commit 9e68bdd850568e3248f42d65e55a09fa603a6bd2
parent 0c6c776df87168a47343a50d3d502ca607fa40cd
Author: miksa234 <milutin@popovic.xyz>
Date: Wed, 11 Aug 2021 12:08:42 +0200
DONE
Diffstat:
7 files changed, 32 insertions(+), 26 deletions(-)
diff --git a/pdfs/thesis.pdf b/pdfs/thesis.pdf
Binary files differ.
diff --git a/src/thesis/chapters/1_basics.tex b/src/thesis/chapters/1_basics.tex
@@ -526,7 +526,7 @@ 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 $H \simeq \mathbb{C}$.
+irreducibly $H \simeq \mathbb{C}$.
\end{proof}
%---------------- EXAMPLE
diff --git a/src/thesis/chapters/5_twopointspace.tex b/src/thesis/chapters/5_twopointspace.tex
@@ -3,7 +3,7 @@
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).
+ F_X := \left(C(X) = \mathbb{C}^2, H_F, D_F; J_F, \gamma _f\right).
\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
@@ -15,7 +15,7 @@ $H_F$ into
\end{align}
where
\begin{align}
- H_F^\pm = \{\psi \in H_F |\; \gamma_F\psi = \pm \psi\},
+ H_F^\pm = \left\{\psi \in H_F |\; \gamma_F\psi = \pm \psi\right\},
\end{align}
are two eigenspaces. And lastly the Dirac operator $D_F$ lets us
interchange between the two eigenspaces $H_F^\pm$,
@@ -118,7 +118,7 @@ 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
+ 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}
@@ -232,7 +232,7 @@ where $Y_\mu$ the $U(1)$ Gauge field is defined as
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
+ 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
diff --git a/src/thesis/chapters/6_electroncg.tex b/src/thesis/chapters/6_electroncg.tex
@@ -113,8 +113,8 @@ note that we are still left with $D_F = 0$ and the following spectral triple
\end{pmatrix}
\right).
\end{align}
-It can be represented in the following Krajewski diagram,
-with two nodes of multiplicity two bellow
+It can be represented in the following Krajewski diagram \cite{ncgwatler},
+with two nodes of multiplicity two, in figure \ref{fig:krajewski} bellow.
\begin{figure}[H] \centering
\begin{tikzpicture}[
dot/.style = {draw, circle, inner sep=0.06cm},
@@ -130,11 +130,14 @@ with two nodes of multiplicity two bellow
\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}}
+ \caption{Krajewski diagram of the spectral triple from equation
+ \ref{eq:fedfail}
+ \label{fig:krajewski}
+ }
\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
+closer look at the Krajewski diagram in figure \ref{fig:krajewski} 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}
diff --git a/src/thesis/chapters/conclusion.tex b/src/thesis/chapters/conclusion.tex
@@ -8,10 +8,11 @@ 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
+With a similar but more 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.
+correspondence of the gauge theory of an almost commutative manifold, a
+spectral triple, which brings noncommutative geometry to the interest of
+physicists in the first place.
diff --git a/src/thesis/chapters/intro.tex b/src/thesis/chapters/intro.tex
@@ -5,12 +5,12 @@ Differential Geometry, Representation Theory and many more. The origins can
be dated back to the 1940s where two Russian mathematicians Gelfand and
Naimark proved a theorem that connects (in the sense of duality) (classical)
geometry and algebras. From the beginning it was obvious that noncommutative
-geometry has physical applications, explicitly with gauge theories, since a
+geometry has physical applications, explicitly with gauge theories. A
nontrivial gauge group arises naturally from the main structure of
noncommutative geometry called the spectral triple. We will naturally use
this property to present how to derive the Lagrangian of electrodynamics
\ref{sec:5}, and additionally get a purely gravitational Lagrangian.
-In regards to this to get to the action principles in terms of geometrical
+In regards to this, to get to the action principles in terms of geometrical
invariants, a method called the heat kernel expansion is used.
The aim of this thesis is to give a basic foundation of noncommutative
@@ -29,14 +29,14 @@ 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
+combining this insight we work out the Lagrangian of electrodynamics. Thereby
+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,
+finite spectral triples. It is important to understand these basics, since
+they build up the ground work for 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
@@ -46,9 +46,11 @@ 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.
+electrodynamics. We again atone, that this chapter is based on Vassilevich's
+paper \cite{heatkernel}.
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.
+the process of constructing the almost commutative manifold. With this
+information we can calculate the action principles corresponding to the
+almostcommutative manifold, that will give rise to the Lagrangian of
+electrodynamics and an additional purely gravitational Lagrangian.
diff --git a/src/thesis/main.pdf b/src/thesis/main.pdf
Binary files differ.
