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Noncommutative geometry and particle physics
Suijlekom, Walter D. van.,
اطلاعات کتابشناختی
Noncommutative geometry and particle physics
Author :
Suijlekom, Walter D. van.,
Publisher :
Springer,
Pub. Year :
2015
Subjects :
Mathematical physics. Noncommutative differential geometry. Particles (Nuclear physics)...
Call Number :
QC 20 .S83 2015
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Preface
(5)
Contents
(7)
Abbreviations and Symbols
(11)
1 Introduction
(17)
References
(21)
Part INoncommutative Geometric Spaces
(22)
2 Finite Noncommutative Spaces
(23)
2.1 Finite Spaces and Matrix Algebras
(23)
2.1.1 Commutative Matrix Algebras
(27)
2.1.2 Noncommutative Matrix Algebras
(27)
2.2 Noncommutative Geometric Finite Spaces
(32)
2.2.1 Morphisms Between Finite Spectral Triples
(37)
2.3 Classification of Finite Spectral Triples
(39)
References
(43)
3 Finite Real Noncommutative Spaces
(45)
3.1 Finite Real Spectral Triples
(45)
3.1.1 Morphisms Between Finite Real Spectral Triples
(47)
3.2 Classification of Finite Real Spectral Triples
(49)
3.3 Real Algebras and Krajewski Diagrams
(56)
3.4 Classification of Irreducible Geometries
(59)
References
(61)
4 Noncommutative Riemannian Spin Manifolds
(62)
4.1 Clifford Algebras
(62)
4.1.1 Representation Theory of Clifford Algebras
(66)
4.2 Riemannian Spin Geometry
(68)
4.2.1 Spin Manifolds
(69)
4.2.2 Spin Connection and Dirac Operator
(72)
4.2.3 Lichnerowicz Formula
(76)
4.3 Noncommutative Riemannian Spin Manifolds: Spectral Triples
(77)
4.3.1 Commutative Subalgebra
(83)
References
(86)
5 The Local Index Formula in Noncommutative Geometry
(88)
5.1 Local Index Formula on the Circle and on the Torus
(88)
5.1.1 The Winding Number on the Circle
(88)
5.1.2 The Winding Number on the Torus
(90)
5.2 Hochschild and Cyclic Cohomology
(94)
5.3 Abstract Differential Calculus
(98)
5.4 Residues and the Local (b,B)-Cocycle
(103)
5.5 The Local Index Formula
(106)
References
(111)
Part IINoncommutative Geometryand Gauge Theories
(113)
6 Gauge Theories from Noncommutative Manifolds
(114)
6.1 `Inner' Unitary Equivalences as the Gauge Group
(114)
6.1.1 The Gauge Algebra
(117)
6.2 Morita Self-Equivalences as Gauge Fields
(118)
6.2.1 Morita Equivalence
(118)
6.2.2 Morita Equivalence and Spectral Triples
(123)
6.3 Localization
(125)
6.3.1 Localization of Gauge Fields
(128)
References
(130)
7 Spectral Invariants
(131)
7.1 Spectral Action Functional
(131)
7.2 Expansions of the Spectral Action
(133)
7.2.1 Asymptotic Expansion
(133)
7.2.2 Perturbative Expansion in the Gauge Field
(135)
7.A Divided Differences
(140)
References
(144)
8 Almost-Commutative Manifolds and Gauge Theories
(146)
8.1 Gauge Symmetries of AC Manifolds
(146)
8.1.1 Unimodularity
(148)
8.2 Gauge Fields and Scalar Fields
(150)
8.2.1 Gauge Transformations
(152)
8.3 The Heat Expansion of the Spectral Action
(153)
8.3.1 A Generalized Lichnerowicz Formula
(153)
8.3.2 The Heat Expansion
(157)
8.4 The Spectral Action on AC Manifolds
(160)
References
(166)
9 The Noncommutative Geometry of Electrodynamics
(168)
9.1 The Two-Point Space
(168)
9.1.1 The Product Space
(169)
9.1.2 U(1) Gauge Theory
(171)
9.2 Electrodynamics
(172)
9.2.1 The Finite Space
(173)
9.2.2 A Non-trivial Finite Dirac Operator
(174)
9.2.3 The Almost-Commutative Manifold
(175)
9.2.4 The Spectral Action
(176)
9.2.5 The Fermionic Action
(177)
9.2.6 Fermionic Degrees of Freedom
(179)
9.A Grassmann Variables, Grassmann Integration and Pfaffians
(180)
References
(182)
10 The Noncommutative Geometry of Yang--Mills Fields
(184)
10.1 Spectral Triple Obtained from an Algebra Bundle
(184)
10.2 Yang--Mills Theory as a Noncommutative Manifold
(188)
10.2.1 From Algebra Bundles to Principal Bundles
(188)
10.2.2 Inner Fluctuations and Spectral Action
(189)
10.2.3 Topological Spectral Action
(192)
References
(193)
11 The Noncommutative Geometry of the Standard Model
(194)
11.1 The Finite Space
(194)
11.2 The Gauge Theory
(198)
11.2.1 The Gauge Group
(198)
11.2.2 The Gauge and Scalar Fields
(200)
11.3 The Spectral Action
(203)
11.3.1 Coupling Constants and Unification
(209)
11.3.2 The Higgs Mechanism
(210)
11.4 The Fermionic Action
(214)
References
(221)
12 Phenomenology of the Noncommutative Standard Model
(222)
12.1 Mass Relations
(222)
12.1.1 Fermion Masses
(222)
12.1.2 The Higgs Mass
(223)
12.1.3 The Seesaw Mechanism
(224)
12.2 Renormalization Group Flow
(225)
12.2.1 Coupling Constants
(225)
12.2.2 Renormalization Group Equations
(227)
12.2.3 Running Masses
(228)
12.3 Higgs Mass: Comparison to Experimental Results
(231)
12.4 Noncommutative Geometry Beyond the Standard Model
(233)
References
(237)
Subject Index
(240)
Notation Index
(244)
