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gobbles

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## Homework Statement

I am trying to solve problem 5.4(a) in Peskin & Schroeder. I am requested to calculate the decay rate of a ##1S_0## positronium state into two photons. Obviously, we have to sum over all photon polarizations eventually to get the total decay rate. However, I do not understand why the summation is only over circular polarizations, according to the solution given here

http://learn.tsinghua.edu.cn:8080/2010310800/Webpage/files/Peskin_Schroeder/Peskin_Chap05.pdf.

## Homework Equations

For two outgoing photons with polarizations

##\epsilon_\nu^*(k)=(0, \alpha, \beta, 0),

\epsilon_\mu^*(k^\prime)=(0, \alpha^\prime, \beta^\prime, 0),

##

where ##k,k^\prime## are the 4-momenta of the photons, along ##\hat{z}## and ##-\hat{z}##, respectively, I got the following expression for the amplitude of decay of ##1S_0## positronium into two photons:

##\mathcal{M}=\sqrt{2}2ie^2\big[\beta^\prime\alpha-\beta\alpha^\prime\big].##

## The Attempt at a Solution

Obviously, if both photons are right circularly polarized or if both are left circularly polarized, this expression is not zero. But it's not zero also for, say,

##\epsilon_\mu^*(k^\prime)=(0, 1, 0, 0)## and ##\epsilon_\nu^*(k)=(0, 0, 1, 0)##. Why don't we sum over that polarization also? Am I missing something? Can a photon be anything but circularly polarized?