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{\bf SOLUTIONS FINAL ADVANCED DYNAMICS}\\ 
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\centerline{\bf PROBLEM 4}\bigskip

\noindent Total energy: $p^0+q^0=2\,E\,,~~E=0.511\,\rm MeV\,$.
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\noindent Momentum conservation: $\vec{p}=-\vec{q}\,$.
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\noindent Zero photon rest mass: $p_{\alpha}p^{\alpha}=
q_{\alpha}q^{\alpha}=0\,$.
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\begin{enumerate}
\item CM frame (rest from of the positronium, $c=1$):
$$ (p^{\alpha}) = \left( \matrix{E\cr E\cr 0\cr 0} \right)\,,~~~
   (q^{\alpha}) = \left( \matrix{E\cr -E\cr 0\cr 0} \right)\,.$$
\item The transformation to the moving frame is 
$$ (p'^{\,\alpha}) = \left( \matrix{ 
   \ \gamma        & -\gamma\,\beta&\ 0 &\ 0 \cr
    -\gamma\,\beta &\ \gamma       &\ 0 &\ 0 \cr
     0             &\ 0            &\ 1 &\ 0 \cr
     0             &\ 0            &\ 0 &\ 1 \cr}\right)\,(p^{\alpha}) = 
   \left( \matrix{~~\gamma\,E - \gamma\,\beta\,E \cr
                  - \gamma\,E + \gamma\,\beta\,E \cr
                    0 \cr 0 \cr}\right)\,. $$
\item The observed energy follows from $\beta=4/5$ and $\gamma=1/\sqrt{1-\beta^2}
      =5/3$ to be $E'=\gamma\,(1-\beta)\,E=E/3=0.17\,\rm MeV$.
\end{enumerate}
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