# Download e-book for kindle: Noncommutative Geometry and Particle Physics by Walter D. van Suijlekom

By Walter D. van Suijlekom

This publication presents an advent to noncommutative geometry and offers a couple of its fresh purposes to particle physics. it really is meant for graduate scholars in mathematics/theoretical physics who're new to the sphere of noncommutative geometry, in addition to for researchers in mathematics/theoretical physics with an curiosity within the actual functions of noncommutative geometry. within the first half, we introduce the most recommendations and strategies by means of learning finite noncommutative areas, offering a “light” method of noncommutative geometry. We then continue with the overall framework through defining and reading noncommutative spin manifolds and deriving a few major effects on them, akin to the neighborhood index formulation. within the moment half, we exhibit how noncommutative spin manifolds obviously provide upward thrust to gauge theories, utilising this precept to precise examples. We as a consequence geometrically derive abelian and non-abelian Yang-Mills gauge theories, and at last the entire commonplace version of particle physics, and finish through explaining how noncommutative geometry may point out the right way to continue past the normal Model.

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J. Geom. Phys. 75, 71–91 (2014) Chapter 3 Finite Real Noncommutative Spaces In this chapter, we will enrich the finite noncommutative spaces as analyzed in the previous chapter with a real structure. For one thing, this makes the definition of a finite spectral triple more symmetric by demanding the inner product space H to be an A− A-bimodule, rather than just a left A-module. The implementation of this bimodule structure by an anti-unitary operator has close ties with the Tomita–Takesaki theory of Von Neumann algebras, as well as with physics through charge conjugation, as will become clear in the applications in the later chapters of this book.

12 There is a one-to-one correspondence between finite real spectral triples of K O-dimension k modulo unitary equivalence and Krajewski diagrams of KO-dimension k. Specifically, one associates a real spectral triple (A, H, D; J, γ) to a Krajewski diagram in the following way: A= ⎜ Mn (C); n∈ H= ⎜ Cn(v) ∼ Cm(v)◦ ; v∈φ (0) De + De∗ . 9, with the basis dictated by the graph automorphism j : φ → φ. Finally, a grading γ on H is defined by setting γ to be ±1 on Cn(v) ∼ Cm(v)◦ ⊂ H according to the labeling by ±1 of the vertex v.

Bivariant K -theory of groupoids and the noncommutative geometry of limit sets. PhD thesis, Universität Bonn (2009) 17. : Unbounded bivariant K-theory and correspondences in noncommutative geometry. J. Reine Angew. Math. 691, 101–172 (2014) 18. : Gauge networks in noncommutative geometry. J. Geom. Phys. 75, 71–91 (2014) Chapter 3 Finite Real Noncommutative Spaces In this chapter, we will enrich the finite noncommutative spaces as analyzed in the previous chapter with a real structure. For one thing, this makes the definition of a finite spectral triple more symmetric by demanding the inner product space H to be an A− A-bimodule, rather than just a left A-module.