commit 676f03b1300224c2aef5fe512da0d86aa0c46988
parent f4c253f98f2b65a21125c0aeb306bce96475dcdc
Author: miksa234 <milutin@popovic.xyz>
Date: Mon, 17 May 2021 15:21:39 +0200
add some more content to week8 changed cover page
Diffstat:
16 files changed, 204 insertions(+), 46 deletions(-)
diff --git a/pdfs/week1.pdf b/pdfs/week1.pdf
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diff --git a/pdfs/week2.pdf b/pdfs/week2.pdf
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diff --git a/pdfs/week3.pdf b/pdfs/week3.pdf
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diff --git a/pdfs/week4.pdf b/pdfs/week4.pdf
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diff --git a/pdfs/week5.pdf b/pdfs/week5.pdf
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diff --git a/pdfs/week6.pdf b/pdfs/week6.pdf
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diff --git a/pdfs/week7.pdf b/pdfs/week7.pdf
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diff --git a/pdfs/week8.pdf b/pdfs/week8.pdf
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diff --git a/src/week1.tex b/src/week1.tex
@@ -73,16 +73,17 @@
\newtheorem*{idea}{Proof Idea}
-
-
-\title{Notes on \\ Noncommutative Geometry and Particle Physics}
-\author{Popovic Milutin}
+\title{University of Vienna\\ Faculty of Physics\\ \vspace{1.25cm}
+Notes on\\ Noncommutative Geometry and Particle Phyiscs}
\date{Week 1: 05.02 - 12.02}
+\author{Milutin Popovic \\ Supervisor: Dr. Lisa
+Glaser}
\begin{document}
-\maketitle
-\tableofcontents
+ \maketitle
+ \tableofcontents
+ \newpage
\section{Noncommutative Geometric Spaces}
\subsection{Matrix Algebras and Finite Spaces}
diff --git a/src/week2.tex b/src/week2.tex
@@ -73,14 +73,18 @@
\newtheorem*{idea}{Proof Idea}
-\title{Notes on \\ Noncommutative Geometry and Particle Physics}
-\author{Popovic Milutin}
+
+\title{University of Vienna\\ Faculty of Physics\\ \vspace{1.25cm}
+Notes on\\ Noncommutative Geometry and Particle Phyiscs}
+\author{Milutin Popovic \\ Supervisor: Dr. Lisa
+Glaser}
\date{Week 2: 12.02 - 19.02}
\begin{document}
-\maketitle
-\tableofcontents
+ \maketitle
+ \tableofcontents
+ \newpage
\section{Noncommutative Geometric Spaces}
\subsection{Noncommutative Matrix Algebras}
diff --git a/src/week3.tex b/src/week3.tex
@@ -37,15 +37,17 @@
\newtheorem*{idea}{Proof Idea}
-
-\title{Notes on \\ Noncommutative Geometry and Particle Physics}
-\author{Popovic Milutin}
-\date{Week 2: 26.02 - 4.03}
+\title{University of Vienna\\ Faculty of Physics\\ \vspace{1.25cm}
+Notes on\\ Noncommutative Geometry and Particle Phyiscs}
+\author{Milutin Popovic \\ Supervisor: Dr. Lisa
+Glaser}
+\date{Week 3: 26.02 - 4.03}
\begin{document}
-\maketitle
-\tableofcontents
+ \maketitle
+ \tableofcontents
+ \newpage
\section{Excurse to Group Theory and Lie Groups}
\subsection{Groups and Representations}
\begin{definition}
diff --git a/src/week4.tex b/src/week4.tex
@@ -37,15 +37,17 @@
\newtheorem*{idea}{Proof Idea}
-
-\title{Notes on \\ Noncommutative Geometry and Particle Physics}
-\author{Popovic Milutin}
+\title{University of Vienna\\ Faculty of Physics\\ \vspace{1.25cm}
+Notes on\\ Noncommutative Geometry and Particle Phyiscs}
+\author{Milutin Popovic \\ Supervisor: Dr. Lisa
+Glaser}
\date{Week 4: 05.03 - 12.03}
\begin{document}
-\maketitle
-\tableofcontents
+ \maketitle
+ \tableofcontents
+ \newpage
\section{Characters}
\begin{definition}
diff --git a/src/week5.tex b/src/week5.tex
@@ -74,14 +74,18 @@
\newtheorem*{idea}{Proof Idea}
-\title{Notes on \\ Noncommutative Geometry and Particle Physics}
-\author{Popovic Milutin}
+
+\title{University of Vienna\\ Faculty of Physics\\ \vspace{1.25cm}
+Notes on\\ Noncommutative Geometry and Particle Phyiscs}
+\author{Milutin Popovic \\ Supervisor: Dr. Lisa
+Glaser}
\date{Week 5: 12.03 - 19.03}
\begin{document}
-\maketitle
-\tableofcontents
+ \maketitle
+ \tableofcontents
+ \newpage
\section{Noncommutative Geometric Spaces }
\subsection{Exercises}
diff --git a/src/week6.tex b/src/week6.tex
@@ -73,16 +73,20 @@
\newtheorem*{idea}{Proof Idea}
+\date{Week 6: 19.03 - 26.03}
-\title{Notes on \\ Noncommutative Geometry and Particle Physics}
-\author{Popovic Milutin}
+\title{University of Vienna\\ Faculty of Physics\\ \vspace{1.25cm}
+Notes on\\ Noncommutative Geometry and Particle Phyiscs}
+\author{Milutin Popovic \\ Supervisor: Dr. Lisa
+Glaser}
\date{Week 6: 19.03 - 26.03}
-\begin{document}
-\maketitle
-\tableofcontents
+\begin{document}
+ \maketitle
+ \tableofcontents
+ \newpage
\section{Finite Real Noncommutative Spaces}
\subsection{Finite Real Spectral Triples}
Add on to finite real spectral triples a \textit{real structure}. The
diff --git a/src/week7.tex b/src/week7.tex
@@ -79,16 +79,17 @@
\newtheorem*{idea}{Proof Idea}
-
-\title{Notes on \\ Noncommutative Geometry and Particle Physics}
-\author{Popovic Milutin}
+\title{University of Vienna\\ Faculty of Physics\\ \vspace{1.25cm}
+Notes on\\ Noncommutative Geometry and Particle Phyiscs}
+\author{Milutin Popovic \\ Supervisor: Dr. Lisa
+Glaser}
\date{Week 7: 23.04 - 27.04}
\begin{document}
-\maketitle
-\tableofcontents
-
+ \maketitle
+ \tableofcontents
+ \newpage
\section{Classification of Finite Real Spectral Triples}
Here we classify finite real spectral triples modulo unitary equivalence with
diff --git a/src/week8.tex b/src/week8.tex
@@ -79,18 +79,22 @@
\newtheorem*{idea}{Proof Idea}
-\title{Notes on \\ Noncommutative Geometry and Particle Physics}
-\author{Popovic Milutin}
-\date{Week 6: 19.03 - 26.03}
+\title{University of Vienna\\ Faculty of Physics\\ \vspace{1.25cm}
+Notes on\\ Noncommutative Geometry and Particle Phyiscs}
+\author{Milutin Popovic \\ Supervisor: Dr. Lisa
+Glaser}
+\date{Week 8: 8.05 - 18.05}
\begin{document}
\maketitle
\tableofcontents
+ \newpage
+
\section{Excurse}
\textbf{Manifold:} A topological space that is locally Euclidean.
\newline
- \textbf{Riemannian Manifold:}A Manifold equipped with a riemannian
+ \textbf{Riemannian Manifold:}A Manifold equipped with a Riemannian
Metric, a
symmetric bilinear form on Vector Fields $\Gamma(TM)$
\begin{align}
@@ -136,7 +140,7 @@
$(S, J_M)$ is called the \textbf{spin Structure on $M$}
\newline
$J_M$ is called the \textbf{charge conjugation}.
- \section{Noncommutative Geomtery of Electrodynamics}
+ \section{Noncommutative Geometry of Electrodynamics}
\subsection{The Two-Point Space}
Consider a two point space $X := \{x, y\}$. This space=an be described
with
@@ -288,7 +292,7 @@
\begin{align}
\alpha(a) = u a u^* \;\;\; \text{for} \;\; U(A)
\end{align}
- where $U(\mathfrak{A})$ is
+ where $U(A)$ is
\begin{align}
U(A) = \{ u\in A|\;\; uu^* = u^*u=1\} \;\;\;
\text{(unitary)}
@@ -337,7 +341,9 @@
Now we need to find the exact from of the field $B_\mu$ to calculate the
spectral action of a spectral triple. Since $(A_F)_{J_F} \simeq
\mathbb{C}$ we find that $\mathfrak{h}(F) = \mathfrak{u}((A_F)_{J_F})
- \simeq i\mathbb{R}$.\newline
+ \simeq i\mathbb{R}$. Where $\mathfrak{h}(F)$ is the Lie Algebra on $F$
+ and $\mathfrak{u}((A_F)_{J_F})$ is the Lie algebra of the unitary group
+ $(A_F)_{J_F}$.\newline
An arbitrary hermitian field $A_\mu = -ia\partial _\mu b$ is given by
two
@@ -381,7 +387,7 @@ gravity and electrodynamics although in the classical level.
The almost commutative Manifold $M\times F_X$ describes a local gauge group
$U(1)$. The inner fluctuations of the Dirac operator describe $Y_\mu$ the
-gauge field of $U(1)$. There arrise two Problems:
+gauge field of $U(1)$. There arise two Problems:
\newline
(1): With $F_X$, $D_F$ must vanish, however this implies that the electrons
are massless (this we do not want)
@@ -505,7 +511,7 @@ with two nodes of multiplicity two
Add a non-zero Dirac Operator to $F_{ED}$. From the Krajewski Diagram, we see
that edges only exist between the multiple vertices. So we construct a Dirac
operator mapping between the two vertices.
-\begin{align}
+\begin{align}\label{dirac}
D_F =
\begin{pmatrix}
0 & d & 0 & 0 \\
@@ -545,7 +551,7 @@ and the other one acting on $L^2(S) \otimes H_{\bar{e}}$
The derivation of the gauge theory is the same for $F_{ED}$ as for $F_X$, we
have $\mathfrak{B}(F) \simeq U(1)$ and for $B_\mu = A_\mu - J_F A_\mu
J_F^{-1}$
-\begin{align}
+\begin{align} \label{field}
B_\mu =
\begin{pmatrix}
Y_\mu & 0 & 0 & 0 \\
@@ -569,5 +575,139 @@ distance.
even thought we have infinite distance between the same manifold? What do
we get if we fix this?
\end{question}
+\subsection{The Spectral Action}
+Here we calculate the Lagrangian of the almost commutative Manifold $M\times
+F_{ED}$, which corresponds to the Lagrangian of Electrodynamics on a curved
+background Manifold (+ gravitational Lagrangian). It consists of the spectral
+action $S_b$ (bosonic) and of the fermionic action $S_f$.
+
+The simples spectral action of a spectral triple $(A, H, D)$ is given by the
+trace of some function of $D$, we also allow inner fluctuations of the Dirac
+operator $D_\omega = D + \omega + \varepsilon' J\omega J^{-1}$ where $\omega =
+\omega ^* \in \Omega_D^1(A)$.
+\begin{definition}
+ 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}f(\frac{D_\omega}{\Lambda})
+ \end{align}
+ where $\Lambda$ is a real cutoff parameter. The minimal condition on $f$
+ is that $f(\frac{D_\omega}{\Lambda})$ is a traclass operator, which mean
+ that it should be compact operator with well defined finite trace
+ independent of the basis. The subscript $b$ of $S_b$ refers to bosonic,
+ because in physical applications $\omega$ will describe bosonic fields.
+
+ Furthermore there is a topological spectral action, defined with the
+ grading $\gamma$
+ \begin{align}
+ S_{\text{top}}[\omega] := \text{Tr}(\gamma\
+ f(\frac{D_\omega}{\Lambda})).
+ \end{align}
+\end{definition}
+\begin{definition}
+ The fermionic action is defined by
+ \begin{align}
+ S_f[\omega, \psi] = (J\tilde{\psi}, D_\omega \tilde{\psi})
+ \end{align}
+ with $\tilde{\psi} \in H_{cl}^+ := \{\tilde{\psi}: \psi \in H^+\}$.
+ $H_{cl}^+$ is the set of Grassmann variables in $H$ in the +1-eigenspace
+ of the grading $\gamma$.
+\end{definition}
+The grasmann variables are a set of Basis vectors of a vector space, they
+form a unital algebra over a vector field say $V$ where the generators are anti commuting, that is for
+$\theta _i, \theta _j$ some Grassmann variables 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}
+\begin{proposition}
+ 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}
+ with
+ \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}
+ where $N=4$ and $\mathcal{L}_M$ is the Lagrangian of the spectral triple
+ $(C^\infty(M) , L^2(S), D_M)$
+ \begin{align}\label{lagr}
+ \mathcal{L}_M(g_{\mu\nu}) := \frac{f_4 \Lambda ^4}{2\pi^2} -
+ \frac{f_2 \Lambda^2}{24\pi ^2}s - \frac{f(0)}{320\pi^2} C_{\mu\nu
+ \varrho \sigma}C^{\mu\nu \varrho \sigma}.
+ \end{align}
+ Here $C^{\mu\nu \varrho \sigma}$ is defined in terms of the Riemannian
+ curvature tensor $R_{\mu\nu \varrho \sigma}$ and the Ricci tensor
+ $R_{\nu\sigma} = g^{\mu\varrho} R_{\mu\nu \varrho\sigma}$.
+
+ Furthermore $\mathcal{L}_B$ describes the kinetic term of the gauge field
+ \begin{align}
+ \mathcal{L}_B(B_\mu) := \frac{f(0)}{24\pi^2}
+ \text{Tr}(F_{\mu\nu}F^{\mu\nu}).
+ \end{align}
+ Last $\mathcal{L}_\phi$ is the scalar-field Lagrangian with a boundary
+ term.
+ \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))\\
+ &+ \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{proposition}
+\begin{proof}
+ Will maybe be filled in if I go through the last two chapters in the
+ book and understand the proof.
+\end{proof}
+
+Here on we go and calculate the spectral action of $M\times F_{ED}$
+\begin{proposition}
+ The Spectral action of $M\times F_{ED}$ is
+ \begin{align}
+ \text{Tr}(f\frac{D_\omega}{\Lambda}) \sim \int_M \mathcal{L}(g_{\mu\nu},
+ Y_\mu) \sqrt{g}\ d^4x + O(\Lambda^{-1})
+ \end{align}
+ where the Lagrangian is
+ \begin{align}
+ \mathcal{L}(g_{\mu\nu}, Y_\mu) =
+ 4\mathcal{L}_M(g_{\mu\nu})+
+ \mathcal{L}_Y(Y_\mu)+
+ \mathcal{L}_\phi(g_{\mu\nu}, d)
+ \end{align}
+ here the $d$ in $\mathcal{L}_\phi$ is from $D_F$ in equation
+ \ref{dirac}. The Lagrangian $\mathcal{L}_M$ is like in equation
+ \ref{lagr}. The Lagrangian $\mathcal{L}_Y$ is the kinetic term of the
+ $U(1)$ gauge field $Y_\mu$
+ \begin{align}
+ \mathcal{L}_Y(Y_\mu):= \frac{f(0)}{6\pi^2}
+ Y_{\mu\nu}Y^{\mu\nu}\;\;\;\;\;\;\;\;\text{with}\;\;\; Y_{\mu\nu} =
+ \partial_\mu Y_\nu -
+ \partial_\nu Y_\mu.
+ \end{align}
+ Then there is $\mathcal{L}_\phi$, which has two constant terms
+ (disregarding the boundary term) that add up to the Cosmological Constant
+ and a term that for the Einstein-Hilbert action
+ \begin{align}
+ \mathcal{L}_\phi(g_{\mu\nu}, d) := \frac{2f_2 \Lambda ^2}{\pi^2}
+ |d|^2 + \frac{f(0)}{2\pi^2} |d|^4 + \frac{f(0)}{12\pi ^2} s |d|^2.
+ \end{align}
+\end{proposition}
+\begin{proof}
+ The Trace of $\mathbb{C}^4$ (the Hilbertspace) gives $N=4$. With $B_\mu$
+ like in equation \ref{field} we have $\text{Tr}(F_{\mu\nu}
+ F^{\mu\nu})=4Y_{\mu\nu}Y^{\mu\nu}$. This provides $\mathcal{L}_Y$.
+ Furthermore we have $\Phi^2 = D_F^2 = |d|^2$ and $\mathcal{L}_\phi$ only
+ give numerical contributions to the cosmological constant and the
+ Einstein-Hilbert action.
+
+ The proof is relying itself on just plugging the terms into the previous
+ proposition, for which I didn't write the proof for.
+\end{proof}
\end{document}
