Download PDF by Walter D. van Suijlekom: Noncommutative Geometry and Particle Physics

By Walter D. van Suijlekom

ISBN-10: 9401791619

ISBN-13: 9789401791618

ISBN-10: 9401791627

ISBN-13: 9789401791625

This e-book presents an creation to noncommutative geometry and offers a couple of its fresh functions to particle physics. it really is meant for graduate scholars in mathematics/theoretical physics who're new to the sector of noncommutative geometry, in addition to for researchers in mathematics/theoretical physics with an curiosity within the actual purposes of noncommutative geometry. within the first half, we introduce the most strategies and methods by way of learning finite noncommutative areas, offering a “light” method of noncommutative geometry. We then continue with the overall framework through defining and studying noncommutative spin manifolds and deriving a few major effects on them, similar to the neighborhood index formulation. within the moment half, we express how noncommutative spin manifolds clearly provide upward push to gauge theories, utilizing this precept to precise examples. We as a consequence geometrically derive abelian and non-abelian Yang-Mills gauge theories, and at last the total average version of particle physics, and finish by means of explaining how noncommutative geometry may possibly point out easy methods to continue past the normal Model.

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Example text

Since a matrix in Mk (C) that commutes with I must have real entries, this gives A = Mk (R). • If I 2 = −1, then there is a basis {ei , f i } of Ck such that I ei = f i (and thus k is even). Since a matrix in Mk (C) that commutes with I must be a k/2 × k/2-matrix with quaternionic entries, we obtain A = Mk/2 (H). We now reconsider the diagrammatic classification of finite spectral triples, with real ∗-algebras represented faithfully on a Hilbert space. In fact, as far as the decomposition of H into irreducible representations is concerned, we can replace A by the complex ∗-algebra N ⎜ Mn i (C).

N N . Thus, the structure space of A is again given by A {n1 , . . , n N } where ni denotes the irreducible representation of A on Cn i . 36 3 Finite Real Noncommutative Spaces Fig. 1 A node at (ni , n◦j ) indicates the presence of the summand Cn i ∼ Cn j ◦ in H ; the double node indicates the presence of (Cn i ∼ Cn i ◦ ) ⊕ (Cn i ∼ Cn i ◦ ) in H n1 ni nj nN n1 ni nj nN The Hilbert space: As before, the irreducible, faithful representations of A = ⎛ N i=1 Mn i (C) are given by corresponding direct sums: N ⎜ Cn i i=1 on which A acts by left block-diagonal matrix multiplication.

Proof The representation π allows to consider A as a real ∗-subalgebra of Mdim H (C), hence A +i A can be considered a complex ∗-subalgebra of Mdim H (C). Thus A +i A is a matrix algebra, and we may restrict to the case A + i A = Mk (C) for some k ⇐ 1. Note that A ⇒ i A is a two-sided ∗-ideal in Mk (C). As such, it must be either the whole of Mk (C), or zero. In the first case, A + i A = A ⇒ i A so that A = Mk (C). If A ⇒ i A = {0}, then we can uniquely write any element in Mk (C) as a + ib with a, b ∈ A.

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Noncommutative Geometry and Particle Physics by Walter D. van Suijlekom


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