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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 14}\\%
\vspace {3mm} (April 8, 2010) \\
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\begin{description}

\item{(51)} An electron and a positron, each with mass equal to 
$0.511~MeV$, annihilate at rest into two photons. Chose the rest 
frame of the positronium, such that one of the photons moves in 
positive $x=x^1$ direction.
\begin{enumerate}
\item For each photon find the momentum 4-vector.

\item Find the 4-momentum in a frame that moves at a velocity 
       $c\beta\hat{x}$ with respect to the rest frame of the
       positronium.

\item Suppose that the annihilation took place $10^9$ years ago in a 
      galaxy that is receding from us at $\beta = {4/5}$. 
      What is the energy of the photon that we observe?
\end{enumerate}
Due April 21 before class (10~points).  \medskip
\medskip

\item{(52)} Let $T^{\alpha\beta}$ be a rank two tensor.
\begin{enumerate}
\item Write down $T_S^{\alpha\beta}$, the symmetric part of
$T^{\alpha\beta}$. Due April~19 in class (1~point).

\item Write down $T_A^{\alpha\beta}$, the antisymmetric part of
$T^{\alpha\beta}$. Due April~19 in class (1~point).

\item Calculate $T_A^{\alpha\beta}\,{T_S}_{\,\alpha\beta}$.
Due April~19 in class (1~point).

\item Derive the continuity equation from the antisymmetry of
      $F^{\alpha\beta}$. Due April~19 in class (1~point).

\item Show $\partial_{\alpha}\,^*F^{\alpha\beta}=0$, where
$$^*F^{\alpha\beta}=\frac{1}{2}\epsilon^{\alpha\beta\gamma\delta}
                   F_{\gamma\delta} $$
is the dual tensor. Here $\epsilon^{\alpha\beta\gamma\delta}$ is 
the completely antisymmetric Levi-Cevita tensors: $+1$ for
$(\alpha,\beta,\gamma,\delta)$ and even permutation of $(0,1,2,3)$,
$-1$ for $(\alpha,\beta,\gamma,\delta)$ and even permutation of 
$(0,1,2,3)$. Due April~19 in class (2~points).
\end{enumerate}
\medskip

\item{(53)} Under a gauge transformation the vector potential
transform according to 
$$A^{\alpha}\to A'^{\,\alpha}=A^{\alpha}+\partial^{\,\alpha}\Lambda\ .$$
Calculate the corresponding transformation of the electromagnetic
field tensor
$$F^{\alpha\beta}\to F'^{\,\alpha\beta}\ .$$
Due April~21 before class (4~points).

\end{description}

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