Geometric aspects of the Standard Model and the mysteries of matter

doi:10.1017/CBO9781139208642.008

7 - Geometric aspects of the Standard Model and the mysteries of matter

Published online by Cambridge University Press: 05 May 2013

Edited by

Iván Contreras and

Florian Scheck: Affiliation:
Johannes Gutenberg–University
Alexander Cardona: Affiliation:
Universidad de los Andes, Colombia
Iván Contreras: Affiliation:
Universität Zürich
Andrés F. Reyes-Lega: Affiliation:
Universidad de los Andes, Colombia

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Summary

Abstract

The basic structure of gauge theories of fundamental interactions distinguishes radiation from matter. Radiation is described by Yang–Mills theories, matter particles (i.e. quarks and leptons) are described by a Dirac operator which contains the full complexity of their classification and state mixing. In quantum field theory the two categories intermingle. While the construction of Yang–Mills theories, in essence, is a classical one, the phenomenon of spontaneous symmetry breaking exhibits facets which are linked to the quantum symmetries of gauge theories. Furthermore, noncommutative geometry offers new routes to the standard model of electroweak interactions and reveals some of its otherwise mysterious structure.

We review these matters with regard to both their phenomenology and their theoretical and geometric background. Many examples are given and exercises are provided which illustrate some of the main results.

Radiation and matter in gauge theories and General Relativity

The basic structure of gauge theories seems to distinguish radiation from matter as two categories of different origin. The massive and massless vector or tensor bosons, the photon, the W±- and Z0-bosons, the gluons, and the graviton, respectively, which are the carriers of the fundamental forces, belong to what may be termed radiation. Here we allude to the analogy to (quantum) electrodynamics described by Maxwell's equations and to Einstein's equations for General Relativity (GR). They are described by geometric theories, i.e. Yang–Mills (YM) theories or, in the case of GR, by semi-Riemannian geometry in dimension 4. To a large extent, they are classical theories.

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Type: Chapter
Information: Geometric and Topological Methods for Quantum Field Theory
Proceedings of the 2009 Villa de Leyva Summer School
, pp. 274 - 306

DOI: https://doi.org/10.1017/CBO9781139208642.008 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2013

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