\centering\section*{Exercises}
\begin{enumerate}
\item[\textbf{I}]\textbf{Cartesian Photons:}
Cartesian tensors are tensors in three-dimensional Euclidean space. The available tensors are: \begin{itemize}
\item Rank 0: 1
\item Rank 1: $k^i$
\item Rank 2: $k^i k^j$, $\delta^{ij}$, $\epsilon^{ijl} k_l$
\item Rank 3: $\epsilon^{ijl}$
\item Rank 4: combinations of all the above
\end{itemize}
\begin{enumerate}
\item Show that $\epsilon^{ijl} k_j kl$ is not a rank 1 tensor.
\item Show that $\nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$, using the following property: $(\mathbf{A} \times \mathbf{B})^i = \epsilon^{ijl} A{j}B{i}$
\textbf{Hint}:
$$\epsilon{ijl} \epsilon{ij'l'} = \delta{jj'} \delta{ll'}- \delta{jl'}\delta{j'l} $$
\item Given $\int d^3 k \, f(k) \, k^i = 0$ for a scalar function $f(k)$. Show that:
$$\int d^3 k \, f(k) \, k^i k^j = \frac{1}{3} \delta^{ij} \int d^3 k \, f(k) \, k^2.$$
\item Show $\int d^3 k \, f(k, \hat{p}) \, k^i k^j = \frac{1}{2} \int d^3 k \, f\left[k^2 - (k \cdot \hat{p})^2\right] \delta^{ij} + \frac{1}{2} \int d^3 k \, f\left[3(k \cdot \hat{p})^2 - k^2\right] \hat{p}^i \hat{p}^j$.
\end{enumerate}
\item[\textbf{II}] \textbf{Photons in Cartesian Basis:}
Consider photons (or gluons) in Cartesian basis so\ $a^+{k{i}} = \sum\lambda a^+{k{\lambda}} \epsilon^i (k{\lambda})$\,.
\begin{enumerate}
\item Show that the ``scalar photon ball'' is just a scalar state $\gamma\gamma$ that can be written as:
$$| \gamma\gamma; 0^+ \rangle \propto \int d^3 k \, \phi(k) \, a^+{k{i}}a^+{-k{i}}| 0 \rangle\,,$$ where $\phi(k)$ is the momentum wave function.
\item Show that: $$| \gamma\gamma; 0^- \rangle \propto \int d^3 k \, \phi(k) \,\epsilon{ijl} k^l a^+{k{i}}a^+{-k{j}}| 0 \rangle\,.$$
\item Prove the Lee-Yang theorem which states that one cannot construct a $J = 1$, $\gamma\gamma$ state.
\item Show that:
$$| \gamma\gamma\gamma; 0^- \rangle= \int d^3 k_1 d^3 k_2 d^3 k_3 \, \phi(k_1k_2k3) \, \epsilon{i_1i_2i_3}\delta(k_1k_2k3) a^+{k_{1}i1}a^+{k_{2}i2}a^+{k_{3}i_3}| 0 \rangle\,,$$ is a viable state.
\item Can we construct a $| \gamma\gamma\gamma; 1^- \rangle$ state?
\documentclass[11pt]{article}
\usepackage{amsmath}
\begin{document}
\centering\section*{Exercises} \begin{enumerate} \item[\textbf{I}]\textbf{Cartesian Photons:} Cartesian tensors are tensors in three-dimensional Euclidean space. The available tensors are: \begin{itemize} \item Rank 0: 1 \item Rank 1: $k^i$ \item Rank 2: $k^i k^j$, $\delta^{ij}$, $\epsilon^{ijl} k_l$ \item Rank 3: $\epsilon^{ijl}$ \item Rank 4: combinations of all the above \end{itemize}
\begin{enumerate} \item Show that $\epsilon^{ijl} k_j kl$ is not a rank 1 tensor. \item Show that $\nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$, using the following property: $(\mathbf{A} \times \mathbf{B})^i = \epsilon^{ijl} A{j}B{i}$
\textbf{Hint}: $$\epsilon{ijl} \epsilon{ij'l'} = \delta{jj'} \delta{ll'}- \delta{jl'}\delta{j'l} $$ \item Given $\int d^3 k \, f(k) \, k^i = 0$ for a scalar function $f(k)$. Show that: $$\int d^3 k \, f(k) \, k^i k^j = \frac{1}{3} \delta^{ij} \int d^3 k \, f(k) \, k^2.$$ \item Show $\int d^3 k \, f(k, \hat{p}) \, k^i k^j = \frac{1}{2} \int d^3 k \, f\left[k^2 - (k \cdot \hat{p})^2\right] \delta^{ij} + \frac{1}{2} \int d^3 k \, f\left[3(k \cdot \hat{p})^2 - k^2\right] \hat{p}^i \hat{p}^j$. \end{enumerate} \item[\textbf{II}] \textbf{Photons in Cartesian Basis:} Consider photons (or gluons) in Cartesian basis so\ $a^+{k{i}} = \sum\lambda a^+{k{\lambda}} \epsilon^i (k{\lambda})$\,. \begin{enumerate} \item Show that the ``scalar photon ball'' is just a scalar state $\gamma\gamma$ that can be written as: $$| \gamma\gamma; 0^+ \rangle \propto \int d^3 k \, \phi(k) \, a^+{k{i}}a^+{-k{i}}| 0 \rangle\,,$$ where $\phi(k)$ is the momentum wave function. \item Show that: $$| \gamma\gamma; 0^- \rangle \propto \int d^3 k \, \phi(k) \,\epsilon{ijl} k^l a^+{k{i}}a^+{-k{j}}| 0 \rangle\,.$$ \item Prove the Lee-Yang theorem which states that one cannot construct a $J = 1$, $\gamma\gamma$ state. \item Show that: $$| \gamma\gamma\gamma; 0^- \rangle= \int d^3 k_1 d^3 k_2 d^3 k_3 \, \phi(k_1k_2k3) \, \epsilon{i_1i_2i_3}\delta(k_1k_2k3) a^+{k_{1}i1}a^+{k_{2}i2}a^+{k_{3}i_3}| 0 \rangle\,,$$ is a viable state. \item Can we construct a $| \gamma\gamma\gamma; 1^- \rangle$ state?
\textbf{Hint:} $$Pa^+_{ki}P^+=-a^+{-k_i}$$ \end{enumerate} \end{enumerate} \end{document}