commit bd23e8a8ad7160a22748ebfc347dfa3fab149c5b
parent cd1bb3dff4c8467640610d2da5c80cc4a4ec0d07
Author: miksa234 <milutin@popovic.xyz>
Date: Mon, 9 Aug 2021 16:32:59 +0200
checkpoint 6/6, nextup introduction
Diffstat:
6 files changed, 116 insertions(+), 73 deletions(-)
diff --git a/src/thesis/back/packages.tex b/src/thesis/back/packages.tex
@@ -68,6 +68,11 @@
([xshift=0.6cm, yshift=-0.5pt]frame.south
west)--([xshift=0.6cm,yshift=-1pt]frame.north west)
--([xshift=0.6cm]frame.south west)--([xshift=-13cm]frame.south east); },
+ overlay first={
+ \draw[colexam,line width=1pt]
+ ([xshift=0.6cm, yshift=-0.5pt]frame.south
+ west)--([xshift=0.6cm,yshift=-1pt]frame.north west)
+ --([xshift=0.6cm]frame.south west); },
overlay last={
\draw[colexam,line width=1pt]
([xshift=0.6cm, yshift=-0.5pt]frame.south
@@ -105,6 +110,11 @@
([xshift=0.6cm, yshift=-0.5pt]frame.south
west)--([xshift=0.6cm,yshift=-1pt]frame.north west)
--([xshift=0.6cm]frame.south west)--([xshift=-13cm]frame.south east); },
+ overlay first={
+ \draw[colexam,line width=1pt]
+ ([xshift=0.6cm, yshift=-0.5pt]frame.south
+ west)--([xshift=0.6cm,yshift=-1pt]frame.north west)
+ --([xshift=0.6cm]frame.south west); },
overlay last={
\draw[colexam,line width=1pt]
([xshift=0.6cm, yshift=-0.5pt]frame.south
@@ -142,6 +152,11 @@
([xshift=0.6cm, yshift=-0.5pt]frame.south
west)--([xshift=0.6cm,yshift=-1pt]frame.north west)
--([xshift=0.6cm]frame.south west)--([xshift=-13cm]frame.south east); },
+ overlay first={
+ \draw[colexam,line width=1pt]
+ ([xshift=0.6cm, yshift=-0.5pt]frame.south
+ west)--([xshift=0.6cm,yshift=-1pt]frame.north west)
+ --([xshift=0.6cm]frame.south west); },
overlay last={
\draw[colexam,line width=1pt]
([xshift=0.6cm, yshift=-0.5pt]frame.south
@@ -179,6 +194,11 @@
([xshift=0.6cm, yshift=-0.5pt]frame.south
west)--([xshift=0.6cm,yshift=-1pt]frame.north west)
--([xshift=0.6cm]frame.south west)--([xshift=-13cm]frame.south east); },
+ overlay first={
+ \draw[colexam,line width=1pt]
+ ([xshift=0.6cm, yshift=-0.5pt]frame.south
+ west)--([xshift=0.6cm,yshift=-1pt]frame.north west)
+ --([xshift=0.6cm]frame.south west); },
overlay last={
\draw[colexam,line width=1pt]
([xshift=0.6cm, yshift=-0.5pt]frame.south
diff --git a/src/thesis/chapters/basics.tex b/src/thesis/chapters/basics.tex
@@ -3,8 +3,8 @@
To grasp the idea of encoding geometrical data into a spectral triple we
introduce the first ingredient of a spectral triple, an unital $*$ algebra.
\begin{mydefinition}
- A \textit{vector space} $A$ over $\mathbb{C}$ is called a \textit{complex, unital Algebra} if, \\
- $\forall a,b \in A$ :
+ A \textit{vector space} $A$ over $\mathbb{C}$ is called a
+ \textit{complex, unital Algebra} if for all $a,b \in A$:
\begin{align}
A \times A \rightarrow A\\
(a,\ b)\ &\mapsto \ a\cdot b,
@@ -64,11 +64,11 @@ which `pulls back' values even if $\phi$ is not bijective.
Note that the pullback does not map points back, but maps functions on an $*$-algebra $C(X)$.
The pullback, in literature often called a $*$-homomorphism or a $*$-algebra map under
pointwise product has the following properties
-\begin{itemize}
- \item $\phi ^*(f\ g) = \phi ^*(f)\ \phi ^*(g)$,
- \item $\phi ^*(\overline{f}) = \overline{\phi ^*(f)}$,
- \item $\phi ^*(\lambda\ f + g) = \lambda\ \phi ^*(f) + \phi ^*(g)$.
-\end{itemize}
+\begin{align}
+ \phi ^*(f\ g) = \phi ^*(f)\ \phi ^*(g),
+ \phi ^*(\overline{f}) = \overline{\phi ^*(f)},
+ \phi ^*(\lambda\ f + g) = \lambda\ \phi ^*(f) + \phi ^*(g).
+\end{align}
%------------ Exercise
The map $\phi :X_1\ \rightarrow \ X_2$ is an injective (surjective) map,
if only and if the corresponding pullback $\phi ^* :C(X_2)\ \rightarrow \
diff --git a/src/thesis/chapters/electroncg.tex b/src/thesis/chapters/electroncg.tex
@@ -1,26 +1,27 @@
\subsection{Noncommutative Geometry of Electrodynamics}
-In this chapter we describe Electrodynamics with the almost commutative
-manifold $M\times F_X$ and the abelian gauge group $U(1)$.
-We arrive at a unified description of gravity and electrodynamics although in the classical level.
+In this chapter we go through a derivation Electrodynamics with
+the almost commutative manifold $M\times F_X$ and the abelian gauge group
+$U(1)$. The conclusion is an unified description of gravity and
+electrodynamics although in the classical level.
-The almost commutative Manifold $M\times F_X$ describes a local gauge group
+The almost commutative Manifold $M\times F_X$ outlines a local gauge group
$U(1)$. The inner fluctuations of the Dirac operator relate to $Y_\mu$ the
gauge field of $U(1)$. According to the setup we ultimately arrive at two
serious problems.
-First of all in the Two-Point space $F_X$, the operator $D_F$ must vanish for
-us to have a real structure. However this implies that the electrons
-are massless, which would be absurd.
+First of all the operator $D_F$, in the Two-Point space $F_X$, must vanish
+such that a real structure can exists. However this implies that the electrons
+are massless.
The second problem arises when looking at the Euclidean action for a free
Dirac field
\begin{align}
S = - \int i \bar{\psi}(\gamma ^\mu\partial _\mu - m) \psi d^4x,
\end{align}
-where $\psi,\ \bar{\psi}$ must be considered as independent variables, which
-means that the fermionic action $S_f$ needs two independent Dirac spinors.
-Let us try and construct two independent Dirac spinors with our data. To do
-this we take a look at the decomposition of the basis and of the total
+where $\psi,\ \bar{\psi}$ must be considered as two independent variables.
+This means that the fermionic action $S_f$ needs two independent Dirac spinors.
+Let us try and construct two independent Dirac spinors with our data, first
+take a look at the decomposition of the basis and of the total
Hilbertspace $H = L^2(S) \otimes H_F$. For the orthonormal basis of $H_F$ we
can write $\{e, \bar{e}\}$ , where $\{e\}$ is the orthonormal basis of
$H_F^+$ and $\{\bar{e}\}$ the orthonormal basis of $H_F^-$. Accompanied with
@@ -32,31 +33,31 @@ the real structure we arrive at the following relations
Along with the decomposition of $L^2(S) = L^2(S)^+ \oplus L^2(S)^-$ and $\gamma = \gamma _M
\otimes \gamma _F$ we can obtain the positive eigenspace
\begin{align}
- H^+ = L^2(S)^+ \otimes H_F^+ \oplus L^(S)^- \otimes H_F^-.
+ H^+ = L^2(S)^+ \otimes H_F^+ \oplus L(S)^- \otimes H_F^-.
\end{align}
-So, for a $\xi \in H^+$ we can write
+So, for an $\xi \in H^+$ we can write
\begin{align}
- \xi = \psi _L \otimes e + \psi _R \otimes \bar{e}
+ \xi = \psi _L \otimes e + \psi _R \otimes \bar{e},
\end{align}
where $\psi_L \in L^2(S)^+$ and $\psi _R \in L^2(S)^-$ are the two Wheyl
spinors. We denote that $\xi$ is only determined by one Dirac spinor $\psi :=
-\psi_L + \psi _R$, \textbf{but we require two independent spinors}. Our
+\psi_L + \psi _R$. Since \textbf{we require two independent spinors}, our
conclusion is that the definition of the fermionic action gives too much
restrictions to the Two-Point space $F_X$.
\subsubsection{The Finite Space}
To solve the two problems we simply enlarge (double) the Hilbertspace. This
-is visualized by introducing multiplicities in Krajewski Diagrams which will also
-allow us to choose a nonzero Dirac operator that will connect the two
-vertices and preserve real structure making our particles massive and
-bringing anti-particles into the mix.
+is visualized by introducing multiplicities in Krajewski Diagrams
+\cite{ncgwalter} which will also allow us to choose a nonzero Dirac operator
+that will connect the two vertices and preserve real structure making our
+particles massive and bringing anti-particles into the mix.
We start of with the same algebra $C^\infty(M, \mathbb{C}^2)$, corresponding
to space $N= M\times X$. The Hilbertspace describes four particles, meaning
it has four orthonormal basis elements. It describes \textbf{left handed
-electrons} and \textbf{right handed positrons}. Pointing this out, we have
+electrons} and \textbf{right handed positrons}. This way we have
$\{ \underbrace{e_R, e_L}_{\text{left-handed}}, \underbrace{\bar{e}_R,
-\bar{e}_L}_{\text{right-handed}}\}$ the orthonormal basis for $H_F =
-\mathbb{C}^4$. Accompanied with the real structure $J_F$, which allows us to
+\bar{e}_L}_{\text{right-handed}}\}$ an orthonormal basis for $H_F =
+\mathbb{C}^4$. Accompanied with the real structure $J_F$ allowing us to
interchange particles with antiparticles by the following equations
\begin{align}
&J_F e_R = \bar{e}_R, \\
@@ -70,7 +71,7 @@ where $J_F$ and $\gamma_F$ have to following properties
&J_F^2 = 1,\\
& J_F \gamma_F = - \gamma_F J_F.
\end{align}
-By means of $\gamma_F$ we have two options to decompose the total
+By the means of $\gamma_F$ we have two options to decompose the total
Hilbertspace $H$, firstly into
\begin{align}
H_F = \underbrace{H_F^+}_{\text{ONB } \{e_L, \bar{e}_L\}}
@@ -78,12 +79,12 @@ Hilbertspace $H$, firstly into
\end{align}
or alternatively into the eigenspace of particles and their
antiparticles (electrons and positrons) which is preferred in literature and
-which we will use going further
+which will be used further out
\begin{align}
H_F = \underbrace{H_{e}}_{\text{ONB } \{e_L, e_R\}} \oplus
- \underbrace{H_{\bar{e}}}_{\text{ONB } \{\bar{e}_L, \bar{e}_R\}}.
+ \underbrace{H_{\bar{e}}}_{\text{ONB } \{\bar{e}_L, \bar{e}_R\}},
\end{align}
-Here ONB means orthonormal basis.
+the shortening `ONB' means orthonormal basis.
The action of $a \in A = \mathbb{C}^2$ on $H$ with respect to the ONB
$\{e_L, e_R, \bar{e}_L, \bar{e}_R\}$ is represented by
@@ -97,11 +98,10 @@ $\{e_L, e_R, \bar{e}_L, \bar{e}_R\}$ is represented by
0 &0 &0 &a_2\\
\end{pmatrix}
\end{align}
-Do note that this action commutes wit the grading and that
-$[a, b^\circ] = 0$ with $b:= J_F b^*J_F$ because both the left and the right
-action is given by diagonal matrices by equation \eqref{eq:leftrightrepr}. Note
-that we are still left with $D_F = 0$ and the following spectral
-triple
+Do note that this action commutes wit the grading and that $[a, b^\circ] = 0$
+with $b:= J_F b^*J_F$ because both the left and the right action are given by
+diagonal matrices according to equation \eqref{eq:leftrightrepr}. Furthermore
+note that we are still left with $D_F = 0$ and the following spectral triple
\begin{align}\label{eq:fedfail}
\left( \mathbb{C}^2, \mathbb{C}^2, D_F=0; J_F =
\begin{pmatrix}
@@ -114,7 +114,7 @@ triple
\right).
\end{align}
It can be represented in the following Krajewski diagram,
-with two nodes of multiplicity two
+with two nodes of multiplicity two bellow
\begin{figure}[H] \centering
\begin{tikzpicture}[
dot/.style = {draw, circle, inner sep=0.06cm},
@@ -130,6 +130,7 @@ with two nodes of multiplicity two
\node[bigdot](d0) at (1.5,0) [] {};
\node[bigdot](d0) at (0.5,-1) [] {};
\end{tikzpicture}
+ \caption{Krajewski diagram of the spectral triple from equation \ref{eq:fedfail}}
\end{figure}
\subsubsection{A noncommutative Finite Dirac Operator}
To extend our spectral triple with a non-zero Operator, we need to take a
@@ -149,7 +150,7 @@ We can now define the finite space $F_{ED}$.
\begin{align}
F_{ED} := (\mathbb{C}^2, \mathbb{C}^4, D_F; J_F, \gamma_F)
\end{align}
-where $J_F$ and $\gamma_F$ are like in equation \eqref{eq:fedfail} and $D_F$
+where $J_F$ and $\gamma_F$ are as in equation \eqref{eq:fedfail} and $D_F$
from equation \eqref{eq:feddirac}.
\subsubsection{Almost commutative Manifold of Electrodynamics}
@@ -159,11 +160,11 @@ represented by the following spectral triple
M\times F_{ED} := \big(C^\infty(M,\mathbb{C}^2),\ L^2(S)\otimes
\mathbb{C}^4,\
D_M\otimes 1 +\gamma _M \otimes D_F;\; J_M\otimes J_F,\ \gamma_M\otimes
- \gamma _F\big)
+ \gamma _F\big).
\end{align}
The algebra didn't change, thus we can decompose it like before
\begin{align}
- C^\infty(M, \mathbb{C}^2) = C^\infty (M) \oplus C^\infty (M)
+ C^\infty(M, \mathbb{C}^2) = C^\infty (M) \oplus C^\infty (M).
\end{align}
As for the Hilbertspace, we can decomposition it in the following way
\begin{align}
@@ -186,15 +187,15 @@ arbitrary gauge field $B_\mu = A_\mu - J_F A_\mu J_F^{-1}$ we can write
0 & 0 & 0 & Y_\mu
\end{pmatrix} \;\;\;\;\;\ \text{for} \;\;\ Y_\mu (x) \in \mathbb{R}.
\end{align}
-We have one single $U(1)$ gauge field $Y_\mu$, carrying the action of the
+There is one single $U(1)$ gauge field $Y_\mu$, carrying the action of the
gauge group
\begin{align}
\text{$\mathfrak{B}$}(M\times F_{ED}) \simeq C^\infty(M, U(1))
\end{align}
The space $N = M\times X$ consists of two copies of $M$.
-If $D_F = 0$ we have infinite distance between the two copies. Now have
-hacked the spectral triple to have nonzero Dirac operator $D_F$. The new
+If $D_F = 0$ we have infinite distance between the two copies, yet now we have
+adjusted the spectral triple to have a nonzero Dirac operator. The new
Dirac operator still has a commuting relation with the algebra $[D_F, a] = 0$
$\forall a \in A$, and we should note that the distance between the two
copies of $M$ is still infinite. This is purely an mathematically abstract
@@ -210,8 +211,11 @@ action $S_b$ (bosonic) and of the fermionic action $S_f$.
The simplest spectral action of a spectral triple $(A, H, D)$ is given by the
trace of a function of $D$. We also consider inner fluctuations of the Dirac
-operator $D_\omega = D + \omega + \varepsilon' J\omega J^{-1}$ where $\omega =
-\omega ^* \in \Omega_D^1(A)$.
+operator
+\begin{align}
+ D_\omega = D + \omega + \varepsilon' J\omega J^{-1},
+\end{align}
+where $\omega = \omega ^* \in \Omega_D^1(A)$.
\begin{mydefinition}
Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a suitable function
\textbf{positive and even}. The spectral action is then
@@ -224,7 +228,7 @@ operator $D_\omega = D + \omega + \varepsilon' J\omega J^{-1}$ where $\omega =
independent of the basis. The subscript $b$ in $S_b$ stands for bosonic,
because in physical applications $\omega$ will describe bosonic fields.
- In addition to the bosonic action $S_b$ we can define a topological spectral
+ In addition to the bosonic action $S_b$, we can define a topological spectral
action $S_{top}$. Leaning on the grading $\gamma$ the topological spectral action is
\begin{align}
S_{\text{top}}[\omega] := \text{Tr}(\gamma\
@@ -237,19 +241,18 @@ operator $D_\omega = D + \omega + \varepsilon' J\omega J^{-1}$ where $\omega =
S_f[\omega, \psi] = (J\tilde{\psi}, D_\omega \tilde{\psi})
\end{align}
with $\tilde{\psi} \in H_{cl}^+ := \{\tilde{\psi}: \psi \in H^+\}$, where
- $H_{cl}^+$ is a set of Grassmann variables in $H$ in the +1-eigenspace
+ $H_{cl}^+$ is a set of Grassmann variables in $H$ in the $+1$-eigenspace
of the grading $\gamma$.
\end{mydefinition}
%---------------------- APPENDIX ?????????????--------------------
-\textbf{APPENDIX??}
Grassmann variables are a set of Basis vectors of a vector space, they
form a unital algebra over a vector field $V$, where the generators are
anti commuting, that is for Grassmann variables $\theta _i, \theta _j$ we have
\begin{align}
- &\theta _i \theta _j = -\theta _j \theta _i \\
- &\theta _i x = x\theta _j \;\;\;\; x\in V \\
- &(\theta_i)^2 = 0 \;\;\; (\theta _i \theta _i = -\theta _i \theta _i)
+ &\theta _i \theta _j = -\theta _j \theta _i, \\
+ &\theta _i x = x\theta _j \;\;\;\; x\in V, \\
+ &(\theta_i)^2 = 0 \;\;\; (\theta _i \theta _i = -\theta _i \theta _i).
\end{align}
%---------------------- APPENDIX ?????????????--------------------
\begin{myproposition}
@@ -266,17 +269,21 @@ anti commuting, that is for Grassmann variables $\theta _i, \theta _j$ we have
\mathcal{L}_B(B_\mu)+
\mathcal{L}_\phi(g_{\mu\nu}, B_\mu, \Phi).
\end{align}
- The Lagrangian $\mathcal{L}_M$ is of the spectral triple, represented by
- the following term
- $(C^\infty(M) , L^2(S), D_M)$
+ The Lagrangian $\mathcal{L}_M$ is of the spectral triple $(C^\infty(M) ,
+ L^2(S), D_M)$, represented by the following term
\begin{align}\label{lagr}
\mathcal{L}_M(g_{\mu\nu}) := \frac{f_4 \Lambda ^4}{2\pi^2} -
\frac{f_2 \Lambda^2}{24\pi ^2}s - \frac{f(0)}{320\pi^2} C_{\mu\nu
\varrho \sigma}C^{\mu\nu \varrho \sigma},
\end{align}
- here $C^{\mu\nu \varrho \sigma}$ is defined in terms of the Riemannian
+ where $C^{\mu\nu \varrho \sigma}$ is the Weyl tensor defined in terms of the Riemannian
curvature tensor $R_{\mu\nu \varrho \sigma}$ and the Ricci tensor
- $R_{\nu\sigma} = g^{\mu\varrho} R_{\mu\nu \varrho\sigma}$.
+ $R_{\nu\sigma} = g^{\mu\varrho} R_{\mu\nu \varrho\sigma}$ such that
+ \begin{align}
+ C^{\mu\nu\varrho\sigma}C_{\mu\nu\varrho\sigma}=
+ R_{\mu\nu\varrho\sigma}R^{\mu\nu\varrho\sigma} -
+ 2R_{\nu\sigma}R^{\nu\sigma} + \frac{1}{2}s^2.
+ \end{align}
The kinetic term of the gauge field is described by the Lagrangian
$\mathcal{L}_B$, which takes the following shape
\begin{align}
@@ -308,7 +315,7 @@ anti commuting, that is for Grassmann variables $\theta _i, \theta _j$ we have
and they are dependent on the fluctuated Dirac operator
$D_\omega$. We can rewrite the heat kernel coefficients in terms of $D_M$,
for the first two terms $a_0$ and $a_2$ we use $N:=
- \text{Tr}\mathbbm{1_{H_F}})$ and write
+ \text{Tr}(\mathbbm{1}_{H_F})$ and one obtains
\begin{align}
a_0(D_\omega^2) &= Na_0(D_M^2),\\
a_2(D_\omega^2 &= Na_2(D_M^2) - \frac{1}{4\pi^2}\int_M
@@ -323,7 +330,7 @@ anti commuting, that is for Grassmann variables $\theta _i, \theta _j$ we have
&F^2 = \frac{1}{16}s^2\otimes 1 + 1\otimes \Phi^4 - \frac{1}{4}
\gamma^\mu\gamma^\nu \gamma^\varrho\gamma^\sigma F_{\mu\nu}F^{\mu\nu}+\\
&\;\;\;\;\;\;\;+\gamma^\mu\gamma^\nu\otimes(D_\mu\Phi)(D_\nu
- \Phi)+\frac{1}{2}s\otimes \Phi^2 + \ \text{traceless terms}\\
+ \Phi)+\frac{1}{2}s\otimes \Phi^2 + \ \text{traceless terms},\\
\nonumber\\
&\frac{1}{360}\text{Tr}(180F^2) = \frac{1}{8}s^2N + 2\text{Tr}(\Phi^4)
+ \text{Tr}(F_{\mu\nu}F^{\mu\nu}) +\\
@@ -335,7 +342,7 @@ anti commuting, that is for Grassmann variables $\theta _i, \theta _j$ we have
\end{align}
The cross terms of the trace in $\Omega_{\mu\nu}^E\Omega^{E\mu\nu}$
vanishes because of the antisymmetric property of the Riemannian
- curvature tensor, thus we can write
+ curvature tensor, reading
\begin{align}
\Omega_{\mu\nu}^E\Omega^{E\mu\nu} = \Omega_{\mu\nu}^S\Omega^{S\mu\nu}
\otimes 1 - 1\otimes F_{\mu\nu}F^{\mu\nu} + 2i\Omega_{\mu\nu}^S
@@ -353,7 +360,8 @@ anti commuting, that is for Grassmann variables $\theta _i, \theta _j$ we have
\frac{N}{24}R_{\mu\nu\varrho\sigma}R^{\mu\nu\varrho\sigma}
-\frac{1}{3}\text{Tr}(F_{\mu\nu}F^{\mu\nu}).
\end{align}
- Finally plugging the results into the coefficient $a_4$ and simplifying we get
+ Finally plugging the results into the coefficient $a_4$ and simplifying
+ one gets
\begin{align}
a_4(x, D_\omega^4) &= Na_4(x, D_M^2) + \frac{1}{4\pi^2}\bigg(\frac{1}{12} s
\text{Tr}(\Phi^2) + \frac{1}{2}\text{Tr}(\Phi^4) \nonumber \\
@@ -372,9 +380,9 @@ We remind ourselves the definition of the fermionic action in definition
\eqref{eq:almost commutative manifold}. The Hilbertspace $H_F$ is separated
into the particle-antiparticle states with ONB $\{e_R, e_L, \bar{e}_R,
\bar{e}_L\}$. The orthonormal basis of $H_F^+$ is $\{e_L, \bar{e}_R\}$ and
-consequently for $H_F^-$, $\{e_R, \bar{e}_L\}$. We can decompose a spinor
-$\psi \in L^2(S)$ in each of the eigenspaces $H_F^\pm$, $\psi = \psi_R+
-\psi_L$. That means for an arbitrary $\psi \in H^+$ we can write
+consequently for $H_F^-$, $\{e_R, \bar{e}_L\}$. The decomposition of a spinor
+$\psi \in L^2(S)$ in each of the eigenspaces $H_F^\pm$ is $\psi = \psi_R+
+\psi_L$. Meaning for an arbitrary $\psi \in H^+$ we can write
\begin{align}
\psi = \chi_R \otimes e_R + \chi_L \otimes e_L + \psi_L \otimes
\bar{e}_R+
@@ -383,7 +391,7 @@ $\psi \in L^2(S)$ in each of the eigenspaces $H_F^\pm$, $\psi = \psi_R+
where $\chi_L, \psi_L \in L^2(S)^+$ and $\chi_R, \psi_R \in L^2(S)^-$.
Since the fermionic action yields too much restriction on $F_{ED}$ (modified
-Two-Point space $F_X$) we redefine it by taking account the fluctuated Dirac
+Two-Point space $F_X$) one redefines it by taking into account the fluctuated Dirac
operator
\begin{align}
D_\omega = D_M \otimes i + \gamma^\mu \otimes B_\mu + \gamma_M \otimes
@@ -439,8 +447,9 @@ Finally the fermionic action of $M\times F_{ED}$ takes the form
\tilde{\Psi}\big) + \big(S_M\tilde{\chi}_L, \bar{d}\tilde{\psi}_L\big) -
\big(J_M\tilde{\chi}_R, d \tilde{\psi}_R\big).
\end{align}
-Ultimately we arrive at the full Lagrangian of $M\times F_{ED}$, which is the
-sum of purely gravitational Lagrangian
+Ultimately we arrive at the full Lagrangian of the almost commutative
+manifold $M\times F_{ED}$, which is the sum of the purely gravitational
+Lagrangian
\begin{align}
\mathcal{L}_{grav}(g_{\mu\nu})=4\mathcal{L}_M(g_{\mu\nu})+
\mathcal{L}_\phi (g_{\mu\nu}),
diff --git a/src/thesis/main.pdf b/src/thesis/main.pdf
Binary files differ.
diff --git a/src/thesis/main.tex b/src/thesis/main.tex
@@ -24,15 +24,15 @@
\input{chapters/main_sec}
-%\input{chapters/basics} % ausgearbeitet ohne exercises, ohne examples
+\input{chapters/basics} % ausgearbeitet ohne exercises, ohne examples
-%\input{chapters/finitencg} % ausgearbeitet ohne exercises, ohne examples
+\input{chapters/finitencg} % ausgearbeitet ohne exercises, ohne examples
-%\input{chapters/realncg} % ausgearbeitet ohne exercises, ohne examples
+\input{chapters/realncg} % ausgearbeitet ohne exercises, ohne examples
-%\input{chapters/heatkernel} % ausgearbeitet ohne exercises, ohne examples
+\input{chapters/heatkernel} % ausgearbeitet ohne exercises, ohne examples
-%\input{chapters/twopointspace} % ausgearbeitet ohne exercises, ohne examples
+\input{chapters/twopointspace} % ausgearbeitet ohne exercises, ohne examples
\input{chapters/electroncg}
diff --git a/src/thesis/todo.md b/src/thesis/todo.md
@@ -3,3 +3,17 @@
* rewrite geometrical invariants $E$ in heatkernel.tex into the right one
from electroncg.tex !
* figures need caption and numbering
+
+# NORMAL TODO
+
+ 1. go through the main part and rewrite the chapters make sure the context
+ is followed, the equations have a comma or a dot at the sentence end
+ etc. DONE
+ 2. rethink the chapters
+ 3. write introduction
+ 4. write conclusion
+ 5. cut out exercises and examples in the main part if necessary, read
+ through the not cut out and write them up nicely
+ 6. write abstract
+ 7. read through
+ 8. submit
