commit 426279727d81800c0e37a92c527120538732a9a4
parent 30fec419bdd434928e5155667b35d4f5a9080a6b
Author: miksa <milutin@popovic.xyz>
Date: Wed, 9 Jun 2021 11:01:52 +0200
week10.tex initial
Diffstat:
2 files changed, 167 insertions(+), 2 deletions(-)
diff --git a/src/week10.tex b/src/week10.tex
@@ -0,0 +1,164 @@
+\documentclass[a4paper]{article}
+
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+
+\usepackage{mathptmx}
+
+\usepackage{subcaption}
+\usepackage[shortlabels]{enumitem}
+\usepackage{amssymb}
+\usepackage{amsthm}
+\usepackage{mathtools}
+\usepackage{bbm}
+\usepackage{graphicx}
+\usepackage[colorlinks=true,naturalnames=true,plainpages=false,pdfpagelabels=true]{hyperref}
+\usepackage[parfill]{parskip}
+
+\usepackage{tikz}
+\usetikzlibrary{patterns,decorations.pathmorphing,positioning}
+
+\usepackage[framemethod=TikZ]{mdframed}
+
+\tikzstyle{titlered} =
+ [draw=black, thick, fill=white,%
+ text=black, rectangle,
+ right, minimum height=.7cm]
+
+\newcounter{exercise}
+
+\renewcommand*\theexercise{Exercise~\arabic{exercise}}
+
+\makeatletter
+\mdfdefinestyle{exercisestyle}{%
+ outerlinewidth=1em,%
+ outerlinecolor=white,%
+ leftmargin=-1em,%
+ rightmargin=-1em,%
+ middlelinewidth=1.2pt,%
+ roundcorner=5pt,%
+ linecolor=black,%
+ backgroundcolor=blue!5,
+ innertopmargin=1.2\baselineskip,
+ skipabove={\dimexpr0.5\baselineskip+\topskip\relax},
+ skipbelow={-1em},
+ needspace=3\baselineskip,
+ frametitlefont=\sffamily\bfseries,
+ settings={\global\stepcounter{exercise}},
+ singleextra={%
+ \node[titlered,xshift=1cm] at (P-|O) %
+ {~\mdf@frametitlefont{\theexercise}~};},%
+ firstextra={%
+ \node[titlered,xshift=1cm] at (P-|O) %
+ {~\mdf@frametitlefont{\theexercise}~};},
+}
+\makeatother
+
+\newenvironment{MyExercise}%
+{\begin{mdframed}[style=exercisestyle]}{\end{mdframed}}
+
+\theoremstyle{definition}
+\newtheorem{definition}{Definition}
+
+\theoremstyle{definition}
+\newtheorem{question}{Question}
+
+\theoremstyle{definition}
+\newtheorem{example}{Example}
+
+\theoremstyle{theorem}
+\newtheorem{theorem}{Theorem}
+
+\theoremstyle{theorem}
+\newtheorem{lemma}{Lemma}
+
+
+\theoremstyle{theorem}
+\newtheorem{proposition}{Proposition}
+
+\newtheorem*{idea}{Proof Idea}
+
+
+\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{Spectral Action of the Fluctuated Dirac Operator}
+\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.
+ \textbf{PROOF IN week10.pdf}
+\end{proof}
+
+\section{Fermionic Action}
+\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}
+
+\begin{align}
+ M\times F_{ED} := \left(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\right)
+\end{align}
+
+
+
+
+\end{document}
diff --git a/src/week8.tex b/src/week8.tex
@@ -665,9 +665,10 @@ $\theta _i, \theta _j$ some Grassmann variables we have
\begin{proof}
Will maybe be filled in if I go through the last two chapters in the
book and understand the proof.
+ \textbf{PROOF IN week10.pdf}
\end{proof}
-Here on we go and calculate the spectral action of $M\times F_{ED}$
+From 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}
@@ -709,5 +710,5 @@ Here on we go and calculate the spectral action of $M\times F_{ED}$
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}
