week3.tex (8481B)
1 \documentclass[a4paper]{article} 2 3 \usepackage[T1]{fontenc} 4 \usepackage[utf8]{inputenc} 5 6 \usepackage{mathptmx} 7 8 \usepackage{subcaption} 9 \usepackage[shortlabels]{enumitem} 10 \usepackage{amsmath,amssymb} 11 \usepackage{amsthm} 12 \usepackage{bbm} 13 \usepackage{graphicx} 14 \usepackage[colorlinks=true,naturalnames=true,plainpages=false,pdfpagelabels=true]{hyperref} 15 \usepackage[parfill]{parskip} 16 17 \theoremstyle{definition} 18 \newtheorem{definition}{Definition} 19 20 \theoremstyle{definition} 21 \newtheorem{question}{Question} 22 23 \theoremstyle{definition} 24 \newtheorem{example}{Example} 25 26 \theoremstyle{theorem} 27 \newtheorem{theorem}{Theorem} 28 29 \theoremstyle{theorem} 30 \newtheorem{exercise}{Exercise} 31 32 \theoremstyle{theorem} 33 \newtheorem{lemma}{Lemma} 34 35 \theoremstyle{definition} 36 \newtheorem{solution}{Solution} 37 38 \newtheorem*{idea}{Proof Idea} 39 40 \title{University of Vienna\\ Faculty of Physics\\ \vspace{1.25cm} 41 Notes on\\ Noncommutative Geometry and Particle Phyiscs} 42 \author{Milutin Popovic \\ Supervisor: Dr. Lisa 43 Glaser} 44 \date{Week 3: 26.02 - 4.03} 45 46 \begin{document} 47 48 \maketitle 49 \tableofcontents 50 \newpage 51 \section{Excurse to Group Theory and Lie Groups} 52 \subsection{Groups and Representations} 53 \begin{definition} 54 A Group $G$ is a set with a binary operation on $G$ satisfying. 55 \begin{enumerate} 56 \item $f, g \in G$ we have $fg = h \in G$. 57 \item $f(gh) = (fg) h$ 58 \item $\exists\ e \in G\ \forall f\in G$ with $ef=fe=f$ 59 \item $\forall f \in G\ \exists\ f^{-1}\in G$ with $ff^{-1}=f^{-1}f=e$ 60 \end{enumerate} 61 \end{definition} 62 63 \begin{definition} 64 A Representation of a Group $G$ is a mapping, $D$ of elements of $G$ onto a set of \textit{linear 65 operators} such that: 66 \begin{enumerate} 67 \item $D(e) = 1$, $1$ is the identity operator in the space on which linear operators act 68 \item $D(g_1)D(g_2) = D(g_1g_2)$, the mapping is linear in group the group operation 69 \end{enumerate} 70 \end{definition} 71 72 Just by looking at symmetries of a Group we can find a nice representation, and if the group is finite we 73 can even find a matrix representation (Cheyley's Theorem). We all ready know a lot about linear algebra 74 which will then allow us to study these Groups very thoroughly and derive physical properties with 75 minimal information. 76 77 78 \subsection{Lie Groups} 79 Group elements now depend \textit{smoothly} on a set \textit{continuous parameters} $g(\alpha) \in G$. 80 We are looking at continuous symmetries, e.g. a Sphere in $\mathbb{R}^3$ can be rotated in any direction 81 without changing. The collection of rotations forms a Lie group because the group elements are smoothly 82 differentiable. 83 84 \subsubsection{Generators} 85 We parameterize $g(\alpha)|_{\alpha=0} = e$ and we assume that near the identity element, the group 86 elements can be described by a finite set of elements $\alpha_a$ for $a = 1,..,N$. For a representation 87 $D$ of this group, linear operators need to be parametrized the same way: 88 \begin{align} 89 D(\alpha)|_{\alpha=0} = 1 90 \end{align} 91 92 Because of the smoothness and continuity we can Taylor expand a representation near the identity: 93 \begin{align} 94 D(\alpha) &= 1 + id\alpha_a X_a + \cdots && \\ 95 \text{with}&\;\; X_a = -i \frac{\partial D(\alpha)}{\partial \alpha_a}\bigg\arrowvert _{\alpha=0} 96 && \text{\footnote{Einstein Summation Convention, summation over repeated indices}} 97 \end{align} 98 99 We call $X_a$ the \textit{generators of the group}. 100 \begin{itemize} 101 \item If the parametrization is \textit{parsimonious}\footnote{parsimonious - 102 All parameters are needed to distinguish between group elements} then all 103 of $X_a$ will be independent. 104 105 \item If the representation is unitary then $X_a$ will be \textit{hermitian}, because of the 106 $i$ in the definition. 107 108 \item Sophus Lie showed how to derive generators without representations. 109 \end{itemize} 110 111 Now let us go in some fixed infinitesimal direction from the identity. 112 \begin{align} 113 D(d\alpha) = 1+ id\alpha _a X_a 114 \end{align} 115 Because of the group property of closure with respect to the group operation we can raise $D(d\alpha)$ 116 to a large power and still get a group element. 117 \begin{align} 118 D(\alpha) = \lim_{k\rightarrow \infty}(1+i\frac{\alpha_a X_a}{k})^k = e^{i\alpha_a X_a} 119 \end{align} 120 This is called the \textit{exponential parameterization}. Looking at the expression we see that 121 group elements can be expressed in terms of generators, and generators form a vector space. 122 They are often referred to any element in the real linear space spanned by $X_a's$. 123 124 \subsubsection{Lie Algebras} 125 Let us consider a parameter family of group elements created by one generator $X_a$: 126 \begin{align} 127 U(\lambda) = e^{i\lambda \alpha _a X_a} 128 \end{align} 129 We know for that for the same generator the group multiplication is linear meaning: 130 \begin{align} 131 U(\lambda _1)U(\lambda _2) = U(\lambda_1 + \lambda_2) 132 \end{align} 133 But if we multiply elements generated by two different generators the general case is 134 \begin{align} 135 e^{i\alpha_a X_a} e^{i\beta_b X_b} \neq e^{i (\alpha _a + \beta_b) X_a} 136 \end{align} 137 Yet because the exponentials are a representation of a group, and a group has closure under 138 group operation we know the above needs to be true for some $\delta _a$ 139 \begin{align} 140 e^{i\alpha_a X_a} e^{i\beta_b X_b} = e^{i \delta _a X_a} 141 \end{align} 142 To further examine the exponent we rewrite the expression and Taylor expand $ln(1+K)$ 143 to the second of $K = e^{i\alpha_a X_a} e^{i\beta_b X_b} -1$ 144 \begin{align} 145 i\delta _a X_a =& ln(1 + K) = K - \frac{K^2}{2} + \cdots \\ 146 \text{and}\;\;\; K =&\ e^{i\alpha_a X_a} e^{i\beta_b X_b} -1 \\ 147 =&\ (1 + i\alpha _a X_a - \frac{1}{2}(\alpha _a X_a)^2 + \cdots) \\ 148 \cdot&\ (1 + i\beta _b X_b - \frac{1}{2}(\beta _b X_b)^2 + \cdots) -1 \\ 149 =&\ i\alpha _a X_a + i\beta _b X_b - \alpha_a X_a \beta _b X_b \\ 150 -&\ \frac{1}{2}(\alpha _a X_a)^2 - \frac{1}{2}(\beta _b X_b)^2 + \cdots 151 \end{align} 152 So: 153 \begin{align} 154 i\delta _a X_a =&\ i\alpha _a X_a + i\beta _b X_b - \alpha_a X_a \beta _b X_b \\ 155 -&\ \frac{1}{2}(\alpha _a X_a)^2 - \frac{1}{2}(\beta _b X_b)^2 \\ 156 +&\ \frac{1}{2}(\alpha _a X_a + i\beta _b X_b - \alpha_a X_a \beta _b X_b)^2 \\ 157 =&\ i\alpha _a X_a + i\beta _b X_b - \alpha_a X_a \beta _b X_b \\ 158 -&\ \frac{1}{2}(\alpha _a X_a)^2 - \frac{1}{2}(\beta _b X_b)^2 \\ 159 +&\ \frac{1}{2}(\alpha _a X_a)^2 + \frac{1}{2}(\beta _b X_b)^2 \\ 160 +& \frac{1}{2}\alpha _a X_a \beta _b X_b + \frac{1}{2}\beta _b X_b \alpha _a X_a 161 \end{align} 162 Because $X$'s are linear operators $\alpha _a X_a \beta _b X_b \neq \beta _b X_b \alpha _a X_a$. 163 These generators form an \textit{algebra under commutation} and we get 164 \begin{align} 165 i\delta _a X_a =&\ i\alpha _a X_a + i\beta _b X_b - \alpha_a X_a \beta _b X_b \\ 166 -&\ \frac{1}{2}[\alpha _a X_a, \beta _b X_b] + \cdots 167 \end{align} 168 Thus rewriting the equation gives us 169 \begin{align} 170 [\alpha _a X_a, \beta _b X_b] = -2i(\delta _c -\alpha _c -\beta _c) X_c \cdots \equiv i\gamma _c X_c 171 \end{align} 172 Because this is true for all $\alpha$ and $\beta$, and considering the group closure, there exists some 173 \textit{real} $f_{abc}$ called the \textit{structure constant} satisfying. 174 \begin{equation} 175 \gamma _c = \alpha _a \beta _b f_{abc} 176 \end{equation} 177 Which is the same as. 178 \begin{equation} 179 [X_a, X_b] = i f_{abc} X_c 180 \end{equation} 181 This is called the \textit{Lie algebra of a group} 182 \newline 183 \newline 184 So $f$ is antisymmetric because $[A, B] = -[B, A]$, which means $f_{abc} = -f_{bac}$. 185 \newline 186 And $\delta$ can now be written as 187 \begin{equation} 188 \delta _a = \alpha _a + \beta _a - \frac{1}{2} \gamma _a \cdots 189 \end{equation} 190 Just by following the properties of Lie Groups (dependence on parameters and smoothness) in a fixed 191 direction near die identity to find physical statements. E.g. 192 $[\hat{r}_i, \hat{p}_j] = i \hslash \delta _{ij}$ tells us that we can't know the position 193 and the momentum of a particle exactly at a given time. 194 195 196 197 198 \end{document}