main.tex (9481B)
1 \documentclass[fleqn]{beamer} 2 \beamertemplatenavigationsymbolsempty 3 4 \usepackage[T1]{fontenc} 5 \usepackage[utf8]{inputenc} 6 7 \usepackage{amsmath,amssymb} 8 \usepackage{graphicx} 9 \usepackage{mathptmx} 10 \usepackage{subcaption} 11 \usepackage{amsthm} 12 \usepackage{tikz} 13 %\usepackage[colorlinks=true,naturalnames=true,plainpages=false,pdfpagelabels=true]{hyperref} 14 \usetikzlibrary{patterns,decorations.pathmorphing,positioning, arrows, chains} 15 16 \usepackage[backend=biber, sorting=none]{biblatex} 17 \addbibresource{uni.bib} 18 19 \setbeamertemplate{endpage}{% 20 \begin{frame} 21 \centering 22 \Large \emph{Thank You!} 23 \end{frame} 24 } 25 26 \AtEndDocument{\usebeamertemplate{endpage}} 27 28 % vertical separator macro 29 \newcommand{\vsep}{ 30 \column{0.0\textwidth} 31 \begin{tikzpicture} 32 \draw[very thick,black!10] (0,0) -- (0,7.3); 33 \end{tikzpicture} 34 } 35 \setlength{\mathindent}{0pt} 36 37 % Beamer theme 38 \usetheme{UniVienna} 39 \usefonttheme[onlysmall]{structurebold} 40 \mode<presentation> 41 \setbeamercovered{transparent=10} 42 43 \title 44 {Noncommutative Geometry} 45 \subtitle{Bachelor's seminar} 46 \author[Popovic Milutin] 47 {Popovic Milutin \newline Supervisor: Dr. Lisa Glaser} 48 \date{16. April 2021} 49 50 \begin{document} 51 \begin{frame} 52 \titlepage 53 \end{frame} 54 55 \begin{frame}{Introduction} 56 \begin{itemize} 57 \item Noncommutative geometry (NCG) brings many\\ 58 mathematical fields together (e.g. K-Theory, Differential Geometry) 59 \item Physics application (spectral Standard Model) 60 \item Gelfand-Naimark-Theorem in Functional Analysis in the 1940s \\ 61 duality between (classical) geometry and Algebra 62 \end{itemize} 63 \end{frame} 64 65 \begin{frame}{Spaces and Algebras} 66 Introduce: 67 \begin{block} 68 {Algebra} 69 \centering 70 Vectorspace with a multiplication operation\\ 71 (associative and possesses an identity element) 72 \end{block} 73 \begin{block} 74 {Finite topological Space $X$ consisting of $N$ points. (discrete topology)} 75 \begin{figure}[h!] 76 \centering 77 \begin{tikzpicture}[ 78 dot/.style = {draw, circle, inner sep=0.05cm, fill}, 79 smalldot/.style = {draw, circle, inner sep=0.015cm,fill}, 80 ] 81 \node[dot] at (-3,0.) [label=below:$1$]{}; 82 \node[dot] at (-1.5,0) [label=below:$2$]{}; 83 \node[dot] at (2.1,0) [label=below:$N$]{}; 84 \node[smalldot] at (-0.4,0) {}; 85 \node[smalldot] at (0.1,0) {}; 86 \node[smalldot] at (0.6,0) {}; 87 \end{tikzpicture} 88 \end{figure} 89 90 \end{block} 91 \begin{block}{Commutative algebra of continuous functions on $X$} 92 \centering 93 $C(X) = \{ f: X \rightarrow \mathbb{C}:\;\; 94 \text{$f$ is continuous}\}$ 95 \end{block} 96 \end{frame} 97 \begin{frame} 98 {Spaces and commutative Algebras} 99 Results of the Theorem: 100 \begin{itemize} 101 \item $X$ and $C(X)$ contain the same information (duality) 102 \item Construct $X$, given $C(X)$. 103 \item Translate geometrical properties of $X$ to algebraic data\\ 104 (metric, differential forms, vector fields, curvature, etc.) 105 \end{itemize} 106 \end{frame} 107 108 \begin{frame}{Geometry as a Spectral Triple} 109 \begin{block} 110 {\centering The Spectral Triple} 111 \centering 112 $(\;A,\;\; H,\;\; D\;)$ 113 \end{block} 114 \begin{itemize} 115 \item $A$ - Algebra 116 \item $H$ - Hilbertspace 117 \item $D$ - self adjoint Operator acting on $H$ 118 \end{itemize} 119 \end{frame} 120 121 \begin{frame}{Geometry as a Spectral Triple} 122 \begin{block} 123 {The Spectral Triple of a Circle $\mathbb{S}^1$} 124 \centering 125 $ (\; C^{\infty}(\mathbb{S}^1),\;\; L^2(\mathbb{S}^1),\;\; -i\frac{d}{dt} \;)$ 126 \end{block} 127 \end{frame} 128 129 \begin{frame}{Introducing the Metric} 130 \begin{itemize} 131 \item The metric describes distances between points on a space 132 \end{itemize} 133 \begin{columns}[T] 134 \column{0.4\textwidth} 135 \begin{block}{\centering Discrete Metric} 136 \centering 137 $ 138 d_{ij} = 139 \begin{cases} 140 0\;\;\; \text{if}\;\;\; i = j \\ 141 1\;\;\; \text{if}\;\;\; i \neq j 142 \end{cases} 143 $ 144 \end{block} 145 146 \column{0.4\textwidth} 147 \begin{block}{\centering Minkowski Metric} 148 \centering 149 $ 150 \eta _{\mu \nu} = 151 \begin{pmatrix} 152 -1 & 0 & 0 & 0 \\ 153 0 & 1 & 0 & 0 \\ 154 0 & 0 & 1 & 0 \\ 155 0 & 0 & 0 & 1 156 \end{pmatrix} 157 $ 158 \end{block} 159 \end{columns} 160 161 \begin{figure}[h!] \centering 162 \begin{tikzpicture}[ 163 dot/.style = {draw, circle, inner sep=0.05cm, fill}, 164 smalldot/.style = {draw, circle, inner sep=0.015cm,fill}, 165 ] 166 \node[dot](m1) at (-3,0.) [label=left:$1$] {}; 167 \node[dot](m2) at (-1.5, 2) [label=above right:$2$] {}; 168 \node[dot](m3) at (2.1,0) [label=right:$3$] {}; 169 170 \draw[<->, >=stealth](m1) -- ++(m2) node [midway, fill=white] {$d_{12}$}; 171 \draw[<->, >=stealth](m2) -- ++(m3) node [midway, fill=white] {$d_{23}$}; 172 \draw[<->, >=stealth](m3) -- ++(m1) node [midway, fill=white] {$d_{13}$}; 173 \end{tikzpicture} 174 \end{figure} 175 176 \end{frame} 177 178 179 \begin{frame}{Algebraic Formulation of the Metric} 180 \begin{itemize} 181 \item Utilize results of the Gelfand-Naimark Theorem 182 \item Characterize the Metric with\\ 183 \begin{itemize} 184 \item[\bullet] commutative Algebra 185 \item[\bullet] finite-dimensional Hilbertspace $H$ 186 \item[\bullet] symmetric operator $D$ 187 \end{itemize} 188 \end{itemize} 189 \begin{block}{Metric with $(A, H, D)$ on finite Space (commutative case)} 190 \centering 191 $d_{ij} = \sup_{a \in A}\{ |a(i) - a(j)| : ||[D, a]|| \leq 1\}$ 192 \end{block} 193 \end{frame} 194 195 \begin{frame}{Algebraic Formulation of the Metric} 196 In the noncommutative Case: 197 \begin{itemize} 198 \item replace Algebra with matrix Algebra (noncommutative) 199 \item define in terms of invariants 200 \end{itemize} 201 \begin{block}{Metric with $(A, H, D)$ on finite Space (noncommutative case)} 202 \centering 203 $d_{ij} = \sup_{a \in A}\{ |\text{Tr}(a(i)) - \text{Tr}(a(j))| :||[D, a]|| \leq 1\}$ 204 \end{block} 205 \end{frame} 206 207 \begin{frame}{Algebraic Formulation of the Metric} 208 \begin{itemize} 209 \item describe the Metric on a Manifold $M$ 210 \item We need \\ 211 \begin{itemize} 212 \item[\bullet] $C^\infty(M)$ - Algebra 213 \item[\bullet] $L^2(S)$ - Hilbertspace 214 \item[\bullet] $D$ - Dirac Operator 215 \end{itemize} 216 \end{itemize} 217 \begin{block}{Metric with $(C^\infty(M),\;\; L^2(S),\;\; D)$ on a Manifod} 218 \centering 219 $d(x, y) = \sup_{f \in C^\infty(M) }\{ |f(x) - f(y)| : 220 ||[D, f]|| \leq 1\}$ 221 \end{block} 222 \begin{figure}[h!] \centering 223 \begin{tikzpicture}[ 224 dot/.style = {draw, circle, inner sep=0.06cm, fill}, 225 smalldot/.style = {draw, circle, inner sep=0.015cm,fill}, 226 ] 227 \node[dot](b) at (0,0) [label=below left:$x$] {}; 228 \node[dot](a) at (2, 0) [label=below right:$y$] {}; 229 \node[dot](a) at (5, 0) [label=below left:$x$] {}; 230 \node[dot](a) at (8, 0) [label=below right:$y$] {}; 231 % \node[dot](m3) at (2.1,0) [label=right:$3$] {}; 232 233 \draw[<->, >=stealth, line width=0.4mm, style=dashed](0, 0.2) -- ++(2, 0) {}; 234 \draw[line width=0.5mm] (-0.3, 0) -- (2.3, 0) {}; 235 236 \draw[line width=0.5mm] (4.7, 0) -- (8.3, 0) {}; 237 \draw[<->, >=stealth, line width=0.4mm, style=dashed](8, 2) -- (8, 0.1) {}; 238 \draw[line width=0.5mm] (5, 0) -- (8, 2) node [pos=.75, label=:$f$] {} ; 239 \end{tikzpicture} 240 \end{figure} 241 \end{frame} 242 243 %\begin{frame}{Noncommutative Case} 244 % \begin{itemize} 245 % \item Introduce a richer geometry 246 % \item From finite topological space to a Manifold with noncommutativity 247 % \item From finite to general spectral triples with a \\ 248 % self adjoint Operator (Dirac Operator) 249 % \end{itemize} 250 % \end{frame} 251 252 \begin{frame}{Applications In Physics} 253 \begin{itemize} 254 \item NCG of the Quantum Hall Effect 255 \item NCG of the Standard Model 256 \begin{itemize} 257 \item[\bullet] going to noncommutative Manifolds 258 \item[\bullet] obtain Standard Model gauge fields (scalar Higgs filed) 259 \item[\bullet] minimal coupling to gravity 260 \item[\bullet] construct the Full Lagrangian 261 \end{itemize} 262 \end{itemize} 263 \end{frame} 264 265 \begin{frame}{Bibliography} 266 \nocite{ncgwalter} 267 \nocite{liealgebra} 268 \nocite{ncg4pages} 269 \nocite{ncgshort} 270 \printbibliography 271 \end{frame} 272 \end{document} 273