6_electroncg.tex (21445B)
1 \subsection{Noncommutative Geometry of Electrodynamics\label{sec:6}} 2 In this chapter we go through a derivation Electrodynamics with 3 the almost commutative manifold $M\times F_X$ and the abelian gauge group 4 $U(1)$. The conclusion is an unified description of gravity and 5 electrodynamics although in the classical level. 6 7 The almost commutative Manifold $M\times F_X$ outlines a local gauge group 8 $U(1)$. The inner fluctuations of the Dirac operator relate to $Y_\mu$ the 9 gauge field of $U(1)$. According to the setup we ultimately arrive at two 10 serious problems. 11 12 First of all the operator $D_F$, in the Two-Point space $F_X$, must vanish 13 such that a real structure can exists. However this implies that the electrons 14 are massless. 15 16 The second problem arises when looking at the Euclidean action for a free 17 Dirac field 18 \begin{align} 19 S = - \int i \bar{\psi}(\gamma ^\mu\partial _\mu - m) \psi d^4x, 20 \end{align} 21 where $\psi,\ \bar{\psi}$ must be considered as two independent variables. 22 This means that the fermionic action $S_f$ needs two independent Dirac spinors. 23 Let us try and construct two independent Dirac spinors with our data, first 24 take a look at the decomposition of the basis and of the total 25 Hilbertspace $H = L^2(S) \otimes H_F$. For the orthonormal basis of $H_F$ we 26 can write $\{e, \bar{e}\}$ , where $\{e\}$ is the orthonormal basis of 27 $H_F^+$ and $\{\bar{e}\}$ the orthonormal basis of $H_F^-$. Accompanied with 28 the real structure we arrive at the following relations 29 \begin{align} 30 J_F e &= \bar{e} \;\;\;\;\;\; J_F \bar{e} = e, \\ 31 \gamma_F e &= e \;\;\;\;\;\; \gamma_F \bar{e} = \bar{e}. 32 \end{align} 33 Along with the decomposition of $L^2(S) = L^2(S)^+ \oplus L^2(S)^-$ and $\gamma = \gamma _M 34 \otimes \gamma _F$ we can obtain the positive eigenspace 35 \begin{align} 36 H^+ = L^2(S)^+ \otimes H_F^+ \oplus L(S)^- \otimes H_F^-. 37 \end{align} 38 So, for an $\xi \in H^+$ we can write 39 \begin{align} 40 \xi = \psi _L \otimes e + \psi _R \otimes \bar{e}, 41 \end{align} 42 where $\psi_L \in L^2(S)^+$ and $\psi _R \in L^2(S)^-$ are the two Wheyl 43 spinors. We denote that $\xi$ is only determined by one Dirac spinor $\psi := 44 \psi_L + \psi _R$. Since \textbf{we require two independent spinors}, our 45 conclusion is that the definition of the fermionic action gives too much 46 restrictions to the Two-Point space $F_X$. 47 \subsubsection{The Finite Space} 48 To solve the two problems we simply enlarge (double) the Hilbertspace. This 49 is visualized by introducing multiplicities in Krajewski Diagrams 50 \cite{ncgwalter} which will also allow us to choose a nonzero Dirac operator 51 that will connect the two vertices and preserve real structure making our 52 particles massive and bringing anti-particles into the mix. 53 54 We start of with the same algebra $C^\infty(M, \mathbb{C}^2)$, corresponding 55 to space $N= M\times X$. The Hilbertspace describes four particles, meaning 56 it has four orthonormal basis elements. It describes \textbf{left handed 57 electrons} and \textbf{right handed positrons}. This way we have 58 $\{ \underbrace{e_R, e_L}_{\text{left-handed}}, \underbrace{\bar{e}_R, 59 \bar{e}_L}_{\text{right-handed}}\}$ an orthonormal basis for $H_F = 60 \mathbb{C}^4$. Accompanied with the real structure $J_F$ allowing us to 61 interchange particles with antiparticles by the following equations 62 \begin{align} 63 &J_F e_R = \bar{e}_R, \\ 64 &J_F e_L = \bar{e_L}, \\ 65 \nonumber \\ 66 &\gamma _F e_R = -e_R,\\ 67 &\gamma_F e_L = e_L, 68 \end{align} 69 where $J_F$ and $\gamma_F$ have to following properties 70 \begin{align} 71 &J_F^2 = 1,\\ 72 & J_F \gamma_F = - \gamma_F J_F. 73 \end{align} 74 By the means of $\gamma_F$ we have two options to decompose the total 75 Hilbertspace $H$, firstly into 76 \begin{align} 77 H_F = \underbrace{H_F^+}_{\text{ONB } \{e_L, \bar{e}_L\}} 78 \oplus \underbrace{H_F^-}_{\text{ONB } \{e_R, \bar{e}_R\}}, 79 \end{align} 80 or alternatively into the eigenspace of particles and their 81 antiparticles (electrons and positrons) which is preferred in literature and 82 which will be used further out 83 \begin{align} 84 H_F = \underbrace{H_{e}}_{\text{ONB } \{e_L, e_R\}} \oplus 85 \underbrace{H_{\bar{e}}}_{\text{ONB } \{\bar{e}_L, \bar{e}_R\}}, 86 \end{align} 87 the shortening `ONB' means orthonormal basis. 88 89 The action of $a \in A = \mathbb{C}^2$ on $H$ with respect to the ONB 90 $\{e_L, e_R, \bar{e}_L, \bar{e}_R\}$ is represented by 91 \begin{align}\label{eq:leftrightrepr} 92 a = 93 (a_1 , a_2 ) \mapsto 94 \begin{pmatrix} 95 a_1 &0 &0 &0\\ 96 0&a_1 &0 &0\\ 97 0 &0 &a_2 &0\\ 98 0 &0 &0 &a_2\\ 99 \end{pmatrix} 100 \end{align} 101 Do note that this action commutes wit the grading and that $[a, b^\circ] = 0$ 102 with $b:= J_F b^*J_F$ because both the left and the right action are given by 103 diagonal matrices according to equation \eqref{eq:leftrightrepr}. Furthermore 104 note that we are still left with $D_F = 0$ and the following spectral triple 105 \begin{align}\label{eq:fedfail} 106 \left( \mathbb{C}^2, \mathbb{C}^2, D_F=0; J_F = 107 \begin{pmatrix} 108 0 & C \\ C &0 109 \end{pmatrix}, 110 \gamma _F = 111 \begin{pmatrix} 112 1 & 0 \\ 0 &-1 113 \end{pmatrix} 114 \right). 115 \end{align} 116 It can be represented in the following Krajewski diagram \cite{ncgwatler}, 117 with two nodes of multiplicity two, in figure \ref{fig:krajewski} bellow. 118 \begin{figure}[H] \centering 119 \begin{tikzpicture}[ 120 dot/.style = {draw, circle, inner sep=0.06cm}, 121 bigdot/.style = {draw, circle, inner sep=0.09cm}, 122 no/.style = {}, 123 ] 124 \node[no](a) at (0,0) [label=left:$\textbf{1}^\circ$] {}; 125 \node[no](b) at (0, -1) [label=left:$\textbf{1}^\circ$] {}; 126 \node[no](c) at (0.5, 0.5) [label=above:$\textbf{1}$] {}; 127 \node[no](d) at (1.5, 0.5) [label=above:$\textbf{1}$] {}; 128 \node[dot](d0) at (1.5,0) [] {}; 129 \node[dot](d0) at (0.5,-1) [] {}; 130 \node[bigdot](d0) at (1.5,0) [] {}; 131 \node[bigdot](d0) at (0.5,-1) [] {}; 132 \end{tikzpicture} 133 \caption{Krajewski diagram of the spectral triple from equation 134 \ref{eq:fedfail} 135 \label{fig:krajewski} 136 } 137 \end{figure} 138 \subsubsection{A noncommutative Finite Dirac Operator} 139 To extend our spectral triple with a non-zero Operator, we need to take a 140 closer look at the Krajewski diagram in figure \ref{fig:krajewski} above. Notice that edges only exist 141 between multiple vertices, meaning we can construct a Dirac operator mapping 142 between the two vertices. The operator can be represented by the following matrix 143 \begin{align}\label{eq:feddirac} 144 D_F = 145 \begin{pmatrix} 146 0 & d & 0 & 0 \\ 147 \bar{d} & 0 & 0 & 0 \\ 148 0 & 0 & 0 & \bar{d} \\ 149 0 & 0 & d & 0 150 \end{pmatrix} 151 \end{align} 152 We can now define the finite space $F_{ED}$. 153 \begin{align} 154 F_{ED} := (\mathbb{C}^2, \mathbb{C}^4, D_F; J_F, \gamma_F) 155 \end{align} 156 where $J_F$ and $\gamma_F$ are as in equation \eqref{eq:fedfail} and $D_F$ 157 from equation \eqref{eq:feddirac}. 158 159 \subsubsection{Almost commutative Manifold of Electrodynamics} 160 The almost commutative manifold $M\times F_{ED}$ has KO-dimension 2, and is 161 represented by the following spectral triple 162 \begin{align}\label{eq:almost commutative manifold} 163 M\times F_{ED} := \big(C^\infty(M,\mathbb{C}^2),\ L^2(S)\otimes 164 \mathbb{C}^4,\ 165 D_M\otimes 1 +\gamma _M \otimes D_F;\; J_M\otimes J_F,\ \gamma_M\otimes 166 \gamma _F\big). 167 \end{align} 168 The algebra didn't change, thus we can decompose it like before 169 \begin{align} 170 C^\infty(M, \mathbb{C}^2) = C^\infty (M) \oplus C^\infty (M). 171 \end{align} 172 As for the Hilbertspace, we can decomposition it in the following way 173 \begin{align} 174 H = (L^2(S) \otimes H_e ) \oplus (L^2(S) \otimes H_{\bar{e}}). 175 \end{align} 176 Note that the one component of the algebra is acting on $L^2(S) \otimes H_e$, 177 and the other one acting on $L^2(S) \otimes H_{\bar{e}}$. In other words the components of 178 the decomposition of both the algebra and the Hilbertspace match by the action of 179 the algebra. 180 181 The derivation of the gauge theory is the same for $F_{ED}$ as for the 182 Two-Point space $F_X$. We have $\mathfrak{B}(F) \simeq U(1)$ and for an 183 arbitrary gauge field $B_\mu = A_\mu - J_F A_\mu J_F^{-1}$ we can write 184 \begin{align} \label{field} 185 B_\mu = 186 \begin{pmatrix} 187 Y_\mu & 0 & 0 & 0 \\ 188 0 & Y_\mu& 0 & 0 \\ 189 0 & 0 & Y_\mu& 0 \\ 190 0 & 0 & 0 & Y_\mu 191 \end{pmatrix} \;\;\;\;\;\ \text{for} \;\;\ Y_\mu (x) \in \mathbb{R}. 192 \end{align} 193 There is one single $U(1)$ gauge field $Y_\mu$, carrying the action of the 194 gauge group 195 \begin{align} 196 \text{$\mathfrak{B}$}(M\times F_{ED}) \simeq C^\infty(M, U(1)) 197 \end{align} 198 199 The space $N = M\times X$ consists of two copies of $M$. 200 If $D_F = 0$ we have infinite distance between the two copies, yet now we have 201 adjusted the spectral triple to have a nonzero Dirac operator. The new 202 Dirac operator still has a commuting relation with the algebra $[D_F, a] = 0$ 203 $\forall a \in A$, and we should note that the distance between the two 204 copies of $M$ is still infinite. This is purely an mathematically abstract 205 observation and doesn't affect physical results. 206 207 \subsubsection{Spectral Action} 208 In this chapter we bring all our results together to establish an 209 Action functional to describe a physical system. It turns out that 210 the Lagrangian of the almost commutative manifold $M\times F_{ED}$ 211 corresponds to the Lagrangian of Electrodynamics on a curved 212 background manifold (+ gravitational Lagrangian), consisting of the spectral 213 action $S_b$ (bosonic) and of the fermionic action $S_f$. 214 215 The simplest spectral action of a spectral triple $(A, H, D)$ is given by the 216 trace of a function of $D$. We also consider inner fluctuations of the Dirac 217 operator 218 \begin{align} 219 D_\omega = D + \omega + \varepsilon' J\omega J^{-1}, 220 \end{align} 221 where $\omega = \omega ^* \in \Omega_D^1(A)$. 222 \begin{mydefinition} 223 Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a suitable function 224 \textbf{positive and even}. The spectral action is then 225 \begin{align} 226 S_b [\omega] := \text{Tr}\big(f(\frac{D_\omega}{\Lambda})\big) 227 \end{align} 228 where $\Lambda$ is a real cutoff parameter. The minimal condition on $f$ 229 is that $f(\frac{D_\omega}{\Lambda})$ is a trace class operator. A trace 230 class operator is a compact operator with a well defined finite trace 231 independent of the basis. The subscript $b$ in $S_b$ stands for bosonic, 232 because in physical applications $\omega$ will describe bosonic fields. 233 234 In addition to the bosonic action $S_b$, we can define a topological spectral 235 action $S_{top}$. Leaning on the grading $\gamma$ the topological spectral action is 236 \begin{align} 237 S_{\text{top}}[\omega] := \text{Tr}(\gamma\ 238 f(\frac{D_\omega}{\Lambda})). 239 \end{align} 240 \end{mydefinition} 241 \begin{mydefinition}\label{def:fermionic action} 242 The fermionic action is defined by 243 \begin{align} 244 S_f[\omega, \psi] = (J\tilde{\psi}, D_\omega \tilde{\psi}) 245 \end{align} 246 with $\tilde{\psi} \in H_{cl}^+ := \{\tilde{\psi}: \psi \in H^+\}$, where 247 $H_{cl}^+$ is a set of Grassmann variables in $H$ in the $+1$-eigenspace 248 of the grading $\gamma$. 249 \end{mydefinition} 250 251 %---------------------- APPENDIX ?????????????-------------------- 252 Grassmann variables are a set of Basis vectors of a vector space, they 253 form a unital algebra over a vector field $V$, where the generators are 254 anti commuting, that is for Grassmann variables $\theta _i, \theta _j$ we have 255 \begin{align} 256 &\theta _i \theta _j = -\theta _j \theta _i, \\ 257 &\theta _i x = x\theta _j \;\;\;\; x\in V, \\ 258 &(\theta_i)^2 = 0 \;\;\; (\theta _i \theta _i = -\theta _i \theta _i). 259 \end{align} 260 %---------------------- APPENDIX ?????????????-------------------- 261 \begin{myproposition} 262 The spectral action of the almost commutative manifold $M$ with $\dim(M) 263 =4$ with a fluctuated Dirac operator is 264 \begin{align} 265 \text{Tr}(f\frac{D_\omega}{\Lambda}) \sim \int_M \mathcal{L}(g_{\mu\nu}, 266 B_\mu, \Phi) \sqrt{g}\ d^4x + O(\Lambda^{-1}), 267 \end{align} 268 where 269 \begin{align} 270 \mathcal{L}(g_{\mu\nu}, B_\mu, \Phi) = 271 N\mathcal{L}_M(g_{\mu\nu}) 272 \mathcal{L}_B(B_\mu)+ 273 \mathcal{L}_\phi(g_{\mu\nu}, B_\mu, \Phi). 274 \end{align} 275 The Lagrangian $\mathcal{L}_M$ is of the spectral triple $(C^\infty(M) , 276 L^2(S), D_M)$, represented by the following term 277 \begin{align}\label{lagr} 278 \mathcal{L}_M(g_{\mu\nu}) := \frac{f_4 \Lambda ^4}{2\pi^2} - 279 \frac{f_2 \Lambda^2}{24\pi ^2}s - \frac{f(0)}{320\pi^2} C_{\mu\nu 280 \varrho \sigma}C^{\mu\nu \varrho \sigma}, 281 \end{align} 282 where $C^{\mu\nu \varrho \sigma}$ is the Weyl tensor defined in terms of the Riemannian 283 curvature tensor $R_{\mu\nu \varrho \sigma}$ and the Ricci tensor 284 $R_{\nu\sigma} = g^{\mu\varrho} R_{\mu\nu \varrho\sigma}$ such that 285 \begin{align} 286 C^{\mu\nu\varrho\sigma}C_{\mu\nu\varrho\sigma}= 287 R_{\mu\nu\varrho\sigma}R^{\mu\nu\varrho\sigma} - 288 2R_{\nu\sigma}R^{\nu\sigma} + \frac{1}{2}s^2. 289 \end{align} 290 The kinetic term of the gauge field is described by the Lagrangian 291 $\mathcal{L}_B$, which takes the following shape 292 \begin{align} 293 \mathcal{L}_B(B_\mu) := \frac{f(0)}{24\pi^2} 294 \text{Tr}(F_{\mu\nu}F^{\mu\nu}). 295 \end{align} 296 Lastly $\mathcal{L}_\phi$ is the scalar-field Lagrangian with a boundary 297 term, given by 298 \begin{align} 299 \mathcal{L}_\phi(g_{\mu\nu}, B_\mu, \Phi) := 300 &-\frac{2f_2\Lambda^2}{4\pi^2}\text{Tr}(\Phi^2) + \frac{f(0)}{8\pi^2} 301 \text{Tr}(\Phi^4) + \frac{f(0)}{24\pi^2} 302 \Delta(\text{Tr}(\Phi^2))\nonumber\\ 303 &+ \frac{f(0)}{48\pi^2}s\text{Tr}(\Phi^2) 304 \frac{f(0)}{8\pi^2}\text{Tr}((D_\mu \Phi)(D^\mu \Phi)). 305 \end{align} 306 \end{myproposition} 307 \begin{proof} 308 The dimension of the manifold $M$ is $\dim(M) = \text{Tr}(id) =4$. For 309 an $x \in M$, we have an asymptotic expansion of the term 310 $\text{Tr}(f(\frac{D_\omega}{\Lambda}))$ as $\Lambda$ goes to infinity, 311 which can be written as 312 \begin{align} 313 \text{Tr}(f(\frac{D_\omega}{\Lambda})) \simeq& \ 2f_4 \Lambda ^4 314 a_0(D_\omega ^2)+ 2f_2\Lambda^2 a_2(D_\omega^2)\nonumber \\&+ f(0) a_4(D_\omega^4) 315 +O(\Lambda^{-1}).\label{eq:trheatkernel} 316 \end{align} 317 We have to note here that the heat kernel coefficients are zero for uneven $k$, 318 and they are dependent on the fluctuated Dirac operator 319 $D_\omega$. We can rewrite the heat kernel coefficients in terms of $D_M$, 320 for the first two terms $a_0$ and $a_2$ we use $N:= 321 \text{Tr}(\mathbbm{1}_{H_F})$ and one obtains 322 \begin{align} 323 a_0(D_\omega^2) &= Na_0(D_M^2),\\ 324 a_2(D_\omega^2) &= Na_2(D_M^2) - \frac{1}{4\pi^2}\int_M 325 \text{Tr}(\Phi^2)\sqrt{g}d^4x. 326 \end{align} 327 For $a_4$ we extend in terms of coefficients of $F$ from equation 328 \eqref{eq: a_4} 329 \begin{align} 330 &\frac{1}{360}\text{Tr}(60RE)= -\frac{1}{6}S(NR + 4 331 \text{Tr}(\Phi^2))\\ 332 \nonumber\\ 333 &E^2 = \frac{1}{16}R^2\otimes 1 + 1\otimes \Phi^4 - \frac{1}{4} 334 \gamma^\mu\gamma^\nu \gamma^\varrho\gamma^\sigma 335 F_{\mu\nu}F^{\mu\nu}+\nonumber\\ 336 &\;\;\;\;\;\;\;+\gamma^\mu\gamma^\nu\otimes(D_\mu\Phi)(D_\nu 337 \Phi)+\frac{1}{2}s\otimes \Phi^2 + \ \text{traceless terms},\\ 338 \nonumber\\ 339 &\frac{1}{360}\text{Tr}(180E^2) = \frac{1}{8}R^2N + 2\text{Tr}(\Phi^4) 340 + \text{Tr}(F_{\mu\nu}F^{\mu\nu}) +\nonumber\\ 341 &\;\;\;\;\;\;\;+2\text{Tr}((D_\mu\Phi)(D^\mu\Phi)) 342 + s\text{Tr}(\Phi^2)\\ 343 \nonumber\\ 344 &\frac{1}{360}\text{Tr}(-60\Delta E)= 345 \frac{1}{6}\Delta(NR+4\text{Tr}(\Phi^2)). 346 \end{align} 347 The cross terms of the trace in $\Omega_{\mu\nu}^E\Omega^{E\mu\nu}$ 348 vanishes because of the antisymmetric property of the Riemannian 349 curvature tensor, reading 350 \begin{align} 351 \Omega_{\mu\nu}^E\Omega^{E\mu\nu} = \Omega_{\mu\nu}^S\Omega^{S\mu\nu} 352 \otimes 1 - 1\otimes F_{\mu\nu}F^{\mu\nu} + 2i\Omega_{\mu\nu}^S 353 \otimes F^{\mu\nu}. 354 \end{align} 355 The trace of the cross term $\Omega^{S}_{\mu\nu}$ vanishes because 356 \begin{align} 357 \text{Tr}(\Omega^{S}_{\mu\nu}) = \frac{1}{4} 358 R_{\mu\nu\varrho\sigma}\text{Tr}(\gamma^\mu\gamma^\nu) = \frac{1}{4} 359 R_{\mu\nu\varrho\sigma}g^{\mu\nu} =0, 360 \end{align} 361 then the trace of the whole term is given by 362 \begin{align} 363 \frac{1}{360}\text{Tr}(30\Omega^E_{\mu\nu}\Omega^{E\mu\nu}) = 364 \frac{N}{24}R_{\mu\nu\varrho\sigma}R^{\mu\nu\varrho\sigma} 365 -\frac{1}{3}\text{Tr}(F_{\mu\nu}F^{\mu\nu}). 366 \end{align} 367 Finally plugging the results into the coefficient $a_4$ and simplifying 368 one gets 369 \begin{align} 370 a_4(x, D_\omega^4) &= Na_4(x, D_M^2) + \frac{1}{4\pi^2}\bigg(\frac{1}{12} s 371 \text{Tr}(\Phi^2) + \frac{1}{2}\text{Tr}(\Phi^4) \nonumber \\ 372 &+ \frac{1}{4} 373 \text{Tr}((D_\mu\Phi)(D^\mu \Phi)) + \frac{1}{6} 374 \Delta\text{Tr}(\Phi^2) + \frac{1}{6} 375 \text{Tr}(F_{\mu\nu}F^{\mu\nu})\bigg). 376 \end{align} 377 The only thing left is to substitute the heat kernel coefficients into the 378 heat kernel expansion in equation \eqref{eq:trheatkernel}. 379 \end{proof} 380 381 \subsubsection{Fermionic Action} 382 We remind ourselves the definition of the fermionic action in definition 383 \ref{def:fermionic action} and the manifold we are dealing with in equation 384 \eqref{eq:almost commutative manifold}. The Hilbertspace $H_F$ is separated 385 into the particle-antiparticle states with ONB $\{e_R, e_L, \bar{e}_R, 386 \bar{e}_L\}$. The orthonormal basis of $H_F^+$ is $\{e_L, \bar{e}_R\}$ and 387 consequently for $H_F^-$, $\{e_R, \bar{e}_L\}$. The decomposition of a spinor 388 $\psi \in L^2(S)$ in each of the eigenspaces $H_F^\pm$ is $\psi = \psi_R+ 389 \psi_L$. Meaning for an arbitrary $\psi \in H^+$ we can write 390 \begin{align} 391 \psi = \chi_R \otimes e_R + \chi_L \otimes e_L + \psi_L \otimes 392 \bar{e}_R+ 393 \psi_R \otimes \bar{e}_L, 394 \end{align} 395 where $\chi_L, \psi_L \in L^2(S)^+$ and $\chi_R, \psi_R \in L^2(S)^-$. 396 397 Since the fermionic action yields too much restriction on $F_{ED}$ (modified 398 Two-Point space $F_X$) one redefines it by taking into account the fluctuated Dirac 399 operator 400 \begin{align} 401 D_\omega = D_M \otimes i + \gamma^\mu \otimes B_\mu + \gamma_M \otimes 402 D_F. 403 \end{align} 404 The Fermionic Action is 405 \begin{align} 406 S_F = (J\tilde{\xi}, D_\omega\tilde{\xi}) 407 \end{align} 408 for a $\xi \in H^+$. Then the straight forward calculation gives \begin{align} 409 \frac{1}{2}(J\tilde{\xi}, D_\omega\tilde{\xi}) 410 &=\frac{1}{2}(J\tilde{\xi}, (D_M \otimes 411 i)\tilde{\xi})\label{eq:fermionic1}\\ 412 &+\frac{1}{2}(J\tilde{\xi}, (\gamma^\mu \otimes B_\mu) 413 \tilde{\xi})\label{eq:fermionic2}\\ 414 &+\frac{1}{2}(J\tilde{\xi}, (\gamma_M\otimes 415 D_F)\tilde{\xi})\label{eq:fermionic3}, 416 \end{align} 417 (note that we add the constant $\frac{1}{2}$ to the action). 418 For the term in \eqref{eq:fermionic1} we calculate 419 \begin{align} 420 \frac{1}{2}(J\tilde{\xi}, (D_M\otimes 1)\tilde{\xi}) &= 421 \frac{1}{2}(J_M\tilde{\chi}_R,D_M\tilde{\psi}_L)+\nonumber 422 \frac{1}{2}(J_M\tilde{\chi}_L,D_M\tilde{\psi}_R)+ 423 \\&+\frac{1}{2}(J_M\tilde{\psi}_L,D_M\tilde{\psi}_R)+\nonumber 424 \frac{1}{2}(J_M\tilde{\chi}_R,D_M\tilde{\chi}_L)\\ 425 &= (J_M\tilde{\chi},D_M\tilde{\chi}). 426 \end{align} 427 For the term in \eqref{eq:fermionic2} we have 428 \begin{align} 429 \frac{1}{2}(J\tilde{\xi}, (\gamma^\mu \otimes B_\mu)\tilde{\xi})&= 430 -\frac{1}{2}(J_M\tilde{\chi}_R, \gamma^\mu Y_\mu\tilde{\psi}_R) 431 -\frac{1}{2}(J_M\tilde{\chi}_L, \gamma^\mu Y_\mu\tilde{\psi}_R)+\nonumber\\ 432 &+\frac{1}{2}(J_M\tilde{\psi}_L, \gamma^\mu Y_\mu\tilde{\chi}_R)+ 433 \frac{1}{2}(J_M\tilde{\psi}_R, \gamma^\mu Y_\mu\tilde{\chi}_L)=\nonumber\\ 434 &= -(J_M\tilde{\chi}, \gamma^\mu Y_\mu\tilde{\psi}). 435 \end{align} 436 And for \eqref{eq:fermionic3} we can write 437 \begin{align} 438 \frac{1}{2}(J\tilde{\xi}, (\gamma_M\otimes D_F)\tilde{\xi})&= 439 +\frac{1}{2}(J_M\tilde{\chi}_R, d\gamma_M\tilde{\chi}_R) 440 +\frac{1}{2}(J_M\tilde{\chi}_L, \bar{d}\gamma_M\tilde{\chi}_L)+\nonumber\\ 441 &+\frac{1}{2}(J_M\tilde{\chi}_L, \bar{d}\gamma_M\tilde{\chi}_L) 442 +\frac{1}{2}(J_M\tilde{\chi}_R, d\gamma_M\tilde{\chi}_R)=\nonumber\\ 443 &= i(J_M\tilde{\chi}, m\tilde{\psi}). 444 \end{align} 445 A small problem arises, we obtain a complex mass parameter $d$, but we can 446 write $d:=im$ for $m\in \mathbb{R}$, which stands for the real mass. 447 448 Finally the fermionic action of $M\times F_{ED}$ takes the form 449 \begin{align} 450 S_f = -i\big(J_M\tilde{\chi}, \gamma(\nabla^S_\mu - i\Gamma_\mu) 451 \tilde{\Psi}\big) + \big(S_M\tilde{\chi}_L, \bar{d}\tilde{\psi}_L\big) - 452 \big(J_M\tilde{\chi}_R, d \tilde{\psi}_R\big). 453 \end{align} 454 Ultimately we arrive at the full Lagrangian of the almost commutative 455 manifold $M\times F_{ED}$, which is the sum of the purely gravitational 456 Lagrangian 457 \begin{align}\label{eq:final1} 458 \mathcal{L}_{grav}(g_{\mu\nu})=4\mathcal{L}_M(g_{\mu\nu})+ 459 \mathcal{L}_\phi (g_{\mu\nu}), 460 \end{align} 461 and the Lagrangian of electrodynamics 462 \begin{align}\label{eq:final2} 463 \mathcal{L}_{ED} = -i\bigg\langle 464 J_M\tilde{\chi},\big(\gamma^\mu(\nabla^S_\mu - iY_\mu) -m\big)\tilde{\psi}) 465 \bigg\rangle 466 +\frac{f(0)}{6\pi^2} Y_{\mu\nu}Y^{\mu\nu}. 467 \end{align} 468