intro.tex (3468B)
1 \section{Introduction} 2 Noncommutative geometry is a branch of mathematics that incorporates many 3 different mathematical fields, e.g. Functional analysis, K-Theory, 4 Differential Geometry, Representation Theory and many more. The origins can 5 be dated back to the 1940s where two Russian mathematicians Gelfand and 6 Naimark proved a theorem that connects (in the sense of duality) (classical) 7 geometry and algebras. From the beginning it was obvious that noncommutative 8 geometry has physical applications, explicitly with gauge theories. A 9 nontrivial gauge group arises naturally from the main structure of 10 noncommutative geometry called the spectral triple. We will naturally use 11 this property to present how to derive the Lagrangian of electrodynamics 12 \ref{sec:5}, and additionally get a purely gravitational Lagrangian. 13 In regards to this, to get to the action principles in terms of geometrical 14 invariants, a method called the heat kernel expansion is used. 15 16 The aim of this thesis is to give a basic foundation of noncommutative 17 geometry and to present a physical application which can be derived from this 18 theory. Additionally we emphasize that this thesis is only literature work, 19 where chapters \ref{sec:1}, \ref{sec:2}, \ref{sec:3}, \ref{sec:5} and 20 \ref{sec:6} are from the work of Walter D. Suijlekom's book 21 `\textit{Noncommutative Geometry and Particle Physics}' \cite{ncgwalter} and 22 chapter \ref{sec:4} from D.V. Vassilevich's paper \cite{heatkernel}. 23 24 The prominent structure of noncommutative geometry is the spectral triple. 25 The most basic form of a spectral triple consists of a unital $C^*$ algebra 26 $A$ acting on a Hilbertspace $H$. Together with a self-adjoint operator $D$ in 27 $H$, with specific conditions coinciding with the Dirac operator on 28 a Riemannian spin$^c$ manifold which square is the Laplacian (up to a scalar 29 term). 30 31 The structure of the thesis is based on first getting the background 32 knowledge of noncommutative geometry and the heat kernel expansion. Then by 33 combining this insight we work out the Lagrangian of electrodynamics. Thereby 34 the first two chapters \ref{sec:1} and \ref{sec:2} go through the basic 35 version of noncommutative geometry, in the sense of finite discrete spaces, 36 finite spectral triples. It is important to understand these basics, since 37 they build up the ground work for constructing the almost commutative 38 manifold of electrodynamics, that is the Two-Point space $F_X$. Additionally 39 the notion of equivalence relations between spectral triples, called Morita 40 equivalence is introduced. 41 42 The next chapter \ref{sec:3} extends the finite spectral triple with a real 43 structure, called the real finite spectral triple, we also examine Morita 44 equivalence within this extension. 45 46 Chapter \ref{sec:4} explains the heat kernel and leads off to the heat kernel 47 expansion, where the famous heat kernel coefficients arise. Hereof we 48 calculate the heat kernel coefficients, which become important when 49 calculating the Lagrangian of the almost commutative manifold of 50 electrodynamics. We again atone, that this chapter is based on Vassilevich's 51 paper \cite{heatkernel}. 52 53 In the last two chapters \ref{sec:5} and \ref{sec:6} we go over the ideas and 54 the process of constructing the almost commutative manifold. With this 55 information we can calculate the action principles corresponding to the 56 almostcommutative manifold, that will give rise to the Lagrangian of 57 electrodynamics and an additional purely gravitational Lagrangian.