ncg

bachelorthesis in physics
git clone git://popovic.xyz/ncg.git
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commit 0532a5a7c45146e2d3498e6b8fa8b169a595fcff
Author: miksa <milutin@popovic.xyz>
Date:   Fri, 12 Feb 2021 18:05:40 +0100

initial

Diffstat:
ANCGBook.pdf | 42123+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Amain.tex | 359+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
2 files changed, 42482 insertions(+), 0 deletions(-)

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+\newtheorem{definition}{Definition} + +\theoremstyle{definition} +\newtheorem{question}{Question} + +\theoremstyle{theorem} +\newtheorem{theorem}{Theorem} + +\theoremstyle{theorem} +\newtheorem{exercise}{Exercise} + +\theoremstyle{definition} +\newtheorem{solution}{Solution} + +\newtheorem*{idea}{Proof Idea} + + +\title{Notes on \\ Noncommutative Geometry and Particle Physics} +\author{Popovic Milutin} +\date{Week 1: 05.02 - 12.02} + +\begin{document} + +\maketitle +\tableofcontents + +\section{Noncommutative Geometric Spaces} +\subsection{Matrix Algebras and Finite Spaces} +\subsubsection{$*$-Algebra} +\begin{definition} + A \textit{vector space} $A$ over $\mathbb{C}$ is called a \textit{complex, unital Algebra} if, \\ + $\forall a,b \in A$ : + \begin{enumerate} + \item + $A \times A \rightarrow A$ \hspace{0.1\textwidth} \textit{bilinear} \\ + $(a, b)\ \mapsto \ a\cdot b$ + \item + $1a = a1 =a$ \hspace{0.08\textwidth} \textit{unitary} \\ + \end{enumerate} +\end{definition} + +\begin{definition} + A $*$-algebra is an algebra $A$ with a \textit{conjugate linear map (involution)} $*:A\ \rightarrow A$, + $\forall a, b \in A$ satisfying: + \begin{enumerate} + \item + $(ab)^* = b^*a^*$ \hspace{0.05\textwidth} \textit{antidistributive} + \item + $(a^*)^* = a$ \hspace{0.1\textwidth} \textit{closure} + \end{enumerate} +\end{definition} +In the following all unitary algebras are referred to as algebras. + +\subsubsection{Functions on Discrete Spaces} +Let $X$ be a \textit{discretized topological} space with $N$ points. +Consider functions of a continuous $*$-algebra $C(X)$ assigning values to $\mathbb{C}$, for $f, g \in C(X)$, +$\lambda \in \mathbb{C}$ and $x \in X$ they provide the following structures: + +\begin{itemize} + \item \textit{pointwise linear} \\ + $(f + g)(x) = f(x) + g(x)$\\ + $(\lambda f)(x) = \lambda (f(x))$ + \item \textit{pointwise multiplication} \\ + $fg(x) = f(x)g(x)$ \hspace{0.1\textwidth} same as $(fg)(x) = f(x)g(x))$? + \item \textit{pointwise involution} \\ + $f^*(x) = \overline{f(x)}$ +\end{itemize} + +\begin{question} + Mathematical difference between Topological Discreet Spaces and just Discreet Spaces? +\end{question} + +The author indicates that $\mathbb{C}$-valued functions on $X$ are automatically continuous. +\begin{idea} + CAN WE USE THE METRIC? >> NO! + We know that $X$ is a \textit{finite discrete space}, meaning in an $\epsilon$-$\delta$ approach + for each $x \in X$ the only $y \in X$, that is small enough is $x$ by itself, which implies + $\epsilon$ is always bigger than zero, thus every function $f:X\ \rightarrow\ \mathbb{C}$ is continuous. +\end{idea} + +\subsubsection{Isomorphism Property} +Furthermore $C(X)$ $*$-algebra is \textit{isomorphic} to a $*$-algebra $\mathbb{C}^N$ with involution +($N$ number of points in $X$), written as $C(X) \simeq \mathbb{C}^N$. +A function $f:X\ \rightarrow\ \mathbb{C}$ can be represented with $N \times N$ diagonal matrices, +where the value $(ii)$ is the value of the function at the corresponding +$i$-th point ($i = 1,...,N$). The structure is preserved because of the definitions of +matrix multiplication and the hermitian conjugate of matrices. + +\begin{question} + Can isomorphisms between $C(X)$ and $\mathbb{C}^N$ be shown with matrix factorization? +\end{question} + isomorphisms are bijective perserve strucure and dont lose physical information + +\subsubsection{Mapping Finite Discrete Spaces} + +\begin{definition} + A \textit{map} between finite discrete spaces $X_1$ and $X_2$ is a function $\phi:\ X_1 \rightarrow\ X_2$ +\end{definition} + +For every map between finite discrete spaces there exists a corresponding map \\ +$\phi ^*:C(X_2)\ \rightarrow C(X_1)$, which `pulls back' values even if $\phi$ is not bijective. + +This map is called a pullback (or a $*$-homomorphism or a $*$-algebra map under pointwise product). +Under the pointwise product: +\begin{itemize} + \item $\phi ^*(fg) = \phi ^*(f) \phi ^*(g)$ + \item $\phi ^*(\overline{f}) = \overline{\phi ^*(f)}$ + \item $\phi ^*(\lambda f + g) = \lambda \phi ^*(f) + \phi ^*(g)$ +\end{itemize} + +\begin{question} + $\phi$ is in most cases not bijective, so how can we prove that there exists such a + pullback for every map between discreet spaces which preserves information? For bijective + it is given by its inverse, which by definition exists because $\phi$ is a map. + Or I didn't understand this correctly? +\end{question} + +\begin{exercise} + Show that $\phi :X_1\ \rightarrow \ X_2$ is injective (surjective) map of finite spaces iff + $\phi ^* :C(X_2)\ \rightarrow \ C(X_1)$ is surjective (injective). +\end{exercise} +\begin{solution} + Consider $X_1$ with $n$ points and $X_2$ with $m$ points. Then there are three cases: + \begin{enumerate} + \item $n=m$ \\ + Obviously $\phi$ is bijective and $\phi ^*$ is given by $\phi ^{-1}$. + \item $n > m$ \\ + $\phi$ assigns $n$ points to $m$ points when $n > m$, + which is by definition surjective. \\ + $\phi ^*$ assigns $m$ points to $n$ points when $n > m$, + which is by definition injective. \\ + \item $n < m $ \\ + analogous + \end{enumerate} +\end{solution} + +\subsubsection{Matrix Algebras} +\begin{definition} + 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{definition} + +So from a topological discreet space $X$, we can construct a $*$-algebra $C(X)$ which is isomorphic +to a matrix algebra $A$. The question is can we construct $X$ given $A$? $A$ is a matrix algebra, +which are in most cases is not commutative, so the answer is generally no. + +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. + +\begin{question} + Why are non-commutative algebras not physically interesting? + Maybe too far fetched,but because physical observables (QM-Operators) are not commutative? +\end{question} + +\subsubsection{Finite Inner Product Spaces and Representations} +Until now we looked at a finite topological discreet space, moreover we can consider a +finite dimensional inner product space $H$ (finite Hilbert-spaces), with inner product +$(\cdot,\cdot)\rightarrow \mathbb{C}$. $L(H)$ is the $*$-algebra of operators on $H$ +with product given by composition and involution given by the adjoint, $T \mapsto T^*$. +$L(H)$ is a \textit{normed vector space} with +\begin{align*} + &\|T\|^2 = \text{sup}_{h \in H}\{(Th,Th): (h,h) \leq 1\} \hspace{0.1\textwidth} T \in L(H) \\ + &\|T\| = \text{sup}\{\sqrt{\lambda}: \lambda \text{ eigenvalue of } T\} +\end{align*} + + +\begin{definition} + The \textit{representation} of a finite dimensional $*$-algebra A is a pair $(H, \pi)$. + $H$ is a finite, dimensional inner product space and $\pi$ is a $*$-\textit{algebra map} + \begin{align*} + \pi:A\ \rightarrow \ L(H) + \end{align*} +\end{definition} +\begin{definition} + $(H, \pi)$ is called \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{definition} + +Examples for reducible and irreducible representations +\begin{itemize} + \item $A = M_n(\mathbb{C})$, representation $H=\mathbb{C}^n$, $A$ acts as matrix multiplication\\ + $H$ is irreducible. + \item $A = M_n(\mathbb{C})$, 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} + +\begin{definition} + Let $(H_1, \pi _1)$ and $(H_2, \pi _2)$ be representations of a $*$-algebra $A$. They are called + \textit{unitarily 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{definition} + +\begin{question} + In matrix representation this is diagonalization condition? (unitary diagonalization) +\end{question} + +\begin{definition} + $A$ a $*$-algebra then, $\hat{A}$ is called the structure space of all \textit{unitary equivalence classes + of irreducible representations of A} +\end{definition} + +\begin{question} + Gelfand duality and the spectrum of $\hat{A}$, examples Fourier-Transform and Laplace-Transform + for simple spaces. +\end{question} + +\begin{exercise} + Given $(H, \pi)$ 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)' = \{T \in L(H):\pi (a)T = T\pi (a) \;\;\; \forall a\in A\} + \end{align*} + \begin{enumerate} + \item Show that $\pi (A)'$ is a $*$-algebra. + \item Show that a representation $(H, \pi)$ of $A$ is irreducible iff the commutant $\pi (A)'$ + consists of multiples of the identity + \end{enumerate} +\end{exercise} +\begin{solution} + 1. To show that $\pi (A)'$ is a $*$-algebra we have to show that it is unital, associative and involute. + And note that $\pi (a) \in L(H)\ \forall a \in A$. + Unity is given by the unitary operator of the $*$-algebra of operators $L(H)$, which exists by definition + because H is a inner product space. Associativity is given by $*$-algebra of $L(H)$, $L(H) \times L(H) \mapsto L(H)$, + which is associative by definition. Involutnes is also given by the $*$-algebra $L(H)$ + with a map $*: L(H) \mapsto L(H)$ only for $T$ that commute with $\pi (a)$. + \\ + 2.? +\end{solution} + +\begin{exercise} + \begin{enumerate} + \item If $A$ is a unital $*$-algebra, show that the $n \times n$ matrices $M_n(A)$ with entries + in $A$ form a unital $*$-algebra. + \item Let $\pi :A\ \rightarrow \ L(H)$ be a representation of a $*$-algebra $A$ and set + $H^n = H \oplus ... \oplus H$, $n$ times. Show that $\tilde{\pi}:M_n(A) \rightarrow L(H^n)$ + of $M_n(A)$ with\\ + $\tilde{\pi}((a_{ij})) = (\tilde{\pi}(a_{ij})) \in M_n(A)$. + \item Let $\tilde{\pi}:M_n(A) \rightarrow L(H^n)$ be a $*$ algebra representation of $M_n(A)$. + Show that $\pi: A \rightarrow L(H^n)$ is a representation of $A$. + \end{enumerate} +\end{exercise} + +\begin{solution} + 1. We know $A$ is a $*$ algebra. Unitary operaton in $M_n(A)$ is given by the identity Matrix, which + has to exists because every entry in $M_n(A)$ has to behave like in $A$. Associativity is given by + matrix multiplication. Involutnes is given by the conjugate transpose.\\ + 2. $A \simeq M_n(A)$ and $H \simeq H^n$ meaning $\tilde{\pi}$ is a valid reducible representation.\\ + 3. $\tilde{\pi}$ and $\pi$ are unitary equivalent, there is a map $U: H^n \rightarrow H^n$ given by + $U=\mathbbm{1}_n$:\\ + $\pi (a) = \mathbbm{1}_n^*\ \tilde{\pi}((a_{ij}))\ \mathbbm{1}_n = \tilde{\pi}((a_{ij})) = \pi (a_{ij}) + \Rightarrow a_{ij} = a\mathbbm{1}_n$. +\end{solution} + +\subsection{Commutative Matrix Algebras} +\begin{itemize} + \item Commutative matrix algebras can be used to reconstruct a discrete space given + a matrix \textit{commutative} matrix algebra. + \item The structure space $\hat{A}$ is used for this. Because $A \simeq \mathbb{C}^N$ we all any + irreducible representation are of the form + $\pi _i:(\lambda_1,...,\lambda_N)\in \mathbb{C}^N \mapsto \lambda_i \in \mathbb{C}$ \\ + for $i = 1,...,N \Rightarrow \hat{A} \simeq \{1,...,N\}.$ + \item Conclusion is that there is a duality between discreet spaces and commutative matrix algebra + this duality is called the \textit{finite dimensional Gelfand duality} +\end{itemize} + +\begin{question} + More about Gelfand duality? +\end{question} + +\subsection{Noncommutative Matrix Algebras} +Aim is to construct duality between finite dimensional spaces and \textit{equivalence classes} +of matrix algebras, to preserve general non-commutivity of matrices. +\begin{itemize} + \item Equivalence classes are described by a generalized notion of ispomorphisms between matrix + algebras (\textit{Morita Equivalence}) +\end{itemize} + +\subsubsection{Algebraic Modules} +\begin{definition} + 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_1a_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 $\exists$ a bilinear map $\gamma: F \times B \rightarrow F$ with + \begin{align*} + f \cdot (b_1b_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{definition} +Notion of A-\textbf{module homomorphism} as linear map $\phi: E\rightarrow F$ which respects the +representation of A, e.g. for left module. +\begin{align*} + \phi (ae) = a \phi (e); \;\;\; a \in A, e \in E. +\end{align*} +Remark on the notation +\begin{itemize} + \item ${}_A E$ left $A$-module $E$; + \item ${}_A E_B$ right $B$-module $F$; + \item ${}_A E_B$ $A$-$B$-bimodule $E$; +\end{itemize} + +\begin{exercise} + Check that a representation of $\pi : A \rightarrow L(H)$ of a $*$-algebra A turns H into a + left module ${}_A H$. +\end{exercise} +\begin{solution} + Not quite sure but \\ + $a \in A$, $h_1, h_2 \in H$, we know $\pi (a) = T \in L(H)$ than + \begin{align*} + <\pi (a) h_1, \pi (a) h_2> = <T h_1, T h_2> = <T^*T h_1, h_2> = <h_1, h_2> + \end{align*} +\end{solution} + +\begin{exercise} + Show that $A$ is a bimodule ${}_A A_A$ with itself. +\end{exercise} + +\begin{solution} + $\gamma: A\times A\times A \rightarrow A$ which is trivially given by the inner product of the $*$-algebra. +\end{solution} + +\end{document}