week8.tex (27853B)
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 \section{Excurse} 95 \textbf{Manifold:} A topological space that is locally Euclidean. 96 \newline 97 \textbf{Riemannian Manifold:}A Manifold equipped with a Riemannian 98 Metric, a 99 symmetric bilinear form on Vector Fields $\Gamma(TM)$ 100 \begin{align} 101 &g: \Gamma(TM) \times \Gamma(TM) \rightarrow C(M) \\ 102 \text{with}& \nonumber\\ 103 &g(X, Y) \in \mathbb{R} \;\;\; \text{if $X, Y \in \mathbb{R}$}\\ 104 &\text{$g$ is $C(M)$-bilinear } \forall f\in C(M):\;\; g(fX, Y) = 105 g(X, 106 fY) = fg(X,Y)\\ 107 &g(X,X) \begin{cases}\geq 0 \;\;\; \forall X \\ = 0 \;\;\; \forall X 108 =0 109 \end{cases} 110 \end{align} 111 $g$ on $M$ gives rise to a distance function on $M$ 112 \begin{align} 113 d_g(x, y) = \inf_\gamma \left\{\int_0^1(\dot{\gamma}(t), 114 \dot{\gamma}(t))dt;\;\; \gamma(0) = x, \gamma(1) = y \right\} 115 \end{align} 116 Riemannian Manifold is called spin$^c$ if there exists a vector bundle $S 117 \rightarrow M$ with an algebra bundle isomorphism 118 \begin{align} 119 \mathbb{C}\text{I}(TM) &\simeq \text{End}(S)\;\;\; &\text{($dim(M)$ 120 even)}\\ 121 \mathbb{C}\text{I}(TM)^\circ &\simeq \text{End}(S)\;\;\; 122 &\text{($dim(M)$ odd)}\\ 123 \end{align} 124 $(M,S)$ is called the \textbf{spin$^c$ structure on $M$}. 125 \newline 126 $S$ is called the \textbf{spinor Bundle}. 127 \newline 128 $\Gamma(S)$ are the \textbf{spinors}. 129 130 Riemannian spin$^c$ Manifold is called spin if there exists an 131 anti-unitary 132 operator $J_M:\Gamma(S) \rightarrow \Gamma(S)$ such that: 133 \begin{enumerate} 134 \item $J_M$ commutes with the action of real-valued continuous 135 functions 136 on $\Gamma(S)$. 137 \item $J_M$ commutes with $\text{Cliff}^-(M)$ (even case)\\ 138 $J_M$ commutes with $\text{Cliff}^-(M)^\circ$ (odd case) 139 \end{enumerate} 140 $(S, J_M)$ is called the \textbf{spin Structure on $M$} 141 \newline 142 $J_M$ is called the \textbf{charge conjugation}. 143 \section{Noncommutative Geometry of Electrodynamics} 144 \subsection{The Two-Point Space} 145 Consider a two point space $X := \{x, y\}$. This space=an be described 146 with 147 the following spectral triple 148 \begin{align} 149 F_x := (C(X) = \mathbb{C}^2, H_F, D_F; J_F, \gamma _f). 150 \end{align} 151 152 Notes on the spectral triple: 153 \begin{itemize} 154 \item Action of $C(X)$ on $H_F$ is faithful ($\dim (H_F) \geq 2$)\\ 155 we choose $H_F = \mathbb{C}^2$ 156 \item $\gamma_F$ is the $\mathbb{Z}_2$ grading, which allows us to 157 decompose $H_F = H_F^+ \oplus H_F^- = \mathbb{C} \oplus \mathbb{C}$\\ 158 where $H_F^{\pm} = \{ \psi \in H_F |\;\; \gamma _F \psi = \pm \psi\}$ 159 are the two eigenspaces 160 \item $D_F$ interchanges between $H_F^\pm$, $D_F = 161 \begin{pmatrix}0 & t \\ \bar{t} & 0\end{pmatrix}$ where $t \in 162 \mathbb{C}$ 163 \end{itemize} 164 165 \begin{proposition} 166 $F_x$ can only have a real structure if $D_F = 0$ in that case we 167 have 168 $KO-dim = 0, 2, 6$ 169 \end{proposition} 170 \begin{proof} 171 There are two diagram representations of $F_x$ at 172 $\underbrace{\mathbb{C} \oplus \mathbb{C}}_{C(X)}$ 173 on $\underbrace{\mathbb{C} \oplus\mathbb{C}}_{H_F}$ 174 175 \begin{figure}[h!] \centering 176 \begin{tikzpicture}[ 177 dot/.style = {draw, circle, inner sep=0.06cm}, 178 no/.style = {}, 179 ] 180 \node[no](a) at (0,0) [label=left:$\textbf{1}^\circ$] {}; 181 \node[no](b) at (0, -1) [label=left:$\textbf{1}^\circ$] {}; 182 \node[no](c) at (1, 0.5) [label=above:$\textbf{1}$] {}; 183 \node[no](d) at (2, 0.5) [label=above:$\textbf{1}$] {}; 184 \node[dot](d0) at (2,0) [] {}; 185 \node[dot](d0) at (1,-1) [] {}; 186 187 \node[no](a1) at (6,0) [label=left:$\textbf{1}^\circ$] {}; 188 \node[no](b2) at (6, -1) [label=left:$\textbf{1}^\circ$] {}; 189 \node[no](c2) at (7, 0.5) [label=above:$\textbf{1}$] {}; 190 \node[no](d2) at (8, 0.5) [label=above:$\textbf{1}$] {}; 191 \node[dot](d0) at (7,0) [] {}; 192 \node[dot](d0) at (8,-1) [] {}; 193 \end{tikzpicture} 194 \end{figure} 195 If $F_x$ a real spectral triple then $D_F$ can only go vertically or 196 horizontally $\Rightarrow D_F = 0$. Furthermore the diagram on the 197 left has KO-dimension 2 and 6, diagram on the right has KO-dimension 198 0 and 4. Yet KO-dimension 4 is not allowed because 199 $dim(H_F^\pm) = 1$ (see Lemma 3.8 Book), so $J_F^2 = -1$ is not 200 allowed. 201 \end{proof} 202 \subsection{The product Space} 203 Let $M$ be a 4-dim Riemannian spin Manifold, then we have the almost 204 commutative manifold $M\times F_x$ 205 \begin{align} 206 M\times F_x = (C^\infty(M, \mathbb{C}^2, L^2(S)\otimes \mathbb{C}^2, 207 D_M\otimes 1 ; J_M\otimes J_F, \gamma_M \otimes \gamma_F) 208 \end{align} 209 ($J_M$ is missing need to choose)\newline 210 $C^\infty(M, \mathbb{C}^2) \simeq C^\infty(M) \oplus C^\infty(M)$ 211 (decomposition) and from Gelfand duality we we have 212 \begin{align} 213 N:= M\otimes X \simeq M\sqcup X 214 \end{align} 215 $H = L^2(S) \oplus L^2(S)$ (decomposition), such that for 216 $\underbrace{a,b 217 \in C^\infty(M)}_{(a, b) \in C^\infty(N)}$ 218 and $\underbrace{\psi, \phi \in L^2(S)}_{(\psi, \phi) \in H}$ we have 219 \begin{align} 220 (a, b)(\psi, \phi) = (a\psi, b\phi) 221 \end{align} 222 We can consider a distance formula on $M\times F_x$ by 223 \begin{align} 224 d_{D_F}(x,y) = \sup\left\{ |a(x) - a(y)|:a\in A_F, ||[D_F, a]|| \leq 225 1 \right\} 226 \end{align} 227 Now lets calculate the distance between two points on the two point space 228 $X= 229 \{x, y\}$, between $x$ and $y$. Let $a \in \mathbb{C}^2 = C(X)$, $a$ is 230 specified with two complex numbers $a(x)$ and $a(y)$ 231 \begin{align} 232 &||[D_F , a]|| = ||(a(y) - a(x))\begin{pmatrix}0 &t\\\bar{t} &0 233 \end{pmatrix}|| \leq 1\\ 234 &\Rightarrow |a(y) - a(x)|\leq \frac{1}{|t|} 235 \end{align} 236 Therefore the distance between two points $x$ and $y$ is 237 \begin{align} 238 d_{D_F} (x,y) = \frac{1}{|t|} 239 \end{align} 240 Note that if there exists $J_M$ (real structure) $\Rightarrow t=0$ then 241 $d_{D_F}(x,y) \rightarrow \infty$! 242 \newline 243 244 Now let $p \in M$, then take two points on $N=M\times X$, $(p, x)$ and 245 $(p,y)$ and $a \in C^\infty(N)$ is determined by $a_x(p):=a(p, x)$ and 246 $a_y(p):=a(p, y)$. The distance between these two points is then 247 \begin{align} 248 d_{D_F\otimes 1}(n_1, n_2) = \sup \left\{ |a(n_1) - a(n_2)|: a\in 249 A, ||[D\otimes 1, a]||\right\} 250 \end{align} 251 \textbf{Remark}: If $n_1 = (p,x)$ and $n_2 = (q, x)$ for $p,q \in M$ then 252 \begin{align} 253 d_{D_M \otimes 1} (n_1, n_2) = |a_x(p) - a_x(q)| \;\;\; a_x\in 254 C^\infty(M) \;\; \text{with} \;\; ||[D\otimes 1, a_x]|| \leq 1 255 \end{align} 256 The distance turns to the geodestic distance formula 257 \begin{align} 258 d_{D_M\otimes1}(n_1, n_2) = d_g(p, q) 259 \end{align} 260 261 However if $n_1 = (p, x)$ and $n_2 = (q, y)$ then the two conditions are 262 $||[D_M, a_x]|| \leq 1$ and $||[D_M, a_y|| \leq 1$. They have no 263 restriction which results in the distance being infinite! And $N = 264 M\times X$ is given by two disjoint copies of M which are separated by 265 infinite distance 266 267 \textbf{Note}: distance is only finite if $[D_F, a] \neq 1$. The 268 commutator 269 generates a scalar field say $\phi$ and the finiteness of the distance is 270 related to the existence of scalar fields. 271 \subsection{$U(1)$ Gauge Group} 272 Here we determine the Gauge theory corresponding to the almost 273 commutative 274 Manifold $M\times F_x$. 275 276 \textbf{Gauge Group of a Spectral Triple}: 277 \begin{align} 278 \mathfrak{B}(A, H; J) := \{ U = uJuJ^{-1} | u\in U(A)\} 279 \end{align} 280 \begin{definition} 281 A *-automorphism of a *-algebra $A$ is a linear invertible 282 map 283 \begin{align} 284 &\alpha:A \rightarrow A\;\;\; \text{with}\\ 285 \nonumber\\ 286 &\alpha(ab) = \alpha(a)\alpha(b)\\ 287 &\alpha(a)^* = \alpha(a^*) 288 \end{align} 289 The \textbf{Group of automorphisms of the *-Algebra $A$} is 290 $(A)$.\newline 291 The automorphism $\alpha$ is called \textbf{inner} if 292 \begin{align} 293 \alpha(a) = u a u^* \;\;\; \text{for} \;\; U(A) 294 \end{align} 295 where $U(A)$ is 296 \begin{align} 297 U(A) = \{ u\in A|\;\; uu^* = u^*u=1\} \;\;\; 298 \text{(unitary)} 299 \end{align} 300 \end{definition} 301 The Gauge group is given by the quotient $U(A)/U(A_J)$. 302 We want a nontrivial Gauge group so we need to choose $U(A_J) \neq 303 U(A)$ which is the same as $U((A_F)_{J_F}) \neq 304 U(A_F)$. 305 We consider $F_x$ to be 306 \begin{align} 307 F_x := \left(\mathbb{C}^2,\mathbb{C}^2, D_F =\begin{pmatrix} 308 0&0\\0&0\end{pmatrix}; J_f =\begin{pmatrix} 309 0&C\\C&0\end{pmatrix}, 310 \gamma_F = \begin{pmatrix}1&0\\0&-1\end{pmatrix}\right). 311 \end{align} 312 Here $C$ is the complex conjugation, and $F_X$ is a real even finite 313 spectral triple (space) with $KO-dim=6$ 314 315 \begin{proposition} 316 The Gauge group $\mathfrak{B}(F)$ of the two point space is given by 317 $U(1)$. 318 \end{proposition} 319 \begin{proof} 320 Note that $U(A_F) = U(1) \times U(1)$. We need to show that 321 $U(\mathcal{A}_F) 322 \cap U(A_F)_{J_F}) \simeq U(1)$, such that $\mathfrak{B}(F) 323 \simeq U(1)$.\newline 324 325 So for $a \in \mathbb{C}^2$ to be in $(A_F)_{J_F}$ it has 326 to satisfy $J_F a^* J_F = a$. 327 \begin{align} 328 J_F a^* J^{-1} = 329 \begin{pmatrix}0&C\\C&0\end{pmatrix} 330 \begin{pmatrix}\bar{a}_1&0\\0&\bar{a}_2\end{pmatrix} 331 \begin{pmatrix}0&C\\C&0\end{pmatrix} 332 = 333 \begin{pmatrix}a_2&0\\0&a_1\end{pmatrix} 334 \end{align} 335 Which is only the case if $a_1 = a_2$. So we have 336 $(A_F)_{J_F} \simeq \mathbb{C}$, whose unitary elements 337 from $U(1)$ are contained in the diagonal subgroup of 338 $U(\mathcal{A}_F)$. 339 \end{proof} 340 341 Now we need to find the exact from of the field $B_\mu$ to calculate the 342 spectral action of a spectral triple. Since $(A_F)_{J_F} \simeq 343 \mathbb{C}$ we find that $\mathfrak{h}(F) = \mathfrak{u}((A_F)_{J_F}) 344 \simeq i\mathbb{R}$. Where $\mathfrak{h}(F)$ is the Lie Algebra on $F$ 345 and $\mathfrak{u}((A_F)_{J_F})$ is the Lie algebra of the unitary group 346 $(A_F)_{J_F}$.\newline 347 348 An arbitrary hermitian field $A_\mu = -ia\partial _\mu b$ is given by 349 two 350 $U(1)$ Gauge fields $X_\mu^1, X_\mu^2 \in C^\infty(M, \mathbb{R})$. 351 However $A_\mu$ appears in combination $A_\mu - J_F A_\mu J_F^{-1}$: 352 \begin{align} 353 B_\mu = A_\mu - J_F A_\mu J_F^{-1} = 354 \begin{pmatrix}X_\mu^1&0\\0&X_\mu^2 \end{pmatrix} 355 - 356 \begin{pmatrix}X_\mu^2&0\\0&X_\mu^1 \end{pmatrix} 357 =: 358 \begin{pmatrix}Y_\mu&0\\0&-Y_\mu \end{pmatrix}= 359 Y_\mu \otimes \gamma _F 360 \end{align} 361 where $Y_\mu$ the $U(1)$ Gauge field is defined as 362 \begin{align} 363 Y_\mu := X_\mu^1 - X_\mu^2 \in C^\infty(M, \mathbb{R}) = C^\infty(M, 364 i\ u(1)). 365 \end{align} 366 367 \begin{proposition} 368 The inner fluctuations of the almost-commutative manifold $M\times 369 F_x$ described above are parametrized by a $U(1)$-gauge field $Y_\mu$ 370 as 371 \begin{align} 372 D \mapsto D' = D + \gamma ^\mu Y_\mu \otimes \gamma_F 373 \end{align} 374 The action of the gauge group $\mathfrak{B}(M\times F_X) \simeq 375 C^\infty (M, U(1))$ on $D'$ is implemented by 376 \begin{align} 377 Y_\mu \mapsto Y_\mu - i\ u\partial_\mu u^*; \;\;\;\;\; (u\in 378 \mathfrak{B}(M\times F_X)). 379 \end{align} 380 \end{proposition} 381 382 \section{Electrodynamics} 383 Now we use the almost commutative Manifold and the abelian gauge group 384 $U(1)$ to describe Electrodynamics. We arrive at a unified description of 385 gravity and electrodynamics although in the classical level. 386 \newline 387 388 The almost commutative Manifold $M\times F_X$ describes a local gauge group 389 $U(1)$. The inner fluctuations of the Dirac operator describe $Y_\mu$ the 390 gauge field of $U(1)$. There arise two Problems: 391 \newline 392 (1): With $F_X$, $D_F$ must vanish, however this implies that the electrons 393 are massless (this we do not want) 394 \newline 395 396 (2): The Euclidean action for a free Dirac field is 397 \begin{align} 398 S = - \int i \bar{\psi}(\gamma ^\mu\partial _\mu - m) \psi d^4x, 399 \end{align} 400 $\psi,\ \bar{\psi}$ must be considered as independent variables, which means 401 $S_F$ need two independent Dirac Spinors. We write $\{e, \bar{e}\}$ for the 402 ONB of $H_F$, where $\{e\}$ is the ONB of $H_F^+$ and $\{\bar{e}\}$ the ONB 403 of $H_F^-$ with the real structure this gives us the following relations 404 \begin{align} 405 J_F e &= \bar{e} \;\;\;\;\;\; J_F \bar{e} = e \\ 406 \gamma_F e &= e \;\;\;\;\;\; \gamma_F \bar{e} = \bar{e}. 407 \end{align} 408 The total Hilbertspace is $H = L^2(S) \otimes H_F$, with $\gamma _F$ we can 409 decompose $L^2(S) = L^2(S)^+ \oplus L^2(S)^-$, so with $\gamma = \gamma _M 410 \otimes \gamma _F$ we can obtain the positive eigenspace $H^+$ 411 \begin{align} 412 H^+ = L^2(S)^+ \otimes H_F^+ \oplus L^(S)^- \otimes H_F^-. 413 \end{align} 414 For a $\xi \i H^+$ we can write 415 \begin{align} 416 \xi = \psi _L \otimes e + \psi _R \otimes \bar{e} 417 \end{align} 418 where $\psi _L \in L^2(S)^+$ and $\psi _R \in L^2(S)^-$ are the two Wheyl 419 spinors. We denote that $\xi$ is only determined by one Dirac spinor $\psi := 420 \psi_L + \psi _R$, \textbf{but we require two independent spinors}. This is 421 too much restriction for $F_X$. 422 \subsection{The Finite Space} 423 Here we solve the two problems by enlarging(doubling) the Hilbertspace. This 424 is done by introducing multiplicities in Krajewski Diagrams which will also 425 allow us to choose a nonzero Dirac operator which will connect the two 426 vertices (next chapter). 427 \newline 428 429 We start of with the same algebra $C^\infty(M, \mathbb{C}^2)$, corresponding 430 to space $N= M\times X \simeq M\sqcup M$. 431 \newline 432 433 The Hilbertspace will describe four particles, 434 \begin{itemize} 435 \item left handed electrons 436 \item right handed positrons 437 \end{itemize} 438 Thus we have $\{ \underbrace{e_R, e_L}_{\text{left-handed}}, 439 \underbrace{\bar{e}_R, \bar{e}_L}_{\text{right-handed}}\}$ the ONB for $H_F 440 \mathbb{C}^4$. 441 \newline 442 Then with $J_F$ we interchange particles with antiparticles we have the 443 following properties 444 \begin{align} 445 &J_F e_R = \bar{e}_R \;\;\;\;\; &J_F e_L = \bar{e_L} \\ 446 &\gamma _F e_R = -e_R \;\;\;\;\; &\gamma_F e_L = e_L \\ 447 \text{and}& \nonumber \\ 448 &J_F^2 = 1 \;\;\;\;\; & J_F \gamma_F = - \gamma_F J_F 449 \end{align} 450 This corresponds to KO-dim$= 6$. Then $\gamma_F$ allows us to can decompose 451 $H$ 452 \begin{align} 453 H_F = \underbrace{H_F^+}_{\text{ONB } \{e_L, \bar{e}_L\}} 454 \oplus \underbrace{H_F^-}_{\text{ONB } \{e_R, \bar{e}_R\}}. 455 \end{align} 456 Alternatively we can decompose $H$ into the eigenspace of particles and their 457 antiparticles (electrons and positrons) which we will use going further. 458 \begin{align} 459 H_F = \underbrace{H_{e}}_{\text{ONB } \{e_L, e_R\}} \oplus 460 \underbrace{H_{\bar{e}}}_{\text{ONB } \{\bar{e}_L, \bar{e}_R\}} 461 \end{align} 462 Now the action of $a \in A = \mathbb{C}^2$ on $H$ with respect to the ONB 463 $\{e_L, e_R, \bar{e}_L, \bar{e}_R\}$ is represented by 464 \begin{align} 465 a = 466 \begin{pmatrix}a_1 & a_2 \end{pmatrix} \mapsto 467 \begin{pmatrix} 468 a_1 &0 &0 &0\\ 469 0&a_1 &0 &0\\ 470 0 &0 &a_2 &0\\ 471 0 &0 &0 &a_2\\ 472 \end{pmatrix} 473 \end{align} 474 Do note that this action commutes wit the grading and that 475 $[a, b^\circ] = 0$ with $b:= J_F b^*J_F$ because both the left and the right 476 action is given by diagonal matrices. 477 \begin{proposition} 478 The data 479 \begin{align} 480 \left( \mathbb{C}^2, \mathbb{C}^2, D_F=0; J_F = 481 \begin{pmatrix} 482 0 & C \\ C &0 483 \end{pmatrix}, 484 \gamma _F = 485 \begin{pmatrix} 486 1 & 0 \\ 0 &-1 487 \end{pmatrix} 488 \right) 489 \end{align} 490 defines a real even spectral triple of KO-dimension 6. 491 \end{proposition} 492 This spectral triple can be represented in the following Krajewski diagram, 493 with two nodes of multiplicity two 494 \begin{figure}[h!] \centering 495 \begin{tikzpicture}[ 496 dot/.style = {draw, circle, inner sep=0.06cm}, 497 bigdot/.style = {draw, circle, inner sep=0.09cm}, 498 no/.style = {}, 499 ] 500 \node[no](a) at (0,0) [label=left:$\textbf{1}^\circ$] {}; 501 \node[no](b) at (0, -1) [label=left:$\textbf{1}^\circ$] {}; 502 \node[no](c) at (0.5, 0.5) [label=above:$\textbf{1}$] {}; 503 \node[no](d) at (1.5, 0.5) [label=above:$\textbf{1}$] {}; 504 \node[dot](d0) at (1.5,0) [] {}; 505 \node[dot](d0) at (0.5,-1) [] {}; 506 \node[bigdot](d0) at (1.5,0) [] {}; 507 \node[bigdot](d0) at (0.5,-1) [] {}; 508 \end{tikzpicture} 509 \end{figure} 510 \subsection{A noncommutative Finite Dirac Operator} 511 Add a non-zero Dirac Operator to $F_{ED}$. From the Krajewski Diagram, we see 512 that edges only exist between the multiple vertices. So we construct a Dirac 513 operator mapping between the two vertices. 514 \begin{align}\label{dirac} 515 D_F = 516 \begin{pmatrix} 517 0 & d & 0 & 0 \\ 518 \bar{d} & 0 & 0 & 0 \\ 519 0 & 0 & 0 & \bar{d} \\ 520 0 & 0 & d & 0 521 \end{pmatrix} 522 \end{align} 523 We can now consider the finite space $F_{ED}$. 524 \begin{align} 525 F_{ED} := (\mathbb{C}^2, \mathbb{C}^4, D_F; J_F, \gamma_F) 526 \end{align} 527 where $J_F$ and $\gamma_F$ like before, $D_F$ like above. 528 \subsection{The almost-commutative Manifold} 529 The almost commutative manifold $M\times F_{ED}$ has KO-dim$=2$, it is the 530 following spectral triple 531 \begin{align} 532 M\times F_{ED} := \left(C^\infty(M,\mathbb{C}^2),\ L^2(S)\otimes 533 \mathbb{C}^4,\ 534 D_M\otimes 1 +\gamma _M \otimes D_F;\; J_M\otimes J_F,\ \gamma_M\otimes 535 \gamma _F\right) 536 \end{align} 537 538 The algebra decomposition is like before 539 \begin{align} 540 C^\infty(M, \mathbb{C}^2) = C^\infty (M) \oplus C^\infty (M) 541 \end{align} 542 543 The Hilbertspace decomposition is 544 \begin{align} 545 H = (L^2(S) \otimes H_e ) \oplus (L^2(S) \otimes H_{\bar{e}}). 546 \end{align} 547 Here we have the one component of the algebra acting on $L^2(S) \otimes H_e$, 548 and the other one acting on $L^2(S) \otimes H_{\bar{e}}$ 549 \newline 550 551 The derivation of the gauge theory is the same for $F_{ED}$ as for $F_X$, we 552 have $\mathfrak{B}(F) \simeq U(1)$ and for $B_\mu = A_\mu - J_F A_\mu 553 J_F^{-1}$ 554 \begin{align} \label{field} 555 B_\mu = 556 \begin{pmatrix} 557 Y_\mu & 0 & 0 & 0 \\ 558 0 & Y_\mu& 0 & 0 \\ 559 0 & 0 & Y_\mu& 0 \\ 560 0 & 0 & 0 & Y_\mu 561 \end{pmatrix} \;\;\;\;\;\ \text{for} \;\;\ Y_\mu (x) \in \mathbb{R}. 562 \end{align} 563 We have one single $U(1)$ gauge field $Y_\mu$, carrying the action of the 564 gauge group 565 \begin{align} 566 \text{$\mathfrak{B}$}(M\times F_{ED}) \simeq C^\infty(M, U(1)) 567 \end{align} 568 569 Our space $N = M\times X \simeq M\sqcup M$ consists of two compies of $M$. 570 If $D_F = 0$ we have infinite distance between the two copies. Now we have $D_F$ 571 nonzero but $[D_F, a] = 0$ $\forall a \in A$ which still yields infinite 572 distance. 573 \begin{question} 574 What does this imply (physically, mathematically)? Why can we continue 575 even thought we have infinite distance between the same manifold? What do 576 we get if we fix this? 577 \end{question} 578 \subsection{The Spectral Action} 579 Here we calculate the Lagrangian of the almost commutative Manifold $M\times 580 F_{ED}$, which corresponds to the Lagrangian of Electrodynamics on a curved 581 background Manifold (+ gravitational Lagrangian). It consists of the spectral 582 action $S_b$ (bosonic) and of the fermionic action $S_f$. 583 584 The simples spectral action of a spectral triple $(A, H, D)$ is given by the 585 trace of some function of $D$, we also allow inner fluctuations of the Dirac 586 operator $D_\omega = D + \omega + \varepsilon' J\omega J^{-1}$ where $\omega = 587 \omega ^* \in \Omega_D^1(A)$. 588 \begin{definition} 589 Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a suitable function 590 \textbf{positive and even}. The spectral action is then 591 \begin{align} 592 S_b [\omega] := \text{Tr}f(\frac{D_\omega}{\Lambda}) 593 \end{align} 594 where $\Lambda$ is a real cutoff parameter. The minimal condition on $f$ 595 is that $f(\frac{D_\omega}{\Lambda})$ is a traclass operator, which mean 596 that it should be compact operator with well defined finite trace 597 independent of the basis. The subscript $b$ of $S_b$ refers to bosonic, 598 because in physical applications $\omega$ will describe bosonic fields. 599 600 Furthermore there is a topological spectral action, defined with the 601 grading $\gamma$ 602 \begin{align} 603 S_{\text{top}}[\omega] := \text{Tr}(\gamma\ 604 f(\frac{D_\omega}{\Lambda})). 605 \end{align} 606 \end{definition} 607 \begin{definition} 608 The fermionic action is defined by 609 \begin{align} 610 S_f[\omega, \psi] = (J\tilde{\psi}, D_\omega \tilde{\psi}) 611 \end{align} 612 with $\tilde{\psi} \in H_{cl}^+ := \{\tilde{\psi}: \psi \in H^+\}$. 613 $H_{cl}^+$ is the set of Grassmann variables in $H$ in the +1-eigenspace 614 of the grading $\gamma$. 615 \end{definition} 616 The grasmann variables are a set of Basis vectors of a vector space, they 617 form a unital algebra over a vector field say $V$ where the generators are anti commuting, that is for 618 $\theta _i, \theta _j$ some Grassmann variables we have 619 \begin{align} 620 &\theta _i \theta _j = -\theta _j \theta _i \\ 621 &\theta _i x = x\theta _j \;\;\;\; x\in V \\ 622 &(\theta_i)^2 = 0 \;\;\; (\theta _i \theta _i = -\theta _i \theta _i) 623 \end{align} 624 \begin{proposition} 625 The spectral action of the almost commutative manifold $M$ with $\dim(M) 626 =4$ with a fluctuated Dirac operator is. 627 \begin{align} 628 \text{Tr}(f\frac{D_\omega}{\Lambda}) \sim \int_M \mathcal{L}(g_{\mu\nu}, 629 B_\mu, \Phi) \sqrt{g}\ d^4x + O(\Lambda^{-1}) 630 \end{align} 631 with 632 \begin{align} 633 \mathcal{L}(g_{\mu\nu}, B_\mu, \Phi) = 634 N\mathcal{L}_M(g_{\mu\nu}) 635 \mathcal{L}_B(B_\mu)+ 636 \mathcal{L}_\phi(g_{\mu\nu}, B_\mu, \Phi) 637 \end{align} 638 where $N=4$ and $\mathcal{L}_M$ is the Lagrangian of the spectral triple 639 $(C^\infty(M) , L^2(S), D_M)$ 640 \begin{align}\label{lagr} 641 \mathcal{L}_M(g_{\mu\nu}) := \frac{f_4 \Lambda ^4}{2\pi^2} - 642 \frac{f_2 \Lambda^2}{24\pi ^2}s - \frac{f(0)}{320\pi^2} C_{\mu\nu 643 \varrho \sigma}C^{\mu\nu \varrho \sigma}. 644 \end{align} 645 Here $C^{\mu\nu \varrho \sigma}$ is defined in terms of the Riemannian 646 curvature tensor $R_{\mu\nu \varrho \sigma}$ and the Ricci tensor 647 $R_{\nu\sigma} = g^{\mu\varrho} R_{\mu\nu \varrho\sigma}$. 648 649 650 Furthermore $\mathcal{L}_B$ describes the kinetic term of the gauge field 651 \begin{align} 652 \mathcal{L}_B(B_\mu) := \frac{f(0)}{24\pi^2} 653 \text{Tr}(F_{\mu\nu}F^{\mu\nu}). 654 \end{align} 655 Last $\mathcal{L}_\phi$ is the scalar-field Lagrangian with a boundary 656 term. 657 \begin{align} 658 \mathcal{L}_\phi(g_{\mu\nu}, B_\mu, \Phi) := 659 &-\frac{2f_2\Lambda^2}{4\pi^2}\text{Tr}(\Phi^2) + \frac{f(0)}{8\pi^2} 660 \text{Tr}(\Phi^4) + \frac{f(0)}{24\pi^2} \Delta(\text{Tr}(\Phi^2))\\ 661 &+ \frac{f(0)}{48\pi^2}s\text{Tr}(\Phi^2) 662 \frac{f(0)}{8\pi^2}\text{Tr}((D_\mu \Phi)(D^\mu \Phi)). 663 \end{align} 664 \end{proposition} 665 \begin{proof} 666 Will maybe be filled in if I go through the last two chapters in the 667 book and understand the proof. 668 \textbf{PROOF: in week10.pdf} 669 \end{proof} 670 671 Here on we go and calculate the spectral action of $M\times F_{ED}$ 672 \begin{proposition} 673 The Spectral action of $M\times F_{ED}$ is 674 \begin{align} 675 \text{Tr}(f\frac{D_\omega}{\Lambda}) \sim \int_M \mathcal{L}(g_{\mu\nu}, 676 Y_\mu) \sqrt{g}\ d^4x + O(\Lambda^{-1}) 677 \end{align} 678 where the Lagrangian is 679 \begin{align} 680 \mathcal{L}(g_{\mu\nu}, Y_\mu) = 681 4\mathcal{L}_M(g_{\mu\nu})+ 682 \mathcal{L}_Y(Y_\mu)+ 683 \mathcal{L}_\phi(g_{\mu\nu}, d) 684 \end{align} 685 here the $d$ in $\mathcal{L}_\phi$ is from $D_F$ in equation 686 \ref{dirac}. The Lagrangian $\mathcal{L}_M$ is like in equation 687 \ref{lagr}. The Lagrangian $\mathcal{L}_Y$ is the kinetic term of the 688 $U(1)$ gauge field $Y_\mu$ 689 \begin{align} 690 \mathcal{L}_Y(Y_\mu):= \frac{f(0)}{6\pi^2} 691 Y_{\mu\nu}Y^{\mu\nu}\;\;\;\;\;\;\;\;\text{with}\;\;\; Y_{\mu\nu} = 692 \partial_\mu Y_\nu - 693 \partial_\nu Y_\mu. 694 \end{align} 695 Then there is $\mathcal{L}_\phi$, which has two constant terms 696 (disregarding the boundary term) that add up to the Cosmological Constant 697 and a term that for the Einstein-Hilbert action 698 \begin{align} 699 \mathcal{L}_\phi(g_{\mu\nu}, d) := \frac{2f_2 \Lambda ^2}{\pi^2} 700 |d|^2 + \frac{f(0)}{2\pi^2} |d|^4 + \frac{f(0)}{12\pi ^2} s |d|^2. 701 \end{align} 702 \end{proposition} 703 \begin{proof} 704 The Trace of $\mathbb{C}^4$ (the Hilbertspace) gives $N=4$. With $B_\mu$ 705 like in equation \ref{field} we have $\text{Tr}(F_{\mu\nu} 706 F^{\mu\nu})=4Y_{\mu\nu}Y^{\mu\nu}$. This provides $\mathcal{L}_Y$. 707 Furthermore we have $\Phi^2 = D_F^2 = |d|^2$ and $\mathcal{L}_\phi$ only 708 give numerical contributions to the cosmological constant and the 709 Einstein-Hilbert action. 710 711 The proof is relying itself on just plugging the terms into the previous 712 proposition, for which I didn't write the proof for. 713 \end{proof} 714 \end{document}