% chapters/chap2.tex
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\chapter{Seigel-Bargmann Space}

In this chapter, we will collect ideas and other perspective we have understanding the concentration of measure phenomenon. Especially with symmetric product of $\C P^1$ and see how it relates to Riemman surfaces and Seigel-Bargmann spaces.

\begin{figure}[h]
    \centering
    \begin{tikzpicture}[node distance=40mm, thick,
        main/.style={draw, draw=white},
        towards/.style={->},
        towards_imp/.style={<->,red},
        mutual/.style={<->}
        ]
        \node[main] (cp) {$\mathbb{C}P^{n}$};
        \node[main] (c) [below of=cp] {$\mathbb{C}^{n+1}$};
        \node[main] (p) [right of=cp] {$\mathbb{P}^n$};
        \node[main] (sym) [below of=p] {$\operatorname{Sym}_n(\mathbb{C}P^1)$};
        % draw edges
        \draw[towards] (c) -- (cp) node[midway, left] {$z\sim \lambda z$};
        \draw[towards] (c) -- (p) node[midway, fill=white] {$w(z)=\sum_{i=0}^n Z_i z^i$};
        \draw[towards_imp] (cp) -- (p) node[midway, above] {$w(z)\sim w(\lambda z)$};
        \draw[mutual] (p) -- (sym) node[midway, right] {root of $w(z)$};
    \end{tikzpicture}
    \caption{Majorana stellar representation}
    \label{fig:majorana_stellar_representation}
\end{figure}

Basically, there is a bijection between the complex projective space $\mathbb{C}P^n$ and the set of roots of a polynomial of degree $n$.

We can use a symmetric group of permutations of $n$ complex numbers (or $S^2$) to represent the $\mathbb{C}P^n$, that is, $\mathbb{C}P^n=S^2\times S^2\times \cdots \times S^2/S_n$.

One might be interested in the random sampling over the $\operatorname{Sym}_n(\mathbb{C}P^1)$ and the concentration of measure phenomenon on that.

\section{Majorana stellar representation of the quantum state}

\begin{defn}
    Let $n$ be a positive integer. The Majorana stellar representation of the quantum state is the set of all roots of a polynomial of degree $n$ in $\mathbb{C}$.

    
\end{defn}
\section{Space of complex valued functions and pure states}



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