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[[Spontaneous symmetry breaking]], a vacuum [[Higgs field]], a [[Higgs boson]] are quantum phenomena. A vacuum '''Higgs field''' is responsible for spontaneous symmetry breaking the [[gauge theory|gauge symmetries]] of fundamental interactions and provides the [[Higgs mechanism]] of generating mass of elementary particles. However, no adequate mathematical model of this Higgs vacuum has been suggested in the framework of [[gauge theory|quantum gauge theory]], though somebody treats it as ''sui generis'' a condensate by analogy with that of [[Cooper pair]]s in [[condensed matter physics]].

At the same time, [[gauge theory|classical gauge theory]] admits comprehensive geometric formulation where [[gauge theory|gauge fields]] are represented by [[connection (principal bundle)|connections]] on [[principal bundle]]s. In this framework, spontaneous symmetry breaking is characterized as a [[reduction of the structure group]] <math>G</math> of a principal bundle <math>P\to X</math> to its closed subgroup <math>H</math>. By the well-known theorem, such a reduction takes place if and only if there exists a global section <math>h</math> of the quotient bundle <math>P/G\to X</math>. This section is treated as a '''classical Higgs field'''.

A key point is that there exists a composite bundle <math>P\to P/G\to X</math> where <math>P\to P/G</math> is a principal bundle with the structure group <math>H</math>. Then matter fields, possessing an exact symmetry group <math>H</math>, in the presence of classical Higgs fields are described by sections of some [[Connection (composite bundle)|composite bundle]] <math>E\to P/G\to X</math>, where <math>E\to P/G</math> is some [[associated bundle]] to <math>P\to P/G</math>. Herewith, a [[Lagrangian system|Lagrangian]] of these matter fields is gauge invariant only if it factorizes through the vertical covariant differential of some connection on a principal bundle <math>P\to P/G</math>, but not <math>P\to X</math>.

An example of a classical Higgs field is a classical [[gravitational field]] identified with a [[pseudo-Riemannian manifold|pseudo-Riemannian metric]] on a [[world manifold]] <math> X</math>. In the framework of [[gauge gravitation theory]], it is described as a global section of the quotient bundle <math>FX/O(1,3)\to X</math> where <math>FX</math> is a principal bundle of the tangent frames to <math>X</math> with the structure group <math>GL(4,\mathbb R)</math>.

== References ==

* [[Dmitri Ivanenko|D. Ivanenko]] and [[Sardanashvily|G. Sardanashvily]], The gauge treatment of gravity, Phys. Rep. '''94''' (1983) 1.
*A. Trautman, ‘’Differential Geometry for Physicists’’’ (Bibliopolis, Naples, 1984).
*L. Nikolova and V. Rizov, V. (1984). Geometrical approach to the reduction of gauge theories with spontaneous broken symmetries, Rep. Math. Phys. '''20''' (1984) 287.
* M. Keyl, About the geometric structure of symmetry breaking, J. Math. Phys. '''32''' (1991) 1065.
* G. Giachetta, L. Mangiarotti and [[Gennadi Sardanashvily|G. Sardanashvily]], "Advanced Classical Field Theory", World Scientific, 2009, ISBN 978-981-283-895-7.

==External links==
* [[Gennadi Sardanashvily|G. Sardanashvily]], Geometry of classical Higgs fields, Int. J. Geom. Methods Mod. Phys. '''3''' (2006) 139; [http://xxx.lanl.gov/abs/hep-th/0510168 arXiv: hep-th/0510168].

== See also ==

*[[Spontaneous symmetry breaking]]
*[[Higgs boson]]
*[[Reduction of the structure group]]
*[[Gauge gravitation theory]]

[[Category:Theoretical physics]]
[[Category:Gauge theories]]
[[Category:Symmetry]]

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en>Wcherowi: refactored - see talk page