ncg

commit 4495052f7eb47c650ed0f894868f2fd1e3fa38ae
parent 5b023921cca4f36e1f6268fd94ed761dd7dab504
Author: miksa234 <milutin@popovic.xyz>
Date:   Tue, 10 Aug 2021 14:56:36 +0200

almost done

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diff --git a/src/thesis/back/abstract.tex b/src/thesis/back/abstract.tex
@@ -1,3 +1,4 @@
+\vspace*{\fill}
 \begin{abstract}
     Noncommutative geometry is a branch of mathematics that has deep
     connections to applications in physics. From reconstructing the
@@ -11,3 +12,4 @@
     the basic backbone of noncommutative geometry and to further out derive
     the Lagrangian of electrodynamics.
 \end{abstract}
+\vspace*{\fill}
diff --git a/src/thesis/chapters/1_basics.tex b/src/thesis/chapters/1_basics.tex
@@ -0,0 +1,564 @@
+\subsection{Noncommutative Geometric Spaces\label{sec:1}}
+\subsubsection{$*$-Algebra}
+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 for all $a,b \in A$:
+            \begin{align}
+            A \times A \rightarrow A\\
+            (a,\ b)\ &\mapsto \ a\cdot b,
+            \end{align}
+    with an identity element:
+            \begin{align}
+            1a = a1 =a.
+            \end{align}
+    Extending the definition, a $*$-algebra is an algebra $A$ with a \textit{conjugate linear map (involution)} $*:A\ \rightarrow  A$,
+    $\forall a, b \in A$ satisfying
+    \begin{align}
+        (a\ b)^* &= b^*a^*,\\
+        (a^*)^* &= a.
+    \end{align}
+\end{mydefinition}
+In the following all unital algebras are referred to as algebras.
+
+\subsubsection{Finite Discrete Space}
+Let us consider an example, a $*$-algebra of continuous functions $C(X)$
+on a discrete topological space $X$ with $N$ points. Functions of a
+continuous $*$-algebra $C(X)$ assign values to $\mathbb{C}$ and for $f,\ g \in
+C(X)$, $\lambda \in \mathbb{C}$ and $x \in X$ they provide the following structure:
+\begin{itemize}
+    \item \textit{pointwise linear}
+        \begin{align}
+            (f + g)(x) &= f(x) + g(x),\\
+            (\lambda\ f)(x) &= \lambda (f(x)),
+        \end{align}
+    \item \textit{pointwise multiplication}
+        \begin{align}
+        f\ g\ (x) = f(x)g(x),
+        \end{align}
+    \item \textit{pointwise involution}
+        \begin{align}
+        f^*(x) = \overline{f(x)}.
+        \end{align}
+\end{itemize}
+The $*$-algebra $C(X)$ is \textit{isomorphic} to a $*$-algebra $\mathbb{C}^N$
+with involution ($N$ number of points in $X$), we write $C(X) \simeq
+\mathbb{C}^N$.  Isomorphisms are bijective maps that preserve structure and
+don't lose physical information.  A function $f:X\ \rightarrow\ \mathbb{C}$
+can be represented with $N \times N$ diagonal matrices, where each diagonal
+value represents the function value at the corresponding $i$-th point for $i
+= 1,...,N$. Matrix multiplication and hermitian conjugation of
+matrices we have a preserving structure.
+
+Moreover we can \textit{map} between finite discrete spaces $X_1$ and $X_2$ with a
+function
+\begin{align}
+    \phi:\ X_1 \rightarrow\ X_2.
+\end{align}
+For every such map there exists a corresponding map
+\begin{align}
+    \phi ^*:C(X_2)\ \rightarrow C(X_1),
+\end{align}
+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{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 $\phi ^* :C(X_2)\ \rightarrow \
+    C(X_1)$ is surjective (injective). To clarify let us say that $X_1$ has $n$ points and
+    $X_2$ with $m$ points. Then there are three different cases, first $n=m$ and
+    obviously $\phi$ is bijective and $\phi ^*$ too. Then $n >  m$, in this case
+    $\phi$ assigns $n$ points to $m$ points when $n >  m$, which is by definition
+    surjective. On the other hand $\phi ^*$ assigns $m$ points to $n$ points when
+    $n >  m$, which is by definition injective. Lastly $n < m $, which is
+    completely analogous to the case $n > m$.
+%------------ Exercise
+
+\begin{mydefinition}
+    A \textit{(complex) matrix algebra} A is a direct sum, for $n_i, N \in
+    \mathbb{N}$
+    \begin{align}
+        A = \bigoplus _{i=1}^{N} M_{n_i}(\mathbb{C}).
+    \end{align}
+    The involution is the hermitian conjugate. A $*$ algebra with involution is referred to as
+    a matrix algebra
+\end{mydefinition}
+
+To summarize, from a topological discrete space $X$, we can construct a
+$*$-algebra $C(X)$ which is isomorphic to a matrix algebra $A$. Then the
+question instantly arises, if we can construct $X$ given $A$? For a matrix
+algebra $A$, which in most cases is not commutative, the answer is generally
+no. Hence there are two options. We can restrict ourselves to commutative
+matrix algebras, which are the vast minority and not physically interesting.
+Or we can allow more morphisms (isomorphisms) between matrix algebras.
+
+\subsubsection{Finite Inner Product Spaces and Representations}
+Until now we have looked at finite topological discrete spaces, moreover we can consider a
+finite dimensional inner product space $H$ (finite Hilbertspaces), with inner product
+$(\cdot,\cdot)\rightarrow \mathbb{C}$. We denote $L(H)$ as the $*$-algebra of operators on $H$
+equipped with a product given by composition and involution of the adjoint, $T \mapsto T^*$.
+Then $L(H)$ is a \textit{normed vector space} with
+\begin{align}
+    \|T\|^2 &= \sup_{h \in H}\big\{(T\ h,\ T\ h): (h,\ h) \leq 1\big|\ T
+    \in L(H)\big \},\\
+    \|T\| &= \sup\big\{\sqrt{\lambda}:\; \lambda \text{ eigenvalue of } T\big\}.
+\end{align}
+The Hilbert space allows us to define representations of $*$-algebras.
+\begin{mydefinition}
+    The \textit{representation} of a finite dimensional $*$-algebra $A$ is a
+    pair $(H, \pi)$, where $H$ is a finite dimensional inner product space
+    and $\pi$ is a $*$-\textit{algebra map}
+    \begin{align}
+        \pi:A\ \rightarrow \ L(H).
+    \end{align}
+    We call the representation $(H, \pi)$ \textit{irreducible} if
+    \begin{itemize}
+        \item $H \neq \emptyset$,
+        \item only $\emptyset$ or $H$ is invariant under the action of $A$ on
+            $H$.
+    \end{itemize}
+\end{mydefinition}
+Here are some examples of reducible and irreducible representations
+\begin{itemize}
+    \item For $A = M_n(\mathbb{C})$ the representation $H=\mathbb{C}^n$, $A$ acts as matrix multiplication\\
+            $H$ is irreducible.
+    \item For $A = M_n(\mathbb{C})$ the representation $H=\mathbb{C}^n\oplus \mathbb{C}^n$, with $a \in A$ acting
+        in block form \\ $\pi: a \mapsto \big(\begin{smallmatrix} a & 0\\ 0 & a \end{smallmatrix}\big)$ is
+            reducible.
+\end{itemize}
+Naturally there are also certain equivalences between different
+representations.
+\begin{mydefinition}
+Two representations of a $*$-algebra $A$, $(H_1, \pi _1)$ and
+$(H_2, \pi _2)$  are called \textit{unitary equivalent} if there exists a map
+$U: H_1 \rightarrow H_2$ such that.
+    \begin{align}
+        \pi _1(a) = U^* \pi _2(a) U
+    \end{align}
+\end{mydefinition}
+
+Furthermore we define a mathematical structure called the structure space,
+which will become important later when speaking of the duality between a
+spectral triple and a geometrical space.
+\begin{mydefinition}
+    Let $A$ be a $*$-algebra then, $\hat{A}$ is called the structure space of all \textit{unitary equivalence classes
+    of irreducible representations of A}.
+\end{mydefinition}
+%------------- EXERCISE
+    Given a representation $(H, \pi)$ of a $*$-algebra $A$, the \textbf{commutant} $\pi (A)'$ of $\pi (A)$ is defined as a set
+    of operators in $L(H)$ that commute with all $\pi (a)$
+    \begin{align}
+        \pi (A)' = \big\{T \in L(H):\ \pi(a)\ T = T\ \pi(a) \;\; \forall a\in
+        A\big\}
+    \end{align}
+    The commutant $\pi (A)'$ is also a $*$-algebra, since it has unital,
+    associative and involutive properties.  The unitary property is given by
+    the unital operator of the $*$-algebra of operators $L(H)$, which exists
+    by definition because $H$ is a inner product space. Associativity is
+    given by the $*$-algebra of $L(H)$, where $L(H) \times L(H)~\mapsto
+    L(H)$, which is associative by definition. The involutive property is
+    also given by the $*$-algebra $L(H)$ with a map $*: L(H) \mapsto L(H)$
+    only for a $T \in H$ that commutes with $\pi (a)$.
+%------------- EXERCISE
+
+%------------- EXERCISE
+    For a unital algebra $*$-algebra $A$, the matrices $M_n(A)$ with entries
+    in $A$ form a unital $*$-algebra, because the unitary operation in
+    $M_n(A)$ is given by the identity Matrix, which exists in every
+    entry in $M_n(A)$ and behaves like in $A$. Associativity is given by
+    matrix multiplication. Lastly, involution is given by the conjugate
+    transpose.
+
+    Consider a representation $\pi :A\ \rightarrow \ L(H)$ of a $*$-algebra
+    $A$ and set $H^n = H \oplus ... \oplus H$, $n$ times. Then we have the following
+    representation $\tilde{\pi}:M_n(A) \rightarrow L(H^n)$ for the Matrix
+    Algebra with $\tilde{\pi}((a_{ij})) = (\tilde{\pi}(a_{ij})) \in M_n(A)$,
+    since a direct isomorphisms of $A \simeq M_n(A)$ and $H \simeq H^n$
+    exists. Meaning $\tilde{\pi}$ is a valid reducible representation.
+
+    By looking at $\tilde{\pi}:M_n(A) \rightarrow L(H^n)$ a $*$ algebra
+    representation of $M_n(A)$. We see that $\pi: A \rightarrow L(H^n)$ is a representation of $A$.
+    The fact that $\tilde{\pi}$ and $\pi$ are unitary equivalent, there is
+    a map $U: H^n \rightarrow H^n$ given by $U=\mathbbm{1}_n$, thus
+    \begin{align}
+        \pi (a) &= \mathbbm{1}_n^*\ \tilde{\pi}((a_{ij})), \\
+        \mathbbm{1}_n &= \tilde{\pi}((a_{ij})) = \pi (a_{ij})
+    \Rightarrow a_{ij} = a\ \mathbbm{1}_n.
+    \end{align}
+%------------- EXERCISE
+
+
+With help of the structure space $\hat{A}$, a commutative matrix algebra can be used to reconstruct a discrete space.
+Since $A \simeq \mathbb{C}^N$ all irreducible representation are of the form
+\begin{align}
+   \pi _i:(\lambda_1,...,\lambda_N)\in \mathbb{C}^N \mapsto \lambda_i \in
+   \mathbb{C}
+\end{align}
+for $i = 1,...,N$, and thus $\hat{A} \simeq \{1,...,N\}$.
+We can conclude that there is a duality between discrete spaces and
+commutative matrix algebras. This duality is called the \textit{finite
+dimensional Gelfand duality}
+
+Our aim is to make a further generalization by constructing a duality between
+finite dimensional spaces and \textit{equivalence classes} of matrix
+algebras that preserves general non-commutativity of matrices. Equivalence
+classes are described by a concept of isomorphisms between matrix
+algebras called \textit{Morita Equivalence}.
+
+\subsubsection{Algebraic Modules}
+An important part of the Morita Equivalence are algebraic modules, later
+extended by Hilbert bimodules.
+\begin{mydefinition}
+    Let $A$, $B$ be algebras (need not be matrix algebras)
+    \begin{enumerate}
+        \item \textit{left} A-module is a vector space $E$, that carries a left
+            representation of $A$, that is $\exists$ a bilinear map $\gamma: A
+            \times E \rightarrow E$ with
+            \begin{align}
+                (a_1\ a_2)\cdot e = a_1 \cdot (a_2 \cdot e);\;\;\; a_1, a_2 \in
+                A, e \in E.
+            \end{align}
+        \item \textit{right} B-module is a vector space $F$, that carries a
+            right representation of $A$, that is there exists a bilinear map
+            $\gamma: F \times B \rightarrow F$ with
+            \begin{align}
+                f \cdot (b_1\ b_2)= (f \cdot b_1) \cdot b_2;\;\;\; b_1, b_2 \in B, f \in F
+            \end{align}
+        \item \textit{left} A-module and \textit{right} B-module is a
+            \textit{bimodule}, a vector space $E$ satisfying
+            \begin{align}
+                a \cdot (e \cdot b)= (a \cdot e) \cdot b;\;\;\;  a \in A, b \in B, e \in E
+            \end{align}
+    \end{enumerate}
+\end{mydefinition}
+An $A$-\textbf{module homomorphism} is linear map $\phi: E\rightarrow F$ which respects the
+representation of A, e.g.\ for left module.
+\begin{align}
+    \phi (a\ e) = a \phi (e); \;\;\; a \in A, e \in E.
+\end{align}
+We will use the notation
+\begin{itemize}
+    \item ${}_A E$, for left $A$-module $E$;
+    \item ${}_A E_B$, for right $B$-module $F$;
+    \item ${}_A E_B$, for $A$-$B$-bimodule $E$, simply bimodule.
+\end{itemize}
+%------------------- EXERCISE
+From a simple observation, we see that an arbitrary representation $\pi : A
+\rightarrow L(H)$ of a $*$-algebra A, turns H into a left module ${}_A H$.  If
+$_A H$ than $(a_1\ a_2) h = a_1 (a_2\ h)$ for $a_1, a_2 \in A$ and $h \in H$. We
+take the representation of an $a \in A$, $\pi (a)$, and write
+\begin{align}
+    \big(\pi(a_1)\ \pi(a_2)\big)h = \pi(a_1)\big(\pi(a_2)\ h\big) =
+    \big(T_1\ T_2\big) h = T_1 \big(T_2\ h\big)
+\end{align}
+For $T_1, T_2 \in L(H)$, which operate naturally from the left on $h$.
+
+%------------------- EXERCISE
+%------------------- EXERCISE
+
+Furthermore notice that that an $*$-algebra $A$ is a bimodule ${}_A A_A$ with
+itself, given by the map
+\begin{align}
+    \gamma: A\times A\times A \rightarrow A,
+\end{align}
+which is the inner product of a $*$-algebra.
+%------------------- EXERCISE
+
+\subsubsection{Balanced Tensor Product and Hilbert Bimodules}
+In this chapter we introduce the balanced tensor product later called the
+Kasparov product. This operation allows us to naturally construct a bimodule
+of a third algebra in chapter \ref{chap: kasparov product}.
+\begin{mydefinition}
+    Let $A$ be an algebra, $E$ be a \textit{right} $A$-module and $F$ be a
+    \textit{left} $A$-module.  The \textit{balanced tensor product} of $E$ and
+    $F$ forms a $A$-bimodule.
+    \begin{align}
+        E \otimes _A F := E \otimes F / \left\{\sum _i e_i a_i \otimes f_i -
+        e_i \otimes a_i f_i : \;\;\; a_i \in A,\ e_i \in E,\ f_i \in F
+    \right\}.
+    \end{align}
+\end{mydefinition}
+The symbol $/$ denotes the quotient space. By careful examination we can say
+that the operation $\otimes _A$ takes two left/right modules and makes a
+bimodule. Additionally with the help of the tensor product of the two modules and the quotient
+space which takes out all the elements from the tensor product that don't
+preserver the left/right representation and that are duplicates.
+\begin{mydefinition}
+    Let $A$, $B$ be \textit{matrix algebras}. The \textit{Hilbert bimodule} for
+    $(A, B)$ is given by an $A$-$B$-bimodue $E$ and by an $B$-valued
+    \textit{inner product} $\langle \cdot,\cdot\rangle_E: E\times E \rightarrow
+    B$, which satisfies the following conditions for $e, e_1, e_2 \in
+    E,\ a \in A$ and $b \in B$
+\begin{align}
+    \langle e_1,\ a\cdot e_2\rangle_E &= \langle a^*\cdot e_1,\ e_2\rangle_E
+    \;\;\;\; & \text{sesquilinear in $A$},\\
+    \langle e_1,\ e_2 \cdot b\rangle_E
+             &= \langle e_1,\ e_2\rangle_E b \;\;\;\; & \text{scalar in $B$},\\
+    \langle e_1,\ e_2\rangle_E &= \langle e_2,\ e_1\rangle^*_E \;\;\;\; &
+    \text{hermitian}, \\
+    \langle e,\ e\rangle_E &\ge 0 \;\;\;\; & \text{equality
+    holds iff $e=0$}.
+\end{align}
+We denote $KK_f(A,\ B)$ as the set of all \textit{Hilbert bimodules} of $(A,\ B)$.
+\end{mydefinition}
+%-------------- EXERCISE
+
+And indeed the Hilbert bimodule extension takes a representation $\pi:\ A \
+\rightarrow L(H)$ of a matrix algebra $A$ and turns $H$ into a Hilbert bimodule for
+$(A, \mathbb{C})$, because the representation for a $a \in A$, $\pi(a)=T \in L(H)$ fulfills
+the conditions of the $\mathbb{C}$-valued inner product for $h_1, h_2 \in H$
+\begin{itemize}
+    \item $\langle h_1,\ \pi(a)\ h_2\rangle _\mathbb{C} = \langle h_1,\ T\ h_2\rangle _\mathbb{C} =
+        \langle T^* h_1, h_2\rangle _\mathbb{C}$, $T^*$ given by the adjoint,
+    \item $\langle h_1,\ h_2\ \pi(a)\rangle _\mathbb{C} = \langle h_1,\ h_2\
+        T\rangle _\mathbb{C} = \langle h_1,\ h_2\rangle _\mathbb{C}$ , $T$ acts
+        from the left,
+    \item $\langle h_1,\ h_2\rangle _\mathbb{C}^* = \langle h_2,\ h_1\rangle _\mathbb{C}$, hermitian because of the
+        $\mathbb{C}$-valued inner product
+    \item $\langle h_1,\ h_2\rangle  \ge 0$, $\mathbb{C}$-valued inner product.
+\end{itemize}
+%-------------- EXERCISE
+
+%-------------- EXERCISE
+Take again the $A-A$ bimodule given by an $*$-algebra $A$. By looking at the
+following inner product $\langle \cdot,\cdot\rangle_A:A \times A \rightarrow A$
+\begin{align}
+    \langle a,\ a\rangle_A = a^*a' \;\;\;\; a,a'\in A.
+    \label{eq:inner-product},
+\end{align}
+it becomes clear that $A \in KK_f(A,\ A)$.
+Simply checking the conditions in $\langle \cdot, \cdot\rangle _A$ for
+$a, a_1, a_2 \in~A$
+\begin{align}
+    &\langle a_1,\ a\cdot a_2\rangle _A = a^* a\cdot a_2 =
+        (a^*a_1)^*\ a_2 = \langle  a^*\ a_1,\ a_2\rangle, \\
+   &\langle a_1,\ a_2 \cdot a\rangle _A = a^*_1\ (a_2\cdot a) =
+    (a^*a_2)\cdot a = \langle a_1,\ a_2\rangle _A\ a,\\
+   &\langle a_1,\ a_2\rangle _A^* = (a_1^*\ a_2)^* = a_2^*\
+        (a_1^*)^* = a_2^*\ a_1 = \langle a_2,\ a_1\rangle.
+\end{align}
+
+%-------------- EXERCISE
+
+%-------------- EXAMPLE
+%As an for overview consider a $*$ homomorphism between two matrix
+%algebras $\phi:A\rightarrow B$, we can construct a Hilbert bimodule
+%$E_{\phi} \in KK_f(A, B)$ in the following way. We let $E_{\phi}$ be $B$ in
+%as an vector space and an inner product from above in equation
+%\eqref{eq:inner-product}, with $A$ acting on the left with $\phi$.
+%\begin{align}
+%    a\cdot b = \phi(a)\ b
+%\end{align}
+%for $a\in A, b\in E_{\phi}$.
+%-------------- EXAMPLE
+
+\subsubsection{Kasparov Product and Morita Equivalence\label{chap: kasparov
+product}}
+\begin{mydefinition}
+    Let $E \in KK_f(A, B)$ and $F \in KK_F(B, D)$ the \textit{Kasparov product} is defined as
+    with the balanced tensor product
+    \begin{align}
+        F \circ E := E \otimes _B F.
+    \end{align}
+    Then $F\circ E \in KK_f(A,D)$ is equipped with a $D$-valued inner product
+    \begin{align}
+        \langle e_1 \otimes f_1,\ e_2 \otimes f_2\rangle _{E\otimes _B F} =
+        \langle f_1,\langle e_1,\ e_2\rangle _E f_2\rangle _F
+    \end{align}
+\end{mydefinition}
+
+%-------------- EXERCISE
+The Kasparov product for $*$-algebra homomorphism $\phi: A \rightarrow B$ and
+$\psi: B \rightarrow C$ are isomorphisms in the sense that
+\begin{align}
+                E_{\psi} \circ E_{\phi}\ \equiv\ E_{\phi} \otimes _B E_{\psi}\
+                \simeq\
+                E_{\psi \circ \phi} \in KK_f(A,C).
+\end{align}
+
+The direct computation for $a \in A$, $b\in B$, and $c\in C$ which is $\psi
+\circ \phi$ shows us
+\begin{align}
+a \cdot b \cdot c = \psi(\phi (a) \cdot b) \cdot c
+\end{align}
+An interesting case arises when looking at $E_{\text{id}_A} \simeq A \in
+KK_f(A,A)$, where $\text{id}_A$ is the identity in $A$. Let $E_{\phi}$ be $A$
+with a natural right representation. It follows that $E_{\phi}\simeq A$, where
+an inner product, acting from the left on $A$ for $\phi$, $a', a\in A$ reads
+\begin{align}
+    a'\ a = (\phi(a')\ a) \in A,
+\end{align}
+which is satisfied only by $\phi = \text{id}_A$.
+
+\begin{mydefinition}
+    Let $A$, $B$ be \textit{matrix algebras}. They are called \textit{Morita equivalent} if there
+    exists an $E \in KK_f(A, B)$ and an $F \in KK_f(B, A)$ such that
+    \begin{align}
+        E \otimes _B F \simeq A \;\;\; \text{and} \;\;\; F \otimes _A E \simeq
+        B,
+    \end{align}
+    where $\simeq$ denotes the isomorphism between Hilbert bimodules and note
+    that $A$ or $B$ is a bimodule by itself.
+\end{mydefinition}
+
+Since we land in the same space as we started, the modules $E$ and $F$ are
+each others inverse in regards to the Kasparov Product. More clearly, in the
+definition we have $E \in KK_f(A, B)$. Naturally we start from $A$ and $E
+\otimes _B F$, which lands in $A$. On the other hand we have $F \in KK_f(B,
+D)$ and start from $B$, $F \otimes _A E$, which lands in $B$.
+
+%------------- EXERCISE
+By definition  $E \otimes _B F$ is a $A-D$ bimodule. Since
+\begin{align}
+    E \otimes _B F = E \otimes F / \bigg\{\sum_i\ e_i\ b_i \otimes f_i - e_i
+        \otimes b_i\ f_i\ \big|\;\; e_i \in E_i,\ b_i \in B,\ f_i \in F\bigg\},
+\end{align}
+the last part takes out all tensor product elements of $E$ and $F$ that don't
+preserver the left/right representation and that are duplicates.
+
+Additionally $\langle \cdot,\cdot\rangle _{E\oplus _B F}$ defines a $D$ valued
+inner product, as $\langle e_1,\ e_2\rangle _E \in B$ and $\langle f_1,\ f_2\rangle _F \in C$ by
+definition. So for $\langle e_1,\ e_2\rangle _E =b$ we have
+\begin{align}
+    \langle e_1 \otimes f_1,\ e_2 \otimes f_2\rangle _{E\otimes _B F} = \langle
+    f_1,\ \langle e_1,\ e_2\rangle _E\ f_2\rangle _F = \langle f_1,\ b\ f_2\rangle _F \in C
+\end{align}
+%------------- EXERCISE
+%------------- EXAMPLE
+Picking up the example of $(A, A)$, the Hilbert bimodule $A$, we can
+consider an $E \in KK_f(A,B)$ for
+\begin{align}
+    E \circ A = A\oplus _A E \simeq E.
+\end{align}
+We conclude, that $_A A_A$ is the identity element in the Kasparov product (up
+to isomorphism).
+%------------- EXAMPLE
+%------------- EXAMPLE
+Let us examine another example for $E = \mathbb{C}^n$, which is a
+$(M_n(\mathbb{C}), \mathbb{C})$ Hilbert bimodule with the standard $\mathbb{C}$
+inner product. Further let $F = \mathbb{C}^n$, which is a $(\mathbb{C},
+M_n(\mathbb{C}))$ Hilbert bimodule by right matrix multiplication with
+$M_n(\mathbb{C})$ valued inner product, we can write
+    \begin{align}
+        \langle v_1, v_2\rangle =\bar{v_1}v_2^t \;\; \in M_n(\mathbb{C}).
+    \end{align}
+If we take the Kasparov product of $E$ and $F$
+   \begin{align}
+       F\circ E\ &=\  E\otimes _{\mathbb{C}}F\ \;\;\;\;\;\; \simeq \
+        M_n(\mathbb{C}),\\
+        E\circ F\ &=\ F\otimes _{M_n(\mathbb{C})}E\ \simeq\ \mathbb{C},
+    \end{align}
+we see that $M_n(\mathbb{C})$ and $\mathbb{C}$ are Morita equivalent!
+%------------- EXAMPLE
+
+\begin{mylemma}
+    Two matrix algebras are Morita Equivalent if, and only if their their structure spaces
+    are isomorphic as discreet spaces (have the same cardinality / same number
+    of elements).
+\end{mylemma}
+\begin{proof}
+    Let $A$, $B$ be \textit{Morita equivalent}. Then there exist the modules
+    $_A E_B$ and $_B F_A$ with
+    \begin{align}
+        E \otimes _B F \simeq A \;\;\; \text{and} \;\;\; F \otimes _A E \simeq
+        B.
+    \end{align}
+    Also consider $[(\pi _B, H)] \in \hat{B}$. We can construct a
+    representation of $A$, which reads
+    \begin{align}
+        \pi _A \rightarrow L(E \otimes _B H)\;\;\; \text{with} \;\;\; \pi _A(a)
+        (e \otimes v) = a e \otimes w
+    \end{align}
+    Vice versa, we have $[(\pi _A, W)] \in \hat{A}$ and we can construct $\pi _B$
+    as
+    \begin{align}
+        \pi _B: B \rightarrow L(F \otimes _A W) \;\;\; \text{and}\;\;\; \pi
+        _B(b) (f\otimes w) = bf\otimes w.
+    \end{align}
+    Now we need to show that the representation $\pi _A$ is irreducible if and
+    only if $\pi _B$ is irreducible. For $(\pi _B, H)$ to be irreducible, we
+    need $H \neq \emptyset$ and only $\emptyset$ or $H$ to be invariant under
+    the Action of $B$ on $H$. Than $E\otimes _B H$ and $E\otimes _B H \simeq A$
+    cannot be empty, because $E$ preserves left representation of $A$.
+
+    Lastly we need to check if the association of the class $[\pi _A]$ to $[\pi
+    _B]$ is independent of the choice of representatives $\pi _A$ and $\pi _B$.
+    The important thing is that $[\pi _A] \in \hat{A}$ respectively $[\pi _B] \in
+    \hat{B}$, hence any choice of representation is irreducible, because the
+    structure space denotes all unitary equivalence classes of irreducible
+    representations.
+
+    Note that the statements $E \simeq H$ and $F \simeq W$ are not particularly
+    true, since all infinite dimensional Hilbert spaces are isomorphic.  Here
+    we are looking at finite dimensional Hilbert spaces. Another thing to keep
+    in mind, is that for $[\pi _B, H] \in \hat{B}$ and looking at algebraic
+    bimodules, we know that $H$ is a bimodule of $B$, hence $E \otimes _B
+    H\simeq A$, and for $[\pi _A, W]$, which is the same.
+    Finally we can conclude, that these maps are each others inverses, thus
+    $\hat{A} \simeq \hat{B}$.
+\end{proof}
+
+\begin{mylemma}
+    The matrix algebra $M_n(\mathbb{C})$ has a unique irreducible
+    representation (up to isomorphism) given by the defining representation on
+    $\mathbb{C}^n$.
+\end{mylemma}
+\begin{proof}
+    We know $\mathbb{C}^n$ is a irreducible representation of $A=
+    M_n(\mathbb{C})$. Let $H$ be irreducible and of dimension $k$, then we
+    define a map
+    \begin{align}
+        \phi : A\oplus...\oplus A &\rightarrow H^* \\
+    (a_1,...,a_k)&\mapsto e^1\circ a_1^t+...+e^k\circ a_k^t,
+    \end{align}
+where $\{e^1,...,e^k\}$ is the basis of the dual space $H^*$ and
+$(\circ)$ being the pre-composition of elements in $H^*$ and $A$ acting on $H$.
+This forms a morphism of $M_n(\mathbb{C})$ modules, provided a matrix $a \in A$
+acts on $H^*$ with $v\mapsto v\circ a^t$ ($v\in H^*$).  Furthermore this
+morphism is surjective, thus making the pullback $\phi ^*:H\mapsto (A^k)^*$
+injective.  Now identify $(A^k)^*$ with $A^k$ as a $A$-module and note that
+$A=M_n(\mathbb{C}) \simeq \oplus ^n \mathbb{C}^n$ as a n A module.  It follows
+that $H$ is a submodule of $A^k \simeq \oplus ^{nk}\mathbb{C}$. By
+irreducibility $H \simeq \mathbb{C}$.
+\end{proof}
+
+%---------------- EXAMPLE
+Let us look at an example, two matrix algebras $A$, and $B$.
+\begin{align}
+    A = \bigoplus ^N_{i=1} M_{n_i}(\mathbb{C}), \;\;\;
+    B = \bigoplus ^M_{j=1} M_{m_j}(\mathbb{C}).
+\end{align}
+Let $\hat{A} \simeq \hat{B}$, this implies $N=M$. Further define $E$ with $A$
+acting by block-diagonal matrices on the first tensor and B acting in the same
+manner on the second tensor. Define $F$ vice versa, ultimately reading
+\begin{align}
+    E:= \bigoplus _{i=1}^N \mathbb{C}^{n_i} \otimes \mathbb{C}^{m_i}, \;\;\;
+    F:= \bigoplus _{i=1}^N \mathbb{C}^{m_i} \otimes \mathbb{C}^{n_i}.
+\end{align}
+When we calculate the Kasparov product we get the following
+\begin{align}
+    E \otimes _B F &\simeq \bigoplus _{i=1}^N (\mathbb{C}^{n_i}\otimes\mathbb{C}^{m_i})
+        \otimes _{M_{m_i}(\mathbb{C})} (\mathbb{C}^{m_i}\otimes\mathbb{C}^{n_i}) \\
+                   &\simeq \bigoplus _{i=1}^N \mathbb{C}^{n_i}\otimes
+                   \left(\mathbb{C}^{m_i}\otimes _{M_{m_i}(\mathbb{C})}\mathbb{C}^{m_i}\right)
+                    \oplus \mathbb{C}^{n_i} \\
+                   &\simeq \bigoplus _{i=1}^N
+                   \mathbb{C}^{m_i}\otimes\mathbb{C}^{n_i} \simeq A.
+\end{align}
+On the other hand we get
+\begin{align}
+    F \otimes _A E \simeq B.
+\end{align}
+%---------------- EXAMPLE
+
+To summarize, there is a duality between finite spaces and Morita equivalence
+classes of matrix algebras. Furthermore by replacing $*$-homomorphism $A\rightarrow B$
+with Hilbert bimodules $(A,B)$ we introduce a richer structure of morphism
+between matrix algebras.
diff --git a/src/thesis/chapters/finitencg.tex b/src/thesis/chapters/2_finitencg.tex
diff --git a/src/thesis/chapters/3_realncg.tex b/src/thesis/chapters/3_realncg.tex
@@ -0,0 +1,292 @@
+\subsection{Finite Real Noncommutative Spaces\label{sec:3}}
+\subsubsection{Finite Real Spectral Triples}
+In this chapter we supplement the finite spectral triples with a \textit{real
+structure}. We additionally require a symmetry condition that that $H$ is an
+$A$-$A$-bimodule rather than only a $A$-left module. This ansatz has tight
+bounds with physical properties such as charge conjugation, into which we will
+dive in deeper in later chapters. In regards to this we will need to set a basis
+of definitions to get an overview.
+First we introduce a $\mathbb{Z}_2$-grading $\gamma$ with the following
+properties
+\begin{align}
+    \gamma ^* &= \gamma, \\
+    \gamma ^2 &= 1, \\
+    \gamma D &= - D \gamma,\\
+    \gamma a &= a \gamma, \;\;\;\; a\in A.
+\end{align}
+Then we can define a finite real spectral triple.
+\begin{mydefinition}
+    A \textit{finite real spectral triple} is given by a finite spectral
+    triple $(A, H, D)$ and a anti-unitary operator $J:H\rightarrow H$ called
+    the \textit{real structure}, such that
+    \begin{align}
+        a^\circ := J\ a^*\ J^{-1},
+    \end{align}
+    is a right representation of $A$ on $H$, that is $(ab)^\circ = b^\circ
+    a^\circ$. With two requirements
+    \begin{align}
+        &[a, b^\circ] = 0,\\
+        &[[D, a],\ b^\circ] = 0.
+    \end{align}
+    The two properties are called the \textit{commutant property}, they
+    require that the left action of an element in $A$ and $\Omega _D^1(A)$ commutes with the right
+    action on $A$.
+\end{mydefinition}
+\begin{mydefinition}
+    The $KO$-dimension of a real spectral triple is determined by the sings
+    $\epsilon, \epsilon ' ,\epsilon '' \in \{-1, 1\}$ appearing in
+    \begin{align}
+        J^2 &= \epsilon, \\
+        J\ D &= \epsilon \ D\ J,\\
+        J\ \gamma &= \epsilon''\ \gamma\ J.
+    \end{align}
+\end{mydefinition}
+\begin{table}[h!]
+    \centering
+    \caption{$KO$-dimension $k$ modulo $8$ of a real spectral triple}
+    \begin{tabular}{ c | c c c c c c c c}
+        \hline
+        $k$        & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
+        \hline
+     $\epsilon$    & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\
+     $\epsilon '$  & 1 & -1 & 1 & 1 & 1 & -1 & 1 & 1 \\
+     $\epsilon ''$ & 1 &  & -1 &  & 1 &  & -1 &  \\
+     \hline
+    \end{tabular}
+\end{table}
+\noindent
+Even thought the KO-dimension of a real spectral triple is important, we will
+not be doing in-depth introduction of the KO-dimension, for this we reference
+again to \cite{ncgwalter}.
+
+\begin{mydefinition}
+An opposite-algebra $A^\circ$ of a $A$ is defined to be equal to $A$ as a
+vector space with the opposite product
+\begin{align}
+    &a\circ b := ba\\
+    &\Rightarrow a^\circ = Ja^* J^{-1},
+\end{align}
+which defines the left representation of $A^\circ$ on $H$
+\end{mydefinition}
+
+
+%------------EXAMPLE EXERCISE
+Let us examine an example of a matrix algebra $M_N(\mathbb{C})$ acting on
+$H=M_N(\mathbb{C})$ by left matrix multiplication with the Hilbert Schmidt
+inner product.
+\begin{align}
+    \langle a , b \rangle = \text{Tr}(a^* b).
+\end{align}
+We can define $\gamma (a) = a$ and $J(a) = a^*$ with $a\in H$.  Since $D$
+must be odd with respect to $\gamma$ it vanishes identically.  Furthermore we
+know the multiplicity space is $V_i = \mathbb{C}^{m_i}$, and also we know
+that for $T\in H$ and$a\in A'$ to work we need $a\ T=T\ a$. Thus by laws of
+matrix multiplication we need $A' \simeq \bigoplus _i M_{m_i}(\mathbb{C})$. For
+this to work we naturally need $H = \bigoplus_i \mathbb{C}^{n_i} \otimes
+\mathbb{C}^{m_i}$.  Hence the right action of $M_N(\mathbb{C})$ on $H =
+M_N(\mathbb{C})$ as defined by $a \mapsto a^\circ$ is given by right matrix
+multiplication
+\begin{align}
+    a^\circ \xi = J a^* J^{-1}\xi = Ja^* \xi^* = J\xi a=\xi^* a
+\end{align}
+
+%------------EXAMPLE EXERCISE
+
+\begin{mydefinition}
+    We call $\xi \in H$ \textbf{cyclic vector} in $A$ if:
+    \begin{align}
+        A\xi := { a\xi:\;\; a\in A} = H
+    \end{align}
+    We call $\xi \in H$ \textbf{separating vector} in $A$ if:
+    \begin{align}
+        a\xi = 0\;\; \Rightarrow \;\; a=0;\;\;\; a\in A
+    \end{align}
+\end{mydefinition}
+%------------------- EXERCISE
+Suppose $(A, H, D = 0)$ is a finite spectral triple such that $H$ possesses a
+cyclic and separating vector for $A$ and let
+\begin{align}
+    J: H \rightarrow H
+\end{align}
+be the operator in $S = J \Delta ^{1/2}$ with $\Delta = S^*S$. By composition
+$S(a\xi) = a*\xi$ this is literally anti-linearity, then $S(a \xi) = a* \xi$
+defines a anti-linear operator. Furthermore the operator $S$ is invertible
+because, if a $\xi \in H$ is cyclic then we have $S(A\xi) = A^*\xi = A\xi =
+H$. Vice versa the same has to work for $S^{-1}$, otherwise $\xi$ wouldn't
+exist. And hence $S^{-1}(A^*\xi) = S^{-1}(H) = H$. Additionally $J$ is
+anti-unitary because firstly, $S$ is bijective thus $\Delta ^{1/2}$ and $J$ need to be bijective.
+Also have $J = S \Delta^{-1/2}$ and $\Delta^* = \Delta$, so for a $\xi _1 ,
+\xi _2 \in H$ we can write
+\begin{align}
+    \langle J \xi _1 , J \xi _2 \rangle  &= \langle  J^*J\xi_1 , \xi_2\rangle ^* =\nonumber\\
+    &= \langle (\Delta ^{-1/2})^* S^* S \Delta ^{-1/2} \xi_1, \xi_2\rangle ^* =\nonumber \\
+    &= \langle (\Delta^{-1/2})^* \Delta \Delta^{-1/2} \xi_1, \xi_2\rangle ^* =\nonumber\\
+    &= \langle \Delta^{-1/2} \Delta^{1/2}\Delta^{1/2} \Delta^{-1/2} \xi_1, \xi_2\rangle ^*
+    =\nonumber\\
+    &= \langle \xi _1, \xi_2\rangle ^* = \langle \xi_2 , \xi_1\rangle ,
+\end{align}
+which concludes the anti-unitarity by definition.
+%------------------- EXERCISE
+\subsubsection{Morphisms Between Finite Real Spectral Triples}
+Like the unitary equivalence relation for finite spectral triples, we can it
+extend it to finite real spectral triples.
+\begin{mydefinition}
+    We call two finite real spectral triples $(A_1, H_1 ,D_1 ; J_1 , \gamma_1)$
+    and $(A_2, H_2, D_2; J_2, \gamma _2)$ unitarily equivalent if $A_1 =
+    A_2$ and if there exists a unitary operator $U: H_1 \rightarrow H_2$ such
+    that
+    \begin{align}
+        U\ \pi_1(a)\ U^* &= \pi _2(a),\\
+        U\ D_1\ U^* &= D_2,\\
+        U \gamma _1\ U^*  &= \gamma _2,\\
+        U\ J_1\ U^* &= J_2.
+    \end{align}
+\end{mydefinition}
+\begin{mydefinition}
+    Let $E$ be a $B$-$A$ bimodule. The \textit{conjugate Module} $E^\circ$ is
+    given by the $A$-$B$-bimodule.
+    \begin{align}
+        E^\circ = \{\bar{e} : e\in E\},
+    \end{align}
+    with
+    \begin{align}
+    a \cdot \bar{e} \cdot b = b^*\ \bar{e}\ a^*, \;\;\;\; \forall a\in A, b \in
+        B.
+    \end{align}
+\end{mydefinition}
+We bear in mind that $E^\circ$ is not a Hilbert bimodule for $(A, B)$ because
+it doesn't have a natural $B$-valued inner product. But there is a $A$-valued
+inner product on the left $A$-module $E^\circ$ with
+\begin{align}
+    \langle \bar{e}_1, \bar{e}_2 \rangle = \langle e_2 , e_1 \rangle,
+    \;\;\;\; e_1, e_2 \in E.
+\end{align}
+And linearity in $A$ by the terms
+\begin{align}
+    \langle a\ \bar{e}_1, \bar{e}_2 \rangle = a \langle \bar{e}_1, \bar{e}_2
+    \rangle, \;\;\;\; \forall a \in A.
+\end{align}
+
+%------------- EXERCISE
+It turns out that $E^\circ$ is a Hilbert bimodule
+of $(B^{\circ}, A^{\circ})$. A straightforward calculation of the properties of the Hilbert bimodule and its $B^{\circ}$
+valued inner product gives the results. For $\bar{e}_1, \bar{e}_2 \in E^{\circ}$ and $a^\circ \in A,
+b^\circ \in B$ we write
+\begin{align}
+    \langle\bar{e}_1, a^\circ \bar{e}_2\rangle &= \langle\bar{e}_1, Ja^*J^{-1}
+    \bar{e}_2\rangle=\nonumber\\
+    &= \langle\bar{e}_1 , J a^* e_2\rangle \nonumber \\
+    &= \langle J^{-1} e_1, a^* e_2\rangle \nonumber\\
+    & = \langle a^* e_1, e_2\rangle= \langle J^{-1}(a^\circ)^* J e_1, e_2\rangle  \nonumber\\
+    & = \langle J^{-1} (a^\circ)^* \bar{e}_1, e_2\rangle \nonumber\\
+    & = \langle (a^\circ)^* \bar{e}_1 , \bar{e}_2\rangle.
+\end{align}
+Next for $\langle\bar{e}_1, \bar{e}_2 b^\circ\rangle = \langle\bar{e}_1,
+\bar{e_2}\rangle b^\circ$ we obtain
+\begin{align}
+    \langle\bar{e}_1, \bar{e}_2 b^\circ\rangle  &= \langle\bar{e}_1, \bar{e}_2 Jb^*J^{-1}\rangle
+    \nonumber\\
+    &= \langle\bar{e}_1, \bar{e_2}\rangle Jb^*J^{-1} \nonumber \\
+    &= \langle\bar{e}_1, \bar{e}_2\rangle b^\circ.
+\end{align}
+Additionally we get
+\begin{align}
+    (\langle\bar{e}_1, \bar{e}_2)\rangle_{E^\circ})^* &= (\langle e_2, e_1\rangle_E)^*\nonumber\\
+                                          &= \langle e_1, e_2\rangle_E^* \nonumber\\
+                                          &= \langle\bar{e}_2, \bar{e}_2\rangle_{E^\circ}.
+\end{align}
+And finally we have
+\begin{align}
+    \langle\bar{e}, \bar{e}\rangle = \langle e, e\rangle \geq 0
+\end{align}
+%------------- EXERCISE
+
+Given the results thus far, given a Hilbert bimodule $E$ for $(B, A)$ one can
+construct a spectral triple $(B, H', D'; J', \gamma ')$ from $(A, H, D; J,
+\gamma)$. For $H'$ we make a $\mathbb{C}$-valued inner product on $H'$ by combining
+the $A$ valued inner product on $E$ and $E^\circ$ with the
+$\mathbb{C}$-valued inner product on $H$ by defining
+\begin{align}
+    H' := E\otimes _A H \otimes _A E^\circ.
+\end{align}
+Then the action of $B$ on $H'$ takes the following form
+\begin{align}
+    b(e_2 \otimes \xi \otimes \bar{e}_2 ) = (be_1) \otimes \xi \otimes
+    \bar{e}_2.
+\end{align}
+The right action of $B$ on $H'$ defined by action on the right components of
+$E^\circ$ is
+\begin{align}
+    J'(e_1 \otimes \xi \otimes \bar{e}_2) = e_2 \otimes J \xi \otimes
+    \bar{e}_1,
+\end{align}
+where $b^\circ = J' b^* (J')^{-1}$ and $b^* \in B$ is the action on $H'$.
+Hence the connection reads
+\begin{align}
+    &\nabla: E \rightarrow E\otimes _A \Omega _D ^1(A) \\
+    &\bar{\nabla}:E^\circ \rightarrow \Omega _D^1(A) \otimes _A E^\circ,
+\end{align}
+which gives the Dirac operator on $H' = E \otimes _A H \otimes _A
+E^\circ$ as
+\begin{align}
+    D'(e_1 \otimes \xi \otimes \bar{e}_2) = (\nabla e_1) \xi \otimes
+    \bar{e_2}+ e_1 \otimes D\xi \otimes \bar{e}_2 + e_1 \otimes
+    \xi(\bar{\nabla}\bar{e}_2).
+\end{align}
+And the right action of $\omega \in \Omega _D ^1(A)$ on $\xi \in H$ is
+defined by
+\begin{align}
+    \xi \mapsto \epsilon' J \omega ^* J^{-1}\xi.
+\end{align}
+Finally for the grading one obtains
+\begin{align}
+    \gamma ' = 1 \otimes \gamma \otimes 1.
+\end{align}
+
+Summarizing we can write down the following theorem
+\begin{mytheorem}
+    Suppose $(A, H, D; J, \gamma)$ is a finite spectral triple of
+    $KO$-dimension $k$, let $\nabla$ be a connection satisfying the
+    compatibility condition (same as with finite spectral triples).
+    Then $(B, H',D'; J', \gamma')$ is a finite spectral triple of
+    $KO$-Dimension $k$. ($H', D', J', \gamma'$)
+\end{mytheorem}
+
+\begin{proof}
+    The only thing left is to check is, if the $KO$-dimension is preserved.
+    That is one needs to check if if the $\epsilon$'s are the same.
+    \begin{align}
+        &(J')^2 = 1 \otimes J^2 \otimes 1 = \epsilon,\\
+        &J' \gamma '= \epsilon ''\gamma'J'.
+    \end{align}
+    Lastly for $\epsilon '$ one obtains
+    \begin{align}
+        J'D'(e_1 \otimes \xi \otimes \bar{e}_2)&=J'\big((\nabla e_1) \xi \otimes
+        \bar{e_2} + e_1 \otimes D\xi \otimes \bar{e}_2 + e_1 \otimes \xi (\tau
+        \nabla e_2)\big)\nonumber \\
+        &= \epsilon' D'\left(e_2 \otimes J\xi \otimes \bar{e}_2\right)\nonumber\\
+        &= \epsilon' D'J'\left(e_1 \otimes \xi \bar{e}_2\right)
+    \end{align}
+\end{proof}
+
+Let us take a look at $\nabla : E \Rightarrow E \otimes _A \Omega _d^1 (A)$,
+the right connection on $E$ and consider the following anti-linear map
+\begin{align}
+    \tau : E \otimes_A \Omega _D^1 (A) &\rightarrow \Omega _D^1 (A) \otimes_A E^\circ\\
+            e \otimes \omega &\mapsto -\omega ^* \otimes \bar{e}.
+\end{align}
+Interestingly the map $\bar{\nabla} : E^\circ \rightarrow \Omega _D^1(A) \otimes E^\circ$
+with $\bar{\nabla}(\bar{e}) = \tau \circ \nabla(e)$ is a left connection, that means
+show that it satisfied the left Leibniz rule, for one
+\begin{align}
+    \tau \circ \nabla(ae) = \bar{\nabla}(a\bar{e}) = \bar{\nabla}(a^*
+    \bar{e}).
+\end{align}
+And for two
+\begin{align}
+    \tau \circ \nabla(ae) &= \tau(\nabla(e)a) + \tau \circ(e \otimes
+     d(a))\nonumber \\
+     &=a^*\bar{\nabla}(\bar{e}) - d(a)^* \otimes \bar{e}. \nonumber\\
+     &= a^*\bar{\nabla}(\bar{e}) + d(a^*) \otimes \bar{e}.
+\end{align}
+
diff --git a/src/thesis/chapters/heatkernel.tex b/src/thesis/chapters/4_heatkernel.tex
diff --git a/src/thesis/chapters/5_twopointspace.tex b/src/thesis/chapters/5_twopointspace.tex
@@ -0,0 +1,244 @@
+\subsection{Almost-commutative Manifold\label{sec:5}}
+\subsubsection{Two-Point Space}
+One of the basics forms of noncommutative space is the Two-Point space $X
+:= \{x, y\}$. The Two-Point space can be represented by the following spectral triple
+\begin{align}
+    F_X := (C(X) = \mathbb{C}^2, H_F, D_F; J_F, \gamma _f).
+\end{align}
+Three properties of $F_X$ stand out. First of all the action of $C(X)$ on
+$H_F$ is faithful for $dim(H_F) \geq 2$, thus a simple choice for the
+Hilbertspace can be made, for instance $H_F = \mathbb{C}^2$. Furthermore
+$\gamma_F$ is the $\mathbb{Z}_2$ grading, which allows for a decomposition of
+$H_F$ into
+\begin{align}
+   H_F = H_F^+ \otimes H_F^- = \mathbb{C} \otimes \mathbb{C},
+\end{align}
+where
+\begin{align}
+    H_F^\pm = \{\psi \in H_F |\; \gamma_F\psi = \pm \psi\},
+\end{align}
+are two eigenspaces. And lastly the Dirac operator $D_F$ lets us
+interchange between the two eigenspaces $H_F^\pm$,
+\begin{align}
+    D_F =
+        \begin{pmatrix}0 & t \\ \bar{t} & 0\end{pmatrix},  \;\;\;\;\;
+            \text{with} \;\; t\in\mathbb{C}.
+\end{align}
+
+The Two-Point space $F_X$ can only have a real structure if the Dirac
+operator vanishes, i.e. $D_F = 0$. In that case the KO-dimension is 0,
+2 or 6. To elaborate further, we draw the only two diagram representations of
+$F_X$ at $\underbrace{\mathbb{C} \oplus \mathbb{C}}_{C(X)}$ on
+$\underbrace{\mathbb{C} \oplus\mathbb{C}}_{H_F}$, which are
+\begin{figure}[h!] \centering
+\begin{tikzpicture}[
+    dot/.style = {draw, circle, inner sep=0.06cm},
+    no/.style = {},
+    ]
+    \node[no](a) at (0,0) [label=left:$\textbf{1}^\circ$] {};
+    \node[no](b) at (0, -1) [label=left:$\textbf{1}^\circ$] {};
+    \node[no](c) at (1, 0.5) [label=above:$\textbf{1}$] {};
+    \node[no](d) at (2, 0.5) [label=above:$\textbf{1}$] {};
+    \node[dot](d0) at (2,0) [] {};
+    \node[dot](d0) at (1,-1) [] {};
+
+    \node[no](a1) at (6,0) [label=left:$\textbf{1}^\circ$] {};
+    \node[no](b2) at (6, -1) [label=left:$\textbf{1}^\circ$] {};
+    \node[no](c2) at (7, 0.5) [label=above:$\textbf{1}$] {};
+    \node[no](d2) at (8, 0.5) [label=above:$\textbf{1}$] {};
+    \node[dot](d0) at (7,0) [] {};
+    \node[dot](d0) at (8,-1) [] {};
+    \end{tikzpicture}
+    \caption{Two diagram representations of $F_X$}
+\end{figure}\newline
+If the Two-Point space $F_X$ would be a real spectral triple then $D_F$ can
+only go vertically or horizontally. This would mean that $D_F$ vanishes.
+As for the KO-dimension The diagram on the left has KO-dimension 2 and 6, the diagram on the
+right 0 and 4. Yet KO-dimension 4 is ruled out because
+$dim(H_F^\pm) = 1$ (Lemma 3.8 in \cite{ncgwalter}) , which ultimately means $J_F^2 = -1$ is
+not allowed.
+\subsubsection{Product Space}
+By Extending the Two-Point space with a four dimensional Riemannian spin
+manifold, we get an almost commutative manifold $M\times F_X$, given by
+\begin{align}
+    M\times F_X = \big(C^\infty(M, \mathbb{C}^2), L^2(S)\otimes \mathbb{C}^2,
+    D_M\otimes 1 ; J_M\otimes J_F, \gamma_M \otimes \gamma_F\big),
+\end{align}
+where
+\begin{align}
+   C^\infty(M, \mathbb{C}^2) \simeq C^\infty(M) \oplus  C^\infty(M).
+\end{align}
+According to Gelfand duality the algebra $C^\infty(M, \mathbb{C}^2)$ of the
+spectral triple corresponds to the space
+\begin{align}
+    N:= M\otimes X.
+\end{align}
+Keep in mind that we still need to find an appropriate real structure on the
+Riemannian spin manifold, $J_M$. Furthermore the total Hilbertspace can be
+decomposed into $H = L^2(S) \oplus L^2(S)$, such that for $\underbrace{a,b\in
+C^\infty(M)}_{(a, b) \in C^\infty(N)}$ and $\underbrace{\psi, \phi \in
+L^2(S)}_{(\psi, \phi) \in H}$ we have
+\begin{align}
+    (a, b)(\psi, \phi) = (a\psi, b\phi).
+\end{align}
+Along with the decomposition of the total Hilbertspace a
+distance formula on $M\times F_X$ can be considered with
+\begin{align}\label{eq:commutator inequality}
+    d_{D_F}(x,y) = \sup\left\{  |a(x) - a(y)|:a\in A_F, ||[D_F, a]|| \leq
+    1 \right\}.
+\end{align}
+To calculate the distance between two points on the Two-Point space $X= \{x,
+y\}$, between $x$ and $y$, we consider an $a \in \mathbb{C}^2 = C(X)$, which is
+specified by two complex numbers $a(x)$ and $a(y)$. Then we simplify the
+commutator inequality in \eqref{eq:commutator inequality}
+\begin{align}
+    &||[D_F , a]|| = ||(a(y) - a(x))\begin{pmatrix}0 &t\\\bar{t} &0
+    \end{pmatrix}|| \leq 1,\\
+    &\Leftrightarrow |a(y) - a(x)|\leq \frac{1}{|t|}.
+\end{align}
+The supremum then gives us the distance
+\begin{align}
+    d_{D_F} (x,y) = \frac{1}{|t|}.
+\end{align}
+An interesting observation here is that, if the Riemannian spin manifold can be
+represented by a real spectral triple then a real structure $J_M$ exists,
+along the lines it follows that $t=0$ and the distance becomes infinite. This is a
+purely mathematical observation and has no physical meaning.
+
+We can also construct a distance formula on $N$ (in reference to a point $p
+\in M$) between two points on $N=M\times X$, $(p, x)$ and $(p,y)$. Then an $a
+\in C^\infty(N)$ is determined by $a_x(p):=a(p, x)$ and $a_y(p):=a(p, y)$.
+The distance between these two points is
+\begin{align}
+    d_{D_F\otimes 1}(n_1, n_2) =  \sup \left\{ |a(n_1) - a(n_2)|: a\in
+    A, ||[D\otimes 1, a]||\right\}.
+\end{align}
+On the other hand if we consider $n_1 = (p,x)$ and $n_2 = (q, x)$
+for $p,q \in M$ then
+\begin{align}
+    d_{D_M \otimes 1} (n_1, n_2) = |a_x(p) - a_x(q)| \;\;\;\text{for}\;\;
+    a_x\in
+    C^\infty(M) \;\; \text{with} \;\; ||[D\otimes 1, a_x]|| \leq 1
+\end{align}
+The distance formula turns to out to be the geodesic distance formula
+\begin{align}
+    d_{D_M\otimes1}(n_1, n_2) = d_g(p, q),
+\end{align}
+which is to be expected since we are only looking at the manifold.
+However if $n_1 = (p, x)$ and $n_2 = (q, y)$ then the two conditions are
+\begin{align}
+    &||[D_M, a_x]|| \leq 1, \;\;\; \text{and}\\
+    &||[D_M, a_y|| \leq 1.
+\end{align}
+These conditions have no restriction which results in the distance being
+infinite! And $N = M\times X$ is given by two disjoint copies of M  which are
+separated by infinite distance
+
+The distance is only finite if $[D_F, a] < 1$. In this case the commutator
+generates a scalar field and the finiteness of the distance is
+related to the existence of scalar fields.
+
+\subsubsection{$U(1)$ Gauge Group}
+To get a insight into the physical properties of the almost commutative
+manifold $M\times F_X$, that is to calculate the spectral action, we need to
+determine the corresponding Gauge group.
+For this we set of with simple definitions and important propositions to
+help us break down and search for the gauge group of the Two-Point $F_X$
+space which we then extend to $M\times F_X$. We will only be diving
+superficially into this chapter, for further reading we refer to
+\cite{ncgwalter}.
+\begin{mydefinition}
+Gauge Group of a real spectral triple is given by
+\begin{align}
+    \mathfrak{B}(A, H; J) := \{ U = uJuJ^{-1} | u\in U(A)\}.
+\end{align}
+\end{mydefinition}
+\begin{mydefinition}
+    A *-automorphism of a *-algebra $A$ is a linear invertible
+    map
+    \begin{align}
+        &\alpha:A \rightarrow A,\;\;\; \text{with}\\
+        \nonumber\\
+        &\alpha(ab) = \alpha(a)\alpha(b),\\
+        &\alpha(a)^* = \alpha(a^*).
+    \end{align}
+    The \textbf{Group of automorphisms of the *-Algebra $A$} is denoted by
+    $(A)$.\newline
+    The automorphism $\alpha$ is called \textbf{inner} if
+    \begin{align}
+        \alpha(a) = u a u^* \;\;\; \text{for} \;\; U(A),
+    \end{align}
+    where $U(A)$ is
+    \begin{align}
+        U(A) = \{ u\in A|\;\; uu^* = u^*u=1\}. \;\;\;
+        \text{(unitary)}
+    \end{align}
+\end{mydefinition}
+The Gauge group of $F_X$ is given by the quotient $U(A)/U(A_J)$.
+To get a nontrivial Gauge group so we need to choose a $U(A_J) \neq
+U(A)$ and $U((A_F)_{J_F}) \neq U(A_F)$.
+We consider our Two-Point space $F_X$ to be equipped with a real structure,
+which means the operator vanishes, and the spectral triple representation is
+\begin{align}
+    F_X := \left(\mathbb{C}^2,\mathbb{C}^2, D_F =\begin{pmatrix}
+        0&0\\0&0\end{pmatrix}; J_f =\begin{pmatrix}
+    0&C\\C&0\end{pmatrix},
+            \gamma_F = \begin{pmatrix}1&0\\0&-1\end{pmatrix}\right).
+\end{align}
+Here $C$ is the complex conjugation, and $F_X$ is a real even finite
+spectral triple (space) of KO-dimension 6.
+
+\begin{myproposition}
+The Gauge group of the Two-Point space $\mathfrak{B}(F_X)$ is $U(1)$.
+\end{myproposition}
+\begin{proof}
+    Note that $U(A_F) = U(1) \times U(1)$. We need to show that $U(A_F) \cap
+    U(A_F)_{J_F}) \simeq U(1)$, such that $\mathfrak{B}(F) \simeq U(1)$. So
+    for an element $a \in \mathbb{C}^2$ to be in $(A_F)_{J_F}$, it has to
+    satisfy $J_F a^* J_F = a$,
+    \begin{align}
+        J_F a^* J^{-1} =
+        \begin{pmatrix}0&C\\C&0\end{pmatrix}
+            \begin{pmatrix}\bar{a}_1&0\\0&\bar{a}_2\end{pmatrix}
+        \begin{pmatrix}0&C\\C&0\end{pmatrix}
+            =
+            \begin{pmatrix}a_2&0\\0&a_1\end{pmatrix}.
+    \end{align}
+    This can only be the case if $a_1 = a_2$. So we have
+    $(A_F)_{J_F} \simeq \mathbb{C}$, whose unitary elements
+    from $U(1)$ are contained in the diagonal subgroup of
+    $U(A_F)$.
+\end{proof}
+
+An arbitrary hermitian field $A_\mu = -ia\partial _\mu b$  is given by
+two $U(1)$ Gauge fields $X_\mu^1, X_\mu^2 \in C^\infty(M, \mathbb{R})$.
+However $A_\mu$ appears in combination $A_\mu - J_F A_\mu J_F^{-1}$:
+\begin{align}
+ A_\mu - J_F A_\mu J_F^{-1} =
+    \begin{pmatrix}X_\mu^1&0\\0&X_\mu^2 \end{pmatrix}
+        -
+    \begin{pmatrix}X_\mu^2&0\\0&X_\mu^1 \end{pmatrix}
+        =:
+    \begin{pmatrix}Y_\mu&0\\0&-Y_\mu \end{pmatrix}
+    = Y_\mu \otimes \gamma _F,
+\end{align}
+where $Y_\mu$ the $U(1)$ Gauge field is defined as
+\begin{align}
+    Y_\mu := X_\mu^1 - X_\mu^2 \in C^\infty(M, \mathbb{R}) = C^\infty(M,
+    i\ u(1)).
+\end{align}
+
+\begin{myproposition}
+    The inner fluctuations of the almost-commutative manifold $M\times
+    F_X$ are parameterized by a $U(1)$-gauge field $Y_\mu$ as
+    \begin{align}
+        D \mapsto D' = D + \gamma ^\mu Y_\mu \otimes \gamma_F
+    \end{align}
+    The action of the gauge group $\mathfrak{B}(M\times F_X) \simeq
+    C^\infty (M, U(1))$ on $D'$ is implemented by
+    \begin{align}
+        Y_\mu \mapsto Y_\mu - i\ u\partial_\mu u^*; \;\;\;\;\; (u\in
+        \mathfrak{B}(M\times F_X)).
+    \end{align}
+\end{myproposition}
+
diff --git a/src/thesis/chapters/6_electroncg.tex b/src/thesis/chapters/6_electroncg.tex
@@ -0,0 +1,465 @@
+\subsection{Noncommutative Geometry of Electrodynamics\label{sec:5}}
+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 $M\times F_X$ 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 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 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
+the real structure we arrive at the following relations
+\begin{align}
+    J_F e &= \bar{e} \;\;\;\;\;\; J_F \bar{e} = e, \\
+    \gamma_F e &= e  \;\;\;\;\;\;   \gamma_F \bar{e} = \bar{e}.
+\end{align}
+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^-.
+\end{align}
+So, for an $\xi \in H^+$ we can write
+\begin{align}
+    \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 $\psi :=
+\psi_L + \psi _R$. 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
+\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}. This way we have
+$\{ \underbrace{e_R, e_L}_{\text{left-handed}}, \underbrace{\bar{e}_R,
+\bar{e}_L}_{\text{right-handed}}\}$ 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, \\
+    &J_F e_L = \bar{e_L}, \\
+    \nonumber \\
+    &\gamma _F e_R = -e_R,\\
+    &\gamma_F e_L = e_L,
+\end{align}
+where $J_F$ and $\gamma_F$ have to following properties
+\begin{align}
+    &J_F^2 = 1,\\
+    & J_F \gamma_F  = - \gamma_F J_F.
+\end{align}
+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\}}
+    \oplus \underbrace{H_F^-}_{\text{ONB } \{e_R, \bar{e}_R\}},
+\end{align}
+or alternatively into the eigenspace of particles and their
+antiparticles (electrons and positrons) which is preferred in literature and
+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\}},
+\end{align}
+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
+\begin{align}\label{eq:leftrightrepr}
+    a =
+    (a_1 , a_2 ) \mapsto
+        \begin{pmatrix}
+            a_1 &0 &0 &0\\
+             0&a_1 &0 &0\\
+            0 &0 &a_2 &0\\
+            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 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}
+            0 & C \\ C &0
+        \end{pmatrix},
+        \gamma _F =
+        \begin{pmatrix}
+            1 & 0 \\ 0 &-1
+        \end{pmatrix}
+        \right).
+    \end{align}
+It can be represented in the following Krajewski diagram,
+with two nodes of multiplicity two bellow
+    \begin{figure}[H] \centering
+    \begin{tikzpicture}[
+        dot/.style = {draw, circle, inner sep=0.06cm},
+        bigdot/.style = {draw, circle, inner sep=0.09cm},
+        no/.style = {},
+        ]
+        \node[no](a) at (0,0) [label=left:$\textbf{1}^\circ$] {};
+        \node[no](b) at (0, -1) [label=left:$\textbf{1}^\circ$] {};
+        \node[no](c) at (0.5, 0.5) [label=above:$\textbf{1}$] {};
+        \node[no](d) at (1.5, 0.5) [label=above:$\textbf{1}$] {};
+        \node[dot](d0) at (1.5,0) [] {};
+        \node[dot](d0) at (0.5,-1) [] {};
+        \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
+closer look at the Krajewski diagram above. Notice that edges only exist
+between multiple vertices, meaning we can construct a Dirac operator mapping
+between the two vertices. The operator can be represented by the following matrix
+\begin{align}\label{eq:feddirac}
+    D_F =
+    \begin{pmatrix}
+    0 & d & 0 & 0 \\
+    \bar{d} & 0 & 0 & 0 \\
+    0 & 0 & 0 & \bar{d} \\
+    0 & 0 & d & 0
+    \end{pmatrix}
+\end{align}
+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 as in equation \eqref{eq:fedfail} and $D_F$
+from equation \eqref{eq:feddirac}.
+
+\subsubsection{Almost commutative Manifold of Electrodynamics}
+The almost commutative manifold $M\times F_{ED}$ has KO-dimension 2, and is
+represented by the following spectral triple
+\begin{align}\label{eq:almost commutative manifold}
+    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).
+\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).
+\end{align}
+As for the Hilbertspace, we can decomposition it in the following way
+\begin{align}
+    H = (L^2(S) \otimes H_e ) \oplus (L^2(S) \otimes H_{\bar{e}}).
+\end{align}
+Note that the one component of the algebra is acting on $L^2(S) \otimes H_e$,
+and the other one acting on $L^2(S) \otimes H_{\bar{e}}$. In other words the components of
+the decomposition of both the algebra and the Hilbertspace match by the action of
+the algebra.
+
+The derivation of the gauge theory is the same for $F_{ED}$ as for the
+Two-Point space $F_X$. We have $\mathfrak{B}(F) \simeq U(1)$ and for an
+arbitrary gauge field $B_\mu = A_\mu - J_F A_\mu J_F^{-1}$ we can write
+\begin{align} \label{field}
+    B_\mu =
+    \begin{pmatrix}
+        Y_\mu & 0 & 0 & 0 \\
+        0 & Y_\mu& 0 & 0 \\
+        0 & 0 & Y_\mu& 0 \\
+        0 & 0 & 0 & Y_\mu
+    \end{pmatrix} \;\;\;\;\;\ \text{for} \;\;\ Y_\mu (x) \in \mathbb{R}.
+\end{align}
+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, 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
+observation and doesn't affect physical results.
+
+\subsubsection{Spectral Action}
+In this chapter we bring all our results together to establish an
+Action functional to describe a physical system. It turns out that
+the Lagrangian of the almost commutative manifold $M\times F_{ED}$
+corresponds to the Lagrangian of Electrodynamics on a curved
+background manifold (+ gravitational Lagrangian), consisting of the spectral
+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
+\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
+    \begin{align}
+        S_b [\omega] := \text{Tr}\big(f(\frac{D_\omega}{\Lambda})\big)
+    \end{align}
+    where $\Lambda$ is a real cutoff parameter. The minimal condition on $f$
+    is that $f(\frac{D_\omega}{\Lambda})$ is  a trace class operator. A trace
+    class operator is a compact operator with a well defined finite trace
+    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
+    action $S_{top}$. Leaning on the grading $\gamma$ the topological spectral action is
+    \begin{align}
+        S_{\text{top}}[\omega] := \text{Tr}(\gamma\
+        f(\frac{D_\omega}{\Lambda})).
+    \end{align}
+\end{mydefinition}
+\begin{mydefinition}\label{def:fermionic action}
+    The fermionic action is defined by
+    \begin{align}
+        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
+    of the grading $\gamma$.
+\end{mydefinition}
+
+%---------------------- 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).
+\end{align}
+%---------------------- APPENDIX ?????????????--------------------
+\begin{myproposition}
+    The spectral action of the almost commutative manifold $M$ with $\dim(M)
+    =4$ with a fluctuated Dirac operator is
+    \begin{align}
+        \text{Tr}(f\frac{D_\omega}{\Lambda}) \sim \int_M \mathcal{L}(g_{\mu\nu},
+         B_\mu, \Phi) \sqrt{g}\ d^4x + O(\Lambda^{-1}),
+    \end{align}
+    where
+    \begin{align}
+        \mathcal{L}(g_{\mu\nu}, B_\mu, \Phi) =
+        N\mathcal{L}_M(g_{\mu\nu})
+        \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 $(C^\infty(M) ,
+    L^2(S), D_M)$, 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}
+    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
+    $R_{\nu\sigma} = g^{\mu\varrho} R_{\mu\nu \varrho\sigma}$ 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}
+        \mathcal{L}_B(B_\mu) := \frac{f(0)}{24\pi^2}
+        \text{Tr}(F_{\mu\nu}F^{\mu\nu}).
+    \end{align}
+    Lastly $\mathcal{L}_\phi$ is the scalar-field Lagrangian with a boundary
+    term, given by
+    \begin{align}
+        \mathcal{L}_\phi(g_{\mu\nu}, B_\mu, \Phi) :=
+        &-\frac{2f_2\Lambda^2}{4\pi^2}\text{Tr}(\Phi^2) + \frac{f(0)}{8\pi^2}
+        \text{Tr}(\Phi^4) + \frac{f(0)}{24\pi^2}
+        \Delta(\text{Tr}(\Phi^2))\nonumber\\
+        &+ \frac{f(0)}{48\pi^2}s\text{Tr}(\Phi^2)
+        \frac{f(0)}{8\pi^2}\text{Tr}((D_\mu \Phi)(D^\mu \Phi)).
+    \end{align}
+\end{myproposition}
+\begin{proof}
+     The dimension of the manifold $M$ is $\dim(M) = \text{Tr}(id) =4$. For
+     an $x \in M$, we have an asymptotic expansion of the term
+     $\text{Tr}(f(\frac{D_\omega}{\Lambda}))$ as $\Lambda$ goes to infinity,
+     which can be written as
+     \begin{align}
+         \text{Tr}(f(\frac{D_\omega}{\Lambda})) \simeq& \ 2f_4 \Lambda ^4
+         a_0(D_\omega ^2)+ 2f_2\Lambda^2 a_2(D_\omega^2)\nonumber \\&+ f(0) a_4(D_\omega^4)
+         +O(\Lambda^{-1}).\label{eq:trheatkernel}
+     \end{align}
+     We have to note here that the heat kernel coefficients are zero for uneven $k$,
+     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 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
+         \text{Tr}(\Phi^2)\sqrt{g}d^4x.
+     \end{align}
+     For $a_4$ we extend in terms of coefficients of $F$ from equation
+     \eqref{eq: a_4}
+     \begin{align}
+         &\frac{1}{360}\text{Tr}(60RE)= -\frac{1}{6}S(NR + 4
+         \text{Tr}(\Phi^2))\\
+        \nonumber\\
+         &E^2 = \frac{1}{16}R^2\otimes 1 + 1\otimes \Phi^4 - \frac{1}{4}
+         \gamma^\mu\gamma^\nu \gamma^\varrho\gamma^\sigma
+         F_{\mu\nu}F^{\mu\nu}+\nonumber\\
+         &\;\;\;\;\;\;\;+\gamma^\mu\gamma^\nu\otimes(D_\mu\Phi)(D_\nu
+         \Phi)+\frac{1}{2}s\otimes \Phi^2 + \ \text{traceless terms},\\
+         \nonumber\\
+         &\frac{1}{360}\text{Tr}(180E^2) = \frac{1}{8}R^2N + 2\text{Tr}(\Phi^4)
+         + \text{Tr}(F_{\mu\nu}F^{\mu\nu}) +\nonumber\\
+         &\;\;\;\;\;\;\;+2\text{Tr}((D_\mu\Phi)(D^\mu\Phi))
+         + s\text{Tr}(\Phi^2)\\
+         \nonumber\\
+         &\frac{1}{360}\text{Tr}(-60\Delta E)=
+         \frac{1}{6}\Delta(NR+4\text{Tr}(\Phi^2)).
+     \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, 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
+         \otimes F^{\mu\nu}.
+     \end{align}
+     The trace  of the cross term $\Omega^{S}_{\mu\nu}$ vanishes because
+     \begin{align}
+         \text{Tr}(\Omega^{S}_{\mu\nu}) = \frac{1}{4}
+         R_{\mu\nu\varrho\sigma}\text{Tr}(\gamma^\mu\gamma^\nu) = \frac{1}{4}
+         R_{\mu\nu\varrho\sigma}g^{\mu\nu} =0,
+     \end{align}
+     then the trace of the whole term is given by
+     \begin{align}
+         \frac{1}{360}\text{Tr}(30\Omega^E_{\mu\nu}\Omega^{E\mu\nu}) =
+         \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
+     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 \\
+         &+ \frac{1}{4}
+         \text{Tr}((D_\mu\Phi)(D^\mu \Phi)) + \frac{1}{6}
+         \Delta\text{Tr}(\Phi^2) + \frac{1}{6}
+         \text{Tr}(F_{\mu\nu}F^{\mu\nu})\bigg).
+     \end{align}
+     The only thing left is to substitute the heat kernel coefficients into the
+     heat kernel expansion in equation \eqref{eq:trheatkernel}.
+\end{proof}
+
+\subsubsection{Fermionic Action}
+We remind ourselves the definition of the fermionic action in definition
+\ref{def:fermionic action} and the manifold we are dealing with in equation
+\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 $H_F^-$, $\{e_R, \bar{e}_L\}$. 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+
+    \psi_R \otimes \bar{e}_L,
+\end{align}
+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 $F_X$) 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
+    D_F.
+\end{align}
+The Fermionic Action is
+\begin{align}
+S_F = (J\tilde{\xi}, D_\omega\tilde{\xi})
+\end{align}
+for a $\xi \in H^+$. Then the straight forward calculation gives \begin{align}
+    \frac{1}{2}(J\tilde{\xi}, D_\omega\tilde{\xi})
+        &=\frac{1}{2}(J\tilde{\xi}, (D_M \otimes
+        i)\tilde{\xi})\label{eq:fermionic1}\\
+        &+\frac{1}{2}(J\tilde{\xi}, (\gamma^\mu \otimes B_\mu)
+        \tilde{\xi})\label{eq:fermionic2}\\
+        &+\frac{1}{2}(J\tilde{\xi}, (\gamma_M\otimes
+        D_F)\tilde{\xi})\label{eq:fermionic3},
+\end{align}
+(note that we add the constant $\frac{1}{2}$ to the action).
+For the term in \eqref{eq:fermionic1} we calculate
+\begin{align}
+    \frac{1}{2}(J\tilde{\xi}, (D_M\otimes 1)\tilde{\xi}) &=
+    \frac{1}{2}(J_M\tilde{\chi}_R,D_M\tilde{\psi}_L)+\nonumber
+    \frac{1}{2}(J_M\tilde{\chi}_L,D_M\tilde{\psi}_R)+
+    \\&+\frac{1}{2}(J_M\tilde{\psi}_L,D_M\tilde{\psi}_R)+\nonumber
+    \frac{1}{2}(J_M\tilde{\chi}_R,D_M\tilde{\chi}_L)\\
+    &= (J_M\tilde{\chi},D_M\tilde{\chi}).
+\end{align}
+For the term in \eqref{eq:fermionic2} we have
+\begin{align}
+    \frac{1}{2}(J\tilde{\xi}, (\gamma^\mu \otimes B_\mu)\tilde{\xi})&=
+    -\frac{1}{2}(J_M\tilde{\chi}_R, \gamma^\mu Y_\mu\tilde{\psi}_R)
+    -\frac{1}{2}(J_M\tilde{\chi}_L, \gamma^\mu Y_\mu\tilde{\psi}_R)+\nonumber\\
+    &+\frac{1}{2}(J_M\tilde{\psi}_L, \gamma^\mu Y_\mu\tilde{\chi}_R)+
+    \frac{1}{2}(J_M\tilde{\psi}_R, \gamma^\mu Y_\mu\tilde{\chi}_L)=\nonumber\\
+    &= -(J_M\tilde{\chi}, \gamma^\mu Y_\mu\tilde{\psi}).
+\end{align}
+And for \eqref{eq:fermionic3} we can write
+\begin{align}
+    \frac{1}{2}(J\tilde{\xi}, (\gamma_M\otimes D_F)\tilde{\xi})&=
+    +\frac{1}{2}(J_M\tilde{\chi}_R, d\gamma_M\tilde{\chi}_R)
+    +\frac{1}{2}(J_M\tilde{\chi}_L, \bar{d}\gamma_M\tilde{\chi}_L)+\nonumber\\
+    &+\frac{1}{2}(J_M\tilde{\chi}_L, \bar{d}\gamma_M\tilde{\chi}_L)
+    +\frac{1}{2}(J_M\tilde{\chi}_R, d\gamma_M\tilde{\chi}_R)=\nonumber\\
+    &= i(J_M\tilde{\chi}, m\tilde{\psi}).
+\end{align}
+A small problem arises, we obtain a complex mass parameter $d$, but we can
+write $d:=im$ for $m\in \mathbb{R}$, which stands for the real mass.
+
+Finally the fermionic action of $M\times F_{ED}$ takes the form
+    \begin{align}
+        S_f = -i\big(J_M\tilde{\chi}, \gamma(\nabla^S_\mu - i\Gamma_\mu)
+        \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 the almost commutative
+manifold $M\times F_{ED}$, which is the sum of the purely gravitational
+Lagrangian
+\begin{align}\label{eq:final1}
+        \mathcal{L}_{grav}(g_{\mu\nu})=4\mathcal{L}_M(g_{\mu\nu})+
+        \mathcal{L}_\phi (g_{\mu\nu}),
+    \end{align}
+and the Lagrangian of electrodynamics
+\begin{align}\label{eq:final2}
+        \mathcal{L}_{ED} = -i\bigg\langle
+        J_M\tilde{\chi},\big(\gamma^\mu(\nabla^S_\mu - iY_\mu) -m\big)\tilde{\psi})
+        \bigg\rangle
+        +\frac{f(0)}{6\pi^2} Y_{\mu\nu}Y^{\mu\nu}.
+    \end{align}
+
diff --git a/src/thesis/chapters/acknowledgment.tex b/src/thesis/chapters/acknowledgment.tex
@@ -1,3 +1,9 @@
-
 \section{Acknowledgment}
-\lipsum[1]
+First and foremost, I thank my supervisor Lisa Glaser for introducing me to
+this rich field of study, where I had the chance to learn from a broad variety of
+different mathematical and physical topics accompanied with great reflection sessions with
+my supervisor every week. I thank my fellow students for the great time,
+for the learning groups in preparation for exams and for on topic discussions
+during my studies. And lastly I thank my friends and family for their continuous
+love and support, for healthy breaks, and for discussions in different topics
+outside of physics.
diff --git a/src/thesis/chapters/basics.tex b/src/thesis/chapters/basics.tex
@@ -1,564 +0,0 @@
-\subsection{Noncommutative Geometric Spaces\label{sec:1}}
-\subsubsection{$*$-Algebra}
-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 for all $a,b \in A$:
-            \begin{align}
-            A \times A \rightarrow A\\
-            (a,\ b)\ &\mapsto \ a\cdot b,
-            \end{align}
-    with an identity element:
-            \begin{align}
-            1a = a1 =a.
-            \end{align}
-    Extending the definition, a $*$-algebra is an algebra $A$ with a \textit{conjugate linear map (involution)} $*:A\ \rightarrow  A$,
-    $\forall a, b \in A$ satisfying
-    \begin{align}
-        (a\ b)^* &= b^*a^*,\\
-        (a^*)^* &= a.
-    \end{align}
-\end{mydefinition}
-In the following all unital algebras are referred to as algebras.
-
-\subsubsection{Finite Discrete Space}
-Let us consider an example, a $*$-algebra of continuous functions $C(X)$
-on a discrete topological space $X$ with $N$ points. Functions of a
-continuous $*$-algebra $C(X)$ assign values to $\mathbb{C}$ and for $f,\ g \in
-C(X)$, $\lambda \in \mathbb{C}$ and $x \in X$ they provide the following structure:
-\begin{itemize}
-    \item \textit{pointwise linear}
-        \begin{align}
-            (f + g)(x) &= f(x) + g(x),\\
-            (\lambda\ f)(x) &= \lambda (f(x)),
-        \end{align}
-    \item \textit{pointwise multiplication}
-        \begin{align}
-        f\ g\ (x) = f(x)g(x),
-        \end{align}
-    \item \textit{pointwise involution}
-        \begin{align}
-        f^*(x) = \overline{f(x)}.
-        \end{align}
-\end{itemize}
-The $*$-algebra $C(X)$ is \textit{isomorphic} to a $*$-algebra $\mathbb{C}^N$
-with involution ($N$ number of points in $X$), we write $C(X) \simeq
-\mathbb{C}^N$.  Isomorphisms are bijective maps that preserve structure and
-don't lose physical information.  A function $f:X\ \rightarrow\ \mathbb{C}$
-can be represented with $N \times N$ diagonal matrices, where each diagonal
-value represents the function value at the corresponding $i$-th point for $i
-= 1,...,N$. Matrix multiplication and hermitian conjugation of
-matrices we have a preserving structure.
-
-Moreover we can \textit{map} between finite discrete spaces $X_1$ and $X_2$ with a
-function
-\begin{align}
-    \phi:\ X_1 \rightarrow\ X_2.
-\end{align}
-For every such map there exists a corresponding map
-\begin{align}
-    \phi ^*:C(X_2)\ \rightarrow C(X_1),
-\end{align}
-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{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 $\phi ^* :C(X_2)\ \rightarrow \
-    C(X_1)$ is surjective (injective). To clarify let us say that $X_1$ has $n$ points and
-    $X_2$ with $m$ points. Then there are three different cases, first $n=m$ and
-    obviously $\phi$ is bijective and $\phi ^*$ too. Then $n >  m$, in this case
-    $\phi$ assigns $n$ points to $m$ points when $n >  m$, which is by definition
-    surjective. On the other hand $\phi ^*$ assigns $m$ points to $n$ points when
-    $n >  m$, which is by definition injective. Lastly $n < m $, which is
-    completely analogous to the case $n > m$.
-%------------ Exercise
-
-\begin{mydefinition}
-    A \textit{(complex) matrix algebra} A is a direct sum, for $n_i, N \in
-    \mathbb{N}$
-    \begin{align}
-        A = \bigoplus _{i=1}^{N} M_{n_i}(\mathbb{C}).
-    \end{align}
-    The involution is the hermitian conjugate. A $*$ algebra with involution is referred to as
-    a matrix algebra
-\end{mydefinition}
-
-To summarize, from a topological discrete space $X$, we can construct a
-$*$-algebra $C(X)$ which is isomorphic to a matrix algebra $A$. Then the
-question instantly arises, if we can construct $X$ given $A$? For a matrix
-algebra $A$, which in most cases is not commutative, the answer is generally
-no. Hence there are two options. We can restrict ourselves to commutative
-matrix algebras, which are the vast minority and not physically interesting.
-Or we can allow more morphisms (isomorphisms) between matrix algebras.
-
-\subsubsection{Finite Inner Product Spaces and Representations}
-Until now we have looked at finite topological discrete spaces, moreover we can consider a
-finite dimensional inner product space $H$ (finite Hilbertspaces), with inner product
-$(\cdot,\cdot)\rightarrow \mathbb{C}$. We denote $L(H)$ as the $*$-algebra of operators on $H$
-equipped with a product given by composition and involution of the adjoint, $T \mapsto T^*$.
-Then $L(H)$ is a \textit{normed vector space} with
-\begin{align}
-    \|T\|^2 &= \sup_{h \in H}\big\{(T\ h,\ T\ h): (h,\ h) \leq 1\big|\ T
-    \in L(H)\big \},\\
-    \|T\| &= \sup\big\{\sqrt{\lambda}:\; \lambda \text{ eigenvalue of } T\big\}.
-\end{align}
-The Hilbert space allows us to define representations of $*$-algebras.
-\begin{mydefinition}
-    The \textit{representation} of a finite dimensional $*$-algebra $A$ is a
-    pair $(H, \pi)$, where $H$ is a finite dimensional inner product space
-    and $\pi$ is a $*$-\textit{algebra map}
-    \begin{align}
-        \pi:A\ \rightarrow \ L(H).
-    \end{align}
-    We call the representation $(H, \pi)$ \textit{irreducible} if
-    \begin{itemize}
-        \item $H \neq \emptyset$,
-        \item only $\emptyset$ or $H$ is invariant under the action of $A$ on
-            $H$.
-    \end{itemize}
-\end{mydefinition}
-Here are some examples of reducible and irreducible representations
-\begin{itemize}
-    \item For $A = M_n(\mathbb{C})$ the representation $H=\mathbb{C}^n$, $A$ acts as matrix multiplication\\
-            $H$ is irreducible.
-    \item For $A = M_n(\mathbb{C})$ the representation $H=\mathbb{C}^n\oplus \mathbb{C}^n$, with $a \in A$ acting
-        in block form \\ $\pi: a \mapsto \big(\begin{smallmatrix} a & 0\\ 0 & a \end{smallmatrix}\big)$ is
-            reducible.
-\end{itemize}
-Naturally there are also certain equivalences between different
-representations.
-\begin{mydefinition}
-Two representations of a $*$-algebra $A$, $(H_1, \pi _1)$ and
-$(H_2, \pi _2)$  are called \textit{unitary equivalent} if there exists a map
-$U: H_1 \rightarrow H_2$ such that.
-    \begin{align}
-        \pi _1(a) = U^* \pi _2(a) U
-    \end{align}
-\end{mydefinition}
-
-Furthermore we define a mathematical structure called the structure space,
-which will become important later when speaking of the duality between a
-spectral triple and a geometrical space.
-\begin{mydefinition}
-    Let $A$ be a $*$-algebra then, $\hat{A}$ is called the structure space of all \textit{unitary equivalence classes
-    of irreducible representations of A}.
-\end{mydefinition}
-%------------- EXERCISE
-    Given a representation $(H, \pi)$ of a $*$-algebra $A$, the \textbf{commutant} $\pi (A)'$ of $\pi (A)$ is defined as a set
-    of operators in $L(H)$ that commute with all $\pi (a)$
-    \begin{align}
-        \pi (A)' = \big\{T \in L(H):\ \pi(a)\ T = T\ \pi(a) \;\; \forall a\in
-        A\big\}
-    \end{align}
-    The commutant $\pi (A)'$ is also a $*$-algebra, since it has unital,
-    associative and involutive properties.  The unitary property is given by
-    the unital operator of the $*$-algebra of operators $L(H)$, which exists
-    by definition because $H$ is a inner product space. Associativity is
-    given by the $*$-algebra of $L(H)$, where $L(H) \times L(H)~\mapsto
-    L(H)$, which is associative by definition. The involutive property is
-    also given by the $*$-algebra $L(H)$ with a map $*: L(H) \mapsto L(H)$
-    only for a $T \in H$ that commutes with $\pi (a)$.
-%------------- EXERCISE
-
-%------------- EXERCISE
-    For a unital algebra $*$-algebra $A$, the matrices $M_n(A)$ with entries
-    in $A$ form a unital $*$-algebra, because the unitary operation in
-    $M_n(A)$ is given by the identity Matrix, which exists in every
-    entry in $M_n(A)$ and behaves like in $A$. Associativity is given by
-    matrix multiplication. Lastly, involution is given by the conjugate
-    transpose.
-
-    Consider a representation $\pi :A\ \rightarrow \ L(H)$ of a $*$-algebra
-    $A$ and set $H^n = H \oplus ... \oplus H$, $n$ times. Then we have the following
-    representation $\tilde{\pi}:M_n(A) \rightarrow L(H^n)$ for the Matrix
-    Algebra with $\tilde{\pi}((a_{ij})) = (\tilde{\pi}(a_{ij})) \in M_n(A)$,
-    since a direct isomorphisms of $A \simeq M_n(A)$ and $H \simeq H^n$
-    exists. Meaning $\tilde{\pi}$ is a valid reducible representation.
-
-    By looking at $\tilde{\pi}:M_n(A) \rightarrow L(H^n)$ a $*$ algebra
-    representation of $M_n(A)$. We see that $\pi: A \rightarrow L(H^n)$ is a representation of $A$.
-    The fact that $\tilde{\pi}$ and $\pi$ are unitary equivalent, there is
-    a map $U: H^n \rightarrow H^n$ given by $U=\mathbbm{1}_n$, thus
-    \begin{align}
-        \pi (a) &= \mathbbm{1}_n^*\ \tilde{\pi}((a_{ij})), \\
-        \mathbbm{1}_n &= \tilde{\pi}((a_{ij})) = \pi (a_{ij})
-    \Rightarrow a_{ij} = a\ \mathbbm{1}_n.
-    \end{align}
-%------------- EXERCISE
-
-
-With help of the structure space $\hat{A}$, a commutative matrix algebra can be used to reconstruct a discrete space.
-Since $A \simeq \mathbb{C}^N$ all irreducible representation are of the form
-\begin{align}
-   \pi _i:(\lambda_1,...,\lambda_N)\in \mathbb{C}^N \mapsto \lambda_i \in
-   \mathbb{C}
-\end{align}
-for $i = 1,...,N$, and thus $\hat{A} \simeq \{1,...,N\}$.
-We can conclude that there is a duality between discrete spaces and
-commutative matrix algebras. This duality is called the \textit{finite
-dimensional Gelfand duality}
-
-Our aim is to make a further generalization by constructing a duality between
-finite dimensional spaces and \textit{equivalence classes} of matrix
-algebras that preserves general non-commutativity of matrices. Equivalence
-classes are described by a concept of isomorphisms between matrix
-algebras called \textit{Morita Equivalence}.
-
-\subsubsection{Algebraic Modules}
-An important part of the Morita Equivalence are algebraic modules, later
-extended by Hilbert bimodules.
-\begin{mydefinition}
-    Let $A$, $B$ be algebras (need not be matrix algebras)
-    \begin{enumerate}
-        \item \textit{left} A-module is a vector space $E$, that carries a left
-            representation of $A$, that is $\exists$ a bilinear map $\gamma: A
-            \times E \rightarrow E$ with
-            \begin{align}
-                (a_1\ a_2)\cdot e = a_1 \cdot (a_2 \cdot e);\;\;\; a_1, a_2 \in
-                A, e \in E.
-            \end{align}
-        \item \textit{right} B-module is a vector space $F$, that carries a
-            right representation of $A$, that is there exists a bilinear map
-            $\gamma: F \times B \rightarrow F$ with
-            \begin{align}
-                f \cdot (b_1\ b_2)= (f \cdot b_1) \cdot b_2;\;\;\; b_1, b_2 \in B, f \in F
-            \end{align}
-        \item \textit{left} A-module and \textit{right} B-module is a
-            \textit{bimodule}, a vector space $E$ satisfying
-            \begin{align}
-                a \cdot (e \cdot b)= (a \cdot e) \cdot b;\;\;\;  a \in A, b \in B, e \in E
-            \end{align}
-    \end{enumerate}
-\end{mydefinition}
-An $A$-\textbf{module homomorphism} is linear map $\phi: E\rightarrow F$ which respects the
-representation of A, e.g.\ for left module.
-\begin{align}
-    \phi (a\ e) = a \phi (e); \;\;\; a \in A, e \in E.
-\end{align}
-We will use the notation
-\begin{itemize}
-    \item ${}_A E$, for left $A$-module $E$;
-    \item ${}_A E_B$, for right $B$-module $F$;
-    \item ${}_A E_B$, for $A$-$B$-bimodule $E$, simply bimodule.
-\end{itemize}
-%------------------- EXERCISE
-From a simple observation, we see that an arbitrary representation $\pi : A
-\rightarrow L(H)$ of a $*$-algebra A, turns H into a left module ${}_A H$.  If
-$_A H$ than $(a_1\ a_2) h = a_1 (a_2\ h)$ for $a_1, a_2 \in A$ and $h \in H$. We
-take the representation of an $a \in A$, $\pi (a)$, and write
-\begin{align}
-    \big(\pi(a_1)\ \pi(a_2)\big)h = \pi(a_1)\big(\pi(a_2)\ h\big) =
-    \big(T_1\ T_2\big) h = T_1 \big(T_2\ h\big)
-\end{align}
-For $T_1, T_2 \in L(H)$, which operate naturally from the left on $h$.
-
-%------------------- EXERCISE
-%------------------- EXERCISE
-
-Furthermore notice that that an $*$-algebra $A$ is a bimodule ${}_A A_A$ with
-itself, given by the map
-\begin{align}
-    \gamma: A\times A\times A \rightarrow A,
-\end{align}
-which is the inner product of a $*$-algebra.
-%------------------- EXERCISE
-
-\subsubsection{Balanced Tensor Product and Hilbert Bimodules}
-In this chapter we introduce the balanced tensor product later called the
-Kasparov product. This operation allows us to naturally construct a bimodule
-of a third algebra in chapter \ref{chap: kasparov product}.
-\begin{mydefinition}
-    Let $A$ be an algebra, $E$ be a \textit{right} $A$-module and $F$ be a
-    \textit{left} $A$-module.  The \textit{balanced tensor product} of $E$ and
-    $F$ forms a $A$-bimodule.
-    \begin{align}
-        E \otimes _A F := E \otimes F / \left\{\sum _i e_i a_i \otimes f_i -
-        e_i \otimes a_i f_i : \;\;\; a_i \in A,\ e_i \in E,\ f_i \in F
-    \right\}.
-    \end{align}
-\end{mydefinition}
-The symbol $/$ denotes the quotient space. By careful examination we can say
-that the operation $\otimes _A$ takes two left/right modules and makes a
-bimodule. Additionally with the help of the tensor product of the two modules and the quotient
-space which takes out all the elements from the tensor product that don't
-preserver the left/right representation and that are duplicates.
-\begin{mydefinition}
-    Let $A$, $B$ be \textit{matrix algebras}. The \textit{Hilbert bimodule} for
-    $(A, B)$ is given by an $A$-$B$-bimodue $E$ and by an $B$-valued
-    \textit{inner product} $\langle \cdot,\cdot\rangle_E: E\times E \rightarrow
-    B$, which satisfies the following conditions for $e, e_1, e_2 \in
-    E,\ a \in A$ and $b \in B$
-\begin{align}
-    \langle e_1,\ a\cdot e_2\rangle_E &= \langle a^*\cdot e_1,\ e_2\rangle_E
-    \;\;\;\; & \text{sesquilinear in $A$},\\
-    \langle e_1,\ e_2 \cdot b\rangle_E
-             &= \langle e_1,\ e_2\rangle_E b \;\;\;\; & \text{scalar in $B$},\\
-    \langle e_1,\ e_2\rangle_E &= \langle e_2,\ e_1\rangle^*_E \;\;\;\; &
-    \text{hermitian}, \\
-    \langle e,\ e\rangle_E &\ge 0 \;\;\;\; & \text{equality
-    holds iff $e=0$}.
-\end{align}
-We denote $KK_f(A,\ B)$ as the set of all \textit{Hilbert bimodules} of $(A,\ B)$.
-\end{mydefinition}
-%-------------- EXERCISE
-
-And indeed the Hilbert bimodule extension takes a representation $\pi:\ A \
-\rightarrow L(H)$ of a matrix algebra $A$ and turns $H$ into a Hilbert bimodule for
-$(A, \mathbb{C})$, because the representation for a $a \in A$, $\pi(a)=T \in L(H)$ fulfills
-the conditions of the $\mathbb{C}$-valued inner product for $h_1, h_2 \in H$
-\begin{itemize}
-    \item $\langle h_1,\ \pi(a)\ h_2\rangle _\mathbb{C} = \langle h_1,\ T\ h_2\rangle _\mathbb{C} =
-        \langle T^* h_1, h_2\rangle _\mathbb{C}$, $T^*$ given by the adjoint,
-    \item $\langle h_1,\ h_2\ \pi(a)\rangle _\mathbb{C} = \langle h_1,\ h_2\
-        T\rangle _\mathbb{C} = \langle h_1,\ h_2\rangle _\mathbb{C}$ , $T$ acts
-        from the left,
-    \item $\langle h_1,\ h_2\rangle _\mathbb{C}^* = \langle h_2,\ h_1\rangle _\mathbb{C}$, hermitian because of the
-        $\mathbb{C}$-valued inner product
-    \item $\langle h_1,\ h_2\rangle  \ge 0$, $\mathbb{C}$-valued inner product.
-\end{itemize}
-%-------------- EXERCISE
-
-%-------------- EXERCISE
-Take again the $A-A$ bimodule given by an $*$-algebra $A$. By looking at the
-following inner product $\langle \cdot,\cdot\rangle_A:A \times A \rightarrow A$
-\begin{align}
-    \langle a,\ a\rangle_A = a^*a' \;\;\;\; a,a'\in A.
-    \label{eq:inner-product},
-\end{align}
-it becomes clear that $A \in KK_f(A,\ A)$.
-Simply checking the conditions in $\langle \cdot, \cdot\rangle _A$ for
-$a, a_1, a_2 \in~A$
-\begin{align}
-    &\langle a_1,\ a\cdot a_2\rangle _A = a^* a\cdot a_2 =
-        (a^*a_1)^*\ a_2 = \langle  a^*\ a_1,\ a_2\rangle, \\
-   &\langle a_1,\ a_2 \cdot a\rangle _A = a^*_1\ (a_2\cdot a) =
-    (a^*a_2)\cdot a = \langle a_1,\ a_2\rangle _A\ a,\\
-   &\langle a_1,\ a_2\rangle _A^* = (a_1^*\ a_2)^* = a_2^*\
-        (a_1^*)^* = a_2^*\ a_1 = \langle a_2,\ a_1\rangle.
-\end{align}
-
-%-------------- EXERCISE
-
-%-------------- EXAMPLE
-%As an for overview consider a $*$ homomorphism between two matrix
-%algebras $\phi:A\rightarrow B$, we can construct a Hilbert bimodule
-%$E_{\phi} \in KK_f(A, B)$ in the following way. We let $E_{\phi}$ be $B$ in
-%as an vector space and an inner product from above in equation
-%\eqref{eq:inner-product}, with $A$ acting on the left with $\phi$.
-%\begin{align}
-%    a\cdot b = \phi(a)\ b
-%\end{align}
-%for $a\in A, b\in E_{\phi}$.
-%-------------- EXAMPLE
-
-\subsubsection{Kasparov Product and Morita Equivalence\label{chap: kasparov
-product}}
-\begin{mydefinition}
-    Let $E \in KK_f(A, B)$ and $F \in KK_F(B, D)$ the \textit{Kasparov product} is defined as
-    with the balanced tensor product
-    \begin{align}
-        F \circ E := E \otimes _B F.
-    \end{align}
-    Then $F\circ E \in KK_f(A,D)$ is equipped with a $D$-valued inner product
-    \begin{align}
-        \langle e_1 \otimes f_1,\ e_2 \otimes f_2\rangle _{E\otimes _B F} =
-        \langle f_1,\langle e_1,\ e_2\rangle _E f_2\rangle _F
-    \end{align}
-\end{mydefinition}
-
-%-------------- EXERCISE
-The Kasparov product for $*$-algebra homomorphism $\phi: A \rightarrow B$ and
-$\psi: B \rightarrow C$ are isomorphisms in the sense that
-\begin{align}
-                E_{\psi} \circ E_{\phi}\ \equiv\ E_{\phi} \otimes _B E_{\psi}\
-                \simeq\
-                E_{\psi \circ \phi} \in KK_f(A,C).
-\end{align}
-
-The direct computation for $a \in A$, $b\in B$, and $c\in C$ which is $\psi
-\circ \phi$ shows us
-\begin{align}
-a \cdot b \cdot c = \psi(\phi (a) \cdot b) \cdot c
-\end{align}
-An interesting case arises when looking at $E_{\text{id}_A} \simeq A \in
-KK_f(A,A)$, where $\text{id}_A$ is the identity in $A$. Let $E_{\phi}$ be $A$
-with a natural right representation. It follows that $E_{\phi}\simeq A$, where
-an inner product, acting from the left on $A$ for $\phi$, $a', a\in A$ reads
-\begin{align}
-    a'\ a = (\phi(a')\ a) \in A,
-\end{align}
-which is satisfied only by $\phi = \text{id}_A$.
-
-\begin{mydefinition}
-    Let $A$, $B$ be \textit{matrix algebras}. They are called \textit{Morita equivalent} if there
-    exists an $E \in KK_f(A, B)$ and an $F \in KK_f(B, A)$ such that
-    \begin{align}
-        E \otimes _B F \simeq A \;\;\; \text{and} \;\;\; F \otimes _A E \simeq
-        B,
-    \end{align}
-    where $\simeq$ denotes the isomorphism between Hilbert bimodules and note
-    that $A$ or $B$ is a bimodule by itself.
-\end{mydefinition}
-
-Since we land in the same space as we started, the modules $E$ and $F$ are
-each others inverse in regards to the Kasparov Product. More clearly, in the
-definition we have $E \in KK_f(A, B)$. Naturally we start from $A$ and $E
-\otimes _B F$, which lands in $A$. On the other hand we have $F \in KK_f(B,
-D)$ and start from $B$, $F \otimes _A E$, which lands in $B$.
-
-%------------- EXERCISE
-By definition  $E \otimes _B F$ is a $A-D$ bimodule. Since
-\begin{align}
-    E \otimes _B F = E \otimes F / \bigg\{\sum_i\ e_i\ b_i \otimes f_i - e_i
-        \otimes b_i\ f_i\ \big|\;\; e_i \in E_i,\ b_i \in B,\ f_i \in F\bigg\},
-\end{align}
-the last part takes out all tensor product elements of $E$ and $F$ that don't
-preserver the left/right representation and that are duplicates.
-
-Additionally $\langle \cdot,\cdot\rangle _{E\oplus _B F}$ defines a $D$ valued
-inner product, as $\langle e_1,\ e_2\rangle _E \in B$ and $\langle f_1,\ f_2\rangle _F \in C$ by
-definition. So for $\langle e_1,\ e_2\rangle _E =b$ we have
-\begin{align}
-    \langle e_1 \otimes f_1,\ e_2 \otimes f_2\rangle _{E\otimes _B F} = \langle
-    f_1,\ \langle e_1,\ e_2\rangle _E\ f_2\rangle _F = \langle f_1,\ b\ f_2\rangle _F \in C
-\end{align}
-%------------- EXERCISE
-%------------- EXAMPLE
-Picking up the example of $(A, A)$, the Hilbert bimodule $A$, we can
-consider an $E \in KK_f(A,B)$ for
-\begin{align}
-    E \circ A = A\oplus _A E \simeq E.
-\end{align}
-We conclude, that $_A A_A$ is the identity element in the Kasparov product (up
-to isomorphism).
-%------------- EXAMPLE
-%------------- EXAMPLE
-Let us examine another example for $E = \mathbb{C}^n$, which is a
-$(M_n(\mathbb{C}), \mathbb{C})$ Hilbert bimodule with the standard $\mathbb{C}$
-inner product. Further let $F = \mathbb{C}^n$, which is a $(\mathbb{C},
-M_n(\mathbb{C}))$ Hilbert bimodule by right matrix multiplication with
-$M_n(\mathbb{C})$ valued inner product, we can write
-    \begin{align}
-        \langle v_1, v_2\rangle =\bar{v_1}v_2^t \;\; \in M_n(\mathbb{C}).
-    \end{align}
-If we take the Kasparov product of $E$ and $F$
-   \begin{align}
-       F\circ E\ &=\  E\otimes _{\mathbb{C}}F\ \;\;\;\;\;\; \simeq \
-        M_n(\mathbb{C}),\\
-        E\circ F\ &=\ F\otimes _{M_n(\mathbb{C})}E\ \simeq\ \mathbb{C},
-    \end{align}
-we see that $M_n(\mathbb{C})$ and $\mathbb{C}$ are Morita equivalent!
-%------------- EXAMPLE
-
-\begin{mylemma}
-    Two matrix algebras are Morita Equivalent if, and only if their their structure spaces
-    are isomorphic as discreet spaces (have the same cardinality / same number
-    of elements).
-\end{mylemma}
-\begin{proof}
-    Let $A$, $B$ be \textit{Morita equivalent}. Then there exist the modules
-    $_A E_B$ and $_B F_A$ with
-    \begin{align}
-        E \otimes _B F \simeq A \;\;\; \text{and} \;\;\; F \otimes _A E \simeq
-        B.
-    \end{align}
-    Also consider $[(\pi _B, H)] \in \hat{B}$. We can construct a
-    representation of $A$, which reads
-    \begin{align}
-        \pi _A \rightarrow L(E \otimes _B H)\;\;\; \text{with} \;\;\; \pi _A(a)
-        (e \otimes v) = a e \otimes w
-    \end{align}
-    Vice versa, we have $[(\pi _A, W)] \in \hat{A}$ and we can construct $\pi _B$
-    as
-    \begin{align}
-        \pi _B: B \rightarrow L(F \otimes _A W) \;\;\; \text{and}\;\;\; \pi
-        _B(b) (f\otimes w) = bf\otimes w.
-    \end{align}
-    Now we need to show that the representation $\pi _A$ is irreducible if and
-    only if $\pi _B$ is irreducible. For $(\pi _B, H)$ to be irreducible, we
-    need $H \neq \emptyset$ and only $\emptyset$ or $H$ to be invariant under
-    the Action of $B$ on $H$. Than $E\otimes _B H$ and $E\otimes _B H \simeq A$
-    cannot be empty, because $E$ preserves left representation of $A$.
-
-    Lastly we need to check if the association of the class $[\pi _A]$ to $[\pi
-    _B]$ is independent of the choice of representatives $\pi _A$ and $\pi _B$.
-    The important thing is that $[\pi _A] \in \hat{A}$ respectively $[\pi _B] \in
-    \hat{B}$, hence any choice of representation is irreducible, because the
-    structure space denotes all unitary equivalence classes of irreducible
-    representations.
-
-    Note that the statements $E \simeq H$ and $F \simeq W$ are not particularly
-    true, since all infinite dimensional Hilbert spaces are isomorphic.  Here
-    we are looking at finite dimensional Hilbert spaces. Another thing to keep
-    in mind, is that for $[\pi _B, H] \in \hat{B}$ and looking at algebraic
-    bimodules, we know that $H$ is a bimodule of $B$, hence $E \otimes _B
-    H\simeq A$, and for $[\pi _A, W]$, which is the same.
-    Finally we can conclude, that these maps are each others inverses, thus
-    $\hat{A} \simeq \hat{B}$.
-\end{proof}
-
-\begin{mylemma}
-    The matrix algebra $M_n(\mathbb{C})$ has a unique irreducible
-    representation (up to isomorphism) given by the defining representation on
-    $\mathbb{C}^n$.
-\end{mylemma}
-\begin{proof}
-    We know $\mathbb{C}^n$ is a irreducible representation of $A=
-    M_n(\mathbb{C})$. Let $H$ be irreducible and of dimension $k$, then we
-    define a map
-    \begin{align}
-        \phi : A\oplus...\oplus A &\rightarrow H^* \\
-    (a_1,...,a_k)&\mapsto e^1\circ a_1^t+...+e^k\circ a_k^t,
-    \end{align}
-where $\{e^1,...,e^k\}$ is the basis of the dual space $H^*$ and
-$(\circ)$ being the pre-composition of elements in $H^*$ and $A$ acting on $H$.
-This forms a morphism of $M_n(\mathbb{C})$ modules, provided a matrix $a \in A$
-acts on $H^*$ with $v\mapsto v\circ a^t$ ($v\in H^*$).  Furthermore this
-morphism is surjective, thus making the pullback $\phi ^*:H\mapsto (A^k)^*$
-injective.  Now identify $(A^k)^*$ with $A^k$ as a $A$-module and note that
-$A=M_n(\mathbb{C}) \simeq \oplus ^n \mathbb{C}^n$ as a n A module.  It follows
-that $H$ is a submodule of $A^k \simeq \oplus ^{nk}\mathbb{C}$. By
-irreducibility $H \simeq \mathbb{C}$.
-\end{proof}
-
-%---------------- EXAMPLE
-Let us look at an example, two matrix algebras $A$, and $B$.
-\begin{align}
-    A = \bigoplus ^N_{i=1} M_{n_i}(\mathbb{C}), \;\;\;
-    B = \bigoplus ^M_{j=1} M_{m_j}(\mathbb{C}).
-\end{align}
-Let $\hat{A} \simeq \hat{B}$, this implies $N=M$. Further define $E$ with $A$
-acting by block-diagonal matrices on the first tensor and B acting in the same
-manner on the second tensor. Define $F$ vice versa, ultimately reading
-\begin{align}
-    E:= \bigoplus _{i=1}^N \mathbb{C}^{n_i} \otimes \mathbb{C}^{m_i}, \;\;\;
-    F:= \bigoplus _{i=1}^N \mathbb{C}^{m_i} \otimes \mathbb{C}^{n_i}.
-\end{align}
-When we calculate the Kasparov product we get the following
-\begin{align}
-    E \otimes _B F &\simeq \bigoplus _{i=1}^N (\mathbb{C}^{n_i}\otimes\mathbb{C}^{m_i})
-        \otimes _{M_{m_i}(\mathbb{C})} (\mathbb{C}^{m_i}\otimes\mathbb{C}^{n_i}) \\
-                   &\simeq \bigoplus _{i=1}^N \mathbb{C}^{n_i}\otimes
-                   \left(\mathbb{C}^{m_i}\otimes _{M_{m_i}(\mathbb{C})}\mathbb{C}^{m_i}\right)
-                    \oplus \mathbb{C}^{n_i} \\
-                   &\simeq \bigoplus _{i=1}^N
-                   \mathbb{C}^{m_i}\otimes\mathbb{C}^{n_i} \simeq A.
-\end{align}
-On the other hand we get
-\begin{align}
-    F \otimes _A E \simeq B.
-\end{align}
-%---------------- EXAMPLE
-
-To summarize, there is a duality between finite spaces and Morita equivalence
-classes of matrix algebras. Furthermore by replacing $*$-homomorphism $A\rightarrow B$
-with Hilbert bimodules $(A,B)$ we introduce a richer structure of morphism
-between matrix algebras.
diff --git a/src/thesis/chapters/conclusion.tex b/src/thesis/chapters/conclusion.tex
@@ -1,2 +1,17 @@
 \section{Conclusion}
-\lipsum
+We conclude that the framework of noncommutative geometry can fully describe
+the physics of electrodynamics. This is done by introducing the spectral and
+fermionic action principles of the almost commutative manifold $M \times F_{ED}$
+constructed from a four dimensional Riemannian spin manifold and a
+modification of the two point space $F_X$. By going through rough
+calculations of the heat kernel coefficients to describe the Lagrangian in
+terms of geometrical invariants we finally arrive at the Lagrangians in
+equations \eqref{eq:final1} and \eqref{eq:final2}.
+
+With a similar complex ansatz Walter D. Suijlekom describes in his book
+\cite{ncgwalter} how to figure out a specific version of a spectral triple
+corresponding the almost commutative manifold which delivers the physics of
+the full Standard Model and with this information accurately calculating the
+mass of the Higgs boson. Moreover he describes more accurately the
+correspondence of the gauge theory of an almost commutative manifold, which
+brings this noncommutative geometry to the interest of physicists in the first place.
diff --git a/src/thesis/chapters/electroncg.tex b/src/thesis/chapters/electroncg.tex
@@ -1,465 +0,0 @@
-\subsection{Noncommutative Geometry of Electrodynamics\label{sec:5}}
-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 $M\times F_X$ 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 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 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
-the real structure we arrive at the following relations
-\begin{align}
-    J_F e &= \bar{e} \;\;\;\;\;\; J_F \bar{e} = e, \\
-    \gamma_F e &= e  \;\;\;\;\;\;   \gamma_F \bar{e} = \bar{e}.
-\end{align}
-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^-.
-\end{align}
-So, for an $\xi \in H^+$ we can write
-\begin{align}
-    \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 $\psi :=
-\psi_L + \psi _R$. 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
-\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}. This way we have
-$\{ \underbrace{e_R, e_L}_{\text{left-handed}}, \underbrace{\bar{e}_R,
-\bar{e}_L}_{\text{right-handed}}\}$ 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, \\
-    &J_F e_L = \bar{e_L}, \\
-    \nonumber \\
-    &\gamma _F e_R = -e_R,\\
-    &\gamma_F e_L = e_L,
-\end{align}
-where $J_F$ and $\gamma_F$ have to following properties
-\begin{align}
-    &J_F^2 = 1,\\
-    & J_F \gamma_F  = - \gamma_F J_F.
-\end{align}
-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\}}
-    \oplus \underbrace{H_F^-}_{\text{ONB } \{e_R, \bar{e}_R\}},
-\end{align}
-or alternatively into the eigenspace of particles and their
-antiparticles (electrons and positrons) which is preferred in literature and
-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\}},
-\end{align}
-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
-\begin{align}\label{eq:leftrightrepr}
-    a =
-    (a_1 , a_2 ) \mapsto
-        \begin{pmatrix}
-            a_1 &0 &0 &0\\
-             0&a_1 &0 &0\\
-            0 &0 &a_2 &0\\
-            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 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}
-            0 & C \\ C &0
-        \end{pmatrix},
-        \gamma _F =
-        \begin{pmatrix}
-            1 & 0 \\ 0 &-1
-        \end{pmatrix}
-        \right).
-    \end{align}
-It can be represented in the following Krajewski diagram,
-with two nodes of multiplicity two bellow
-    \begin{figure}[H] \centering
-    \begin{tikzpicture}[
-        dot/.style = {draw, circle, inner sep=0.06cm},
-        bigdot/.style = {draw, circle, inner sep=0.09cm},
-        no/.style = {},
-        ]
-        \node[no](a) at (0,0) [label=left:$\textbf{1}^\circ$] {};
-        \node[no](b) at (0, -1) [label=left:$\textbf{1}^\circ$] {};
-        \node[no](c) at (0.5, 0.5) [label=above:$\textbf{1}$] {};
-        \node[no](d) at (1.5, 0.5) [label=above:$\textbf{1}$] {};
-        \node[dot](d0) at (1.5,0) [] {};
-        \node[dot](d0) at (0.5,-1) [] {};
-        \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
-closer look at the Krajewski diagram above. Notice that edges only exist
-between multiple vertices, meaning we can construct a Dirac operator mapping
-between the two vertices. The operator can be represented by the following matrix
-\begin{align}\label{eq:feddirac}
-    D_F =
-    \begin{pmatrix}
-    0 & d & 0 & 0 \\
-    \bar{d} & 0 & 0 & 0 \\
-    0 & 0 & 0 & \bar{d} \\
-    0 & 0 & d & 0
-    \end{pmatrix}
-\end{align}
-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 as in equation \eqref{eq:fedfail} and $D_F$
-from equation \eqref{eq:feddirac}.
-
-\subsubsection{Almost commutative Manifold of Electrodynamics}
-The almost commutative manifold $M\times F_{ED}$ has KO-dimension 2, and is
-represented by the following spectral triple
-\begin{align}\label{eq:almost commutative manifold}
-    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).
-\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).
-\end{align}
-As for the Hilbertspace, we can decomposition it in the following way
-\begin{align}
-    H = (L^2(S) \otimes H_e ) \oplus (L^2(S) \otimes H_{\bar{e}}).
-\end{align}
-Note that the one component of the algebra is acting on $L^2(S) \otimes H_e$,
-and the other one acting on $L^2(S) \otimes H_{\bar{e}}$. In other words the components of
-the decomposition of both the algebra and the Hilbertspace match by the action of
-the algebra.
-
-The derivation of the gauge theory is the same for $F_{ED}$ as for the
-Two-Point space $F_X$. We have $\mathfrak{B}(F) \simeq U(1)$ and for an
-arbitrary gauge field $B_\mu = A_\mu - J_F A_\mu J_F^{-1}$ we can write
-\begin{align} \label{field}
-    B_\mu =
-    \begin{pmatrix}
-        Y_\mu & 0 & 0 & 0 \\
-        0 & Y_\mu& 0 & 0 \\
-        0 & 0 & Y_\mu& 0 \\
-        0 & 0 & 0 & Y_\mu
-    \end{pmatrix} \;\;\;\;\;\ \text{for} \;\;\ Y_\mu (x) \in \mathbb{R}.
-\end{align}
-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, 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
-observation and doesn't affect physical results.
-
-\subsubsection{Spectral Action}
-In this chapter we bring all our results together to establish an
-Action functional to describe a physical system. It turns out that
-the Lagrangian of the almost commutative manifold $M\times F_{ED}$
-corresponds to the Lagrangian of Electrodynamics on a curved
-background manifold (+ gravitational Lagrangian), consisting of the spectral
-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
-\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
-    \begin{align}
-        S_b [\omega] := \text{Tr}\big(f(\frac{D_\omega}{\Lambda})\big)
-    \end{align}
-    where $\Lambda$ is a real cutoff parameter. The minimal condition on $f$
-    is that $f(\frac{D_\omega}{\Lambda})$ is  a trace class operator. A trace
-    class operator is a compact operator with a well defined finite trace
-    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
-    action $S_{top}$. Leaning on the grading $\gamma$ the topological spectral action is
-    \begin{align}
-        S_{\text{top}}[\omega] := \text{Tr}(\gamma\
-        f(\frac{D_\omega}{\Lambda})).
-    \end{align}
-\end{mydefinition}
-\begin{mydefinition}\label{def:fermionic action}
-    The fermionic action is defined by
-    \begin{align}
-        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
-    of the grading $\gamma$.
-\end{mydefinition}
-
-%---------------------- 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).
-\end{align}
-%---------------------- APPENDIX ?????????????--------------------
-\begin{myproposition}
-    The spectral action of the almost commutative manifold $M$ with $\dim(M)
-    =4$ with a fluctuated Dirac operator is
-    \begin{align}
-        \text{Tr}(f\frac{D_\omega}{\Lambda}) \sim \int_M \mathcal{L}(g_{\mu\nu},
-         B_\mu, \Phi) \sqrt{g}\ d^4x + O(\Lambda^{-1}),
-    \end{align}
-    where
-    \begin{align}
-        \mathcal{L}(g_{\mu\nu}, B_\mu, \Phi) =
-        N\mathcal{L}_M(g_{\mu\nu})
-        \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 $(C^\infty(M) ,
-    L^2(S), D_M)$, 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}
-    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
-    $R_{\nu\sigma} = g^{\mu\varrho} R_{\mu\nu \varrho\sigma}$ 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}
-        \mathcal{L}_B(B_\mu) := \frac{f(0)}{24\pi^2}
-        \text{Tr}(F_{\mu\nu}F^{\mu\nu}).
-    \end{align}
-    Lastly $\mathcal{L}_\phi$ is the scalar-field Lagrangian with a boundary
-    term, given by
-    \begin{align}
-        \mathcal{L}_\phi(g_{\mu\nu}, B_\mu, \Phi) :=
-        &-\frac{2f_2\Lambda^2}{4\pi^2}\text{Tr}(\Phi^2) + \frac{f(0)}{8\pi^2}
-        \text{Tr}(\Phi^4) + \frac{f(0)}{24\pi^2}
-        \Delta(\text{Tr}(\Phi^2))\nonumber\\
-        &+ \frac{f(0)}{48\pi^2}s\text{Tr}(\Phi^2)
-        \frac{f(0)}{8\pi^2}\text{Tr}((D_\mu \Phi)(D^\mu \Phi)).
-    \end{align}
-\end{myproposition}
-\begin{proof}
-     The dimension of the manifold $M$ is $\dim(M) = \text{Tr}(id) =4$. For
-     an $x \in M$, we have an asymptotic expansion of the term
-     $\text{Tr}(f(\frac{D_\omega}{\Lambda}))$ as $\Lambda$ goes to infinity,
-     which can be written as
-     \begin{align}
-         \text{Tr}(f(\frac{D_\omega}{\Lambda})) \simeq& \ 2f_4 \Lambda ^4
-         a_0(D_\omega ^2)+ 2f_2\Lambda^2 a_2(D_\omega^2)\nonumber \\&+ f(0) a_4(D_\omega^4)
-         +O(\Lambda^{-1}).\label{eq:trheatkernel}
-     \end{align}
-     We have to note here that the heat kernel coefficients are zero for uneven $k$,
-     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 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
-         \text{Tr}(\Phi^2)\sqrt{g}d^4x.
-     \end{align}
-     For $a_4$ we extend in terms of coefficients of $F$ from equation
-     \eqref{eq: a_4}
-     \begin{align}
-         &\frac{1}{360}\text{Tr}(60RE)= -\frac{1}{6}S(NR + 4
-         \text{Tr}(\Phi^2))\\
-        \nonumber\\
-         &E^2 = \frac{1}{16}R^2\otimes 1 + 1\otimes \Phi^4 - \frac{1}{4}
-         \gamma^\mu\gamma^\nu \gamma^\varrho\gamma^\sigma
-         F_{\mu\nu}F^{\mu\nu}+\nonumber\\
-         &\;\;\;\;\;\;\;+\gamma^\mu\gamma^\nu\otimes(D_\mu\Phi)(D_\nu
-         \Phi)+\frac{1}{2}s\otimes \Phi^2 + \ \text{traceless terms},\\
-         \nonumber\\
-         &\frac{1}{360}\text{Tr}(180E^2) = \frac{1}{8}R^2N + 2\text{Tr}(\Phi^4)
-         + \text{Tr}(F_{\mu\nu}F^{\mu\nu}) +\nonumber\\
-         &\;\;\;\;\;\;\;+2\text{Tr}((D_\mu\Phi)(D^\mu\Phi))
-         + s\text{Tr}(\Phi^2)\\
-         \nonumber\\
-         &\frac{1}{360}\text{Tr}(-60\Delta E)=
-         \frac{1}{6}\Delta(NR+4\text{Tr}(\Phi^2)).
-     \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, 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
-         \otimes F^{\mu\nu}.
-     \end{align}
-     The trace  of the cross term $\Omega^{S}_{\mu\nu}$ vanishes because
-     \begin{align}
-         \text{Tr}(\Omega^{S}_{\mu\nu}) = \frac{1}{4}
-         R_{\mu\nu\varrho\sigma}\text{Tr}(\gamma^\mu\gamma^\nu) = \frac{1}{4}
-         R_{\mu\nu\varrho\sigma}g^{\mu\nu} =0,
-     \end{align}
-     then the trace of the whole term is given by
-     \begin{align}
-         \frac{1}{360}\text{Tr}(30\Omega^E_{\mu\nu}\Omega^{E\mu\nu}) =
-         \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
-     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 \\
-         &+ \frac{1}{4}
-         \text{Tr}((D_\mu\Phi)(D^\mu \Phi)) + \frac{1}{6}
-         \Delta\text{Tr}(\Phi^2) + \frac{1}{6}
-         \text{Tr}(F_{\mu\nu}F^{\mu\nu})\bigg).
-     \end{align}
-     The only thing left is to substitute the heat kernel coefficients into the
-     heat kernel expansion in equation \eqref{eq:trheatkernel}.
-\end{proof}
-
-\subsubsection{Fermionic Action}
-We remind ourselves the definition of the fermionic action in definition
-\ref{def:fermionic action} and the manifold we are dealing with in equation
-\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 $H_F^-$, $\{e_R, \bar{e}_L\}$. 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+
-    \psi_R \otimes \bar{e}_L,
-\end{align}
-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 $F_X$) 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
-    D_F.
-\end{align}
-The Fermionic Action is
-\begin{align}
-S_F = (J\tilde{\xi}, D_\omega\tilde{\xi})
-\end{align}
-for a $\xi \in H^+$. Then the straight forward calculation gives \begin{align}
-    \frac{1}{2}(J\tilde{\xi}, D_\omega\tilde{\xi})
-        &=\frac{1}{2}(J\tilde{\xi}, (D_M \otimes
-        i)\tilde{\xi})\label{eq:fermionic1}\\
-        &+\frac{1}{2}(J\tilde{\xi}, (\gamma^\mu \otimes B_\mu)
-        \tilde{\xi})\label{eq:fermionic2}\\
-        &+\frac{1}{2}(J\tilde{\xi}, (\gamma_M\otimes
-        D_F)\tilde{\xi})\label{eq:fermionic3},
-\end{align}
-(note that we add the constant $\frac{1}{2}$ to the action).
-For the term in \eqref{eq:fermionic1} we calculate
-\begin{align}
-    \frac{1}{2}(J\tilde{\xi}, (D_M\otimes 1)\tilde{\xi}) &=
-    \frac{1}{2}(J_M\tilde{\chi}_R,D_M\tilde{\psi}_L)+\nonumber
-    \frac{1}{2}(J_M\tilde{\chi}_L,D_M\tilde{\psi}_R)+
-    \\&+\frac{1}{2}(J_M\tilde{\psi}_L,D_M\tilde{\psi}_R)+\nonumber
-    \frac{1}{2}(J_M\tilde{\chi}_R,D_M\tilde{\chi}_L)\\
-    &= (J_M\tilde{\chi},D_M\tilde{\chi}).
-\end{align}
-For the term in \eqref{eq:fermionic2} we have
-\begin{align}
-    \frac{1}{2}(J\tilde{\xi}, (\gamma^\mu \otimes B_\mu)\tilde{\xi})&=
-    -\frac{1}{2}(J_M\tilde{\chi}_R, \gamma^\mu Y_\mu\tilde{\psi}_R)
-    -\frac{1}{2}(J_M\tilde{\chi}_L, \gamma^\mu Y_\mu\tilde{\psi}_R)+\nonumber\\
-    &+\frac{1}{2}(J_M\tilde{\psi}_L, \gamma^\mu Y_\mu\tilde{\chi}_R)+
-    \frac{1}{2}(J_M\tilde{\psi}_R, \gamma^\mu Y_\mu\tilde{\chi}_L)=\nonumber\\
-    &= -(J_M\tilde{\chi}, \gamma^\mu Y_\mu\tilde{\psi}).
-\end{align}
-And for \eqref{eq:fermionic3} we can write
-\begin{align}
-    \frac{1}{2}(J\tilde{\xi}, (\gamma_M\otimes D_F)\tilde{\xi})&=
-    +\frac{1}{2}(J_M\tilde{\chi}_R, d\gamma_M\tilde{\chi}_R)
-    +\frac{1}{2}(J_M\tilde{\chi}_L, \bar{d}\gamma_M\tilde{\chi}_L)+\nonumber\\
-    &+\frac{1}{2}(J_M\tilde{\chi}_L, \bar{d}\gamma_M\tilde{\chi}_L)
-    +\frac{1}{2}(J_M\tilde{\chi}_R, d\gamma_M\tilde{\chi}_R)=\nonumber\\
-    &= i(J_M\tilde{\chi}, m\tilde{\psi}).
-\end{align}
-A small problem arises, we obtain a complex mass parameter $d$, but we can
-write $d:=im$ for $m\in \mathbb{R}$, which stands for the real mass.
-
-Finally the fermionic action of $M\times F_{ED}$ takes the form
-    \begin{align}
-        S_f = -i\big(J_M\tilde{\chi}, \gamma(\nabla^S_\mu - i\Gamma_\mu)
-        \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 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}),
-    \end{align}
-and the Lagrangian of electrodynamics
-    \begin{align}
-        \mathcal{L}_{ED} = -i\bigg\langle
-        J_M\tilde{\chi},\big(\gamma^\mu(\nabla^S_\mu - iY_\mu) -m\big)\tilde{\psi})
-        \bigg\rangle
-        +\frac{f(0)}{6\pi^2} Y_{\mu\nu}Y^{\mu\nu}.
-    \end{align}
-
diff --git a/src/thesis/chapters/intro.tex b/src/thesis/chapters/intro.tex
@@ -16,8 +16,39 @@ invariants, a method called the heat kernel expansion is used.
 The aim of this thesis is to give a basic foundation of noncommutative
 geometry and to present a physical application which can be derived from this
 theory. Additionally we emphasize that this thesis is only literature work,
-where chapters \ref{sec:1}-\ref{sec:3} and \ref{sec:5} are from
+where chapters \ref{sec:1}, \ref{sec:2}, \ref{sec:3}, \ref{sec:5} and \ref{sec:6} are from
 the work of Walter D. Suijlekom's book \cite{ncgwalter} and chapter
 \ref{sec:4} from D.V. Vassilevich's paper \cite{heatkernel}.
 
-\textbf{NOW:CHAPTER OVERVIEW}
+The prominent structure of noncommutative geometry is the spectral triple.
+The most basic form of a spectral triple consists of an unital $C^*$ algebra
+$A$ acting on a Hilbertspace $H$. Together with a self-adjoint operator $D$ in
+$H$, with specific conditions coinciding with the Dirac operator on
+Riemannian spin$^c$ manifold which square is the Laplacian (up to a scalar
+term).
+
+The structure of the thesis is based on first getting the background
+knowledge of noncommutative geometry and the heat kernel expansion. Then by
+combining this insight we work out the Lagrangian of electrodynamics. In this
+regard the first two chapters \ref{sec:1} and \ref{sec:2} go through the
+basic version of noncommutative geometry, in the sense of finite discrete
+spaces. It is important to understand these basics, since the they build up
+the ground work of constructing the almost commutative manifold of
+electrodynamics, that is the Two-Point space $F_X$. Additionally the notion
+of equivalence relations between spectral triples, called Morita equivalence is
+introduced
+
+The next chapter \ref{sec:3} extends the finite spectral triple with a real
+structure, called the real finite spectral triple, we also examine Morita
+equivalence within this extension.
+
+Chapter \ref{sec:4} explains the heat kernel and leads off to the heat kernel
+expansion, where the famous heat kernel coefficients arise. Hereof we
+calculate the heat kernel coefficients, which become important when
+calculating the Lagrangian of the almost commutative manifold of
+electrodynamics.
+
+In the last two chapters \ref{sec:5} and \ref{sec:6} we go over the ideas and
+the process of constructing the almost commutative manifold, that will give
+rise to the Lagrangian of electrodynamics and an additional purely
+gravitational Lagrangian.
diff --git a/src/thesis/chapters/realncg.tex b/src/thesis/chapters/realncg.tex
@@ -1,292 +0,0 @@
-\subsection{Finite Real Noncommutative Spaces\label{sec:3}}
-\subsubsection{Finite Real Spectral Triples}
-In this chapter we supplement the finite spectral triples with a \textit{real
-structure}. We additionally require a symmetry condition that that $H$ is an
-$A$-$A$-bimodule rather than only a $A$-left module. This ansatz has tight
-bounds with physical properties such as charge conjugation, into which we will
-dive in deeper in later chapters. In regards to this we will need to set a basis
-of definitions to get an overview.
-First we introduce a $\mathbb{Z}_2$-grading $\gamma$ with the following
-properties
-\begin{align}
-    \gamma ^* &= \gamma, \\
-    \gamma ^2 &= 1, \\
-    \gamma D &= - D \gamma,\\
-    \gamma a &= a \gamma, \;\;\;\; a\in A.
-\end{align}
-Then we can define a finite real spectral triple.
-\begin{mydefinition}
-    A \textit{finite real spectral triple} is given by a finite spectral
-    triple $(A, H, D)$ and a anti-unitary operator $J:H\rightarrow H$ called
-    the \textit{real structure}, such that
-    \begin{align}
-        a^\circ := J\ a^*\ J^{-1},
-    \end{align}
-    is a right representation of $A$ on $H$, that is $(ab)^\circ = b^\circ
-    a^\circ$. With two requirements
-    \begin{align}
-        &[a, b^\circ] = 0,\\
-        &[[D, a],\ b^\circ] = 0.
-    \end{align}
-    The two properties are called the \textit{commutant property}, they
-    require that the left action of an element in $A$ and $\Omega _D^1(A)$ commutes with the right
-    action on $A$.
-\end{mydefinition}
-\begin{mydefinition}
-    The $KO$-dimension of a real spectral triple is determined by the sings
-    $\epsilon, \epsilon ' ,\epsilon '' \in \{-1, 1\}$ appearing in
-    \begin{align}
-        J^2 &= \epsilon, \\
-        J\ D &= \epsilon \ D\ J,\\
-        J\ \gamma &= \epsilon''\ \gamma\ J.
-    \end{align}
-\end{mydefinition}
-\begin{table}[h!]
-    \centering
-    \caption{$KO$-dimension $k$ modulo $8$ of a real spectral triple}
-    \begin{tabular}{ c | c c c c c c c c}
-        \hline
-        $k$        & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
-        \hline
-     $\epsilon$    & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\
-     $\epsilon '$  & 1 & -1 & 1 & 1 & 1 & -1 & 1 & 1 \\
-     $\epsilon ''$ & 1 &  & -1 &  & 1 &  & -1 &  \\
-     \hline
-    \end{tabular}
-\end{table}
-\noindent
-Even thought the KO-dimension of a real spectral triple is important, we will
-not be doing in-depth introduction of the KO-dimension, for this we reference
-again to \cite{ncgwalter}.
-
-\begin{mydefinition}
-An opposite-algebra $A^\circ$ of a $A$ is defined to be equal to $A$ as a
-vector space with the opposite product
-\begin{align}
-    &a\circ b := ba\\
-    &\Rightarrow a^\circ = Ja^* J^{-1},
-\end{align}
-which defines the left representation of $A^\circ$ on $H$
-\end{mydefinition}
-
-
-%------------EXAMPLE EXERCISE
-Let us examine an example of a matrix algebra $M_N(\mathbb{C})$ acting on
-$H=M_N(\mathbb{C})$ by left matrix multiplication with the Hilbert Schmidt
-inner product.
-\begin{align}
-    \langle a , b \rangle = \text{Tr}(a^* b).
-\end{align}
-We can define $\gamma (a) = a$ and $J(a) = a^*$ with $a\in H$.  Since $D$
-must be odd with respect to $\gamma$ it vanishes identically.  Furthermore we
-know the multiplicity space is $V_i = \mathbb{C}^{m_i}$, and also we know
-that for $T\in H$ and$a\in A'$ to work we need $a\ T=T\ a$. Thus by laws of
-matrix multiplication we need $A' \simeq \bigoplus _i M_{m_i}(\mathbb{C})$. For
-this to work we naturally need $H = \bigoplus_i \mathbb{C}^{n_i} \otimes
-\mathbb{C}^{m_i}$.  Hence the right action of $M_N(\mathbb{C})$ on $H =
-M_N(\mathbb{C})$ as defined by $a \mapsto a^\circ$ is given by right matrix
-multiplication
-\begin{align}
-    a^\circ \xi = J a^* J^{-1}\xi = Ja^* \xi^* = J\xi a=\xi^* a
-\end{align}
-
-%------------EXAMPLE EXERCISE
-
-\begin{mydefinition}
-    We call $\xi \in H$ \textbf{cyclic vector} in $A$ if:
-    \begin{align}
-        A\xi := { a\xi:\;\; a\in A} = H
-    \end{align}
-    We call $\xi \in H$ \textbf{separating vector} in $A$ if:
-    \begin{align}
-        a\xi = 0\;\; \Rightarrow \;\; a=0;\;\;\; a\in A
-    \end{align}
-\end{mydefinition}
-%------------------- EXERCISE
-Suppose $(A, H, D = 0)$ is a finite spectral triple such that $H$ possesses a
-cyclic and separating vector for $A$ and let
-\begin{align}
-    J: H \rightarrow H
-\end{align}
-be the operator in $S = J \Delta ^{1/2}$ with $\Delta = S^*S$. By composition
-$S(a\xi) = a*\xi$ this is literally anti-linearity, then $S(a \xi) = a* \xi$
-defines a anti-linear operator. Furthermore the operator $S$ is invertible
-because, if a $\xi \in H$ is cyclic then we have $S(A\xi) = A^*\xi = A\xi =
-H$. Vice versa the same has to work for $S^{-1}$, otherwise $\xi$ wouldn't
-exist. And hence $S^{-1}(A^*\xi) = S^{-1}(H) = H$. Additionally $J$ is
-anti-unitary because firstly, $S$ is bijective thus $\Delta ^{1/2}$ and $J$ need to be bijective.
-Also have $J = S \Delta^{-1/2}$ and $\Delta^* = \Delta$, so for a $\xi _1 ,
-\xi _2 \in H$ we can write
-\begin{align}
-    <J \xi _1 , J \xi _2 > &= < J^*J\xi_1 , \xi_2>^* =\nonumber\\
-    &= <(\Delta ^{-1/2})^* S^* S \Delta ^{-1/2} \xi_1, \xi_2>^* =\nonumber \\
-    &= <(\Delta^{-1/2})^* \Delta \Delta^{-1/2} \xi_1, \xi_2>^* =\nonumber\\
-    &= <\Delta^{-1/2} \Delta^{1/2}\Delta^{1/2} \Delta^{-1/2} \xi_1, \xi_2>^*
-    =\nonumber\\
-    &= <\xi _1, \xi_2>^* = <\xi_2 , \xi_1>,
-\end{align}
-which concludes the anti-unitarity by definition.
-%------------------- EXERCISE
-\subsubsection{Morphisms Between Finite Real Spectral Triples}
-Like the unitary equivalence relation for finite spectral triples, we can it
-extend it to finite real spectral triples.
-\begin{mydefinition}
-    We call two finite real spectral triples $(A_1, H_1 ,D_1 ; J_1 , \gamma_1)$
-    and $(A_2, H_2, D_2; J_2, \gamma _2)$ unitarily equivalent if $A_1 =
-    A_2$ and if there exists a unitary operator $U: H_1 \rightarrow H_2$ such
-    that
-    \begin{align}
-        U\ \pi_1(a)\ U^* &= \pi _2(a),\\
-        U\ D_1\ U^* &= D_2,\\
-        U \gamma _1\ U^*  &= \gamma _2,\\
-        U\ J_1\ U^* &= J_2.
-    \end{align}
-\end{mydefinition}
-\begin{mydefinition}
-    Let $E$ be a $B$-$A$ bimodule. The \textit{conjugate Module} $E^\circ$ is
-    given by the $A$-$B$-bimodule.
-    \begin{align}
-        E^\circ = \{\bar{e} : e\in E\},
-    \end{align}
-    with
-    \begin{align}
-    a \cdot \bar{e} \cdot b = b^*\ \bar{e}\ a^*, \;\;\;\; \forall a\in A, b \in
-        B.
-    \end{align}
-\end{mydefinition}
-We bear in mind that $E^\circ$ is not a Hilbert bimodule for $(A, B)$ because
-it doesn't have a natural $B$-valued inner product. But there is a $A$-valued
-inner product on the left $A$-module $E^\circ$ with
-\begin{align}
-    \langle \bar{e}_1, \bar{e}_2 \rangle = \langle e_2 , e_1 \rangle,
-    \;\;\;\; e_1, e_2 \in E.
-\end{align}
-And linearity in $A$ by the terms
-\begin{align}
-    \langle a\ \bar{e}_1, \bar{e}_2 \rangle = a \langle \bar{e}_1, \bar{e}_2
-    \rangle, \;\;\;\; \forall a \in A.
-\end{align}
-
-%------------- EXERCISE
-It turns out that $E^\circ$ is a Hilbert bimodule
-of $(B^{\circ}, A^{\circ})$. A straightforward calculation of the properties of the Hilbert bimodule and its $B^{\circ}$
-valued inner product gives the results. For $\bar{e}_1, \bar{e}_2 \in E^{\circ}$ and $a^\circ \in A,
-b^\circ \in B$ we write
-\begin{align}
-    \langle\bar{e}_1, a^\circ \bar{e}_2\rangle &= \langle\bar{e}_1, Ja^*J^{-1}
-    \bar{e}_2\rangle=\nonumber\\
-    &= \langle\bar{e}_1 , J a^* e_2\rangle \nonumber \\
-    &= \langle J^{-1} e_1, a^* e_2\rangle \nonumber\\
-    & = \langle a^* e_1, e_2\rangle= \langle J^{-1}(a^\circ)^* J e_1, e_2\rangle  \nonumber\\
-    & = \langle J^{-1} (a^\circ)^* \bar{e}_1, e_2\rangle \nonumber\\
-    & = \langle (a^\circ)^* \bar{e}_1 , \bar{e}_2\rangle.
-\end{align}
-Next for $\langle\bar{e}_1, \bar{e}_2 b^\circ\rangle = \langle\bar{e}_1,
-\bar{e_2}\rangle b^\circ$ we obtain
-\begin{align}
-    \langle\bar{e}_1, \bar{e}_2 b^\circ\rangle  &= \langle\bar{e}_1, \bar{e}_2 Jb^*J^{-1}\rangle
-    \nonumber\\
-    &= \langle\bar{e}_1, \bar{e_2}\rangle Jb^*J^{-1} \nonumber \\
-    &= \langle\bar{e}_1, \bar{e}_2\rangle b^\circ.
-\end{align}
-Additionally we get
-\begin{align}
-    (\langle\bar{e}_1, \bar{e}_2)\rangle_{E^\circ})^* &= (\langle e_2, e_1\rangle_E)^*\nonumber\\
-                                          &= \langle e_1, e_2\rangle_E^* \nonumber\\
-                                          &= \langle\bar{e}_2, \bar{e}_2\rangle_{E^\circ}.
-\end{align}
-And finally we have
-\begin{align}
-    \langle\bar{e}, \bar{e}\rangle = \langle e, e\rangle \geq 0
-\end{align}
-%------------- EXERCISE
-
-Given the results thus far, given a Hilbert bimodule $E$ for $(B, A)$ one can
-construct a spectral triple $(B, H', D'; J', \gamma ')$ from $(A, H, D; J,
-\gamma)$. For $H'$ we make a $\mathbb{C}$-valued inner product on $H'$ by combining
-the $A$ valued inner product on $E$ and $E^\circ$ with the
-$\mathbb{C}$-valued inner product on $H$ by defining
-\begin{align}
-    H' := E\otimes _A H \otimes _A E^\circ.
-\end{align}
-Then the action of $B$ on $H'$ takes the following form
-\begin{align}
-    b(e_2 \otimes \xi \otimes \bar{e}_2 ) = (be_1) \otimes \xi \otimes
-    \bar{e}_2.
-\end{align}
-The right action of $B$ on $H'$ defined by action on the right components of
-$E^\circ$ is
-\begin{align}
-    J'(e_1 \otimes \xi \otimes \bar{e}_2) = e_2 \otimes J \xi \otimes
-    \bar{e}_1,
-\end{align}
-where $b^\circ = J' b^* (J')^{-1}$ and $b^* \in B$ is the action on $H'$.
-Hence the connection reads
-\begin{align}
-    &\nabla: E \rightarrow E\otimes _A \Omega _D ^1(A) \\
-    &\bar{\nabla}:E^\circ \rightarrow \Omega _D^1(A) \otimes _A E^\circ,
-\end{align}
-which gives the Dirac operator on $H' = E \otimes _A H \otimes _A
-E^\circ$ as
-\begin{align}
-    D'(e_1 \otimes \xi \otimes \bar{e}_2) = (\nabla e_1) \xi \otimes
-    \bar{e_2}+ e_1 \otimes D\xi \otimes \bar{e}_2 + e_1 \otimes
-    \xi(\bar{\nabla}\bar{e}_2).
-\end{align}
-And the right action of $\omega \in \Omega _D ^1(A)$ on $\xi \in H$ is
-defined by
-\begin{align}
-    \xi \mapsto \epsilon' J \omega ^* J^{-1}\xi.
-\end{align}
-Finally for the grading one obtains
-\begin{align}
-    \gamma ' = 1 \otimes \gamma \otimes 1.
-\end{align}
-
-Summarizing we can write down the following theorem
-\begin{mytheorem}
-    Suppose $(A, H, D; J, \gamma)$ is a finite spectral triple of
-    $KO$-dimension $k$, let $\nabla$ be a connection satisfying the
-    compatibility condition (same as with finite spectral triples).
-    Then $(B, H',D'; J', \gamma')$ is a finite spectral triple of
-    $KO$-Dimension $k$. ($H', D', J', \gamma'$)
-\end{mytheorem}
-
-\begin{proof}
-    The only thing left is to check is, if the $KO$-dimension is preserved.
-    That is one needs to check if if the $\epsilon$'s are the same.
-    \begin{align}
-        &(J')^2 = 1 \otimes J^2 \otimes 1 = \epsilon,\\
-        &J' \gamma '= \epsilon ''\gamma'J'.
-    \end{align}
-    Lastly for $\epsilon '$ one obtains
-    \begin{align}
-        J'D'(e_1 \otimes \xi \otimes \bar{e}_2)&=J'\big((\nabla e_1) \xi \otimes
-        \bar{e_2} + e_1 \otimes D\xi \otimes \bar{e}_2 + e_1 \otimes \xi (\tau
-        \nabla e_2)\big)\nonumber \\
-        &= \epsilon' D'\left(e_2 \otimes J\xi \otimes \bar{e}_2\right)\nonumber\\
-        &= \epsilon' D'J'\left(e_1 \otimes \xi \bar{e}_2\right)
-    \end{align}
-\end{proof}
-
-Let us take a look at $\nabla : E \Rightarrow E \otimes _A \Omega _d^1 (A)$,
-the right connection on $E$ and consider the following anti-linear map
-\begin{align}
-    \tau : E \otimes_A \Omega _D^1 (A) &\rightarrow \Omega _D^1 (A) \otimes_A E^\circ\\
-            e \otimes \omega &\mapsto -\omega ^* \otimes \bar{e}.
-\end{align}
-Interestingly the map $\bar{\nabla} : E^\circ \rightarrow \Omega _D^1(A) \otimes E^\circ$
-with $\bar{\nabla}(\bar{e}) = \tau \circ \nabla(e)$ is a left connection, that means
-show that it satisfied the left Leibniz rule, for one
-\begin{align}
-    \tau \circ \nabla(ae) = \bar{\nabla}(a\bar{e}) = \bar{\nabla}(a^*
-    \bar{e}).
-\end{align}
-And for two
-\begin{align}
-    \tau \circ \nabla(ae) &= \tau(\nabla(e)a) + \tau \circ(e \otimes
-     d(a))\nonumber \\
-     &=a^*\bar{\nabla}(\bar{e}) - d(a)^* \otimes \bar{e}. \nonumber\\
-     &= a^*\bar{\nabla}(\bar{e}) + d(a^*) \otimes \bar{e}.
-\end{align}
-
diff --git a/src/thesis/chapters/twopointspace.tex b/src/thesis/chapters/twopointspace.tex
@@ -1,244 +0,0 @@
-\subsection{Almost-commutative Manifold\label{sec:4}}
-\subsubsection{Two-Point Space}
-One of the basics forms of noncommutative space is the Two-Point space $X
-:= \{x, y\}$. The Two-Point space can be represented by the following spectral triple
-\begin{align}
-    F_X := (C(X) = \mathbb{C}^2, H_F, D_F; J_F, \gamma _f).
-\end{align}
-Three properties of $F_X$ stand out. First of all the action of $C(X)$ on
-$H_F$ is faithful for $dim(H_F) \geq 2$, thus a simple choice for the
-Hilbertspace can be made, for instance $H_F = \mathbb{C}^2$. Furthermore
-$\gamma_F$ is the $\mathbb{Z}_2$ grading, which allows for a decomposition of
-$H_F$ into
-\begin{align}
-   H_F = H_F^+ \otimes H_F^- = \mathbb{C} \otimes \mathbb{C},
-\end{align}
-where
-\begin{align}
-    H_F^\pm = \{\psi \in H_F |\; \gamma_F\psi = \pm \psi\},
-\end{align}
-are two eigenspaces. And lastly the Dirac operator $D_F$ lets us
-interchange between the two eigenspaces $H_F^\pm$,
-\begin{align}
-    D_F =
-        \begin{pmatrix}0 & t \\ \bar{t} & 0\end{pmatrix},  \;\;\;\;\;
-            \text{with} \;\; t\in\mathbb{C}.
-\end{align}
-
-The Two-Point space $F_X$ can only have a real structure if the Dirac
-operator vanishes, i.e. $D_F = 0$. In that case the KO-dimension is 0,
-2 or 6. To elaborate further, we draw the only two diagram representations of
-$F_X$ at $\underbrace{\mathbb{C} \oplus \mathbb{C}}_{C(X)}$ on
-$\underbrace{\mathbb{C} \oplus\mathbb{C}}_{H_F}$, which are
-\begin{figure}[h!] \centering
-\begin{tikzpicture}[
-    dot/.style = {draw, circle, inner sep=0.06cm},
-    no/.style = {},
-    ]
-    \node[no](a) at (0,0) [label=left:$\textbf{1}^\circ$] {};
-    \node[no](b) at (0, -1) [label=left:$\textbf{1}^\circ$] {};
-    \node[no](c) at (1, 0.5) [label=above:$\textbf{1}$] {};
-    \node[no](d) at (2, 0.5) [label=above:$\textbf{1}$] {};
-    \node[dot](d0) at (2,0) [] {};
-    \node[dot](d0) at (1,-1) [] {};
-
-    \node[no](a1) at (6,0) [label=left:$\textbf{1}^\circ$] {};
-    \node[no](b2) at (6, -1) [label=left:$\textbf{1}^\circ$] {};
-    \node[no](c2) at (7, 0.5) [label=above:$\textbf{1}$] {};
-    \node[no](d2) at (8, 0.5) [label=above:$\textbf{1}$] {};
-    \node[dot](d0) at (7,0) [] {};
-    \node[dot](d0) at (8,-1) [] {};
-    \end{tikzpicture}
-    \caption{Two diagram representations of $F_X$}
-\end{figure}\newline
-If the Two-Point space $F_X$ would be a real spectral triple then $D_F$ can
-only go vertically or horizontally. This would mean that $D_F$ vanishes.
-As for the KO-dimension The diagram on the left has KO-dimension 2 and 6, the diagram on the
-right 0 and 4. Yet KO-dimension 4 is ruled out because
-$dim(H_F^\pm) = 1$ (Lemma 3.8 in \cite{ncgwalter}) , which ultimately means $J_F^2 = -1$ is
-not allowed.
-\subsubsection{Product Space}
-By Extending the Two-Point space with a four dimensional Riemannian spin
-manifold, we get an almost commutative manifold $M\times F_X$, given by
-\begin{align}
-    M\times F_X = \big(C^\infty(M, \mathbb{C}^2), L^2(S)\otimes \mathbb{C}^2,
-    D_M\otimes 1 ; J_M\otimes J_F, \gamma_M \otimes \gamma_F\big),
-\end{align}
-where
-\begin{align}
-   C^\infty(M, \mathbb{C}^2) \simeq C^\infty(M) \oplus  C^\infty(M).
-\end{align}
-According to Gelfand duality the algebra $C^\infty(M, \mathbb{C}^2)$ of the
-spectral triple corresponds to the space
-\begin{align}
-    N:= M\otimes X.
-\end{align}
-Keep in mind that we still need to find an appropriate real structure on the
-Riemannian spin manifold, $J_M$. Furthermore the total Hilbertspace can be
-decomposed into $H = L^2(S) \oplus L^2(S)$, such that for $\underbrace{a,b\in
-C^\infty(M)}_{(a, b) \in C^\infty(N)}$ and $\underbrace{\psi, \phi \in
-L^2(S)}_{(\psi, \phi) \in H}$ we have
-\begin{align}
-    (a, b)(\psi, \phi) = (a\psi, b\phi).
-\end{align}
-Along with the decomposition of the total Hilbertspace a
-distance formula on $M\times F_X$ can be considered with
-\begin{align}\label{eq:commutator inequality}
-    d_{D_F}(x,y) = \sup\left\{  |a(x) - a(y)|:a\in A_F, ||[D_F, a]|| \leq
-    1 \right\}.
-\end{align}
-To calculate the distance between two points on the Two-Point space $X= \{x,
-y\}$, between $x$ and $y$, we consider an $a \in \mathbb{C}^2 = C(X)$, which is
-specified by two complex numbers $a(x)$ and $a(y)$. Then we simplify the
-commutator inequality in \eqref{eq:commutator inequality}
-\begin{align}
-    &||[D_F , a]|| = ||(a(y) - a(x))\begin{pmatrix}0 &t\\\bar{t} &0
-    \end{pmatrix}|| \leq 1,\\
-    &\Leftrightarrow |a(y) - a(x)|\leq \frac{1}{|t|}.
-\end{align}
-The supremum then gives us the distance
-\begin{align}
-    d_{D_F} (x,y) = \frac{1}{|t|}.
-\end{align}
-An interesting observation here is that, if the Riemannian spin manifold can be
-represented by a real spectral triple then a real structure $J_M$ exists,
-along the lines it follows that $t=0$ and the distance becomes infinite. This is a
-purely mathematical observation and has no physical meaning.
-
-We can also construct a distance formula on $N$ (in reference to a point $p
-\in M$) between two points on $N=M\times X$, $(p, x)$ and $(p,y)$. Then an $a
-\in C^\infty(N)$ is determined by $a_x(p):=a(p, x)$ and $a_y(p):=a(p, y)$.
-The distance between these two points is
-\begin{align}
-    d_{D_F\otimes 1}(n_1, n_2) =  \sup \left\{ |a(n_1) - a(n_2)|: a\in
-    A, ||[D\otimes 1, a]||\right\}.
-\end{align}
-On the other hand if we consider $n_1 = (p,x)$ and $n_2 = (q, x)$
-for $p,q \in M$ then
-\begin{align}
-    d_{D_M \otimes 1} (n_1, n_2) = |a_x(p) - a_x(q)| \;\;\;\text{for}\;\;
-    a_x\in
-    C^\infty(M) \;\; \text{with} \;\; ||[D\otimes 1, a_x]|| \leq 1
-\end{align}
-The distance formula turns to out to be the geodesic distance formula
-\begin{align}
-    d_{D_M\otimes1}(n_1, n_2) = d_g(p, q),
-\end{align}
-which is to be expected since we are only looking at the manifold.
-However if $n_1 = (p, x)$ and $n_2 = (q, y)$ then the two conditions are
-\begin{align}
-    &||[D_M, a_x]|| \leq 1, \;\;\; \text{and}\\
-    &||[D_M, a_y|| \leq 1.
-\end{align}
-These conditions have no restriction which results in the distance being
-infinite! And $N = M\times X$ is given by two disjoint copies of M  which are
-separated by infinite distance
-
-The distance is only finite if $[D_F, a] < 1$. In this case the commutator
-generates a scalar field and the finiteness of the distance is
-related to the existence of scalar fields.
-
-\subsubsection{$U(1)$ Gauge Group}
-To get a insight into the physical properties of the almost commutative
-manifold $M\times F_X$, that is to calculate the spectral action, we need to
-determine the corresponding Gauge group.
-For this we set of with simple definitions and important propositions to
-help us break down and search for the gauge group of the Two-Point $F_X$
-space which we then extend to $M\times F_X$. We will only be diving
-superficially into this chapter, for further reading we refer to
-\cite{ncgwalter}.
-\begin{mydefinition}
-Gauge Group of a real spectral triple is given by
-\begin{align}
-    \mathfrak{B}(A, H; J) := \{ U = uJuJ^{-1} | u\in U(A)\}.
-\end{align}
-\end{mydefinition}
-\begin{mydefinition}
-    A *-automorphism of a *-algebra $A$ is a linear invertible
-    map
-    \begin{align}
-        &\alpha:A \rightarrow A,\;\;\; \text{with}\\
-        \nonumber\\
-        &\alpha(ab) = \alpha(a)\alpha(b),\\
-        &\alpha(a)^* = \alpha(a^*).
-    \end{align}
-    The \textbf{Group of automorphisms of the *-Algebra $A$} is denoted by
-    $(A)$.\newline
-    The automorphism $\alpha$ is called \textbf{inner} if
-    \begin{align}
-        \alpha(a) = u a u^* \;\;\; \text{for} \;\; U(A),
-    \end{align}
-    where $U(A)$ is
-    \begin{align}
-        U(A) = \{ u\in A|\;\; uu^* = u^*u=1\}. \;\;\;
-        \text{(unitary)}
-    \end{align}
-\end{mydefinition}
-The Gauge group of $F_X$ is given by the quotient $U(A)/U(A_J)$.
-To get a nontrivial Gauge group so we need to choose a $U(A_J) \neq
-U(A)$ and $U((A_F)_{J_F}) \neq U(A_F)$.
-We consider our Two-Point space $F_X$ to be equipped with a real structure,
-which means the operator vanishes, and the spectral triple representation is
-\begin{align}
-    F_X := \left(\mathbb{C}^2,\mathbb{C}^2, D_F =\begin{pmatrix}
-        0&0\\0&0\end{pmatrix}; J_f =\begin{pmatrix}
-    0&C\\C&0\end{pmatrix},
-            \gamma_F = \begin{pmatrix}1&0\\0&-1\end{pmatrix}\right).
-\end{align}
-Here $C$ is the complex conjugation, and $F_X$ is a real even finite
-spectral triple (space) of KO-dimension 6.
-
-\begin{myproposition}
-The Gauge group of the Two-Point space $\mathfrak{B}(F_X)$ is $U(1)$.
-\end{myproposition}
-\begin{proof}
-    Note that $U(A_F) = U(1) \times U(1)$. We need to show that $U(A_F) \cap
-    U(A_F)_{J_F}) \simeq U(1)$, such that $\mathfrak{B}(F) \simeq U(1)$. So
-    for an element $a \in \mathbb{C}^2$ to be in $(A_F)_{J_F}$, it has to
-    satisfy $J_F a^* J_F = a$,
-    \begin{align}
-        J_F a^* J^{-1} =
-        \begin{pmatrix}0&C\\C&0\end{pmatrix}
-            \begin{pmatrix}\bar{a}_1&0\\0&\bar{a}_2\end{pmatrix}
-        \begin{pmatrix}0&C\\C&0\end{pmatrix}
-            =
-            \begin{pmatrix}a_2&0\\0&a_1\end{pmatrix}.
-    \end{align}
-    This can only be the case if $a_1 = a_2$. So we have
-    $(A_F)_{J_F} \simeq \mathbb{C}$, whose unitary elements
-    from $U(1)$ are contained in the diagonal subgroup of
-    $U(A_F)$.
-\end{proof}
-
-An arbitrary hermitian field $A_\mu = -ia\partial _\mu b$  is given by
-two $U(1)$ Gauge fields $X_\mu^1, X_\mu^2 \in C^\infty(M, \mathbb{R})$.
-However $A_\mu$ appears in combination $A_\mu - J_F A_\mu J_F^{-1}$:
-\begin{align}
- A_\mu - J_F A_\mu J_F^{-1} =
-    \begin{pmatrix}X_\mu^1&0\\0&X_\mu^2 \end{pmatrix}
-        -
-    \begin{pmatrix}X_\mu^2&0\\0&X_\mu^1 \end{pmatrix}
-        =:
-    \begin{pmatrix}Y_\mu&0\\0&-Y_\mu \end{pmatrix}
-    = Y_\mu \otimes \gamma _F,
-\end{align}
-where $Y_\mu$ the $U(1)$ Gauge field is defined as
-\begin{align}
-    Y_\mu := X_\mu^1 - X_\mu^2 \in C^\infty(M, \mathbb{R}) = C^\infty(M,
-    i\ u(1)).
-\end{align}
-
-\begin{myproposition}
-    The inner fluctuations of the almost-commutative manifold $M\times
-    F_X$ are parameterized by a $U(1)$-gauge field $Y_\mu$ as
-    \begin{align}
-        D \mapsto D' = D + \gamma ^\mu Y_\mu \otimes \gamma_F
-    \end{align}
-    The action of the gauge group $\mathfrak{B}(M\times F_X) \simeq
-    C^\infty (M, U(1))$ on $D'$ is implemented by
-    \begin{align}
-        Y_\mu \mapsto Y_\mu - i\ u\partial_\mu u^*; \;\;\;\;\; (u\in
-        \mathfrak{B}(M\times F_X)).
-    \end{align}
-\end{myproposition}
-
diff --git a/src/thesis/main.aux b/src/thesis/main.aux
diff --git a/src/thesis/main.bcf b/src/thesis/main.bcf
@@ -0,0 +1,2260 @@
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+  <!-- BIBLATEX OPTION SCOPE -->
+  <bcf:optionscope type="GLOBAL">
+    <bcf:option datatype="xml">datamodel</bcf:option>
+    <bcf:option datatype="xml">labelalphanametemplate</bcf:option>
+    <bcf:option datatype="xml">labelalphatemplate</bcf:option>
+    <bcf:option datatype="xml">inheritance</bcf:option>
+    <bcf:option datatype="xml">translit</bcf:option>
+    <bcf:option datatype="xml">uniquenametemplate</bcf:option>
+    <bcf:option datatype="xml">sortingnamekeytemplate</bcf:option>
+    <bcf:option datatype="xml">sortingtemplate</bcf:option>
+    <bcf:option datatype="xml">extradatespec</bcf:option>
+    <bcf:option datatype="xml">labelnamespec</bcf:option>
+    <bcf:option datatype="xml">labeltitlespec</bcf:option>
+    <bcf:option datatype="xml">labeldatespec</bcf:option>
+    <bcf:option datatype="string">controlversion</bcf:option>
+    <bcf:option datatype="string">alphaothers</bcf:option>
+    <bcf:option datatype="string">sortalphaothers</bcf:option>
+    <bcf:option datatype="string">presort</bcf:option>
+    <bcf:option datatype="string">texencoding</bcf:option>
+    <bcf:option datatype="string">bibencoding</bcf:option>
+    <bcf:option datatype="string">sortingtemplatename</bcf:option>
+    <bcf:option datatype="string">sortlocale</bcf:option>
+    <bcf:option datatype="string">language</bcf:option>
+    <bcf:option datatype="string">autolang</bcf:option>
+    <bcf:option datatype="string">langhook</bcf:option>
+    <bcf:option datatype="string">indexing</bcf:option>
+    <bcf:option datatype="string">hyperref</bcf:option>
+    <bcf:option datatype="string">backrefsetstyle</bcf:option>
+    <bcf:option datatype="string">block</bcf:option>
+    <bcf:option datatype="string">pagetracker</bcf:option>
+    <bcf:option datatype="string">citecounter</bcf:option>
+    <bcf:option datatype="string">citetracker</bcf:option>
+    <bcf:option datatype="string">ibidtracker</bcf:option>
+    <bcf:option datatype="string">idemtracker</bcf:option>
+    <bcf:option datatype="string">opcittracker</bcf:option>
+    <bcf:option datatype="string">loccittracker</bcf:option>
+    <bcf:option datatype="string">labeldate</bcf:option>
+    <bcf:option datatype="string">labeltime</bcf:option>
+    <bcf:option datatype="string">dateera</bcf:option>
+    <bcf:option datatype="string">date</bcf:option>
+    <bcf:option datatype="string">time</bcf:option>
+    <bcf:option datatype="string">eventdate</bcf:option>
+    <bcf:option datatype="string">eventtime</bcf:option>
+    <bcf:option datatype="string">origdate</bcf:option>
+    <bcf:option datatype="string">origtime</bcf:option>
+    <bcf:option datatype="string">urldate</bcf:option>
+    <bcf:option datatype="string">urltime</bcf:option>
+    <bcf:option datatype="string">alldatesusetime</bcf:option>
+    <bcf:option datatype="string">alldates</bcf:option>
+    <bcf:option datatype="string">alltimes</bcf:option>
+    <bcf:option datatype="string">gregorianstart</bcf:option>
+    <bcf:option datatype="string">autocite</bcf:option>
+    <bcf:option datatype="string">notetype</bcf:option>
+    <bcf:option datatype="string">uniquelist</bcf:option>
+    <bcf:option datatype="string">uniquename</bcf:option>
+    <bcf:option datatype="string">refsection</bcf:option>
+    <bcf:option datatype="string">refsegment</bcf:option>
+    <bcf:option datatype="string">citereset</bcf:option>
+    <bcf:option datatype="string">sortlos</bcf:option>
+    <bcf:option datatype="string">babel</bcf:option>
+    <bcf:option datatype="string">datelabel</bcf:option>
+    <bcf:option datatype="string">backrefstyle</bcf:option>
+    <bcf:option datatype="string">arxiv</bcf:option>
+    <bcf:option datatype="boolean">familyinits</bcf:option>
+    <bcf:option datatype="boolean">giveninits</bcf:option>
+    <bcf:option datatype="boolean">prefixinits</bcf:option>
+    <bcf:option datatype="boolean">suffixinits</bcf:option>
+    <bcf:option datatype="boolean">useafterword</bcf:option>
+    <bcf:option datatype="boolean">useannotator</bcf:option>
+    <bcf:option datatype="boolean">useauthor</bcf:option>
+    <bcf:option datatype="boolean">usebookauthor</bcf:option>
+    <bcf:option datatype="boolean">usecommentator</bcf:option>
+    <bcf:option datatype="boolean">useeditor</bcf:option>
+    <bcf:option datatype="boolean">useeditora</bcf:option>
+    <bcf:option datatype="boolean">useeditorb</bcf:option>
+    <bcf:option datatype="boolean">useeditorc</bcf:option>
+    <bcf:option datatype="boolean">useforeword</bcf:option>
+    <bcf:option datatype="boolean">useholder</bcf:option>
+    <bcf:option datatype="boolean">useintroduction</bcf:option>
+    <bcf:option datatype="boolean">usenamea</bcf:option>
+    <bcf:option datatype="boolean">usenameb</bcf:option>
+    <bcf:option datatype="boolean">usenamec</bcf:option>
+    <bcf:option datatype="boolean">usetranslator</bcf:option>
+    <bcf:option datatype="boolean">useshortauthor</bcf:option>
+    <bcf:option datatype="boolean">useshorteditor</bcf:option>
+    <bcf:option datatype="boolean">debug</bcf:option>
+    <bcf:option datatype="boolean">loadfiles</bcf:option>
+    <bcf:option datatype="boolean">safeinputenc</bcf:option>
+    <bcf:option datatype="boolean">sortcase</bcf:option>
+    <bcf:option datatype="boolean">sortupper</bcf:option>
+    <bcf:option datatype="boolean">terseinits</bcf:option>
+    <bcf:option datatype="boolean">abbreviate</bcf:option>
+    <bcf:option datatype="boolean">dateabbrev</bcf:option>
+    <bcf:option datatype="boolean">clearlang</bcf:option>
+    <bcf:option datatype="boolean">sortcites</bcf:option>
+    <bcf:option datatype="boolean">sortsets</bcf:option>
+    <bcf:option datatype="boolean">backref</bcf:option>
+    <bcf:option datatype="boolean">backreffloats</bcf:option>
+    <bcf:option datatype="boolean">trackfloats</bcf:option>
+    <bcf:option datatype="boolean">parentracker</bcf:option>
+    <bcf:option datatype="boolean">labeldateusetime</bcf:option>
+    <bcf:option datatype="boolean">datecirca</bcf:option>
+    <bcf:option datatype="boolean">dateuncertain</bcf:option>
+    <bcf:option datatype="boolean">dateusetime</bcf:option>
+    <bcf:option datatype="boolean">eventdateusetime</bcf:option>
+    <bcf:option datatype="boolean">origdateusetime</bcf:option>
+    <bcf:option datatype="boolean">urldateusetime</bcf:option>
+    <bcf:option datatype="boolean">julian</bcf:option>
+    <bcf:option datatype="boolean">datezeros</bcf:option>
+    <bcf:option datatype="boolean">timezeros</bcf:option>
+    <bcf:option datatype="boolean">timezones</bcf:option>
+    <bcf:option datatype="boolean">seconds</bcf:option>
+    <bcf:option datatype="boolean">autopunct</bcf:option>
+    <bcf:option datatype="boolean">punctfont</bcf:option>
+    <bcf:option datatype="boolean">labelnumber</bcf:option>
+    <bcf:option datatype="boolean">labelalpha</bcf:option>
+    <bcf:option datatype="boolean">labeltitle</bcf:option>
+    <bcf:option datatype="boolean">labeltitleyear</bcf:option>
+    <bcf:option datatype="boolean">labeldateparts</bcf:option>
+    <bcf:option datatype="boolean">nohashothers</bcf:option>
+    <bcf:option datatype="boolean">nosortothers</bcf:option>
+    <bcf:option datatype="boolean">noroman</bcf:option>
+    <bcf:option datatype="boolean">singletitle</bcf:option>
+    <bcf:option datatype="boolean">uniquetitle</bcf:option>
+    <bcf:option datatype="boolean">uniquebaretitle</bcf:option>
+    <bcf:option datatype="boolean">uniquework</bcf:option>
+    <bcf:option datatype="boolean">uniqueprimaryauthor</bcf:option>
+    <bcf:option datatype="boolean">defernumbers</bcf:option>
+    <bcf:option datatype="boolean">locallabelwidth</bcf:option>
+    <bcf:option datatype="boolean">bibwarn</bcf:option>
+    <bcf:option datatype="boolean">useprefix</bcf:option>
+    <bcf:option datatype="boolean">skipbib</bcf:option>
+    <bcf:option datatype="boolean">skipbiblist</bcf:option>
+    <bcf:option datatype="boolean">skiplab</bcf:option>
+    <bcf:option datatype="boolean">dataonly</bcf:option>
+    <bcf:option datatype="boolean">defernums</bcf:option>
+    <bcf:option datatype="boolean">firstinits</bcf:option>
+    <bcf:option datatype="boolean">sortfirstinits</bcf:option>
+    <bcf:option datatype="boolean">sortgiveninits</bcf:option>
+    <bcf:option datatype="boolean">labelyear</bcf:option>
+    <bcf:option datatype="boolean">isbn</bcf:option>
+    <bcf:option datatype="boolean">url</bcf:option>
+    <bcf:option datatype="boolean">doi</bcf:option>
+    <bcf:option datatype="boolean">eprint</bcf:option>
+    <bcf:option datatype="boolean">related</bcf:option>
+    <bcf:option datatype="boolean">subentry</bcf:option>
+    <bcf:option datatype="boolean">bibtexcaseprotection</bcf:option>
+    <bcf:option datatype="integer">mincrossrefs</bcf:option>
+    <bcf:option datatype="integer">minxrefs</bcf:option>
+    <bcf:option datatype="integer">maxnames</bcf:option>
+    <bcf:option datatype="integer">minnames</bcf:option>
+    <bcf:option datatype="integer">maxbibnames</bcf:option>
+    <bcf:option datatype="integer">minbibnames</bcf:option>
+    <bcf:option datatype="integer">maxcitenames</bcf:option>
+    <bcf:option datatype="integer">mincitenames</bcf:option>
+    <bcf:option datatype="integer">maxsortnames</bcf:option>
+    <bcf:option datatype="integer">minsortnames</bcf:option>
+    <bcf:option datatype="integer">maxitems</bcf:option>
+    <bcf:option datatype="integer">minitems</bcf:option>
+    <bcf:option datatype="integer">maxalphanames</bcf:option>
+    <bcf:option datatype="integer">minalphanames</bcf:option>
+    <bcf:option datatype="integer">maxparens</bcf:option>
+    <bcf:option datatype="integer">dateeraauto</bcf:option>
+  </bcf:optionscope>
+  <bcf:optionscope type="ENTRYTYPE">
+    <bcf:option datatype="string">alphaothers</bcf:option>
+    <bcf:option datatype="string">sortalphaothers</bcf:option>
+    <bcf:option datatype="string">presort</bcf:option>
+    <bcf:option datatype="string">indexing</bcf:option>
+    <bcf:option datatype="string">citetracker</bcf:option>
+    <bcf:option datatype="string">ibidtracker</bcf:option>
+    <bcf:option datatype="string">idemtracker</bcf:option>
+    <bcf:option datatype="string">opcittracker</bcf:option>
+    <bcf:option datatype="string">loccittracker</bcf:option>
+    <bcf:option datatype="string">uniquelist</bcf:option>
+    <bcf:option datatype="string">uniquename</bcf:option>
+    <bcf:option datatype="boolean">familyinits</bcf:option>
+    <bcf:option datatype="boolean">giveninits</bcf:option>
+    <bcf:option datatype="boolean">prefixinits</bcf:option>
+    <bcf:option datatype="boolean">suffixinits</bcf:option>
+    <bcf:option datatype="boolean">useafterword</bcf:option>
+    <bcf:option datatype="boolean">useannotator</bcf:option>
+    <bcf:option datatype="boolean">useauthor</bcf:option>
+    <bcf:option datatype="boolean">usebookauthor</bcf:option>
+    <bcf:option datatype="boolean">usecommentator</bcf:option>
+    <bcf:option datatype="boolean">useeditor</bcf:option>
+    <bcf:option datatype="boolean">useeditora</bcf:option>
+    <bcf:option datatype="boolean">useeditorb</bcf:option>
+    <bcf:option datatype="boolean">useeditorc</bcf:option>
+    <bcf:option datatype="boolean">useforeword</bcf:option>
+    <bcf:option datatype="boolean">useholder</bcf:option>
+    <bcf:option datatype="boolean">useintroduction</bcf:option>
+    <bcf:option datatype="boolean">usenamea</bcf:option>
+    <bcf:option datatype="boolean">usenameb</bcf:option>
+    <bcf:option datatype="boolean">usenamec</bcf:option>
+    <bcf:option datatype="boolean">usetranslator</bcf:option>
+    <bcf:option datatype="boolean">useshortauthor</bcf:option>
+    <bcf:option datatype="boolean">useshorteditor</bcf:option>
+    <bcf:option datatype="boolean">terseinits</bcf:option>
+    <bcf:option datatype="boolean">abbreviate</bcf:option>
+    <bcf:option datatype="boolean">dateabbrev</bcf:option>
+    <bcf:option datatype="boolean">clearlang</bcf:option>
+    <bcf:option datatype="boolean">labelnumber</bcf:option>
+    <bcf:option datatype="boolean">labelalpha</bcf:option>
+    <bcf:option datatype="boolean">labeltitle</bcf:option>
+    <bcf:option datatype="boolean">labeltitleyear</bcf:option>
+    <bcf:option datatype="boolean">labeldateparts</bcf:option>
+    <bcf:option datatype="boolean">nohashothers</bcf:option>
+    <bcf:option datatype="boolean">nosortothers</bcf:option>
+    <bcf:option datatype="boolean">noroman</bcf:option>
+    <bcf:option datatype="boolean">singletitle</bcf:option>
+    <bcf:option datatype="boolean">uniquetitle</bcf:option>
+    <bcf:option datatype="boolean">uniquebaretitle</bcf:option>
+    <bcf:option datatype="boolean">uniquework</bcf:option>
+    <bcf:option datatype="boolean">uniqueprimaryauthor</bcf:option>
+    <bcf:option datatype="boolean">useprefix</bcf:option>
+    <bcf:option datatype="boolean">skipbib</bcf:option>
+    <bcf:option datatype="boolean">skipbiblist</bcf:option>
+    <bcf:option datatype="boolean">skiplab</bcf:option>
+    <bcf:option datatype="boolean">dataonly</bcf:option>
+    <bcf:option datatype="boolean">skiplos</bcf:option>
+    <bcf:option datatype="boolean">labelyear</bcf:option>
+    <bcf:option datatype="boolean">isbn</bcf:option>
+    <bcf:option datatype="boolean">url</bcf:option>
+    <bcf:option datatype="boolean">doi</bcf:option>
+    <bcf:option datatype="boolean">eprint</bcf:option>
+    <bcf:option datatype="boolean">related</bcf:option>
+    <bcf:option datatype="boolean">subentry</bcf:option>
+    <bcf:option datatype="boolean">bibtexcaseprotection</bcf:option>
+    <bcf:option datatype="xml">labelalphatemplate</bcf:option>
+    <bcf:option datatype="xml">translit</bcf:option>
+    <bcf:option datatype="xml">sortexclusion</bcf:option>
+    <bcf:option datatype="xml">sortinclusion</bcf:option>
+    <bcf:option datatype="xml">labelnamespec</bcf:option>
+    <bcf:option datatype="xml">labeltitlespec</bcf:option>
+    <bcf:option datatype="xml">labeldatespec</bcf:option>
+    <bcf:option datatype="integer">maxnames</bcf:option>
+    <bcf:option datatype="integer">minnames</bcf:option>
+    <bcf:option datatype="integer">maxbibnames</bcf:option>
+    <bcf:option datatype="integer">minbibnames</bcf:option>
+    <bcf:option datatype="integer">maxcitenames</bcf:option>
+    <bcf:option datatype="integer">mincitenames</bcf:option>
+    <bcf:option datatype="integer">maxsortnames</bcf:option>
+    <bcf:option datatype="integer">minsortnames</bcf:option>
+    <bcf:option datatype="integer">maxitems</bcf:option>
+    <bcf:option datatype="integer">minitems</bcf:option>
+    <bcf:option datatype="integer">maxalphanames</bcf:option>
+    <bcf:option datatype="integer">minalphanames</bcf:option>
+  </bcf:optionscope>
+  <bcf:optionscope type="ENTRY">
+    <bcf:option datatype="string">noinherit</bcf:option>
+    <bcf:option datatype="string" backendin="sortingnamekeytemplatename,uniquenametemplatename,labelalphanametemplatename">nametemplates</bcf:option>
+    <bcf:option datatype="string" backendout="1">labelalphanametemplatename</bcf:option>
+    <bcf:option datatype="string" backendout="1">uniquenametemplatename</bcf:option>
+    <bcf:option datatype="string" backendout="1">sortingnamekeytemplatename</bcf:option>
+    <bcf:option datatype="string">presort</bcf:option>
+    <bcf:option datatype="string" backendout="1">indexing</bcf:option>
+    <bcf:option datatype="string" backendout="1">citetracker</bcf:option>
+    <bcf:option datatype="string" backendout="1">ibidtracker</bcf:option>
+    <bcf:option datatype="string" backendout="1">idemtracker</bcf:option>
+    <bcf:option datatype="string" backendout="1">opcittracker</bcf:option>
+    <bcf:option datatype="string" backendout="1">loccittracker</bcf:option>
+    <bcf:option datatype="string">uniquelist</bcf:option>
+    <bcf:option datatype="string">uniquename</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">familyinits</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">giveninits</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">prefixinits</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">suffixinits</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">useafterword</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">useannotator</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">useauthor</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">usebookauthor</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">usecommentator</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">useeditor</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">useeditora</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">useeditorb</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">useeditorc</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">useforeword</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">useholder</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">useintroduction</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">usenamea</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">usenameb</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">usenamec</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">usetranslator</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">useshortauthor</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">useshorteditor</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">terseinits</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">abbreviate</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">dateabbrev</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">clearlang</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">labelnumber</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">labelalpha</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">labeltitle</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">labeltitleyear</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">labeldateparts</bcf:option>
+    <bcf:option datatype="boolean">nohashothers</bcf:option>
+    <bcf:option datatype="boolean">nosortothers</bcf:option>
+    <bcf:option datatype="boolean">noroman</bcf:option>
+    <bcf:option datatype="boolean">singletitle</bcf:option>
+    <bcf:option datatype="boolean">uniquetitle</bcf:option>
+    <bcf:option datatype="boolean">uniquebaretitle</bcf:option>
+    <bcf:option datatype="boolean">uniquework</bcf:option>
+    <bcf:option datatype="boolean">uniqueprimaryauthor</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">useprefix</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">skipbib</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">skipbiblist</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">skiplab</bcf:option>
+    <bcf:option datatype="boolean" backendin="uniquename=false,uniquelist=false,skipbib=true,skipbiblist=true,skiplab=true">dataonly</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">skiplos</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">isbn</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">url</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">doi</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">eprint</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">related</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">subentry</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">bibtexcaseprotection</bcf:option>
+    <bcf:option datatype="integer" backendin="maxcitenames,maxbibnames,maxsortnames">maxnames</bcf:option>
+    <bcf:option datatype="integer" backendin="mincitenames,minbibnames,minsortnames">minnames</bcf:option>
+    <bcf:option datatype="integer" backendout="1">maxbibnames</bcf:option>
+    <bcf:option datatype="integer" backendout="1">minbibnames</bcf:option>
+    <bcf:option datatype="integer" backendout="1">maxcitenames</bcf:option>
+    <bcf:option datatype="integer" backendout="1">mincitenames</bcf:option>
+    <bcf:option datatype="integer" backendout="1">maxsortnames</bcf:option>
+    <bcf:option datatype="integer" backendout="1">minsortnames</bcf:option>
+    <bcf:option datatype="integer" backendout="1">maxitems</bcf:option>
+    <bcf:option datatype="integer" backendout="1">minitems</bcf:option>
+    <bcf:option datatype="integer" backendout="1">maxalphanames</bcf:option>
+    <bcf:option datatype="integer" backendout="1">minalphanames</bcf:option>
+  </bcf:optionscope>
+  <bcf:optionscope type="NAMELIST">
+    <bcf:option datatype="string" backendin="sortingnamekeytemplatename,uniquenametemplatename,labelalphanametemplatename">nametemplates</bcf:option>
+    <bcf:option datatype="string" backendout="1">labelalphanametemplatename</bcf:option>
+    <bcf:option datatype="string" backendout="1">uniquenametemplatename</bcf:option>
+    <bcf:option datatype="string" backendout="1">sortingnamekeytemplatename</bcf:option>
+    <bcf:option datatype="string">uniquelist</bcf:option>
+    <bcf:option datatype="string">uniquename</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">familyinits</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">giveninits</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">prefixinits</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">suffixinits</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">terseinits</bcf:option>
+    <bcf:option datatype="boolean">nohashothers</bcf:option>
+    <bcf:option datatype="boolean">nosortothers</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">useprefix</bcf:option>
+  </bcf:optionscope>
+  <bcf:optionscope type="NAME">
+    <bcf:option datatype="string" backendin="sortingnamekeytemplatename,uniquenametemplatename,labelalphanametemplatename">nametemplates</bcf:option>
+    <bcf:option datatype="string" backendout="1">labelalphanametemplatename</bcf:option>
+    <bcf:option datatype="string" backendout="1">uniquenametemplatename</bcf:option>
+    <bcf:option datatype="string" backendout="1">sortingnamekeytemplatename</bcf:option>
+    <bcf:option datatype="string">uniquename</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">familyinits</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">giveninits</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">prefixinits</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">suffixinits</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">terseinits</bcf:option>
+    <bcf:option datatype="boolean" backendout="1">useprefix</bcf:option>
+  </bcf:optionscope>
+  <!-- DATAFIELDSETS -->
+  <bcf:datafieldset name="setnames">
+    <bcf:member datatype="name" fieldtype="list"/>
+  </bcf:datafieldset>
+  <bcf:datafieldset name="settitles">
+    <bcf:member field="title"/>
+    <bcf:member field="booktitle"/>
+    <bcf:member field="eventtitle"/>
+    <bcf:member field="issuetitle"/>
+    <bcf:member field="journaltitle"/>
+    <bcf:member field="maintitle"/>
+    <bcf:member field="origtitle"/>
+  </bcf:datafieldset>
+  <!-- SOURCEMAP -->
+  <bcf:sourcemap>
+    <bcf:maps datatype="bibtex" level="driver">
+      <bcf:map>
+        <bcf:map_step map_field_set="day" map_null="1"/>
+      </bcf:map>
+      <bcf:map>
+        <bcf:map_step map_type_source="conference" map_type_target="inproceedings"/>
+        <bcf:map_step map_type_source="electronic" map_type_target="online"/>
+        <bcf:map_step map_type_source="www" map_type_target="online"/>
+      </bcf:map>
+      <bcf:map>
+        <bcf:map_step map_type_source="mastersthesis" map_type_target="thesis" map_final="1"/>
+        <bcf:map_step map_field_set="type" map_field_value="mathesis"/>
+      </bcf:map>
+      <bcf:map>
+        <bcf:map_step map_type_source="phdthesis" map_type_target="thesis" map_final="1"/>
+        <bcf:map_step map_field_set="type" map_field_value="phdthesis"/>
+      </bcf:map>
+      <bcf:map>
+        <bcf:map_step map_type_source="techreport" map_type_target="report" map_final="1"/>
+        <bcf:map_step map_field_set="type" map_field_value="techreport"/>
+      </bcf:map>
+      <bcf:map>
+        <bcf:map_step map_field_source="hyphenation" map_field_target="langid"/>
+        <bcf:map_step map_field_source="address" map_field_target="location"/>
+        <bcf:map_step map_field_source="school" map_field_target="institution"/>
+        <bcf:map_step map_field_source="annote" map_field_target="annotation"/>
+        <bcf:map_step map_field_source="archiveprefix" map_field_target="eprinttype"/>
+        <bcf:map_step map_field_source="journal" map_field_target="journaltitle"/>
+        <bcf:map_step map_field_source="primaryclass" map_field_target="eprintclass"/>
+        <bcf:map_step map_field_source="key" map_field_target="sortkey"/>
+        <bcf:map_step map_field_source="pdf" map_field_target="file"/>
+      </bcf:map>
+    </bcf:maps>
+  </bcf:sourcemap>
+  <!-- LABELALPHA NAME TEMPLATE -->
+  <bcf:labelalphanametemplate name="global">
+    <bcf:namepart order="1" use="1" pre="1" substring_width="1" substring_compound="1">prefix</bcf:namepart>
+    <bcf:namepart order="2">family</bcf:namepart>
+  </bcf:labelalphanametemplate>
+  <!-- LABELALPHA TEMPLATE -->
+  <bcf:labelalphatemplate type="global">
+    <bcf:labelelement order="1">
+      <bcf:labelpart final="1">shorthand</bcf:labelpart>
+      <bcf:labelpart>label</bcf:labelpart>
+      <bcf:labelpart substring_width="3" substring_side="left" ifnames="1">labelname</bcf:labelpart>
+      <bcf:labelpart substring_width="1" substring_side="left">labelname</bcf:labelpart>
+    </bcf:labelelement>
+    <bcf:labelelement order="2">
+      <bcf:labelpart substring_width="2" substring_side="right">year</bcf:labelpart>
+    </bcf:labelelement>
+  </bcf:labelalphatemplate>
+  <!-- EXTRADATE -->
+  <bcf:extradatespec>
+    <bcf:scope>
+      <bcf:field order="1">labelyear</bcf:field>
+      <bcf:field order="2">year</bcf:field>
+    </bcf:scope>
+  </bcf:extradatespec>
+  <!-- INHERITANCE -->
+  <bcf:inheritance>
+    <bcf:defaults inherit_all="true" override_target="false">
+    </bcf:defaults>
+    <bcf:inherit>
+      <bcf:type_pair source="mvbook" target="inbook"/>
+      <bcf:type_pair source="mvbook" target="bookinbook"/>
+      <bcf:type_pair source="mvbook" target="suppbook"/>
+      <bcf:type_pair source="book" target="inbook"/>
+      <bcf:type_pair source="book" target="bookinbook"/>
+      <bcf:type_pair source="book" target="suppbook"/>
+      <bcf:field source="author" target="author"/>
+      <bcf:field source="author" target="bookauthor"/>
+    </bcf:inherit>
+    <bcf:inherit>
+      <bcf:type_pair source="mvbook" target="book"/>
+      <bcf:type_pair source="mvbook" target="inbook"/>
+      <bcf:type_pair source="mvbook" target="bookinbook"/>
+      <bcf:type_pair source="mvbook" target="suppbook"/>
+      <bcf:field source="title" target="maintitle"/>
+      <bcf:field source="subtitle" target="mainsubtitle"/>
+      <bcf:field source="titleaddon" target="maintitleaddon"/>
+      <bcf:field source="shorttitle" skip="true"/>
+      <bcf:field source="sorttitle" skip="true"/>
+      <bcf:field source="indextitle" skip="true"/>
+      <bcf:field source="indexsorttitle" skip="true"/>
+    </bcf:inherit>
+    <bcf:inherit>
+      <bcf:type_pair source="mvcollection" target="collection"/>
+      <bcf:type_pair source="mvcollection" target="reference"/>
+      <bcf:type_pair source="mvcollection" target="incollection"/>
+      <bcf:type_pair source="mvcollection" target="inreference"/>
+      <bcf:type_pair source="mvcollection" target="suppcollection"/>
+      <bcf:type_pair source="mvreference" target="collection"/>
+      <bcf:type_pair source="mvreference" target="reference"/>
+      <bcf:type_pair source="mvreference" target="incollection"/>
+      <bcf:type_pair source="mvreference" target="inreference"/>
+      <bcf:type_pair source="mvreference" target="suppcollection"/>
+      <bcf:field source="title" target="maintitle"/>
+      <bcf:field source="subtitle" target="mainsubtitle"/>
+      <bcf:field source="titleaddon" target="maintitleaddon"/>
+      <bcf:field source="shorttitle" skip="true"/>
+      <bcf:field source="sorttitle" skip="true"/>
+      <bcf:field source="indextitle" skip="true"/>
+      <bcf:field source="indexsorttitle" skip="true"/>
+    </bcf:inherit>
+    <bcf:inherit>
+      <bcf:type_pair source="mvproceedings" target="proceedings"/>
+      <bcf:type_pair source="mvproceedings" target="inproceedings"/>
+      <bcf:field source="title" target="maintitle"/>
+      <bcf:field source="subtitle" target="mainsubtitle"/>
+      <bcf:field source="titleaddon" target="maintitleaddon"/>
+      <bcf:field source="shorttitle" skip="true"/>
+      <bcf:field source="sorttitle" skip="true"/>
+      <bcf:field source="indextitle" skip="true"/>
+      <bcf:field source="indexsorttitle" skip="true"/>
+    </bcf:inherit>
+    <bcf:inherit>
+      <bcf:type_pair source="book" target="inbook"/>
+      <bcf:type_pair source="book" target="bookinbook"/>
+      <bcf:type_pair source="book" target="suppbook"/>
+      <bcf:field source="title" target="booktitle"/>
+      <bcf:field source="subtitle" target="booksubtitle"/>
+      <bcf:field source="titleaddon" target="booktitleaddon"/>
+      <bcf:field source="shorttitle" skip="true"/>
+      <bcf:field source="sorttitle" skip="true"/>
+      <bcf:field source="indextitle" skip="true"/>
+      <bcf:field source="indexsorttitle" skip="true"/>
+    </bcf:inherit>
+    <bcf:inherit>
+      <bcf:type_pair source="collection" target="incollection"/>
+      <bcf:type_pair source="collection" target="inreference"/>
+      <bcf:type_pair source="collection" target="suppcollection"/>
+      <bcf:type_pair source="reference" target="incollection"/>
+      <bcf:type_pair source="reference" target="inreference"/>
+      <bcf:type_pair source="reference" target="suppcollection"/>
+      <bcf:field source="title" target="booktitle"/>
+      <bcf:field source="subtitle" target="booksubtitle"/>
+      <bcf:field source="titleaddon" target="booktitleaddon"/>
+      <bcf:field source="shorttitle" skip="true"/>
+      <bcf:field source="sorttitle" skip="true"/>
+      <bcf:field source="indextitle" skip="true"/>
+      <bcf:field source="indexsorttitle" skip="true"/>
+    </bcf:inherit>
+    <bcf:inherit>
+      <bcf:type_pair source="proceedings" target="inproceedings"/>
+      <bcf:field source="title" target="booktitle"/>
+      <bcf:field source="subtitle" target="booksubtitle"/>
+      <bcf:field source="titleaddon" target="booktitleaddon"/>
+      <bcf:field source="shorttitle" skip="true"/>
+      <bcf:field source="sorttitle" skip="true"/>
+      <bcf:field source="indextitle" skip="true"/>
+      <bcf:field source="indexsorttitle" skip="true"/>
+    </bcf:inherit>
+    <bcf:inherit>
+      <bcf:type_pair source="periodical" target="article"/>
+      <bcf:type_pair source="periodical" target="suppperiodical"/>
+      <bcf:field source="title" target="journaltitle"/>
+      <bcf:field source="subtitle" target="journalsubtitle"/>
+      <bcf:field source="titleaddon" target="journaltitleaddon"/>
+      <bcf:field source="shorttitle" skip="true"/>
+      <bcf:field source="sorttitle" skip="true"/>
+      <bcf:field source="indextitle" skip="true"/>
+      <bcf:field source="indexsorttitle" skip="true"/>
+    </bcf:inherit>
+    <bcf:inherit>
+      <bcf:type_pair source="*" target="*"/>
+      <bcf:field source="ids" skip="true"/>
+      <bcf:field source="crossref" skip="true"/>
+      <bcf:field source="xref" skip="true"/>
+      <bcf:field source="entryset" skip="true"/>
+      <bcf:field source="entrysubtype" skip="true"/>
+      <bcf:field source="execute" skip="true"/>
+      <bcf:field source="label" skip="true"/>
+      <bcf:field source="options" skip="true"/>
+      <bcf:field source="presort" skip="true"/>
+      <bcf:field source="related" skip="true"/>
+      <bcf:field source="relatedoptions" skip="true"/>
+      <bcf:field source="relatedstring" skip="true"/>
+      <bcf:field source="relatedtype" skip="true"/>
+      <bcf:field source="shorthand" skip="true"/>
+      <bcf:field source="shorthandintro" skip="true"/>
+      <bcf:field source="sortkey" skip="true"/>
+    </bcf:inherit>
+  </bcf:inheritance>
+  <!-- UNIQUENAME TEMPLATES -->
+  <bcf:uniquenametemplate name="global">
+    <bcf:namepart order="1" use="1" base="1">prefix</bcf:namepart>
+    <bcf:namepart order="2" base="1">family</bcf:namepart>
+    <bcf:namepart order="3">given</bcf:namepart>
+  </bcf:uniquenametemplate>
+  <!-- SORTING NAME KEY TEMPLATES -->
+  <bcf:sortingnamekeytemplate name="global">
+    <bcf:keypart order="1">
+      <bcf:part type="namepart" order="1" use="1">prefix</bcf:part>
+      <bcf:part type="namepart" order="2">family</bcf:part>
+    </bcf:keypart>
+    <bcf:keypart order="2">
+      <bcf:part type="namepart" order="1">given</bcf:part>
+    </bcf:keypart>
+    <bcf:keypart order="3">
+      <bcf:part type="namepart" order="1">suffix</bcf:part>
+    </bcf:keypart>
+    <bcf:keypart order="4">
+      <bcf:part type="namepart" order="1" use="0">prefix</bcf:part>
+    </bcf:keypart>
+  </bcf:sortingnamekeytemplate>
+  <bcf:presort>mm</bcf:presort>
+  <!-- DATA MODEL -->
+  <bcf:datamodel>
+    <bcf:constants>
+      <bcf:constant type="list" name="gender">sf,sm,sn,pf,pm,pn,pp</bcf:constant>
+      <bcf:constant type="list" name="nameparts">family,given,prefix,suffix</bcf:constant>
+      <bcf:constant type="list" name="optiondatatypes">boolean,integer,string,xml</bcf:constant>
+      <bcf:constant type="list" name="multiscriptforms">default,transliteration,transcription,translation</bcf:constant>
+    </bcf:constants>
+    <bcf:entrytypes>
+      <bcf:entrytype>article</bcf:entrytype>
+      <bcf:entrytype>artwork</bcf:entrytype>
+      <bcf:entrytype>audio</bcf:entrytype>
+      <bcf:entrytype>bibnote</bcf:entrytype>
+      <bcf:entrytype>book</bcf:entrytype>
+      <bcf:entrytype>bookinbook</bcf:entrytype>
+      <bcf:entrytype>booklet</bcf:entrytype>
+      <bcf:entrytype>collection</bcf:entrytype>
+      <bcf:entrytype>commentary</bcf:entrytype>
+      <bcf:entrytype>customa</bcf:entrytype>
+      <bcf:entrytype>customb</bcf:entrytype>
+      <bcf:entrytype>customc</bcf:entrytype>
+      <bcf:entrytype>customd</bcf:entrytype>
+      <bcf:entrytype>custome</bcf:entrytype>
+      <bcf:entrytype>customf</bcf:entrytype>
+      <bcf:entrytype>dataset</bcf:entrytype>
+      <bcf:entrytype>inbook</bcf:entrytype>
+      <bcf:entrytype>incollection</bcf:entrytype>
+      <bcf:entrytype>inproceedings</bcf:entrytype>
+      <bcf:entrytype>inreference</bcf:entrytype>
+      <bcf:entrytype>image</bcf:entrytype>
+      <bcf:entrytype>jurisdiction</bcf:entrytype>
+      <bcf:entrytype>legal</bcf:entrytype>
+      <bcf:entrytype>legislation</bcf:entrytype>
+      <bcf:entrytype>letter</bcf:entrytype>
+      <bcf:entrytype>manual</bcf:entrytype>
+      <bcf:entrytype>misc</bcf:entrytype>
+      <bcf:entrytype>movie</bcf:entrytype>
+      <bcf:entrytype>music</bcf:entrytype>
+      <bcf:entrytype>mvcollection</bcf:entrytype>
+      <bcf:entrytype>mvreference</bcf:entrytype>
+      <bcf:entrytype>mvproceedings</bcf:entrytype>
+      <bcf:entrytype>mvbook</bcf:entrytype>
+      <bcf:entrytype>online</bcf:entrytype>
+      <bcf:entrytype>patent</bcf:entrytype>
+      <bcf:entrytype>performance</bcf:entrytype>
+      <bcf:entrytype>periodical</bcf:entrytype>
+      <bcf:entrytype>proceedings</bcf:entrytype>
+      <bcf:entrytype>reference</bcf:entrytype>
+      <bcf:entrytype>report</bcf:entrytype>
+      <bcf:entrytype>review</bcf:entrytype>
+      <bcf:entrytype>set</bcf:entrytype>
+      <bcf:entrytype>software</bcf:entrytype>
+      <bcf:entrytype>standard</bcf:entrytype>
+      <bcf:entrytype>suppbook</bcf:entrytype>
+      <bcf:entrytype>suppcollection</bcf:entrytype>
+      <bcf:entrytype>suppperiodical</bcf:entrytype>
+      <bcf:entrytype>thesis</bcf:entrytype>
+      <bcf:entrytype>unpublished</bcf:entrytype>
+      <bcf:entrytype>video</bcf:entrytype>
+      <bcf:entrytype skip_output="true">xdata</bcf:entrytype>
+    </bcf:entrytypes>
+    <bcf:fields>
+      <bcf:field fieldtype="field" datatype="integer">sortyear</bcf:field>
+      <bcf:field fieldtype="field" datatype="integer">volume</bcf:field>
+      <bcf:field fieldtype="field" datatype="integer">volumes</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">abstract</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">addendum</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">annotation</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">booksubtitle</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">booktitle</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">booktitleaddon</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">chapter</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">edition</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">eid</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">entrysubtype</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">eprintclass</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">eprinttype</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">eventtitle</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">eventtitleaddon</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">gender</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">howpublished</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">indexsorttitle</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">indextitle</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">isan</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">isbn</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">ismn</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">isrn</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">issn</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">issue</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">issuesubtitle</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">issuetitle</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">issuetitleaddon</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">iswc</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">journalsubtitle</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">journaltitle</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">journaltitleaddon</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">label</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">langid</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">langidopts</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">library</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">mainsubtitle</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">maintitle</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">maintitleaddon</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">nameaddon</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">note</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">number</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">origtitle</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">pagetotal</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">part</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">relatedstring</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">relatedtype</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">reprinttitle</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">series</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">shorthandintro</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">subtitle</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">title</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">titleaddon</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">usera</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">userb</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">userc</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">userd</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">usere</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">userf</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">venue</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal">version</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal" label="true">shorthand</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal" label="true">shortjournal</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal" label="true">shortseries</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal" label="true">shorttitle</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal" skip_output="true">sorttitle</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal" skip_output="true">sortshorthand</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal" skip_output="true">sortkey</bcf:field>
+      <bcf:field fieldtype="field" datatype="literal" skip_output="true">presort</bcf:field>
+      <bcf:field fieldtype="list" datatype="literal">institution</bcf:field>
+      <bcf:field fieldtype="list" datatype="literal">lista</bcf:field>
+      <bcf:field fieldtype="list" datatype="literal">listb</bcf:field>
+      <bcf:field fieldtype="list" datatype="literal">listc</bcf:field>
+      <bcf:field fieldtype="list" datatype="literal">listd</bcf:field>
+      <bcf:field fieldtype="list" datatype="literal">liste</bcf:field>
+      <bcf:field fieldtype="list" datatype="literal">listf</bcf:field>
+      <bcf:field fieldtype="list" datatype="literal">location</bcf:field>
+      <bcf:field fieldtype="list" datatype="literal">organization</bcf:field>
+      <bcf:field fieldtype="list" datatype="literal">origlocation</bcf:field>
+      <bcf:field fieldtype="list" datatype="literal">origpublisher</bcf:field>
+      <bcf:field fieldtype="list" datatype="literal">publisher</bcf:field>
+      <bcf:field fieldtype="list" datatype="name">afterword</bcf:field>
+      <bcf:field fieldtype="list" datatype="name">annotator</bcf:field>
+      <bcf:field fieldtype="list" datatype="name">author</bcf:field>
+      <bcf:field fieldtype="list" datatype="name">bookauthor</bcf:field>
+      <bcf:field fieldtype="list" datatype="name">commentator</bcf:field>
+      <bcf:field fieldtype="list" datatype="name">editor</bcf:field>
+      <bcf:field fieldtype="list" datatype="name">editora</bcf:field>
+      <bcf:field fieldtype="list" datatype="name">editorb</bcf:field>
+      <bcf:field fieldtype="list" datatype="name">editorc</bcf:field>
+      <bcf:field fieldtype="list" datatype="name">foreword</bcf:field>
+      <bcf:field fieldtype="list" datatype="name">holder</bcf:field>
+      <bcf:field fieldtype="list" datatype="name">introduction</bcf:field>
+      <bcf:field fieldtype="list" datatype="name">namea</bcf:field>
+      <bcf:field fieldtype="list" datatype="name">nameb</bcf:field>
+      <bcf:field fieldtype="list" datatype="name">namec</bcf:field>
+      <bcf:field fieldtype="list" datatype="name">translator</bcf:field>
+      <bcf:field fieldtype="list" datatype="name" label="true">shortauthor</bcf:field>
+      <bcf:field fieldtype="list" datatype="name" label="true">shorteditor</bcf:field>
+      <bcf:field fieldtype="list" datatype="name" skip_output="true">sortname</bcf:field>
+      <bcf:field fieldtype="field" datatype="key">authortype</bcf:field>
+      <bcf:field fieldtype="field" datatype="key">editoratype</bcf:field>
+      <bcf:field fieldtype="field" datatype="key">editorbtype</bcf:field>
+      <bcf:field fieldtype="field" datatype="key">editorctype</bcf:field>
+      <bcf:field fieldtype="field" datatype="key">editortype</bcf:field>
+      <bcf:field fieldtype="field" datatype="key">bookpagination</bcf:field>
+      <bcf:field fieldtype="field" datatype="key">nameatype</bcf:field>
+      <bcf:field fieldtype="field" datatype="key">namebtype</bcf:field>
+      <bcf:field fieldtype="field" datatype="key">namectype</bcf:field>
+      <bcf:field fieldtype="field" datatype="key">pagination</bcf:field>
+      <bcf:field fieldtype="field" datatype="key">pubstate</bcf:field>
+      <bcf:field fieldtype="field" datatype="key">type</bcf:field>
+      <bcf:field fieldtype="list" datatype="key">language</bcf:field>
+      <bcf:field fieldtype="list" datatype="key">origlanguage</bcf:field>
+      <bcf:field fieldtype="field" datatype="entrykey">crossref</bcf:field>
+      <bcf:field fieldtype="field" datatype="entrykey">xref</bcf:field>
+      <bcf:field fieldtype="field" datatype="date" skip_output="true">date</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart" nullok="true">endyear</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart" nullok="true">year</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">month</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">day</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">hour</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">minute</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">second</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">timezone</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">season</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">endmonth</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">endday</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">endhour</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">endminute</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">endsecond</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">endtimezone</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">endseason</bcf:field>
+      <bcf:field fieldtype="field" datatype="date" skip_output="true">eventdate</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart" nullok="true">eventendyear</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart" nullok="true">eventyear</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">eventmonth</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">eventday</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">eventhour</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">eventminute</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">eventsecond</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">eventtimezone</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">eventseason</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">eventendmonth</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">eventendday</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">eventendhour</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">eventendminute</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">eventendsecond</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">eventendtimezone</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">eventendseason</bcf:field>
+      <bcf:field fieldtype="field" datatype="date" skip_output="true">origdate</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart" nullok="true">origendyear</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart" nullok="true">origyear</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">origmonth</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">origday</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">orighour</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">origminute</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">origsecond</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">origtimezone</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">origseason</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">origendmonth</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">origendday</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">origendhour</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">origendminute</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">origendsecond</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">origendtimezone</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">origendseason</bcf:field>
+      <bcf:field fieldtype="field" datatype="date" skip_output="true">urldate</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart" nullok="true">urlendyear</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart" nullok="true">urlyear</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">urlmonth</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">urlday</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">urlhour</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">urlminute</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">urlsecond</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">urltimezone</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">urlseason</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">urlendmonth</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">urlendday</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">urlendhour</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">urlendminute</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">urlendsecond</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">urlendtimezone</bcf:field>
+      <bcf:field fieldtype="field" datatype="datepart">urlendseason</bcf:field>
+      <bcf:field fieldtype="field" datatype="verbatim">doi</bcf:field>
+      <bcf:field fieldtype="field" datatype="verbatim">eprint</bcf:field>
+      <bcf:field fieldtype="field" datatype="verbatim">file</bcf:field>
+      <bcf:field fieldtype="field" datatype="verbatim">verba</bcf:field>
+      <bcf:field fieldtype="field" datatype="verbatim">verbb</bcf:field>
+      <bcf:field fieldtype="field" datatype="verbatim">verbc</bcf:field>
+      <bcf:field fieldtype="field" datatype="uri">url</bcf:field>
+      <bcf:field fieldtype="field" format="xsv" datatype="entrykey" skip_output="true">xdata</bcf:field>
+      <bcf:field fieldtype="field" format="xsv" datatype="entrykey" skip_output="true">ids</bcf:field>
+      <bcf:field fieldtype="field" format="xsv" datatype="entrykey" skip_output="true">entryset</bcf:field>
+      <bcf:field fieldtype="field" format="xsv" datatype="entrykey">related</bcf:field>
+      <bcf:field fieldtype="field" format="xsv" datatype="keyword">keywords</bcf:field>
+      <bcf:field fieldtype="field" format="xsv" datatype="option" skip_output="true">options</bcf:field>
+      <bcf:field fieldtype="field" format="xsv" datatype="option" skip_output="true">relatedoptions</bcf:field>
+      <bcf:field fieldtype="field" datatype="range">pages</bcf:field>
+      <bcf:field fieldtype="field" datatype="code">execute</bcf:field>
+    </bcf:fields>
+    <bcf:entryfields>
+      <bcf:field>abstract</bcf:field>
+      <bcf:field>annotation</bcf:field>
+      <bcf:field>authortype</bcf:field>
+      <bcf:field>bookpagination</bcf:field>
+      <bcf:field>crossref</bcf:field>
+      <bcf:field>day</bcf:field>
+      <bcf:field>doi</bcf:field>
+      <bcf:field>eprint</bcf:field>
+      <bcf:field>eprintclass</bcf:field>
+      <bcf:field>eprinttype</bcf:field>
+      <bcf:field>endday</bcf:field>
+      <bcf:field>endhour</bcf:field>
+      <bcf:field>endminute</bcf:field>
+      <bcf:field>endmonth</bcf:field>
+      <bcf:field>endseason</bcf:field>
+      <bcf:field>endsecond</bcf:field>
+      <bcf:field>endtimezone</bcf:field>
+      <bcf:field>endyear</bcf:field>
+      <bcf:field>entryset</bcf:field>
+      <bcf:field>entrysubtype</bcf:field>
+      <bcf:field>execute</bcf:field>
+      <bcf:field>file</bcf:field>
+      <bcf:field>gender</bcf:field>
+      <bcf:field>hour</bcf:field>
+      <bcf:field>ids</bcf:field>
+      <bcf:field>indextitle</bcf:field>
+      <bcf:field>indexsorttitle</bcf:field>
+      <bcf:field>isan</bcf:field>
+      <bcf:field>ismn</bcf:field>
+      <bcf:field>iswc</bcf:field>
+      <bcf:field>keywords</bcf:field>
+      <bcf:field>label</bcf:field>
+      <bcf:field>langid</bcf:field>
+      <bcf:field>langidopts</bcf:field>
+      <bcf:field>library</bcf:field>
+      <bcf:field>lista</bcf:field>
+      <bcf:field>listb</bcf:field>
+      <bcf:field>listc</bcf:field>
+      <bcf:field>listd</bcf:field>
+      <bcf:field>liste</bcf:field>
+      <bcf:field>listf</bcf:field>
+      <bcf:field>minute</bcf:field>
+      <bcf:field>month</bcf:field>
+      <bcf:field>namea</bcf:field>
+      <bcf:field>nameb</bcf:field>
+      <bcf:field>namec</bcf:field>
+      <bcf:field>nameatype</bcf:field>
+      <bcf:field>namebtype</bcf:field>
+      <bcf:field>namectype</bcf:field>
+      <bcf:field>nameaddon</bcf:field>
+      <bcf:field>options</bcf:field>
+      <bcf:field>origday</bcf:field>
+      <bcf:field>origendday</bcf:field>
+      <bcf:field>origendhour</bcf:field>
+      <bcf:field>origendminute</bcf:field>
+      <bcf:field>origendmonth</bcf:field>
+      <bcf:field>origendseason</bcf:field>
+      <bcf:field>origendsecond</bcf:field>
+      <bcf:field>origendtimezone</bcf:field>
+      <bcf:field>origendyear</bcf:field>
+      <bcf:field>orighour</bcf:field>
+      <bcf:field>origminute</bcf:field>
+      <bcf:field>origmonth</bcf:field>
+      <bcf:field>origseason</bcf:field>
+      <bcf:field>origsecond</bcf:field>
+      <bcf:field>origtimezone</bcf:field>
+      <bcf:field>origyear</bcf:field>
+      <bcf:field>origlocation</bcf:field>
+      <bcf:field>origpublisher</bcf:field>
+      <bcf:field>origtitle</bcf:field>
+      <bcf:field>pagination</bcf:field>
+      <bcf:field>presort</bcf:field>
+      <bcf:field>related</bcf:field>
+      <bcf:field>relatedoptions</bcf:field>
+      <bcf:field>relatedstring</bcf:field>
+      <bcf:field>relatedtype</bcf:field>
+      <bcf:field>season</bcf:field>
+      <bcf:field>second</bcf:field>
+      <bcf:field>shortauthor</bcf:field>
+      <bcf:field>shorteditor</bcf:field>
+      <bcf:field>shorthand</bcf:field>
+      <bcf:field>shorthandintro</bcf:field>
+      <bcf:field>shortjournal</bcf:field>
+      <bcf:field>shortseries</bcf:field>
+      <bcf:field>shorttitle</bcf:field>
+      <bcf:field>sortkey</bcf:field>
+      <bcf:field>sortname</bcf:field>
+      <bcf:field>sortshorthand</bcf:field>
+      <bcf:field>sorttitle</bcf:field>
+      <bcf:field>sortyear</bcf:field>
+      <bcf:field>timezone</bcf:field>
+      <bcf:field>url</bcf:field>
+      <bcf:field>urlday</bcf:field>
+      <bcf:field>urlendday</bcf:field>
+      <bcf:field>urlendhour</bcf:field>
+      <bcf:field>urlendminute</bcf:field>
+      <bcf:field>urlendmonth</bcf:field>
+      <bcf:field>urlendsecond</bcf:field>
+      <bcf:field>urlendtimezone</bcf:field>
+      <bcf:field>urlendyear</bcf:field>
+      <bcf:field>urlhour</bcf:field>
+      <bcf:field>urlminute</bcf:field>
+      <bcf:field>urlmonth</bcf:field>
+      <bcf:field>urlsecond</bcf:field>
+      <bcf:field>urltimezone</bcf:field>
+      <bcf:field>urlyear</bcf:field>
+      <bcf:field>usera</bcf:field>
+      <bcf:field>userb</bcf:field>
+      <bcf:field>userc</bcf:field>
+      <bcf:field>userd</bcf:field>
+      <bcf:field>usere</bcf:field>
+      <bcf:field>userf</bcf:field>
+      <bcf:field>verba</bcf:field>
+      <bcf:field>verbb</bcf:field>
+      <bcf:field>verbc</bcf:field>
+      <bcf:field>xdata</bcf:field>
+      <bcf:field>xref</bcf:field>
+      <bcf:field>year</bcf:field>
+    </bcf:entryfields>
+    <bcf:entryfields>
+      <bcf:entrytype>set</bcf:entrytype>
+      <bcf:field>entryset</bcf:field>
+    </bcf:entryfields>
+    <bcf:entryfields>
+      <bcf:entrytype>article</bcf:entrytype>
+      <bcf:field>addendum</bcf:field>
+      <bcf:field>annotator</bcf:field>
+      <bcf:field>author</bcf:field>
+      <bcf:field>commentator</bcf:field>
+      <bcf:field>editor</bcf:field>
+      <bcf:field>editora</bcf:field>
+      <bcf:field>editorb</bcf:field>
+      <bcf:field>editorc</bcf:field>
+      <bcf:field>editortype</bcf:field>
+      <bcf:field>editoratype</bcf:field>
+      <bcf:field>editorbtype</bcf:field>
+      <bcf:field>editorctype</bcf:field>
+      <bcf:field>eid</bcf:field>
+      <bcf:field>issn</bcf:field>
+      <bcf:field>issue</bcf:field>
+      <bcf:field>issuetitle</bcf:field>
+      <bcf:field>issuesubtitle</bcf:field>
+      <bcf:field>issuetitleaddon</bcf:field>
+      <bcf:field>journalsubtitle</bcf:field>
+      <bcf:field>journaltitle</bcf:field>
+      <bcf:field>journaltitleaddon</bcf:field>
+      <bcf:field>language</bcf:field>
+      <bcf:field>note</bcf:field>
+      <bcf:field>number</bcf:field>
+      <bcf:field>origlanguage</bcf:field>
+      <bcf:field>pages</bcf:field>
+      <bcf:field>pubstate</bcf:field>
+      <bcf:field>series</bcf:field>
+      <bcf:field>subtitle</bcf:field>
+      <bcf:field>title</bcf:field>
+      <bcf:field>titleaddon</bcf:field>
+      <bcf:field>translator</bcf:field>
+      <bcf:field>version</bcf:field>
+      <bcf:field>volume</bcf:field>
+    </bcf:entryfields>
+    <bcf:entryfields>
+      <bcf:entrytype>bibnote</bcf:entrytype>
+      <bcf:field>note</bcf:field>
+    </bcf:entryfields>
+    <bcf:entryfields>
+      <bcf:entrytype>book</bcf:entrytype>
+      <bcf:field>author</bcf:field>
+      <bcf:field>addendum</bcf:field>
+      <bcf:field>afterword</bcf:field>
+      <bcf:field>annotator</bcf:field>
+      <bcf:field>chapter</bcf:field>
+      <bcf:field>commentator</bcf:field>
+      <bcf:field>edition</bcf:field>
+      <bcf:field>editor</bcf:field>
+      <bcf:field>editora</bcf:field>
+      <bcf:field>editorb</bcf:field>
+      <bcf:field>editorc</bcf:field>
+      <bcf:field>editortype</bcf:field>
+      <bcf:field>editoratype</bcf:field>
+      <bcf:field>editorbtype</bcf:field>
+      <bcf:field>editorctype</bcf:field>
+      <bcf:field>eid</bcf:field>
+      <bcf:field>foreword</bcf:field>
+      <bcf:field>introduction</bcf:field>
+      <bcf:field>isbn</bcf:field>
+      <bcf:field>language</bcf:field>
+      <bcf:field>location</bcf:field>
+      <bcf:field>maintitle</bcf:field>
+      <bcf:field>maintitleaddon</bcf:field>
+      <bcf:field>mainsubtitle</bcf:field>
+      <bcf:field>note</bcf:field>
+      <bcf:field>number</bcf:field>
+      <bcf:field>origlanguage</bcf:field>
+      <bcf:field>pages</bcf:field>
+      <bcf:field>pagetotal</bcf:field>
+      <bcf:field>part</bcf:field>
+      <bcf:field>publisher</bcf:field>
+      <bcf:field>pubstate</bcf:field>
+      <bcf:field>series</bcf:field>
+      <bcf:field>subtitle</bcf:field>
+      <bcf:field>title</bcf:field>
+      <bcf:field>titleaddon</bcf:field>
+      <bcf:field>translator</bcf:field>
+      <bcf:field>volume</bcf:field>
+      <bcf:field>volumes</bcf:field>
+    </bcf:entryfields>
+    <bcf:entryfields>
+      <bcf:entrytype>mvbook</bcf:entrytype>
+      <bcf:field>addendum</bcf:field>
+      <bcf:field>afterword</bcf:field>
+      <bcf:field>annotator</bcf:field>
+      <bcf:field>author</bcf:field>
+      <bcf:field>commentator</bcf:field>
+      <bcf:field>edition</bcf:field>
+      <bcf:field>editor</bcf:field>
+      <bcf:field>editora</bcf:field>
+      <bcf:field>editorb</bcf:field>
+      <bcf:field>editorc</bcf:field>
+      <bcf:field>editortype</bcf:field>
+      <bcf:field>editoratype</bcf:field>
+      <bcf:field>editorbtype</bcf:field>
+      <bcf:field>editorctype</bcf:field>
+      <bcf:field>foreword</bcf:field>
+      <bcf:field>introduction</bcf:field>
+      <bcf:field>isbn</bcf:field>
+      <bcf:field>language</bcf:field>
+      <bcf:field>location</bcf:field>
+      <bcf:field>note</bcf:field>
+      <bcf:field>number</bcf:field>
+      <bcf:field>origlanguage</bcf:field>
+      <bcf:field>pagetotal</bcf:field>
+      <bcf:field>publisher</bcf:field>
+      <bcf:field>pubstate</bcf:field>
+      <bcf:field>series</bcf:field>
+      <bcf:field>subtitle</bcf:field>
+      <bcf:field>title</bcf:field>
+      <bcf:field>titleaddon</bcf:field>
+      <bcf:field>translator</bcf:field>
+      <bcf:field>volume</bcf:field>
+      <bcf:field>volumes</bcf:field>
+    </bcf:entryfields>
+    <bcf:entryfields>
+      <bcf:entrytype>inbook</bcf:entrytype>
+      <bcf:entrytype>bookinbook</bcf:entrytype>
+      <bcf:entrytype>suppbook</bcf:entrytype>
+      <bcf:field>addendum</bcf:field>
+      <bcf:field>afterword</bcf:field>
+      <bcf:field>annotator</bcf:field>
+      <bcf:field>author</bcf:field>
+      <bcf:field>booktitle</bcf:field>
+      <bcf:field>bookauthor</bcf:field>
+      <bcf:field>booksubtitle</bcf:field>
+      <bcf:field>booktitleaddon</bcf:field>
+      <bcf:field>chapter</bcf:field>
+      <bcf:field>commentator</bcf:field>
+      <bcf:field>edition</bcf:field>
+      <bcf:field>editor</bcf:field>
+      <bcf:field>editora</bcf:field>
+      <bcf:field>editorb</bcf:field>
+      <bcf:field>editorc</bcf:field>
+      <bcf:field>editortype</bcf:field>
+      <bcf:field>editoratype</bcf:field>
+      <bcf:field>editorbtype</bcf:field>
+      <bcf:field>editorctype</bcf:field>
+      <bcf:field>eid</bcf:field>
+      <bcf:field>foreword</bcf:field>
+      <bcf:field>introduction</bcf:field>
+      <bcf:field>isbn</bcf:field>
+      <bcf:field>language</bcf:field>
+      <bcf:field>location</bcf:field>
+      <bcf:field>mainsubtitle</bcf:field>
+      <bcf:field>maintitle</bcf:field>
+      <bcf:field>maintitleaddon</bcf:field>
+      <bcf:field>note</bcf:field>
+      <bcf:field>number</bcf:field>
+      <bcf:field>origlanguage</bcf:field>
+      <bcf:field>part</bcf:field>
+      <bcf:field>publisher</bcf:field>
+      <bcf:field>pages</bcf:field>
+      <bcf:field>pubstate</bcf:field>
+      <bcf:field>series</bcf:field>
+      <bcf:field>subtitle</bcf:field>
+      <bcf:field>title</bcf:field>
+      <bcf:field>titleaddon</bcf:field>
+      <bcf:field>translator</bcf:field>
+      <bcf:field>volume</bcf:field>
+      <bcf:field>volumes</bcf:field>
+    </bcf:entryfields>
+    <bcf:entryfields>
+      <bcf:entrytype>booklet</bcf:entrytype>
+      <bcf:field>addendum</bcf:field>
+      <bcf:field>author</bcf:field>
+      <bcf:field>chapter</bcf:field>
+      <bcf:field>editor</bcf:field>
+      <bcf:field>editortype</bcf:field>
+      <bcf:field>eid</bcf:field>
+      <bcf:field>howpublished</bcf:field>
+      <bcf:field>language</bcf:field>
+      <bcf:field>location</bcf:field>
+      <bcf:field>note</bcf:field>
+      <bcf:field>pages</bcf:field>
+      <bcf:field>pagetotal</bcf:field>
+      <bcf:field>pubstate</bcf:field>
+      <bcf:field>subtitle</bcf:field>
+      <bcf:field>title</bcf:field>
+      <bcf:field>titleaddon</bcf:field>
+      <bcf:field>type</bcf:field>
+    </bcf:entryfields>
+    <bcf:entryfields>
+      <bcf:entrytype>collection</bcf:entrytype>
+      <bcf:entrytype>reference</bcf:entrytype>
+      <bcf:field>addendum</bcf:field>
+      <bcf:field>afterword</bcf:field>
+      <bcf:field>annotator</bcf:field>
+      <bcf:field>chapter</bcf:field>
+      <bcf:field>commentator</bcf:field>
+      <bcf:field>edition</bcf:field>
+      <bcf:field>editor</bcf:field>
+      <bcf:field>editora</bcf:field>
+      <bcf:field>editorb</bcf:field>
+      <bcf:field>editorc</bcf:field>
+      <bcf:field>editortype</bcf:field>
+      <bcf:field>editoratype</bcf:field>
+      <bcf:field>editorbtype</bcf:field>
+      <bcf:field>editorctype</bcf:field>
+      <bcf:field>eid</bcf:field>
+      <bcf:field>foreword</bcf:field>
+      <bcf:field>introduction</bcf:field>
+      <bcf:field>isbn</bcf:field>
+      <bcf:field>language</bcf:field>
+      <bcf:field>location</bcf:field>
+      <bcf:field>mainsubtitle</bcf:field>
+      <bcf:field>maintitle</bcf:field>
+      <bcf:field>maintitleaddon</bcf:field>
+      <bcf:field>note</bcf:field>
+      <bcf:field>number</bcf:field>
+      <bcf:field>origlanguage</bcf:field>
+      <bcf:field>pages</bcf:field>
+      <bcf:field>pagetotal</bcf:field>
+      <bcf:field>part</bcf:field>
+      <bcf:field>publisher</bcf:field>
+      <bcf:field>pubstate</bcf:field>
+      <bcf:field>series</bcf:field>
+      <bcf:field>subtitle</bcf:field>
+      <bcf:field>title</bcf:field>
+      <bcf:field>titleaddon</bcf:field>
+      <bcf:field>translator</bcf:field>
+      <bcf:field>volume</bcf:field>
+      <bcf:field>volumes</bcf:field>
+    </bcf:entryfields>
+    <bcf:entryfields>
+      <bcf:entrytype>mvcollection</bcf:entrytype>
+      <bcf:entrytype>mvreference</bcf:entrytype>
+      <bcf:field>addendum</bcf:field>
+      <bcf:field>afterword</bcf:field>
+      <bcf:field>annotator</bcf:field>
+      <bcf:field>author</bcf:field>
+      <bcf:field>commentator</bcf:field>
+      <bcf:field>edition</bcf:field>
+      <bcf:field>editor</bcf:field>
+      <bcf:field>editora</bcf:field>
+      <bcf:field>editorb</bcf:field>
+      <bcf:field>editorc</bcf:field>
+      <bcf:field>editortype</bcf:field>
+      <bcf:field>editoratype</bcf:field>
+      <bcf:field>editorbtype</bcf:field>
+      <bcf:field>editorctype</bcf:field>
+      <bcf:field>foreword</bcf:field>
+      <bcf:field>introduction</bcf:field>
+      <bcf:field>isbn</bcf:field>
+      <bcf:field>language</bcf:field>
+      <bcf:field>location</bcf:field>
+      <bcf:field>note</bcf:field>
+      <bcf:field>number</bcf:field>
+      <bcf:field>origlanguage</bcf:field>
+      <bcf:field>publisher</bcf:field>
+      <bcf:field>pubstate</bcf:field>
+      <bcf:field>subtitle</bcf:field>
+      <bcf:field>title</bcf:field>
+      <bcf:field>titleaddon</bcf:field>
+      <bcf:field>translator</bcf:field>
+      <bcf:field>volume</bcf:field>
+      <bcf:field>volumes</bcf:field>
+    </bcf:entryfields>
+    <bcf:entryfields>
+      <bcf:entrytype>incollection</bcf:entrytype>
+      <bcf:entrytype>suppcollection</bcf:entrytype>
+      <bcf:entrytype>inreference</bcf:entrytype>
+      <bcf:field>addendum</bcf:field>
+      <bcf:field>afterword</bcf:field>
+      <bcf:field>annotator</bcf:field>
+      <bcf:field>author</bcf:field>
+      <bcf:field>booksubtitle</bcf:field>
+      <bcf:field>booktitle</bcf:field>
+      <bcf:field>booktitleaddon</bcf:field>
+      <bcf:field>chapter</bcf:field>
+      <bcf:field>commentator</bcf:field>
+      <bcf:field>edition</bcf:field>
+      <bcf:field>editor</bcf:field>
+      <bcf:field>editora</bcf:field>
+      <bcf:field>editorb</bcf:field>
+      <bcf:field>editorc</bcf:field>
+      <bcf:field>editortype</bcf:field>
+      <bcf:field>editoratype</bcf:field>
+      <bcf:field>editorbtype</bcf:field>
+      <bcf:field>editorctype</bcf:field>
+      <bcf:field>eid</bcf:field>
+      <bcf:field>foreword</bcf:field>
+      <bcf:field>introduction</bcf:field>
+      <bcf:field>isbn</bcf:field>
+      <bcf:field>language</bcf:field>
+      <bcf:field>location</bcf:field>
+      <bcf:field>mainsubtitle</bcf:field>
+      <bcf:field>maintitle</bcf:field>
+      <bcf:field>maintitleaddon</bcf:field>
+      <bcf:field>note</bcf:field>
+      <bcf:field>number</bcf:field>
+      <bcf:field>origlanguage</bcf:field>
+      <bcf:field>pages</bcf:field>
+      <bcf:field>part</bcf:field>
+      <bcf:field>publisher</bcf:field>
+      <bcf:field>pubstate</bcf:field>
+      <bcf:field>series</bcf:field>
+      <bcf:field>subtitle</bcf:field>
+      <bcf:field>title</bcf:field>
+      <bcf:field>titleaddon</bcf:field>
+      <bcf:field>translator</bcf:field>
+      <bcf:field>volume</bcf:field>
+      <bcf:field>volumes</bcf:field>
+    </bcf:entryfields>
+    <bcf:entryfields>
+      <bcf:entrytype>dataset</bcf:entrytype>
+      <bcf:field>addendum</bcf:field>
+      <bcf:field>author</bcf:field>
+      <bcf:field>edition</bcf:field>
+      <bcf:field>editor</bcf:field>
+      <bcf:field>editortype</bcf:field>
+      <bcf:field>language</bcf:field>
+      <bcf:field>location</bcf:field>
+      <bcf:field>note</bcf:field>
+      <bcf:field>number</bcf:field>
+      <bcf:field>organization</bcf:field>
+      <bcf:field>publisher</bcf:field>
+      <bcf:field>pubstate</bcf:field>
+      <bcf:field>series</bcf:field>
+      <bcf:field>subtitle</bcf:field>
+      <bcf:field>title</bcf:field>
+      <bcf:field>titleaddon</bcf:field>
+      <bcf:field>type</bcf:field>
+      <bcf:field>version</bcf:field>
+    </bcf:entryfields>
+    <bcf:entryfields>
+      <bcf:entrytype>manual</bcf:entrytype>
+      <bcf:field>addendum</bcf:field>
+      <bcf:field>author</bcf:field>
+      <bcf:field>chapter</bcf:field>
+      <bcf:field>edition</bcf:field>
+      <bcf:field>editor</bcf:field>
+      <bcf:field>editortype</bcf:field>
+      <bcf:field>eid</bcf:field>
+      <bcf:field>isbn</bcf:field>
+      <bcf:field>language</bcf:field>
+      <bcf:field>location</bcf:field>
+      <bcf:field>note</bcf:field>
+      <bcf:field>number</bcf:field>
+      <bcf:field>organization</bcf:field>
+      <bcf:field>pages</bcf:field>
+      <bcf:field>pagetotal</bcf:field>
+      <bcf:field>publisher</bcf:field>
+      <bcf:field>pubstate</bcf:field>
+      <bcf:field>series</bcf:field>
+      <bcf:field>subtitle</bcf:field>
+      <bcf:field>title</bcf:field>
+      <bcf:field>titleaddon</bcf:field>
+      <bcf:field>type</bcf:field>
+      <bcf:field>version</bcf:field>
+    </bcf:entryfields>
+    <bcf:entryfields>
+      <bcf:entrytype>misc</bcf:entrytype>
+      <bcf:entrytype>software</bcf:entrytype>
+      <bcf:field>addendum</bcf:field>
+      <bcf:field>author</bcf:field>
+      <bcf:field>editor</bcf:field>
+      <bcf:field>editortype</bcf:field>
+      <bcf:field>howpublished</bcf:field>
+      <bcf:field>language</bcf:field>
+      <bcf:field>location</bcf:field>
+      <bcf:field>note</bcf:field>
+      <bcf:field>organization</bcf:field>
+      <bcf:field>pubstate</bcf:field>
+      <bcf:field>subtitle</bcf:field>
+      <bcf:field>title</bcf:field>
+      <bcf:field>titleaddon</bcf:field>
+      <bcf:field>type</bcf:field>
+      <bcf:field>version</bcf:field>
+    </bcf:entryfields>
+    <bcf:entryfields>
+      <bcf:entrytype>online</bcf:entrytype>
+      <bcf:field>addendum</bcf:field>
+      <bcf:field>author</bcf:field>
+      <bcf:field>editor</bcf:field>
+      <bcf:field>editortype</bcf:field>
+      <bcf:field>language</bcf:field>
+      <bcf:field>note</bcf:field>
+      <bcf:field>organization</bcf:field>
+      <bcf:field>pubstate</bcf:field>
+      <bcf:field>subtitle</bcf:field>
+      <bcf:field>title</bcf:field>
+      <bcf:field>titleaddon</bcf:field>
+      <bcf:field>version</bcf:field>
+    </bcf:entryfields>
+    <bcf:entryfields>
+      <bcf:entrytype>patent</bcf:entrytype>
+      <bcf:field>addendum</bcf:field>
+      <bcf:field>author</bcf:field>
+      <bcf:field>holder</bcf:field>
+      <bcf:field>location</bcf:field>
+      <bcf:field>note</bcf:field>
+      <bcf:field>number</bcf:field>
+      <bcf:field>pubstate</bcf:field>
+      <bcf:field>subtitle</bcf:field>
+      <bcf:field>title</bcf:field>
+      <bcf:field>titleaddon</bcf:field>
+      <bcf:field>type</bcf:field>
+      <bcf:field>version</bcf:field>
+    </bcf:entryfields>
+    <bcf:entryfields>
+      <bcf:entrytype>periodical</bcf:entrytype>
+      <bcf:field>addendum</bcf:field>
+      <bcf:field>editor</bcf:field>
+      <bcf:field>editora</bcf:field>
+      <bcf:field>editorb</bcf:field>
+      <bcf:field>editorc</bcf:field>
+      <bcf:field>editortype</bcf:field>
+      <bcf:field>editoratype</bcf:field>
+      <bcf:field>editorbtype</bcf:field>
+      <bcf:field>editorctype</bcf:field>
+      <bcf:field>issn</bcf:field>
+      <bcf:field>issue</bcf:field>
+      <bcf:field>issuesubtitle</bcf:field>
+      <bcf:field>issuetitle</bcf:field>
+      <bcf:field>issuetitleaddon</bcf:field>
+      <bcf:field>language</bcf:field>
+      <bcf:field>note</bcf:field>
+      <bcf:field>number</bcf:field>
+      <bcf:field>pubstate</bcf:field>
+      <bcf:field>season</bcf:field>
+      <bcf:field>series</bcf:field>
+      <bcf:field>subtitle</bcf:field>
+      <bcf:field>title</bcf:field>
+      <bcf:field>titleaddon</bcf:field>
+      <bcf:field>volume</bcf:field>
+    </bcf:entryfields>
+    <bcf:entryfields>
+      <bcf:entrytype>mvproceedings</bcf:entrytype>
+      <bcf:field>addendum</bcf:field>
+      <bcf:field>editor</bcf:field>
+      <bcf:field>editortype</bcf:field>
+      <bcf:field>eventday</bcf:field>
+      <bcf:field>eventendday</bcf:field>
+      <bcf:field>eventendhour</bcf:field>
+      <bcf:field>eventendminute</bcf:field>
+      <bcf:field>eventendmonth</bcf:field>
+      <bcf:field>eventendseason</bcf:field>
+      <bcf:field>eventendsecond</bcf:field>
+      <bcf:field>eventendtimezone</bcf:field>
+      <bcf:field>eventendyear</bcf:field>
+      <bcf:field>eventhour</bcf:field>
+      <bcf:field>eventminute</bcf:field>
+      <bcf:field>eventmonth</bcf:field>
+      <bcf:field>eventseason</bcf:field>
+      <bcf:field>eventsecond</bcf:field>
+      <bcf:field>eventtimezone</bcf:field>
+      <bcf:field>eventyear</bcf:field>
+      <bcf:field>eventtitle</bcf:field>
+      <bcf:field>eventtitleaddon</bcf:field>
+      <bcf:field>isbn</bcf:field>
+      <bcf:field>language</bcf:field>
+      <bcf:field>location</bcf:field>
+      <bcf:field>note</bcf:field>
+      <bcf:field>number</bcf:field>
+      <bcf:field>organization</bcf:field>
+      <bcf:field>pagetotal</bcf:field>
+      <bcf:field>publisher</bcf:field>
+      <bcf:field>pubstate</bcf:field>
+      <bcf:field>series</bcf:field>
+      <bcf:field>subtitle</bcf:field>
+      <bcf:field>title</bcf:field>
+      <bcf:field>titleaddon</bcf:field>
+      <bcf:field>venue</bcf:field>
+      <bcf:field>volumes</bcf:field>
+    </bcf:entryfields>
+    <bcf:entryfields>
+      <bcf:entrytype>proceedings</bcf:entrytype>
+      <bcf:field>addendum</bcf:field>
+      <bcf:field>chapter</bcf:field>
+      <bcf:field>editor</bcf:field>
+      <bcf:field>editortype</bcf:field>
+      <bcf:field>eid</bcf:field>
+      <bcf:field>eventday</bcf:field>
+      <bcf:field>eventendday</bcf:field>
+      <bcf:field>eventendhour</bcf:field>
+      <bcf:field>eventendminute</bcf:field>
+      <bcf:field>eventendmonth</bcf:field>
+      <bcf:field>eventendseason</bcf:field>
+      <bcf:field>eventendsecond</bcf:field>
+      <bcf:field>eventendtimezone</bcf:field>
+      <bcf:field>eventendyear</bcf:field>
+      <bcf:field>eventhour</bcf:field>
+      <bcf:field>eventminute</bcf:field>
+      <bcf:field>eventmonth</bcf:field>
+      <bcf:field>eventseason</bcf:field>
+      <bcf:field>eventsecond</bcf:field>
+      <bcf:field>eventtimezone</bcf:field>
+      <bcf:field>eventyear</bcf:field>
+      <bcf:field>eventtitle</bcf:field>
+      <bcf:field>eventtitleaddon</bcf:field>
+      <bcf:field>isbn</bcf:field>
+      <bcf:field>language</bcf:field>
+      <bcf:field>location</bcf:field>
+      <bcf:field>mainsubtitle</bcf:field>
+      <bcf:field>maintitle</bcf:field>
+      <bcf:field>maintitleaddon</bcf:field>
+      <bcf:field>note</bcf:field>
+      <bcf:field>number</bcf:field>
+      <bcf:field>organization</bcf:field>
+      <bcf:field>pages</bcf:field>
+      <bcf:field>pagetotal</bcf:field>
+      <bcf:field>part</bcf:field>
+      <bcf:field>publisher</bcf:field>
+      <bcf:field>pubstate</bcf:field>
+      <bcf:field>series</bcf:field>
+      <bcf:field>subtitle</bcf:field>
+      <bcf:field>title</bcf:field>
+      <bcf:field>titleaddon</bcf:field>
+      <bcf:field>venue</bcf:field>
+      <bcf:field>volume</bcf:field>
+      <bcf:field>volumes</bcf:field>
+    </bcf:entryfields>
+    <bcf:entryfields>
+      <bcf:entrytype>inproceedings</bcf:entrytype>
+      <bcf:field>addendum</bcf:field>
+      <bcf:field>author</bcf:field>
+      <bcf:field>booksubtitle</bcf:field>
+      <bcf:field>booktitle</bcf:field>
+      <bcf:field>booktitleaddon</bcf:field>
+      <bcf:field>chapter</bcf:field>
+      <bcf:field>editor</bcf:field>
+      <bcf:field>editortype</bcf:field>
+      <bcf:field>eid</bcf:field>
+      <bcf:field>eventday</bcf:field>
+      <bcf:field>eventendday</bcf:field>
+      <bcf:field>eventendhour</bcf:field>
+      <bcf:field>eventendminute</bcf:field>
+      <bcf:field>eventendmonth</bcf:field>
+      <bcf:field>eventendseason</bcf:field>
+      <bcf:field>eventendsecond</bcf:field>
+      <bcf:field>eventendtimezone</bcf:field>
+      <bcf:field>eventendyear</bcf:field>
+      <bcf:field>eventhour</bcf:field>
+      <bcf:field>eventminute</bcf:field>
+      <bcf:field>eventmonth</bcf:field>
+      <bcf:field>eventseason</bcf:field>
+      <bcf:field>eventsecond</bcf:field>
+      <bcf:field>eventtimezone</bcf:field>
+      <bcf:field>eventyear</bcf:field>
+      <bcf:field>eventtitle</bcf:field>
+      <bcf:field>eventtitleaddon</bcf:field>
+      <bcf:field>isbn</bcf:field>
+      <bcf:field>language</bcf:field>
+      <bcf:field>location</bcf:field>
+      <bcf:field>mainsubtitle</bcf:field>
+      <bcf:field>maintitle</bcf:field>
+      <bcf:field>maintitleaddon</bcf:field>
+      <bcf:field>note</bcf:field>
+      <bcf:field>number</bcf:field>
+      <bcf:field>organization</bcf:field>
+      <bcf:field>pages</bcf:field>
+      <bcf:field>part</bcf:field>
+      <bcf:field>publisher</bcf:field>
+      <bcf:field>pubstate</bcf:field>
+      <bcf:field>series</bcf:field>
+      <bcf:field>subtitle</bcf:field>
+      <bcf:field>title</bcf:field>
+      <bcf:field>titleaddon</bcf:field>
+      <bcf:field>venue</bcf:field>
+      <bcf:field>volume</bcf:field>
+      <bcf:field>volumes</bcf:field>
+    </bcf:entryfields>
+    <bcf:entryfields>
+      <bcf:entrytype>report</bcf:entrytype>
+      <bcf:field>addendum</bcf:field>
+      <bcf:field>author</bcf:field>
+      <bcf:field>chapter</bcf:field>
+      <bcf:field>eid</bcf:field>
+      <bcf:field>institution</bcf:field>
+      <bcf:field>isrn</bcf:field>
+      <bcf:field>language</bcf:field>
+      <bcf:field>location</bcf:field>
+      <bcf:field>note</bcf:field>
+      <bcf:field>number</bcf:field>
+      <bcf:field>pages</bcf:field>
+      <bcf:field>pagetotal</bcf:field>
+      <bcf:field>pubstate</bcf:field>
+      <bcf:field>subtitle</bcf:field>
+      <bcf:field>title</bcf:field>
+      <bcf:field>titleaddon</bcf:field>
+      <bcf:field>type</bcf:field>
+      <bcf:field>version</bcf:field>
+    </bcf:entryfields>
+    <bcf:entryfields>
+      <bcf:entrytype>thesis</bcf:entrytype>
+      <bcf:field>addendum</bcf:field>
+      <bcf:field>author</bcf:field>
+      <bcf:field>chapter</bcf:field>
+      <bcf:field>eid</bcf:field>
+      <bcf:field>institution</bcf:field>
+      <bcf:field>language</bcf:field>
+      <bcf:field>location</bcf:field>
+      <bcf:field>note</bcf:field>
+      <bcf:field>pages</bcf:field>
+      <bcf:field>pagetotal</bcf:field>
+      <bcf:field>pubstate</bcf:field>
+      <bcf:field>subtitle</bcf:field>
+      <bcf:field>title</bcf:field>
+      <bcf:field>titleaddon</bcf:field>
+      <bcf:field>type</bcf:field>
+    </bcf:entryfields>
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+      <bcf:field>addendum</bcf:field>
+      <bcf:field>author</bcf:field>
+      <bcf:field>eventday</bcf:field>
+      <bcf:field>eventendday</bcf:field>
+      <bcf:field>eventendhour</bcf:field>
+      <bcf:field>eventendminute</bcf:field>
+      <bcf:field>eventendmonth</bcf:field>
+      <bcf:field>eventendseason</bcf:field>
+      <bcf:field>eventendsecond</bcf:field>
+      <bcf:field>eventendtimezone</bcf:field>
+      <bcf:field>eventendyear</bcf:field>
+      <bcf:field>eventhour</bcf:field>
+      <bcf:field>eventminute</bcf:field>
+      <bcf:field>eventmonth</bcf:field>
+      <bcf:field>eventseason</bcf:field>
+      <bcf:field>eventsecond</bcf:field>
+      <bcf:field>eventtimezone</bcf:field>
+      <bcf:field>eventyear</bcf:field>
+      <bcf:field>eventtitle</bcf:field>
+      <bcf:field>eventtitleaddon</bcf:field>
+      <bcf:field>howpublished</bcf:field>
+      <bcf:field>language</bcf:field>
+      <bcf:field>location</bcf:field>
+      <bcf:field>note</bcf:field>
+      <bcf:field>pubstate</bcf:field>
+      <bcf:field>subtitle</bcf:field>
+      <bcf:field>title</bcf:field>
+      <bcf:field>titleaddon</bcf:field>
+      <bcf:field>type</bcf:field>
+      <bcf:field>venue</bcf:field>
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+      <bcf:field>abstract</bcf:field>
+      <bcf:field>addendum</bcf:field>
+      <bcf:field>afterword</bcf:field>
+      <bcf:field>annotator</bcf:field>
+      <bcf:field>author</bcf:field>
+      <bcf:field>bookauthor</bcf:field>
+      <bcf:field>booksubtitle</bcf:field>
+      <bcf:field>booktitle</bcf:field>
+      <bcf:field>booktitleaddon</bcf:field>
+      <bcf:field>chapter</bcf:field>
+      <bcf:field>commentator</bcf:field>
+      <bcf:field>editor</bcf:field>
+      <bcf:field>editora</bcf:field>
+      <bcf:field>editorb</bcf:field>
+      <bcf:field>editorc</bcf:field>
+      <bcf:field>foreword</bcf:field>
+      <bcf:field>holder</bcf:field>
+      <bcf:field>institution</bcf:field>
+      <bcf:field>introduction</bcf:field>
+      <bcf:field>issuesubtitle</bcf:field>
+      <bcf:field>issuetitle</bcf:field>
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+      <bcf:field>journaltitle</bcf:field>
+      <bcf:field>journaltitleaddon</bcf:field>
+      <bcf:field>location</bcf:field>
+      <bcf:field>mainsubtitle</bcf:field>
+      <bcf:field>maintitle</bcf:field>
+      <bcf:field>maintitleaddon</bcf:field>
+      <bcf:field>nameaddon</bcf:field>
+      <bcf:field>note</bcf:field>
+      <bcf:field>organization</bcf:field>
+      <bcf:field>origlanguage</bcf:field>
+      <bcf:field>origlocation</bcf:field>
+      <bcf:field>origpublisher</bcf:field>
+      <bcf:field>origtitle</bcf:field>
+      <bcf:field>part</bcf:field>
+      <bcf:field>publisher</bcf:field>
+      <bcf:field>relatedstring</bcf:field>
+      <bcf:field>series</bcf:field>
+      <bcf:field>shortauthor</bcf:field>
+      <bcf:field>shorteditor</bcf:field>
+      <bcf:field>shorthand</bcf:field>
+      <bcf:field>shortjournal</bcf:field>
+      <bcf:field>shortseries</bcf:field>
+      <bcf:field>shorttitle</bcf:field>
+      <bcf:field>sortname</bcf:field>
+      <bcf:field>sortshorthand</bcf:field>
+      <bcf:field>sorttitle</bcf:field>
+      <bcf:field>subtitle</bcf:field>
+      <bcf:field>title</bcf:field>
+      <bcf:field>titleaddon</bcf:field>
+      <bcf:field>translator</bcf:field>
+      <bcf:field>venue</bcf:field>
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+      <bcf:entrytype>book</bcf:entrytype>
+      <bcf:entrytype>inbook</bcf:entrytype>
+      <bcf:entrytype>bookinbook</bcf:entrytype>
+      <bcf:entrytype>suppbook</bcf:entrytype>
+      <bcf:entrytype>booklet</bcf:entrytype>
+      <bcf:entrytype>collection</bcf:entrytype>
+      <bcf:entrytype>incollection</bcf:entrytype>
+      <bcf:entrytype>suppcollection</bcf:entrytype>
+      <bcf:entrytype>manual</bcf:entrytype>
+      <bcf:entrytype>misc</bcf:entrytype>
+      <bcf:entrytype>mvbook</bcf:entrytype>
+      <bcf:entrytype>mvcollection</bcf:entrytype>
+      <bcf:entrytype>online</bcf:entrytype>
+      <bcf:entrytype>patent</bcf:entrytype>
+      <bcf:entrytype>periodical</bcf:entrytype>
+      <bcf:entrytype>suppperiodical</bcf:entrytype>
+      <bcf:entrytype>proceedings</bcf:entrytype>
+      <bcf:entrytype>inproceedings</bcf:entrytype>
+      <bcf:entrytype>reference</bcf:entrytype>
+      <bcf:entrytype>inreference</bcf:entrytype>
+      <bcf:entrytype>report</bcf:entrytype>
+      <bcf:entrytype>set</bcf:entrytype>
+      <bcf:entrytype>thesis</bcf:entrytype>
+      <bcf:entrytype>unpublished</bcf:entrytype>
+      <bcf:constraint type="mandatory">
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+          <bcf:field>date</bcf:field>
+          <bcf:field>year</bcf:field>
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+      <bcf:entrytype>set</bcf:entrytype>
+      <bcf:constraint type="mandatory">
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+      </bcf:constraint>
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+      <bcf:entrytype>article</bcf:entrytype>
+      <bcf:constraint type="mandatory">
+        <bcf:field>author</bcf:field>
+        <bcf:field>journaltitle</bcf:field>
+        <bcf:field>title</bcf:field>
+      </bcf:constraint>
+    </bcf:constraints>
+    <bcf:constraints>
+      <bcf:entrytype>book</bcf:entrytype>
+      <bcf:entrytype>mvbook</bcf:entrytype>
+      <bcf:constraint type="mandatory">
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+        <bcf:field>title</bcf:field>
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+    <bcf:constraints>
+      <bcf:entrytype>inbook</bcf:entrytype>
+      <bcf:entrytype>bookinbook</bcf:entrytype>
+      <bcf:entrytype>suppbook</bcf:entrytype>
+      <bcf:constraint type="mandatory">
+        <bcf:field>author</bcf:field>
+        <bcf:field>title</bcf:field>
+        <bcf:field>booktitle</bcf:field>
+      </bcf:constraint>
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+    <bcf:constraints>
+      <bcf:entrytype>booklet</bcf:entrytype>
+      <bcf:constraint type="mandatory">
+        <bcf:fieldor>
+          <bcf:field>author</bcf:field>
+          <bcf:field>editor</bcf:field>
+        </bcf:fieldor>
+        <bcf:field>title</bcf:field>
+      </bcf:constraint>
+    </bcf:constraints>
+    <bcf:constraints>
+      <bcf:entrytype>collection</bcf:entrytype>
+      <bcf:entrytype>reference</bcf:entrytype>
+      <bcf:entrytype>mvcollection</bcf:entrytype>
+      <bcf:entrytype>mvreference</bcf:entrytype>
+      <bcf:constraint type="mandatory">
+        <bcf:field>editor</bcf:field>
+        <bcf:field>title</bcf:field>
+      </bcf:constraint>
+    </bcf:constraints>
+    <bcf:constraints>
+      <bcf:entrytype>incollection</bcf:entrytype>
+      <bcf:entrytype>suppcollection</bcf:entrytype>
+      <bcf:entrytype>inreference</bcf:entrytype>
+      <bcf:constraint type="mandatory">
+        <bcf:field>author</bcf:field>
+        <bcf:field>editor</bcf:field>
+        <bcf:field>title</bcf:field>
+        <bcf:field>booktitle</bcf:field>
+      </bcf:constraint>
+    </bcf:constraints>
+    <bcf:constraints>
+      <bcf:entrytype>dataset</bcf:entrytype>
+      <bcf:constraint type="mandatory">
+        <bcf:field>title</bcf:field>
+      </bcf:constraint>
+    </bcf:constraints>
+    <bcf:constraints>
+      <bcf:entrytype>manual</bcf:entrytype>
+      <bcf:constraint type="mandatory">
+        <bcf:field>title</bcf:field>
+      </bcf:constraint>
+    </bcf:constraints>
+    <bcf:constraints>
+      <bcf:entrytype>misc</bcf:entrytype>
+      <bcf:entrytype>software</bcf:entrytype>
+      <bcf:constraint type="mandatory">
+        <bcf:field>title</bcf:field>
+      </bcf:constraint>
+    </bcf:constraints>
+    <bcf:constra
+\ No newline at end of file
diff --git a/src/thesis/main.log b/src/thesis/main.log
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diff --git a/src/thesis/main.out b/src/thesis/main.out
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
@@ -16,25 +16,29 @@
 
 \input{back/abstract}
 
+\newpage
+
 %------------------- INTRO -------------------------
 
 \input{chapters/intro}
 
+\newpage
+
 %----------------- MAIN SECTION --------------------
 
 \input{chapters/main_sec}
 
-\input{chapters/basics}
+\input{chapters/1_basics}
 
-\input{chapters/finitencg}
+\input{chapters/2_finitencg}
 
-\input{chapters/realncg}
+\input{chapters/3_realncg}
 
-\input{chapters/heatkernel}
+\input{chapters/4_heatkernel}
 
-\input{chapters/twopointspace}
+\input{chapters/5_twopointspace}
 
-\input{chapters/electroncg}
+\input{chapters/6_electroncg}
 
 %------------------ OUTRO -------------------------
 
diff --git a/src/thesis/main.toc b/src/thesis/main.toc
diff --git a/src/thesis/todo.md b/src/thesis/todo.md
@@ -10,10 +10,10 @@
       is followed, the equations have a comma or a dot at the sentence end
       etc. DONE
     2. rethink the chapters DONE
-    3. write introduction
-    4. write conclusion
+    6. write abstract DONE
+    3. write introduction DONE
+    4. write conclusion DONE
     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 DONE
     7. read through
     8. submit