Photon Spin Griffiths Elementary Particles Book

Question

In the Book "Elementary Particles" by Griffiths, Chapter 7.8 describes pair annihilation amplitude for $e^+e^- to gamma gamma$. To obtain the final amplitude for the singlet state, he needs the appropriate photon polarization vectors. For unknown reasons, he does not describe where these vectors come from, so I would like to know where I can find the source of these vectors (Apparently these vectors correspond to right and left-circular polarization, although there is no explanation for that name in the book. I did find about an operator for $S_z$ not mentioned in the book, for which these vectors are eigenvectors for $S_z$.)

Emilio Pisanty · Answer

The book is slightly mislabelled -- the notation is not explained in a footnote under equation (7.94), but under (7.93) instead. This footnote reads

Photon states with $m_s=pm 1$ correspond to right- and left-circular polarization; the respective polarization vectors are $boldsymbolepsilon_pm = mp (boldsymbolepsilon^{(1)}pm i boldsymbolepsilon^{(2)})/sqrt{2}$.

The full details are explained in §7.4 ('the photon') and §7.5 ('The Feynman rules for QED'), and particularly under 'photons' in the latter. If you want to understand what "circular polarization" means for an electromagnetic wave, on the other hand, maybe you should be looking at Griffiths' textbook on electromagnetism first?
As a general rule, though, note that following this type of internal reference given in the passage that confuses you forms part of the due diligence that you're expected to do on your own before asking. (On the other hand, if you did find this footnote and you're confused about it, then you need to be significantly clearer about that in the text of the question.)

Photon Spin Griffiths Elementary Particles Book

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