46 lines
1.9 KiB
TeX
46 lines
1.9 KiB
TeX
% chapters/chap2.tex
|
|
\documentclass[../main.tex]{subfiles}
|
|
|
|
\ifSubfilesClassLoaded{
|
|
\addbibresource{chap3.bib}
|
|
}
|
|
|
|
\begin{document}
|
|
|
|
\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.
|
|
|
|
\ifSubfilesClassLoaded{
|
|
\printbibliography[title={References for Chapter 2}]
|
|
}
|
|
\end{document}
|