week10.tex (11993B)
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{amssymb} 11 \usepackage{amsthm} 12 \usepackage{mathtools} 13 \usepackage{bbm} 14 \usepackage{graphicx} 15 \usepackage[colorlinks=true,naturalnames=true,plainpages=false,pdfpagelabels=true]{hyperref} 16 \usepackage[parfill]{parskip} 17 18 \usepackage{tikz} 19 \usetikzlibrary{patterns,decorations.pathmorphing,positioning} 20 21 \usepackage[framemethod=TikZ]{mdframed} 22 23 \tikzstyle{titlered} = 24 [draw=black, thick, fill=white,% 25 text=black, rectangle, 26 right, minimum height=.7cm] 27 28 \newcounter{exercise} 29 30 \renewcommand*\theexercise{Exercise~\arabic{exercise}} 31 32 \makeatletter 33 \mdfdefinestyle{exercisestyle}{% 34 outerlinewidth=1em,% 35 outerlinecolor=white,% 36 leftmargin=-1em,% 37 rightmargin=-1em,% 38 middlelinewidth=1.2pt,% 39 roundcorner=5pt,% 40 linecolor=black,% 41 backgroundcolor=blue!5, 42 innertopmargin=1.2\baselineskip, 43 skipabove={\dimexpr0.5\baselineskip+\topskip\relax}, 44 skipbelow={-1em}, 45 needspace=3\baselineskip, 46 frametitlefont=\sffamily\bfseries, 47 settings={\global\stepcounter{exercise}}, 48 singleextra={% 49 \node[titlered,xshift=1cm] at (P-|O) % 50 {~\mdf@frametitlefont{\theexercise}~};},% 51 firstextra={% 52 \node[titlered,xshift=1cm] at (P-|O) % 53 {~\mdf@frametitlefont{\theexercise}~};}, 54 } 55 \makeatother 56 57 \newenvironment{MyExercise}% 58 {\begin{mdframed}[style=exercisestyle]}{\end{mdframed}} 59 60 \theoremstyle{definition} 61 \newtheorem{definition}{Definition} 62 63 \theoremstyle{definition} 64 \newtheorem{question}{Question} 65 66 \theoremstyle{definition} 67 \newtheorem{example}{Example} 68 69 \theoremstyle{theorem} 70 \newtheorem{theorem}{Theorem} 71 72 \theoremstyle{theorem} 73 \newtheorem{lemma}{Lemma} 74 75 76 \theoremstyle{theorem} 77 \newtheorem{proposition}{Proposition} 78 79 \newtheorem*{idea}{Proof Idea} 80 81 82 \title{University of Vienna\\ Faculty of Physics\\ \vspace{1.25cm} 83 Notes on\\ Noncommutative Geometry and Particle Phyiscs} 84 \author{Milutin Popovic \\ Supervisor: Dr. Lisa 85 Glaser} 86 \date{Week 8: 8.05 - 18.05} 87 88 \begin{document} 89 90 \maketitle 91 \tableofcontents 92 \newpage 93 94 95 \section{Spectral Action of the Fluctuated Dirac Operator} 96 \begin{proposition} 97 The spectral action of the almost commutative manifold $M$ with $\dim(M) 98 =4$ with a fluctuated Dirac operator is. 99 \begin{align} 100 \text{Tr}(f\frac{D_\omega}{\Lambda}) \sim \int_M \mathcal{L}(g_{\mu\nu}, 101 B_\mu, \Phi) \sqrt{g}\ d^4x + O(\Lambda^{-1}) 102 \end{align} 103 with 104 \begin{align} 105 \mathcal{L}(g_{\mu\nu}, B_\mu, \Phi) = 106 N\mathcal{L}_M(g_{\mu\nu}) 107 \mathcal{L}_B(B_\mu)+ 108 \mathcal{L}_\phi(g_{\mu\nu}, B_\mu, \Phi) 109 \end{align} 110 where $N=4$ and $\mathcal{L}_M$ is the Lagrangian of the spectral triple 111 $(C^\infty(M) , L^2(S), D_M)$ 112 \begin{align}\label{lagr} 113 \mathcal{L}_M(g_{\mu\nu}) := \frac{f_4 \Lambda ^4}{2\pi^2} - 114 \frac{f_2 \Lambda^2}{24\pi ^2}s - \frac{f(0)}{320\pi^2} C_{\mu\nu 115 \varrho \sigma}C^{\mu\nu \varrho \sigma}. 116 \end{align} 117 Here $C^{\mu\nu \varrho \sigma}$ is defined in terms of the Riemannian 118 curvature tensor $R_{\mu\nu \varrho \sigma}$ and the Ricci tensor 119 $R_{\nu\sigma} = g^{\mu\varrho} R_{\mu\nu \varrho\sigma}$. 120 121 122 Furthermore $\mathcal{L}_B$ describes the kinetic term of the gauge field 123 \begin{align} 124 \mathcal{L}_B(B_\mu) := \frac{f(0)}{24\pi^2} 125 \text{Tr}(F_{\mu\nu}F^{\mu\nu}). 126 \end{align} 127 Last $\mathcal{L}_\phi$ is the scalar-field Lagrangian with a boundary 128 term. 129 \begin{align} 130 \mathcal{L}_\phi(g_{\mu\nu}, B_\mu, \Phi) := 131 &-\frac{2f_2\Lambda^2}{4\pi^2}\text{Tr}(\Phi^2) + \frac{f(0)}{8\pi^2} 132 \text{Tr}(\Phi^4) + \frac{f(0)}{24\pi^2} \Delta(\text{Tr}(\Phi^2))\\ 133 &+ \frac{f(0)}{48\pi^2}s\text{Tr}(\Phi^2) 134 \frac{f(0)}{8\pi^2}\text{Tr}((D_\mu \Phi)(D^\mu \Phi)). 135 \end{align} 136 \end{proposition} 137 \begin{proof} 138 The dimension of our manifold $M$ is $\dim(M) = \text{Tr}(id) =4 $. Let us 139 take a $x \in M$, we have an asymtotic expansion of 140 $\text{Tr}(f(\frac{D_\omega}{\Lambda}))$ as $\Lambda \rightarrow \infty$ 141 \begin{align} 142 \text{Tr}(f(\frac{D_\omega}{\Lambda})) \simeq& \ 2f_4 \Lambda ^4 143 a_0(D_\omega ^2)+ 2f_2\Lambda^2 a_2(D_\omega^2) \\&+ f(0) a_4(D_\omega^4) 144 +O(\Lambda^{-1}). 145 \end{align} 146 Note that the heat kernel coefficients are zero for uneven $k$, 147 furthermore they are dependent on the fluctuated Dirac operator 148 $D_\omega$. We can rewrite the heat kernel coefficients in terms of $D_M$, 149 for the first two we note that $N:= \text{Tr}\mathbbm{1_{H_F}})$ 150 \begin{align} 151 a_0(D_\omega^2) &= Na_0(D_M^2)\\ 152 a_2(D_\omega^2 &= Na_2(D_M^2) - \frac{1}{4\pi^2}\int_M 153 \text{Tr}(\Phi^2)\sqrt{g}d^4x 154 \end{align} 155 For $a_4$ we need to extend in terms of coefficients of $F$, look week9.pdf 156 for the standard version, 157 \begin{align} 158 &\frac{1}{360}\text{Tr}(60sF)= -\frac{1}{6}S(Ns + 4 159 \text{Tr}(\Phi^2))\\ 160 \nonumber\\ 161 &F^2 = \frac{1}{16}s^2\otimes 1 + 1\otimes \Phi^4 - \frac{1}{4} 162 \gamma^\mu\gamma^\nu \gamma^\varrho\gamma^\sigma F_{\mu\nu}F^{\mu\nu}+\\ 163 &\;\;\;\;\;\;\;+\gamma^\mu\gamma^\nu\otimes(D_\mu\Phi)(D_\nu 164 \Phi)+\frac{1}{2}s\otimes \Phi^2 + \ \text{traceless terms}\\ 165 \nonumber\\ 166 &\frac{1}{360}\text{Tr}(180F^2) = \frac{1}{8}s^2N + 2\text{Tr}(\Phi^4) 167 + \text{Tr}(F_{\mu\nu}F^{\mu\nu}) +\\ 168 &\;\;\;\;\;\;\;+2\text{Tr}((D_\mu\Phi)(D^\mu\Phi)) 169 + s\text{Tr}(\Phi^2)\\ 170 \nonumber\\ 171 &\frac{1}{360}\text{Tr}(-60\Delta F)= 172 \frac{1}{6}\Delta(Ns+4\text{Tr}(\Phi^2)). 173 \end{align} 174 Now for the cross terms of $\Omega_{\mu\nu}^E\Omega^{E\mu\nu}$ the trace 175 vanishes because of the anti-symmetric properties of the Riemannian 176 curvature Tensor 177 \begin{align} 178 \Omega_{\mu\nu}^E\Omega^{E\mu\nu} = \Omega_{\mu\nu}^S\Omega^{S\mu\nu} 179 \otimes 1 - 1\otimes F_{\mu\nu}F^{\mu\nu} + 2i\Omega_{\mu\nu}^S 180 \otimes F^{\mu\nu} 181 \end{align} 182 the trace of the cross term vanishes because 183 \begin{align} 184 \text{Tr}(\Omega^{S}_{\mu\nu} = \frac{1}{4} 185 R_{\mu\nu\varrho\sigma}\text{Tr}(\gamma^\mu\gamma^\nu) = \frac{1}{4} 186 R_{\mu\nu\varrho\sigma}g^{\mu\nu} =0 187 \end{align} 188 and the trace of the whole term is 189 \begin{align} 190 \frac{1}{360}\text{Tr}(30\Omega^E_{\mu\nu}\Omega^{E\mu\nu}) = 191 \frac{N}{24}R_{\mu\nu\varrho\sigma}R^{\mu\nu\varrho\sigma} 192 -\frac{1}{3}\text{Tr}(F_{\mu\nu}F^{\mu\nu}). 193 \end{align} 194 Plugging the results into $a_4$ and simplifying we can write 195 \begin{align} 196 a_4(x, D_\omega^4) &= Na_4(x, D_M^2) + \frac{1}{4\pi^2}\bigg(\frac{1}{12} s 197 \text{Tr}(\Phi^2) + \frac{1}{2}\text{Tr}(\Phi^4) \\ 198 &+ \frac{1}{4} 199 \text{Tr}((D_\mu\Phi)(D^\mu \Phi)) + \frac{1}{6} 200 \Delta\text{Tr}(\Phi^2) + \frac{1}{6} 201 \text{Tr}(F_{\mu\nu}F^{\mu\nu})\bigg) 202 \end{align} 203 The only thing left is to plug in the heat kernel coefficients into the 204 heat kernel expansion above. 205 \end{proof} 206 207 \section{Fermionic Action} 208 A quick reminder with what we are dealing with, the fermionic action is defined 209 in the following way. 210 \begin{definition} 211 The fermionic action is defined by 212 \begin{align} 213 S_f[\omega, \psi] = (J\tilde{\psi}, D_\omega \tilde{\psi}) 214 \end{align} 215 with $\tilde{\psi} \in H_{cl}^+ := \{\tilde{\psi}: \psi \in H^+\}$. 216 $H_{cl}^+$ is the set of Grassmann variables in $H$ in the +1-eigenspace 217 of the grading $\gamma$. 218 \end{definition} 219 220 The almostcommutative Manifold we are dealing with is the following 221 \begin{align} 222 &M\times F_{ED} := \left(C^\infty(M,\mathbb{C}^2),\ L^2(S)\otimes 223 \mathbb{C}^4,\ 224 D_M\otimes 1 +\gamma _M \otimes D_F;\; J_M\otimes J_F,\ \gamma_M\otimes 225 \gamma _F\right).\\ 226 \nonumber\\ 227 &\text{where:} \nonumber \\ 228 &C^\infty(M,\mathbb{C}^2) = C^\infty(M) \otimes C^\infty(M) 229 &\mathcal{H} = \mathcal{H}^+ \otimes \mathcal{H}^-\\ 230 &\mathcal{H} = L^2(S)^+ \otimes H_F^+ \oplus L^2(S)^- \otimes H_F^-. 231 \end{align} 232 Where $H_F$ is separated into the particle-anitparticle states with ONB $\{e_R, 233 e_L, \bar{e}_R, \bar{e}_L\}$. The ONB of $H_F^+$ is $\{e_L, \bar{e}_R\}$ and 234 for $H_F^-$ we have $\{e_R, \bar{e}_L\}$. Furthermore we can decompose a spinor 235 $\psi \in L^2(S)$ for each of the eigenspaces $H_F^\pm$, $\psi = \psi_R 236 \psi_L$. Thus we can write for an arbitrary $\psi \in \mathcal{H}^+$ 237 \begin{align} 238 \psi = \chi_R \otimes e_R + \chi_L \otimes e_L + \psi_L \otimes \bar{e}_R 239 \psi_R \otimes \bar{e}_L 240 \end{align} 241 for $\chi_L, \psi_L \in L^2(S)^+$ and $\chi_R, \psi_R \in L^2(S)^-$. 242 \begin{proposition} 243 We can define the action of the fermionic art of $M\times F_{ED}$ in the 244 following way 245 \begin{align} 246 S_f = -i\big(J_M\tilde{\chi}, \gamma(\nabla^S_\mu - i\Gamma_\mu) 247 \tilde{\Psi}\big) + \big(S_M\tilde{\chi}_L, \bar{d}\tilde{\psi}_L\big) - 248 \big(J_M\tilde{\chi}_R, d \tilde{\psi}_R\big) 249 \end{align} 250 \end{proposition} 251 \begin{proof} 252 We take the fluctuated Dirac operator 253 \begin{align} 254 D_\omega = D_M \otimes i + \gamma^\mu \otimes B_\mu + \gamma_M \otimes 255 D_F 256 \end{align} 257 \end{proof} 258 The Fermionic Action is $S_F = (J\tilde{\xi}, D_\omega\tilde{\xi})$ for a $\xi 259 \in \mathcal{H}^+$, we can begin to calculate (note that we add the constant 260 $\frac{1}{2}$ to the action) 261 \begin{align} 262 \frac{1}{2}(J\tilde{\xi}, D_\omega\tilde{\xi}) =&\\ 263 &+\frac{1}{2}(J\tilde{\xi}, (D_M \otimes i)\tilde{\xi})\label{eq:1}\\ 264 &+\frac{1}{2}(J\tilde{\xi}, (\gamma^\mu \otimes B_\mu) 265 \tilde{\xi})\label{eq:2}\\ 266 &+\frac{1}{2}(J\tilde{\xi}, (\gamma_M\otimes 267 D_F)\tilde{\xi})\label{eq:3}. 268 \end{align} 269 For equation \ref{eq:1} we calculate 270 \begin{align} 271 \frac{1}{2}(J\tilde{\xi}, (D_M\otimes 1)\tilde{\xi}) &= 272 \frac{1}{2}(J_M\tilde{\chi}_R,D_M\tilde{\psi}_L)+ 273 \frac{1}{2}(J_M\tilde{\chi}_L,D_M\tilde{\psi}_R)+ 274 \\&+\frac{1}{2}(J_M\tilde{\psi}_L,D_M\tilde{\psi}_R)+ 275 \frac{1}{2}(J_M\tilde{\chi}_R,D_M\tilde{\chi}_L)\\ 276 &= (J_M\tilde{\chi},D_M\tilde{\chi}). 277 \end{align} 278 For equation \ref{eq:2} we have 279 \begin{align} 280 \frac{1}{2}(J\tilde{\xi}, (\gamma^\mu \otimes B_\mu)\tilde{\xi})&= 281 -\frac{1}{2}(J_M\tilde{\chi}_R, \gamma^\mu Y_\mu\tilde{\psi}_R) 282 -\frac{1}{2}(J_M\tilde{\chi}_L, \gamma^\mu Y_\mu\tilde{\psi}_R)+\\ 283 &+\frac{1}{2}(J_M\tilde{\psi}_L, \gamma^\mu Y_\mu\tilde{\chi}_R)+ 284 \frac{1}{2}(J_M\tilde{\psi}_R, \gamma^\mu Y_\mu\tilde{\chi}_L)=\\ 285 &= -(J_M\tilde{\chi}, \gamma^\mu Y_\mu\tilde{\psi}). 286 \end{align} 287 For equation \ref{eq:3} we have 288 \begin{align} 289 \frac{1}{2}(J\tilde{\xi}, (\gamma_M\otimes D_F)\tilde{\xi})&= 290 +\frac{1}{2}(J_M\tilde{\chi}_R, d\gamma_M\tilde{\chi}_R) 291 +\frac{1}{2}(J_M\tilde{\chi}_L, \bar{d}\gamma_M\tilde{\chi}_L)+\\ 292 &+\frac{1}{2}(J_M\tilde{\chi}_L, \bar{d}\gamma_M\tilde{\chi}_L) 293 +\frac{1}{2}(J_M\tilde{\chi}_R, d\gamma_M\tilde{\chi}_R)=\\ 294 &= i(J_M\tilde{\chi}, m\tilde{\psi}) 295 \end{align} 296 Note that we obtain a complex mass parameter $d$, so we write $d:=im$ for $m\in \mathbb{R}$, 297 which stands for the real mass and we obtain a nice result 298 299 \begin{theorem} 300 The full Lagrangian of $M\times F_{ED}$ is the sum of purely gravitational 301 Lagrangian 302 \begin{align} 303 \mathcal{L}_{grav}(g_{\mu\nu})=4\mathcal{L}_M(g_{\mu\nu}) 304 \mathcal{L}_\phi (g_{\mu\nu}) 305 \end{align} 306 and the Lagrangian of electrodynamics 307 \begin{align} 308 \mathcal{L}_{ED} = -i\bigg\langle 309 J_M\tilde{\chi},\big(\gamma^\mu(\nabla^S_\mu - iY_\mu) -m\big)\tilde{\psi}) 310 \bigg\rangle 311 +\frac{f(0)}{6\pi^2} Y_{\mu\nu}Y^{\mu\nu}. 312 \end{align} 313 314 \end{theorem} 315 316 317 \end{document}