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\sect{Chemical equilibrium for elementary particles}

The reaction  ${\gamma+\gamma .\rightarrow. e^{+}+ e^{-}}$ occurs inside a star, where ${\gamma}$ is a
photon and e± are the positron and electron (mass ${\mathsf{m}}$).
In general e± are relativistic. Assume overall charge neutrality,
and that the system is in equilibrium at temperature ${T}$.  Find an
expression for the densities of  ${e^{±}}$. Also find these
densities in the limit ${T << mc^{2}}$. Solve the same problem for
the reaction ${\gamma +\gamma..\rightarrow
\rightarrow\Pi^{+}+\Pi^{-}}$  where ${\Pi^{±}}$ are bosons with mass
${M}$. Can these bosons become Bose-condensed if the temperature is
sufficiently lowered?  Explain the result physically.