✎ Spinor-Helicity Formalism [The Physics Travel Guide]

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====== Spinor-Helicity Formalism ======

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.**

<cite>https://arxiv.org/pdf/1612.02797.pdf</cite>
</blockquote>

<note tip>
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.
</note>
  
<tabbox Student>

<note tip>
In this section things should be explained by analogy and with pictures and, if necessary, some formulas.
</note>
 
<tabbox Researcher>

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

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