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\markright{{\small\bf Standard Model}  
\hfill {\small\it WS 07/08, Sheet \theBlt,}}

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\begin{document}

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\vspace*{-25mm}\parbox{60mm}
{{\sf Universit\"at Heidelberg}\\ 
\small Jan M. Pawlowski, Ulrich Uwer}
\hfill \parbox{48mm}{\rightline{\it
\begin{tabular}{l}
WS 07/08\\ 
%\today\\
Sheet \theBlt\end{tabular}} }
      

                

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\leftline{{\bf Standard Model}\hfill 
- 1 - } 
\vspace{.05cm}
\hrule 

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\leftline{\large\bf 2. Quantisation and Scattering} \medskip 
\begin{itemize} 
\item[\bf 1)] The action of a free real scalar field is given by 
\begin{equation} 
S[\phi]= \frac{1}{2}\int d^4 x 
\left(\partial_\mu \phi(x)\,\partial^\mu \phi(x)- 
m^2 \phi^2(x)\right)\,. 
\label{eq:0}\end{equation} 
The field operator of a real scalar  field operator is given by 
\begin{equation} 
\phi(x)=\int \frac{d^3 k}{(2 \pi)^3}\frac{1}{2 \omega}\left\{ 
e^{i k\,x } a^\dagger (\vec k)+e^{-i k\,x } a(\vec k)\right\}\,, 
\end{equation}
with $\omega=\sqrt{\vec k^2+m^2}$, and annihilation/creation operators 
$a,a^\dagger$ with  
\begin{equation}
[a(\vec k),a^\dagger (\vec k')]=2 \omega (2 \pi)^3\delta(\vec k-\vec k')\,. 
\label{eq:1}\end{equation} 


\begin{itemize}
\item[\bf a)] Show that the Klein-Gordon equation $
(\partial_\mu\partial^\mu+m^2)\phi(x)=0$ follows from 
\begin{equation} 
\frac{\delta S[\phi]}{\delta \phi(x)}=0\,,\qquad {\rm with} \qquad 
\frac{\delta\phi(y)}{\delta \phi(x)}=\delta(x-y)\,.  
\label{eq:2}\end{equation} 
Show that $\phi(x)$ in (\ref{eq:1}) satisfies the Klein-Gordon equation, and
show, that the canonical momentum operator $\Pi(x)$ is given by 
$\Pi(x) =\dot \phi(x)$. \\[-1ex]


\item[\bf b)] Show with (\ref{eq:1}) that $\phi(x)$ and $\dot\phi(x)$ 
satisfy the canonical commutation relations
\begin{equation}
[  \phi(\vec x,t)\,,\,  \dot\phi(\vec y,t)]=i \delta(\vec x-\vec y)\,, 
\qquad [  \phi(\vec x,t)\,,\,  \phi(\vec y,t)]=
[  \dot \phi(\vec x,t)\,,\,  \dot\phi(\vec y,t)]=0\,.
\end{equation}
\end{itemize}
\vspace{.2cm}
\item[\bf 2)] Consider the scattering of one particle in the presence of an
  interaction Lagrangian ${\cal L}'(x)=-1/2 V(\vec x)
  :\phi(x)\phi(x):$, where $:\,.\,:$ stands for normal ordering, e.g., 
\begin{equation} 
: a_1 \cdots a_n a_1^\dagger  \cdots 
a^\dagger_m:=a_1^\dagger  \cdots 
a^\dagger_m a_1 \cdots a_n\,. 
\end{equation}  
With $H'(t)=-\int d^3 x\, {\cal L}'(x)$, the time evolution in the 
interaction picture is given by 
\begin{equation} \label{eq:4}
i \partial_t |t\rangle = H'(t) |t\rangle\,.
\end{equation} 

\begin{itemize}
 \item[\bf a)]
Show by iterating (\ref{eq:4}) 
in its infinitesimal form, $|t+\Delta t \rangle =(\id -i  H'(t)) |t\rangle $, 
that 
\begin{equation} \label{eq:5}
|t\rangle = U(t,t_0) |t_0\rangle\,,\qquad {\rm where} \qquad 
U(t,t_0)= T e^{-i \int_{t_0}^t dt' H'(t')}\,, 
\end{equation} 
with time ordering $T$: $T\,H'(t_1) H'(t_2)=
H'(t_1) H'(t_2)\theta(t_1-t_2)+H'(t_2) H'(t_1)\theta(t_2-t_1)$. 
Expand $U$ up to the second order.  \\[-1ex]




\item[\bf b)] Show that 
\begin{equation} 
\langle \vec k'|:\phi(x)\phi(x):| \vec k\rangle =2
\langle \vec k'|:\phi(x)|0\rangle\langle 0|\phi(x):| \vec k\rangle \,,
\end{equation}
and compute it. This matrix element is relevant for the first order
perturbation theory.
%\end{description}
\end{itemize}
\end{itemize}
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