Polarization Vector for Quantum EM Field

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SUMMARY

The discussion centers on evaluating the dot products of normalized polarization vectors for photons, specifically \(\vec{\epsilon}_{k,s}\cdot\vec{\epsilon}_{-k,s'}\) and \(\vec{\epsilon}_{k,s}^*\cdot\vec{\epsilon}_{-k,s'}^*\). The participants clarify that the spin refers to helicity, resulting in values of plus or minus one based on the relative helicities of the photons. The context assumes that the polarizations are aligned along Cartesian directions, which is crucial for the calculations involved.

PREREQUISITES
  • Understanding of quantum electrodynamics (QED)
  • Familiarity with polarization vectors in quantum mechanics
  • Knowledge of helicity and its implications in particle physics
  • Basic grasp of vector calculus and dot product operations
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  • Research the properties of polarization vectors in quantum optics
  • Study the implications of helicity in photon interactions
  • Explore the mathematical framework of quantum field theory (QFT)
  • Learn about the role of momentum conservation in particle physics
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Physicists, particularly those specializing in quantum mechanics and quantum field theory, as well as students and researchers involved in photon polarization studies.

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I'm doing some calculations and I've run into something rather strange.

I need to evaluate the following dot products

[tex]\vec{\epsilon}_{k,\,s}\cdot\vec{\epsilon}_{-k,\,s'} = ?[/tex]
[tex]\vec{\epsilon}_{k,\,s}^*\cdot\vec{\epsilon}_{-k,\,s'}^* = ?[/tex]​

where [itex]\vec\epsilon_{k,\,s}[/itex] is the normalized polarization vector for a single photon with momentum [itex]k[/itex], and spin [itex]s[/itex]. Does anyone know where I can look for these identities?
 
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Assume that the polarizations are along cartesian directions...
 
If by "spin", you mean helicity, the e.e'=plus or minus 1, depending on the relative helicities. This works because the momenta are = and opposite.
 

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