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{\scriptsize
{\it Torsten Bringmann} ({\tt torsten.bringmann@fys.uio.no})\\
{\tt http://www.uio.no/studier/emner/matnat/fys/FYS4170/h20/}
}
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{\large
Lecture autumn 2020: \vspace{0.3cm} \\
\underline{ \Large Relativistic quantum field theory}
\vspace{0.3cm} \\
{\bf Problem sheet `0'}
}
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\noindent
$\leadsto$ These problems are scheduled for discussion on {\bf Friday, 21 August 2020} 
%So you must send them in by {\bf XXX} if you want them corrected.

\vspace{1cm}
\noindent
\underline{\bf Problem 1} \\
This problem serves as a reminder to practice the use of tensor notation.
\begin{enumerate}
\item[a)] Write the following in index notation:
\begin{itemize}
\item $\nabla S$ (where $S$ is a scalar).
\item $\nabla \cdot \mathbf{A}$, $\nabla \times \mathbf{A}$ (where $ \mathbf{A}$ is a 3D vector).
\item Trace and Transpose of a matrix $M$.
\end{itemize}

\item[b)] Prove the following 3D identitites, using index notation: 
\begin{itemize}
\item $\nabla \cdot\left(\nabla \times\mathbf{A}\right) =0$, where $ \mathbf{A}$ is a 3D vector.
\item $\nabla \times\left(\nabla S\right)=0$, where $S$ is a scalar.
\end{itemize}


\item[c)] Are these equalities valid? Correct where necessary!
\begin{itemize}
\item $\partial_\mu x^\nu =\delta^\nu_\mu$
\item $\partial_\mu x^\mu =1$
\item $\partial^\mu x^\nu =g^{\mu \nu}$
\item ${T_\alpha} {^\beta} _\gamma$ = $g^{\beta \mu}{T_\alpha} {_\mu} {_\gamma}$= $g^{\mu \beta}{T_\alpha} {_\mu} {_\gamma}$
\item ${T_\alpha} {^\beta} _\beta$ = $g_{\alpha \mu}g^{\beta \alpha}{T^\mu} {_\alpha} {_\beta}$
\end{itemize}

\item[d)] Construct (as many as possible)
\begin{itemize}
\item  independent Lorentz scalars from two four-vectors $A$ and $B$
\item  independent Lorentz scalars from a rank-2 tensor $T$
\item  independent Lorentz scalars involving one (copy of a) rank-2 tensor $T$ and some combination of 
two four-vectors $A$ and $B$
\end{itemize}
{\it (NB: `independent' here does not necessarily mean `linearly independent', so any pair $(x,y)$ that 
can be written as $y=f(x)$ is not independent.)}

\end{enumerate}





\vspace{0.5cm}
\noindent
\underline{\bf Problem 2}\\
Write the Schr\"odinger wave function $\psi(\mathbf{x})$ in Dirac (`bra-ket') notation, by expanding the states $|\psi\rangle$ in the real space basis and then projecting it onto a basis vector $|\mathbf{x}\rangle$. Then change from the real space (or position) basis to the basis of momentum eigenstates and show 
that this recovers the Fourier representation of $\psi(\mathbf{x})$.


\vspace{1cm}
\noindent
\underline{\bf Problem 3}\\
These problems are meant to (re-)familiarize you with delta functions, Fourier tranforms and complex integration. Prove the following identities:
\begin{enumerate}
\item[a)] 
\[  \int {d^3x} \int\frac{d^3k}{(2\pi)^3} e^{i\mathbf{k}\cdot (\mathbf{x}-\mathbf{y})} k_j f(\mathbf{x})=i\frac{df}{dx_j}(\mathbf{y}) \]
{\it \underline{Hint:} Get rid of $k_j$ before you perform the $\int d^3 k$ integration!}

\item[b)] 
\[  \int_{-\infty}^\infty dx\,\frac{e^{ikx}}{a^2+x^2} = \frac{\pi}{a}e^{-|k|a}\qquad(a>0)\]
{\it \underline{Hint:} Rewrite this integral as a closed countour integral in the complex plane; choose the upper or lower half-circle as contour such that the integral over the complex part of the contour vanishes. This means that you need to consider the cases $k<0$ and $k>0$ separately.}

\end{enumerate}

% Harmonic oscillator: 1st exercise of new sheet!




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