Undergrad Relation between tensor decomposition and helicity amplitude

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SUMMARY

The discussion focuses on the relationship between tensor decomposition and helicity amplitudes in quantum field theory, specifically regarding the photon two-point function expressed as $$\Pi^{\mu \nu} = (g^{\mu \nu} - q^{\mu} q^{\nu}/q^2)T_T + q^{\mu} q^{\nu}T_L$$. Participants inquire about converting helicity summed expressions, such as ++, --, and 00, into individual helicity components. The notation for helicity amplitudes, particularly the representation of transverse photons and the light cone components, is clarified as essential for accurate calculations.

PREREQUISITES
  • Understanding of quantum field theory concepts, particularly photon interactions.
  • Familiarity with tensor decomposition and Lorentz invariance.
  • Knowledge of helicity amplitudes and their notation in particle physics.
  • Basic proficiency in mathematical notation used in theoretical physics.
NEXT STEPS
  • Study the derivation of the photon two-point function in quantum electrodynamics.
  • Learn about helicity amplitudes and their applications in scattering processes.
  • Explore the light cone gauge and its implications for particle interactions.
  • Investigate the mathematical techniques for converting between helicity summed and individual helicity states.
USEFUL FOR

This discussion is beneficial for theoretical physicists, particle physicists, and graduate students specializing in quantum field theory, particularly those working on photon interactions and helicity amplitudes.

CAF123
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It is common to write e.g photon two point function in terms of manifest transverse and longitudinal form factors with lorentz structure factored out, e.g $$\Pi^{\mu \nu} = (g^{\mu \nu} - q^{\mu} q^{\nu}/q^2)T_T + q^{\mu} q^{\nu}T_L,$$ where mu and nu are polarisation indices.

How do I relate such expressions to helicity amplitudes which are usually given in terms of e.g ++, -- for transverse degrees of freedom or 00 for longitudinal?

If I take two transverse photons, then mu and nu are both +, but then what is ##g^{++}##? Looks like light cone components but I am not sure if this is the correct notation.
 
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Perhaps put another way, how to go from helicity summed expressions to the individual helicity components?
 

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