commit c114fae80c3165461a37def1e20427fbb9cf5a01
parent 5d3c6f911cae56d27532921f25ef57c3d73228c6
Author: miksa234 <milutin@popovic.xyz>
Date: Fri, 23 Jul 2021 11:36:36 +0200
checkpoint
Diffstat:
7 files changed, 407 insertions(+), 399 deletions(-)
diff --git a/src/thesis/back/title.tex b/src/thesis/back/title.tex
@@ -5,9 +5,9 @@
\hspace{8.8cm}
\includegraphics[width=8cm]{pics/uni_logo}
\end{figure}
-\vspace*{1cm}
+\vspace*{2cm}
- \fontsize{22}{0} \fontfamily{lmss}\selectfont \textbf{Bachelor's Thesis}\\
+ \fontsize{22}{0} \fontfamily{phv}\selectfont \textbf{Bachelor's Thesis}\\
\vspace*{2.5cm}
@@ -15,8 +15,7 @@
\vspace*{0.4cm}
-\fontsize{18}{0} \selectfont \textbf{From Noncommutative Geometry to
-Electrodynamics}\\
+\fontsize{18}{0} \selectfont \textbf{Noncommutative Geomtetry and Physics}\\
\vspace*{1.5cm}
@@ -27,31 +26,31 @@ Electrodynamics}\\
\vspace*{2cm}
- {\fontsize{12}{0} \selectfont in partial fulfillment of the requirements for the degree of}\\
+ {\fontsize{12}{0} \selectfont in partial fulfilment of the requirements for the degree of}\\
\vspace*{0.4cm}
{ \fontsize{14}{0} \selectfont Bachelor of Science (BSc)}\\
\vspace*{2cm}
- { \fontsize{12}{0} \selectfont Vienna, July 2021}\\
+ { \fontsize{10}{0} \selectfont Vienna, July 2021}\\
\vspace*{3.5cm}
\begin{tabular}{p{9cm}p{11.25cm}}
- \fontsize{12}{0} \selectfont degree programme code as it appears on / &
- \fontsize{12}{0} \selectfont A 033 676 \\
+ \fontsize{10}{0} \selectfont degree programme code as it appears on / &
+ \fontsize{10}{0} \selectfont A 033 676 \\
- \fontsize{12}{0} \selectfont the student record sheet:\vspace*{0.4cm} &
- \fontsize{12}{0} \selectfont\\
+ \fontsize{10}{0} \selectfont the student record sheet:\vspace*{0.4cm} &
+ \fontsize{10}{0} \selectfont\\
- \fontsize{12}{0} \selectfont degree
- programme as it appears on / & \fontsize{12}{0} \selectfont Physics \\
+ \fontsize{10}{0} \selectfont degree
+ programme a0 it appears on / & \fontsize{10}{0} \selectfont Physics \\
- \fontsize{12}{0} \selectfont the student record sheet:\vspace*{0.4cm} &
- \fontsize{12}{0} \selectfont \\
+ \fontsize{10}{0} \selectfont the student record sheet:\vspace*{0.4cm} &
+ \fontsize{10}{0} \selectfont \\
- \fontsize{12}{0} \selectfont
- Supervisor:\vspace*{0.4cm}& \fontsize{12}{0} \selectfont Lisa Glaser, PhD\\
+ \fontsize{10}{0} \selectfont
+ Supervisor:\vspace*{0.4cm}& \fontsize{10}{0} \selectfont Lisa Glaser, PhD\\
\end{tabular}
\end{center}
\end{titlepage}
diff --git a/src/thesis/chapters/diffgeo.tex b/src/thesis/chapters/diffgeo.tex
@@ -47,3 +47,82 @@ operator $J_M:\Gamma(S) \rightarrow \Gamma(S)$ such that:
$(S, J_M)$ is called the \textbf{spin Structure on $M$}
\newline
$J_M$ is called the \textbf{charge conjugation}.
+
+\subsection{Operators of Laplace Type}
+Let $M$ be a $n$ dimensional compact Riemannian manifold with $\partial M = 0$.
+Then consider a vector bundle $V$ over $M$ (i.e. there is a vector space to
+each point on $M$), so we can define smooth functions. We want to look at
+arbitrary differential operators $D$ of Laplace type on $V$, they have the general
+from
+\begin{align}
+ D = -(g^{\mu\nu} \partial_\mu\partial_\nu + a^\sigma\partial_\sigma +b)
+\end{align}
+where $a^\sigma, b$ are matrix valued functions on $M$ and $g^{\mu\nu}$ is the
+inverse metric on $M$. There is a unique connection on $V$ and a unique
+endomorphism (matrix valued function) $E$ on $V$, then we can rewrite $D$ in
+terms of $E$ and covariant derivatives
+\begin{align}
+ D = -(g^{\mu\nu} \nabla_\mu \nabla_\nu +E)
+\end{align}
+Where the covariant derivative consists of $\nabla = \nabla^{[R]} +\omega$ the
+standard Riemannian covariant derivative $\nabla^{[R]}$ and a "gauge" bundle
+$\omega$ (fluctuations). WE can write $E$ and $\omega$ in terms of geometrical
+identities
+\begin{align}
+ \omega_\delta &= \frac{1}{2}g_{\nu\delta}(a^\nu
+ +g^{\mu\sigma}\Gamma^\nu_{\mu\sigma}I_V)\\
+ E &= b - g^{\nu\mu}(\partial_\mu \omega_\nu + \omega_\nu \omega_\mu -
+ \omega_\sigma \Gamma^\sigma_{\nu\mu})
+\end{align}
+where $I_V$ is the identity in $V$ and the Christoffel symbol
+\begin{align}
+ \Gamma^\sigma_{\mu\nu} = g^{\sigma\varrho} \frac{1}{2} (\partial_\mu
+ g_{\nu\varrho} + \partial_\nu g_{\mu\varrho} - \partial_\varrho g_{\mu\nu})
+\end{align}
+Furthermore we remind ourselves of the Riemmanian curvature tensor, Ricci
+Tensor and the Scalar curavture.
+\begin{align}
+ R^\mu_{\nu\varrho\sigma} &= \partial_\sigma \Gamma^{\mu}_{\nu\varrho}
+ -\partial_\varrho \Gamma^\mu_{\nu\sigma}
+ \Gamma^{\lambda}_{\nu\varrho}\Gamma^{\mu}_{\lambda\sigma}
+ \Gamma^{\lambda}_{\nu\sigma}\Gamma^{\mu}_{\lambda\varrho}\\
+ R_{\mu\nu} &:= R^{\sigma}_{\mu\nu\sigma}\\
+ R &:= R^\mu_{\ \mu}
+\end{align}
+
+The we let $\{e_1, \dots, e_n\}$ be the local orthonormal frame of
+$TM$(tangent bundle $M$), which will be noted with flat indices $i,j,k,l
+\in\{1,\dots, n\}$, we use $e^k_\mu, e^\nu_j$ to transform between flat indices
+and curved indices $\mu, \nu, \varrho$.
+\begin{align}
+ e^\mu_j e^\nu_k g_{\mu\nu} &= \delta_{jk}\\
+ e^\mu_j e^\nu_k \delta^{jk} &= g^{\mu\nu} \\
+ e^j_\mu e^\mu_k &= \delta^j_k
+\end{align}
+
+The Riemannian part of the covariant derivative contains the standard
+Levi-Civita connection, so that for a $v_\nu$ we write
+\begin{align}
+ \nabla_\mu^{[R]} v_\nu = \partial_\mu v_\nu -
+ \Gamma^{\varrho}_{\mu\nu}v_\varrho.
+\end{align}
+The extended covariant derivative reads then
+\begin{align}
+ \nabla_\mu v^j = \partial_\mu v^j + \sigma^{jk}_\mu v_k.
+\end{align}
+the condition $\nabla_\mu e^k_\nu = 0$ gives us the general connection
+\begin{align}
+ \sigma^{kl}_\mu = e^\nu_l\Gamma^{\varrho}_{\mu\nu}e^k_\varrho - e^\nu_l
+ \partial_\mu e^k_\nu
+\end{align}
+The we may define the field strength $\Omega_{\mu\nu}$ of the connection $\omega$
+\begin{align}
+ \Omega_{\mu\nu} = \partial_\mu \omega_\nu -\partial_\nu \omega_\mu
+ +\omega_\mu \omega_\nu -\omega_\nu\omega_\mu.
+\end{align}
+If we apply the covariant derivative on $\Omega$ we get
+\begin{align}
+ \nabla_\varrho\Omega_{\mu\nu} = \partial_\varrho \Omega_{\mu\nu} -
+ \Gamma^{\sigma}_{\varrho \mu} \Omega_{\sigma\mu} + [\omega_\varrho,
+ \Omega_{\mu\nu}]
+\end{align}
diff --git a/src/thesis/chapters/electroncg.tex b/src/thesis/chapters/electroncg.tex
@@ -1,246 +1,4 @@
\subsection{Noncommutative Geometry of Electrodynamics}
-\subsubsection{The 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 we can make a simple
-choice for the Hilbertspace, $H_F = \mathbb{C}^2$. Furthermore $\gamma_F$ is
-the $\mathbb{Z}_2$ grading, which allows us to decompose $H_F$ 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 $H_F^\pm$,
-\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 we have KO-dimension of 0,
- 2 or 6. To elaborate on this, we know that there are 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}
- \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
-$dim(H_F^\pm) = 1$ (see Lemma 3.8 Book), which ultimately means $J_F^2 = -1$ is
-not allowed.
-\subsubsection{The 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 \simeq M\sqcup X.
-\end{align}
-Keep in mind that we still need to find an appropriate real structure on the
-Riemannian spin manifold, $J_M$. Furthermore 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 we can consider a
-distance formula on $M\times F_X$ 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 \ref{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}
-and the supremum 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,
-then 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 theory.
-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{definition}
-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{definition}
-\begin{definition}
- 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{definition}
-The Gauge group of $F_X$ is given by the quotient $U(A)/U(A_J)$.
-We want 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{proposition}
-The Gauge group of the Two-Point space $\mathfrak{B}(F_X)$ is $U(1)$.
-\end{proposition}
-\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{proposition}
- 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{proposition}
-
-
-\subsection{Electrodynamics}
In this chapter we describe Electrodynamics with the almost commutative
manifold $M\times F_X$ and the abelian gauge group $U(1)$.
We arrive at a unified description of gravity and electrodynamics although in the classical level.
@@ -394,7 +152,7 @@ We can now define the finite space $F_{ED}$.
where $J_F$ and $\gamma_F$ are like in equation \ref{eq:fedfail} and $D_F$
from equation \ref{eq:feddirac}.
-\subsubsection{The almost-commutative Manifold}
+\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}
@@ -442,7 +200,7 @@ $\forall a \in A$, 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{The Spectral Action}
+\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 $M\times F_{ED}$
@@ -694,4 +452,3 @@ and the Lagrangian of electrodynamics
+\frac{f(0)}{6\pi^2} Y_{\mu\nu}Y^{\mu\nu}.
\end{align}
-
diff --git a/src/thesis/chapters/heatkernel.tex b/src/thesis/chapters/heatkernel.tex
@@ -1,4 +1,3 @@
-
\subsection{Heat Kernel Expansion}
\subsubsection{The Heat Kernel}
The heat kernel $K(t; x, y; D)$ is the fundamental solution of the heat
@@ -9,205 +8,137 @@ equation. It depends on the operator $D$ of Laplacian type.
For a flat manifold $M = \mathbb{R}^n$ and $D = D_0 := -\Delta_\mu\Delta^\mu +m^2$ the
Laplacian with a mass term and the initial condition
\begin{align}
- K(0;x,y;D) = \delta(x,y)
+ K(0;x,y;D) = \delta(x,y),
\end{align}
-we have the standard fundamental solution
+takes to form of the standard fundamental solution
\begin{align}\label{eq:standard}
- K(t;x,y;D_0) = (4\pi t)^{-n/2}\exp\left(-\frac{(x-y)^2}{4t}-tm^2\right)
+ K(t;x,y;D_0) = (4\pi t)^{-n/2}\exp\left(-\frac{(x-y)^2}{4t}-tm^2\right).
\end{align}
Let us consider now a more general operator $D$ with a potential term or a
-guage field, the heat kernel reads then
+gauge field, the heat kernel reads then
\begin{align}
K(t;x,y;D) = \langle x|e^{-tD}|y\rangle.
\end{align}
-We can expand it it in terms of $D_0$ and we still have the
-singularity from the equation \ref{eq:standard} as $t\rightarrow 0$ thus the
-expansion gives
+We can expand the heat kernel in $t$, still having a
+singularity from the equation \ref{eq:standard} as $t \rightarrow 0$ thus the
+expansion reads
\begin{align}
- K(t;x,y;D) = K(t;x,y;D_0)\left(1 + tb_2(x,y) + t^2b_4(x,y) + \dots \right)
+ K(t;x,y;D) = K(t;x,y;D_0)\left(1 + tb_2(x,y) + t^2b_4(x,y) + \dots
+ \right),
\end{align}
-where $b_k(x,y)$ are regular in $y \rightarrow x$. They are called the heat
+where $b_k(x,y)$ become regular as $y \rightarrow x$. These coefficients are called the heat
kernel coefficients.
-
-\subsubsection{Example}
-Now let us consider a propagator $D^{-1}(x,y)$ defined through the heat kernel
-in an integral representation
+%----------------------- KANN WEGGELASSEN WERDEN
+\newline
+\textbf{KANN WEGELASSEN WERDEN BIS ZUM NÄCHSTEN KAPITEL}
+Let's turn our attention to a propagator $D^{-1}(x,y)$ defined through the
+heat kernel, with an integral representation
\begin{align}
D^{-1} (x,y) = \int_0^\infty dt K(t;x,y;D).
\end{align}
-We can integrate the expression formally if we assume the heat kernel vanishes
-for $t\rightarrow \infty$ we get
+If we assume the heat kernel vanishes for $t\rightarrow \infty$, we can
+integrate formally to get
\begin{align}
D^{-1}(x,y) \simeq
2(4\pi)^{-n/2}\sum_{j=0}\left(\frac{|x-y|}{2m}\right)^{-\frac{n}{2}+j+1}
- K_{-\frac{n}{2}+j+1}(|x-y|m)b_{2j}(x,y).
+ K_{-\frac{n}{2}+j+1}(|x-y|m)b_{2j}(x,y),
\end{align}
where $b_0 = 1$ and $K_\nu (z)$ is the Bessel function
\begin{align}
- K_\nu(z) = \frac{1}{\pi} \int_0^\pi cos(\nu\tau-z\sin(\tau))d\tau
+ K_\nu(z) = \frac{1}{\pi} \int_0^\pi \cos(\nu\tau-z\sin(\tau))d\tau.
\end{align}
-it solves the differential equation
+The Bessel function solves the following differential equation
\begin{align}
z^2 \frac{d^2K}{dz^2} + z \frac{dK}{dz} + (z^2 - \nu^2)=0.
\end{align}
-By looking at integral approximation of the propagator we conclude
-that the singularities of $D^{-1}$ coincide with the singularities of the heat
-kernel coefficients.
-We consider now a generating functional in terns of $\det(D)$ which is called
-the one-loop effective action (quantum fields theory)
+By looking at an integral approximation for the propagator we conclude that
+the singularities of $D^{-1}$ coincide with the singularities of the heat
+kernel coefficients. Thus we can say, that a generating functional in terms of
+$\det(D)$ is called the one-loop effective action (quantum field theory)
\begin{align}
- W = \frac{1}{2}\ln(\det D)
+ W = \frac{1}{2}\ln(\det D).
\end{align}
-we can relate $W$ with the heat kernel. For each eigenvalue $\lambda >0$ of $D$
+We have a direct relation with one-loop effective action $W$ and the
+heat kernel. Furthermore notice that for each eigenvalue $\lambda >0$ of $D$
we can write the identity.
\begin{align}
\ln \lambda = -\int_0^\infty \frac{e^{-t\lambda}}{t}dt
\end{align}
-This expression is correct up to an infinite constant which does not depent on
-$\lambda$, because of this we can ignore it. Further more we use $\ln(\det D) =
-\text{Tr}(\ln D)$ and therefor we can write for $W$
+This expression is correct up to an infinite constant which does not depend
+on the eigenvalue $\lambda$, thus we can ignore it. By substituting
+$\ln(\det D) = \text{Tr}(\ln D)$ we can rewrite the one-loop effective action
+$W$ into
\begin{align}
- W = -\frac{1}{2} \int_0^\infty dt \frac{K(t, D)}{t}
+ W = -\frac{1}{2} \int_0^\infty dt \frac{K(t, D)}{t},
\end{align}
where
\begin{align}
K(t, D) = \text{Tr}(e^{-tD}) = \int d^n x \sqrt{g}K(t;x,x;D).
\end{align}
-The problem is now that the integral of $W$ is divergent at both limits. Yet
+The problem now is that the integral of $W$ is divergent at both limits. Yet
the divergences at $t\rightarrow \infty$ are caused by $\lambda \leq 0$ of $D$
-(infrared divergences) and are just ignored. The divergences at $t\rightarrow 0$
-are cutoff at $t=\Lambda^{-2}$, thus we write
+(infrared divergences) and can be ignored. The divergences at $t\rightarrow 0$
+are cutoff at $t=\Lambda^{-2}$, simply written as
\begin{align}
W_\Lambda = -\frac{1}{2} \int_{\Lambda^{-2}}^\infty dt \frac{K(t, D)}{t}.
\end{align}
-We can calculate $W_\Lambda$ at up to an order of $\lambda ^0$
+We can calculate $W_\Lambda$ up to an order of $\lambda ^0$
\begin{align}
W_\Lambda &= -(4\pi)^{-n/2} \int d^n x\sqrt{g}\bigg(
\sum_{2(j+l)<n}\Lambda^{n-2j-2l}b_{2j}(x,x) \frac{(-m^2)^l l!}{n-2j-2l} +\\
&+ \sum_{2(j+l) =n }\ln(\Lambda) (-m^2)^l l! b_{2j}(x,x)
\mathcal{O}(\lambda^0) \bigg)
\end{align}
-There is an divergence at $b_2(x,x)$ with $k\leq n$. Now we compute the limit
-$\Lambda \rightarrow \infty$
+There is an divergence at $b_2(x,x)$ for $k\leq n$. Computing the limit
+$\Lambda \rightarrow \infty$ we get
\begin{align}
-\frac{1}{2}(4\pi)^{n/2}m^n\int d^n x\sqrt{g} \sum_{2j>n}
- \frac{b_{2j}(x,x)}{m^{2j}}\Gamma(2j-n)
-\end{align}
-here $\Gamma$ is the gamma function.
-\subsubsection{Differential Geometry and Operators of Laplace Type}
-Let $M$ be a $n$ dimensional compact Riemannian manifold with $\partial M = 0$.
-Then consider a vector bundle $V$ over $M$ (i.e. there is a vector space to
-each point on $M$), so we can define smooth functions. We want to look at
-arbitrary differential operators $D$ of Laplace type on $V$, they have the general
-from
-\begin{align}
- D = -(g^{\mu\nu} \partial_\mu\partial_\nu + a^\sigma\partial_\sigma +b)
-\end{align}
-where $a^\sigma, b$ are matrix valued functions on $M$ and $g^{\mu\nu}$ is the
-inverse metric on $M$. There is a unique connection on $V$ and a unique
-endomorphism (matrix valued function) $E$ on $V$, then we can rewrite $D$ in
-terms of $E$ and covariant derivatives
-\begin{align}
- D = -(g^{\mu\nu} \nabla_\mu \nabla_\nu +E)
-\end{align}
-Where the covariant derivative consists of $\nabla = \nabla^{[R]} +\omega$ the
-standard Riemannian covariant derivative $\nabla^{[R]}$ and a "gauge" bundle
-$\omega$ (fluctuations). WE can write $E$ and $\omega$ in terms of geometrical
-identities
-\begin{align}
- \omega_\delta &= \frac{1}{2}g_{\nu\delta}(a^\nu
- +g^{\mu\sigma}\Gamma^\nu_{\mu\sigma}I_V)\\
- E &= b - g^{\nu\mu}(\partial_\mu \omega_\nu + \omega_\nu \omega_\mu -
- \omega_\sigma \Gamma^\sigma_{\nu\mu})
-\end{align}
-where $I_V$ is the identity in $V$ and the Christoffel symbol
-\begin{align}
- \Gamma^\sigma_{\mu\nu} = g^{\sigma\varrho} \frac{1}{2} (\partial_\mu
- g_{\nu\varrho} + \partial_\nu g_{\mu\varrho} - \partial_\varrho g_{\mu\nu})
-\end{align}
-Furthermore we remind ourselves of the Riemmanian curvature tensor, Ricci
-Tensor and the Scalar curavture.
-\begin{align}
- R^\mu_{\nu\varrho\sigma} &= \partial_\sigma \Gamma^{\mu}_{\nu\varrho}
- -\partial_\varrho \Gamma^\mu_{\nu\sigma}
- \Gamma^{\lambda}_{\nu\varrho}\Gamma^{\mu}_{\lambda\sigma}
- \Gamma^{\lambda}_{\nu\sigma}\Gamma^{\mu}_{\lambda\varrho}\\
- R_{\mu\nu} &:= R^{\sigma}_{\mu\nu\sigma}\\
- R &:= R^\mu_{\ \mu}
+ \frac{b_{2j}(x,x)}{m^{2j}}\Gamma(2j-n),
\end{align}
+where $\Gamma$ stands for the gamma function.
+%----------------------- KANN WEGGELASSEN WERDEN
-The we let $\{e_1, \dots, e_n\}$ be the local orthonormal frame of
-$TM$(tangent bundle $M$), which will be noted with flat indices $i,j,k,l
-\in\{1,\dots, n\}$, we use $e^k_\mu, e^\nu_j$ to transform between flat indices
-and curved indices $\mu, \nu, \varrho$.
-\begin{align}
- e^\mu_j e^\nu_k g_{\mu\nu} &= \delta_{jk}\\
- e^\mu_j e^\nu_k \delta^{jk} &= g^{\mu\nu} \\
- e^j_\mu e^\mu_k &= \delta^j_k
-\end{align}
-
-The Riemannian part of the covariant derivative contains the standard
-Levi-Civita connection, so that for a $v_\nu$ we write
-\begin{align}
- \nabla_\mu^{[R]} v_\nu = \partial_\mu v_\nu -
- \Gamma^{\varrho}_{\mu\nu}v_\varrho.
-\end{align}
-The extended covariant derivative reads then
-\begin{align}
- \nabla_\mu v^j = \partial_\mu v^j + \sigma^{jk}_\mu v_k.
-\end{align}
-the condition $\nabla_\mu e^k_\nu = 0$ gives us the general connection
-\begin{align}
- \sigma^{kl}_\mu = e^\nu_l\Gamma^{\varrho}_{\mu\nu}e^k_\varrho - e^\nu_l
- \partial_\mu e^k_\nu
-\end{align}
-The we may define the field strength $\Omega_{\mu\nu}$ of the connection $\omega$
-\begin{align}
- \Omega_{\mu\nu} = \partial_\mu \omega_\nu -\partial_\nu \omega_\mu
- +\omega_\mu \omega_\nu -\omega_\nu\omega_\mu.
-\end{align}
-If we apply the covariant derivative on $\Omega$ we get
-\begin{align}
- \nabla_\varrho\Omega_{\mu\nu} = \partial_\varrho \Omega_{\mu\nu} -
- \Gamma^{\sigma}_{\varrho \mu} \Omega_{\sigma\mu} + [\omega_\varrho,
- \Omega_{\mu\nu}]
-\end{align}
\subsubsection{Spectral Functions}
-Manifolds without $M$ boundary condition for the operator $e^{-tD}$ for $t>0$ is a
-trace class operator on $L^2(V)$, this means that for any smooth function $f$
-on $M$ we can define
+Manifolds $M$ with a disappearing boundary condition for the operator $e^{-tD}$ for $t>0$ is a
+trace class operator on $L^2(V)$. Meaning for any smooth function $f$ on $M$
+we can define
\begin{align}
- K(t,f,D) = \text{Tr}_{L^2}(fe^{-tD})
+ K(t,f,D) := \text{Tr}_{L^2}(fe^{-tD}),
\end{align}
-and we can rewrite
+or alternately write an integral representation
\begin{align}
- K(t, f, D) = \int_M d^n x \sqrt{g} \text{Tr}_V(K(t;x,x;D)f(x)).
+ K(t, f, D) = \int_M d^n x \sqrt{g} \text{Tr}_V(K(t;x,x;D)f(x)),
\end{align}
-in terms of the Heat kernel $K(t;x,y;D)$ in the regular limit $y\rightarrow y$.
-We can write the Heat Kernel in terms of the spectrum of $D$. Say
-$\{\phi_\lambda\}$ is a ONB of eigenfunctions of $D$ corresponding to the
-eigenvalue $\lambda$
+in the regular limit $y\rightarrow y$. We can write the Heat Kernel in terms
+of the spectrum of $D$. So for an orthonormal basis $\{\phi_\lambda\}$ of
+eigenfunctions for $D$, which corresponds to the eigenvalue $\lambda$, we
+can rewrite the heat kernel into
\begin{align}
K(t;x,y;D) = \sum_\lambda \phi^\dagger_\lambda(x)
\phi_\lambda(y)e^{-t\lambda}.
\end{align}
-We have an asymtotic expansion at $t \rightarrow 0$ for the trace
+An asymptotic expansion as $t \rightarrow 0$ for the trace is then
\begin{align}
- Tr_{L^2}(fe^{-tD}) \simeq \sum_{k\geq 0}t^{(k-n)/2}a_k(f,D).
+ \text{Tr}_{L^2}(fe^{-tD}) \simeq \sum_{k\geq 0}t^{(k-n)/2}a_k(f,D),
\end{align}
where
\begin{align}
- a_k(f,D) = (4\pi)^{-n/2} \int_M d^4x \sqrt{g} b_k(x,x) f(x)
+ a_k(f,D) = (4\pi)^{-n/2} \int_M d^4x \sqrt{g} b_k(x,x) f(x).
\end{align}
\subsubsection{General Formulae}
-We consider a compact Riemmanian Manifold $M$ without boundary condition, a
-vector bundle $V$ over $M$ to define functions which carry discrete (spin or
-gauge) indices. An Laplace style operator $D$ over $V$ and smooth function $f$
-on $M$. There is an asymtotic expansion where the heat kernel coefficients
+Summarizing, in the last chapter we considered a compact Riemannian manifold
+$M$ without boundary condition, a vector bundle $V$ over $M$ to define
+functions which carry discrete (spin or gauge) indices, an Laplace style
+operator $D$ over $V$ and smooth function $f$ on $M$. Now there is an asymptotic
+expansion where the heat kernel coefficients are
+%------------------------------- HERE
+\newline
+\textbf{---------------------HERE}
+\newline
+%------------------------------- HERE
\begin{enumerate}
- \item with odd index $k=2j+1$ vanish
- $a_{2j+1}(f,D) = 0$
+ \item with odd index $k=2j+1$ vanish $a_{2j+1}(f,D) = 0$
\item with even index are locally computable in terms of geometric
invariants
\end{enumerate}
@@ -247,7 +178,7 @@ coefficients and that the only nonzero coefficient is $a_0(1, D_1) =
\mathcal{A}^I_n(D_2)\right).
\end{align}
-On the other had all geometric invariants associated with $D$ are in the $D_2$
+On the other had all geometric invariants associated with $D$ are in the $D_2$
part. Thus all invariants are independent of $x_1$, so we can choose for $M_1$.
Say $M_1 = S^1$ with $x\in (0, 2\pi)$ and $D_1=-\partial_{x_1}^2$ we may
rewrite the heat kernel coefficients in
diff --git a/src/thesis/chapters/twopointspace.tex b/src/thesis/chapters/twopointspace.tex
@@ -0,0 +1,241 @@
+\subsection{Almost-commutative Manifold}
+\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 we can make a simple
+choice for the Hilbertspace, $H_F = \mathbb{C}^2$. Furthermore $\gamma_F$ is
+the $\mathbb{Z}_2$ grading, which allows us to decompose $H_F$ 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 $H_F^\pm$,
+\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 we have KO-dimension of 0,
+ 2 or 6. To elaborate on this, we know that there are 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}
+ \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
+$dim(H_F^\pm) = 1$ (see Lemma 3.8 Book), 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 \simeq M\sqcup X.
+\end{align}
+Keep in mind that we still need to find an appropriate real structure on the
+Riemannian spin manifold, $J_M$. Furthermore 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 we can consider a
+distance formula on $M\times F_X$ 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 \ref{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}
+and the supremum 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,
+then 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 theory.
+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{definition}
+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{definition}
+\begin{definition}
+ 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{definition}
+The Gauge group of $F_X$ is given by the quotient $U(A)/U(A_J)$.
+We want 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{proposition}
+The Gauge group of the Two-Point space $\mathfrak{B}(F_X)$ is $U(1)$.
+\end{proposition}
+\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{proposition}
+ 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{proposition}
+
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
@@ -7,7 +7,6 @@
\input{back/title}
\newpage
-
%-------------------- BACKHAND ---------------------
\input{back/abstract}
@@ -19,8 +18,10 @@
\input{chapters/main_sec}
%\input{chapters/basics}
-%
-%\input{chapters/heatkernel}
+
+\input{chapters/heatkernel}
+
+\input{chapters/twopointspace}
\input{chapters/electroncg}
