |\mathcal{M}|^2 \rangle = -2e^4 \left(\frac{u}{s} + \frac{s}{u}\right).$$ The use of two-component spinors and the associated bracket formalism circumvents cumbersome trace calculations typical of the four-component Dirac approach.

4. Magnetic Moment and Physical Content

In the Weyl-spinor-derived equations, the magnetic moment interaction is contained in the term $e \boldsymbol{\sigma}\cdot\mathbf{B}$, making the magnetic dipole coupling explicit at the two-component spinor level. This is consistent with the analogous result from a non-relativistic reduction of the full Dirac equation, where Pauli matrices act as spin operators directly coupled to the magnetic field.

5. Advantages over the Dirac Four-Component Approach

The spinor helicity (or equivalently, Weyl twistor) formalism yields several advantages:

• Massless Limit Clarity: In the limit $m\to0$, the two-component Weyl equations decouple into definite helicity eigenstates; helicity becomes a good quantum number without additional assumptions.
• Computational Efficiency: By working directly with helicity eigenstates and spinor brackets, calculations for multi-particle processes are substantially less cumbersome compared to the Dirac multi-index formalism, especially in gauge theories with large numbers of external legs.
• Direct Physical Interpretation: The helicity operator is given directly by $\boldsymbol{\sigma}\cdot \mathbf{p}/|\mathbf{p}|$, with the spinor's transformation properties mapping straightforwardly onto the observed polarization states of massless fermions.

In high-energy, multi-leg, and massless (or ultrarelativistic) regimes—such as those encountered in collider phenomenology—the formalism provides expressions that are both compact and manifestly gauge invariant, streamlining both analytic and numerical evaluations.

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The spinor helicity formalism, through operator factorization, minimal coupling, and the exploitation of two-component spinors, establishes a versatile and algebraically efficient technical foundation for modern amplitude calculations and underlies much of the on-shell method revolution in perturbative quantum field theory (Diaz-Cruz et al., 2015).