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# QS 16: Relativistic Decay

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$\newcommand{\f}{{\mathcal F}} \newcommand{\intp}{\int \frac{{\rm d}^3p}{2p^0}} \newcommand{\intpp}{\int \frac{{\rm d}^3p'}{2{p'}^0}} \newcommand{\intx}{\int{\rm d}^3{\rm r}} \newcommand{\tp}{\otimes} \newcommand{\tpp}{\tp\cdots\tp} \newcommand{\kk}[1]{|#1\rangle} \newcommand{\bb}[1]{\langle #1} \newcommand{\dd}[1]{\delta_{#1}} \newcommand{\ddd}[1]{\delta^3(#1)} \newcommand{\vv}[1]{{\bf #1}} \newcommand{\molp}{\Omega^{(+)}} \newcommand{\dydxt}[2]{\frac{d#1}{d#2}} \newcommand{\dydx}[2]{\frac{\partial#1}{\partial#2}}$

This case is similar to a bunch of non-relativistic particles scattered by a fixed potential except that the linear momentum is
conserved.

Let the initial bunch of particles be in a state $\phi_i=U(\vv{r}_i)\phi$ corresponding to sharp momentum $\vv{p}$ and spin $\sigma$.

If there are $N$ particles in the bunch, the number of transitions per unit time will be
\begin{eqnarray*} -\dydxt{N}{t}=n_\xi=\frac{2}{\hbar}{\rm Im}\,\sum_i(\phi_i,B\phi_i) \end{eqnarray*}
where, as before, $B={\molp}^\dagger P_\xi V\molp$. By the familiar argument given above in section 11,
\begin{eqnarray*} \sum_i(\phi_i,B\phi_i)=N\frac{1}{2p^0}
\bb{\vv{p}\sigma}|B\kk{\vv{p}\sigma}
\end{eqnarray*}
Therefore
\begin{eqnarray*} n_\xi=-\dydxt{N}{t}=\frac{2N}{\hbar}\frac{1}{2p^0}
{\rm Im}\,\bb{\vv{p}\sigma}|B\kk{\vv{p}\sigma}
\end{eqnarray*}
Therefore
\begin{eqnarray*} N(t)=N(0)\exp[-t/\tau_\xi] \end{eqnarray*}
where $\tau_\xi$ is the decay constant (lifetime").
\begin{eqnarray*} \frac{1}{\tau_\xi}&=&\frac{2}{\hbar}\frac{1}{2p^0}
{\rm Im}\,\bb{\vv{p}\sigma}|B\kk{\vv{p}\sigma}\\
&=&\frac{2\pi}{\hbar}\frac{1}{2p^0}\int d(\xi)\delta^4(P_\xi-p)
|T(\xi,\vv{p}\sigma)|^2
\end{eqnarray*}
where the T-matrix is defined by
\begin{eqnarray*} \bb{\xi}|S\kk{\vv{p}\sigma}=\bb{\xi}\kk{\vv{p}\sigma}
-2\pi i\delta^4(P_\xi-p)T(\xi,\vv{p}\sigma)
\end{eqnarray*}

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