advanced_notions:quantum_field_theory:spinor-helicity_formalism
Differences
This shows you the differences between two versions of the page.
| Next revision | Previous revision | ||
|
advanced_notions:quantum_field_theory:spinor-helicity_formalism [2017/05/12 15:39] jakobadmin created |
advanced_notions:quantum_field_theory:spinor-helicity_formalism [2018/01/02 14:02] (current) jakobadmin ↷ Page moved from theories:quantum_theory:quantum_field_theory:spinor-helicity_formalism to advanced_notions:quantum_field_theory:spinor-helicity_formalism |
||
|---|---|---|---|
| Line 2: | Line 2: | ||
| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| + | |||
| + | <blockquote> | ||
| + | “A method is more important than a discovery, since the right method will lead to new and even more important discoveries.” | ||
| + | |||
| + | <cite>Lev Landau</cite> | ||
| + | </blockquote> | ||
| <blockquote> | <blockquote> | ||
| Line 27: | Line 33: | ||
| of Feynman diagrams**, that makes locality and unitarity as manifest as possible. | 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 $M(t_aλ_a, t^{−1}_a \tilde \lambda_a) = t^{−2h_a} M(λa, \tilde a)$. | + | 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 | **With this invariant description of the fundamental | ||
| Line 46: | Line 52: | ||
| </blockquote> | </blockquote> | ||
| + | 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 [[http://phys.columbia.edu/~nicolis/GR_from_LI.pdf|this article]]. | ||
| <tabbox Layman> | <tabbox Layman> | ||
| Line 55: | Line 61: | ||
| <tabbox Student> | <tabbox Student> | ||
| - | <note tip> | + | For a nice introduction to the spinor helicity formalism, see section 3.1 in [[https://arxiv.org/pdf/1704.05067.pdf|Amplitudes for Astrophysicists I: Known Knowns]] |
| - | In this section things should be explained by analogy and with pictures and, if necessary, some formulas. | + | by Daniel J. Burger et. al. |
| - | </note> | + | |
| + | Other nice "low-level" discussion can be found [[http://www.preposterousuniverse.com/blog/2013/10/03/guest-post-lance-dixon-on-calculating-amplitudes/|here]] and [[http://www.preposterousuniverse.com/blog/2014/03/31/guest-post-jaroslav-trnka-on-the-amplituhedron/|here]]. | ||
| + | |||
| + | See also [[https://arxiv.org/abs/1310.5353|A brief introduction to modern amplitude methods]] by Lance J. Dixon | ||
| + | |||
| + | and [[http://scipp.ucsc.edu/~haber/ph218/ExperimentersGuideToTheHelicityFormalism.pdf|An Experimenter’s Guide to the Helicity Formalism by Jeffrey D. Richman]] | ||
| <tabbox Researcher> | <tabbox Researcher> | ||
| Line 64: | Line 74: | ||
| The motto in this section is: //the higher the level of abstraction, the better//. | The motto in this section is: //the higher the level of abstraction, the better//. | ||
| </note> | </note> | ||
| + | |||
| --> Common Question 1# | --> Common Question 1# | ||
advanced_notions/quantum_field_theory/spinor-helicity_formalism.1494596371.txt.gz · Last modified: 2017/12/04 08:01 (external edit)
Page Tools
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International
