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(Bosonic)mass meets (extrinsic)curvature

✍ Scribed by Jürgen Tolksdorf


Book ID
104343221
Publisher
Elsevier Science
Year
2003
Tongue
English
Weight
179 KB
Volume
48
Category
Article
ISSN
0393-0440

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✦ Synopsis


In this paper, we discuss the mechanism of spontaneous symmetry breaking from the point view of vacuum pairs, considered as ground states of a Yang-Mills-Higgs gauge theory. We treat a vacuum as a section in an appropriate bundle that is naturally associated with a minimum of a (general) Higgs potential. Such a vacuum spontaneously breaks the underlying gauge symmetry if the invariance group of the vacuum is a proper subgroup of the gauge group. We show that each choice of a vacuum admits to geometrically interpret the bosonic mass matrices as "normal" sections. The spectrum of these sections turns out to be constant over the manifold and independent of the chosen vacuum. Since the mass matrices commute with the invariance group of the chosen vacuum one may decompose the Hermitian vector bundles which correspond to the bosons in the eigenbundles of the bosonic mass matrices. This decomposition is the geometrical analog of the physical notion of a "particle multiplet". In this sense, the basic notion of a "free particle" also makes sense within the geometrical context of a gauge theory, provided the gauge symmetry is spontaneously broken by some vacuum.

We also discuss the Higgs-Kibble mechanism ("Higgs Dinner") from a geometrical point of view. It turns out that the "unitary gauge", usually encountered in the context of discussing the Higgs Dinner, is of purely geometrical origin. In particular, we discuss rotationally symmetric Higgs potentials and give a necessary and sufficient condition for the unitary gauge to exist. As a specific example, we discuss in some detail the electroweak sector of the Standard Model of particle physics in this context.


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(Fermionic) mass meets (intrinsic) curva
✍ Jürgen Tolksdorf 📂 Article 📅 2003 🏛 Elsevier Science 🌐 English ⚖ 104 KB

Using the notion of vacuum pairs we show how the (square of the) mass matrix of the fermions can be considered geometrically as curvature. This curvature together with the curvature of space-time, defines the total curvature of the Clifford module bundle representing a "free" fermion within the geom