Spinor-Helicity Formalism

Why is it interesting?

“A method is more important than a discovery, since the right method will lead to new and even more important discoveries.”

Lev Landau

the standard formalism of field theory, the amplitudes are computed using Feynman diagrams, which give us “Feynman amplitudes” that are not the real amplitudes, but are instead Lorentz tensors. We contract them with polarization vectors to get the actual amplitudes–the polarization vectors are supposed to transform as “bi-fundamentals” under the Lorentz and little groups. For massive particles of any spin, there is a canonical way of associating polarization vectors with given spin states. But this is impossible for massless particles. […] In order to guarantee that the Lorentz tensors $M_{µ_1···µn}$ arising from Feynman diagrams from a Lagrangian satisfies this on-shell Ward-identity, the Lagrangian must be carefully chosen to have an (often non-linearly completed) gauge invariance, which is then gauge-fixed. From the modern point of view, then, gauge symmetry is merely a useful redundancy for describing the physics of interacting massless particle of spin 1 or 2, tied to the specific formalism of Feynman diagrams, that makes locality and unitarity as manifest as possible.

But over the past few decades, we have seen entirely different formalisms for computing scattering amplitudes not tied to this formalism, and here gauge redundancy makes no appearance whatsoever. Instead of polarization vectors that only redundantly describe massless particle states, we can use spinor-helicity variables $\lambda_a$, $\tilde \lambda_a$ for the a'th particle, with momentum $p_a^{\alpha\dot \alpha} = \lambda_a^\alpha \tilde \lambda^{\dot \alpha}$. The $λ, \tilde \lambda$'s do transform cleanly as bi-fundamentals under the Lorentz and little groups; under a Lorentz transformation $Λ$ that maps $(Λp) = p$, we have $λ → tλ$, $\tilde \lambda→ t^{-1} \tilde \lambda$. Thus while the description of amplitude using polarization vectors is gauge-redundant, the amplitude is directly a function of spinor-helicity variables, with the helicities encoded in behavior under rescaling $M(t_aλ_a, t^{-1}_a \tilde \lambda_a) = t^{-2h_a} M(λa, \tilde a)$.

With this invariant description of the fundamental symmetries and kinematics of amplitudes at hand, it becomes possible to pursue entirely new strategies for determining the amplitudes. In a first stage, one can speak of a modern incarnation of the S-matrix program, where the fundamental physics of locality and unitarity are imposed to determine the amplitudes from first principles. This has allowed the computation of amplitudes in an enormous range of theories, from Yang-Mills and gravity to goldstone bosons, revealing stunning simplicity and deep new mathematical structures that are completely hidden in the usual, gauge-redundant Feynman diagram formalism.

For more on the fact that polarization vectors are not lorentz vectors, have a look at Vol. 1 of Weinberg's Quantum Field Theory book, section 5.9. A summary of Weinberg's argument with an easier notation can be found in this article.

Layman

Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.

Student

For a nice introduction to the spinor helicity formalism, see section 3.1 in Amplitudes for Astrophysicists I: Known Knowns by Daniel J. Burger et. al.

Other nice "low-level" discussion can be found here and here.

See also A brief introduction to modern amplitude methods by Lance J. Dixon

Researcher

The motto in this section is: the higher the level of abstraction, the better.
Common Question 1
Common Question 2

Example1
Example2: