View source for Higgs field (classical)

{{Short description|Principal bundle formulation of the Higgs field}}
{{See also|Higgs boson}}
[[Spontaneous symmetry breaking]], a vacuum [[Higgs field]], and its associated [[fundamental particle]] the 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.

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/H\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/H\to X</math> where <math>P\to P/H</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/H\to X</math>, where <math>E\to P/H</math> is some [[associated bundle]] to <math>P\to P/H</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/H</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>.

== See also ==
* [[Gauge gravitation theory]]
* [[Reduction of the structure group]]
* [[Spontaneous symmetry breaking]]
== Bibliography ==
* {{cite journal
 |first1=D. |last1=Ivanenko       |author1-link=Dmitri Ivanenko   
 |first2=G. |last2=Sardanashvily  |author2-link=Gennadi Sardanashvily
 |year=1983
 |title=The gauge treatment of gravity
 |journal=Phys. Rep.
 |volume=94 |issue=1 |page=1
|doi=10.1016/0370-1573(83)90046-7 |bibcode=1983PhR....94....1I }}

* {{cite book
 |first=A. |last=Trautman
 |title=Differential Geometry for Physicists
 |publisher=Bibliopolis
 |place=Naples, IT
 |year=1984
}}

* {{cite journal
 |first1=L. |last1=Nikolova 
 |first2=V. |last2=Rizov
 |year=1984
 |title=Geometrical approach to the reduction of gauge theories with spontaneous broken symmetries
 |journal=Rep. Math. Phys.
 |volume=20 |page=287
|doi=10.1016/0034-4877(84)90039-9 
 }}

* {{cite journal
 |first=M. |last=Keyl
 |year=1991
 |title=About the geometric structure of symmetry breaking
 |journal=J. Math. Phys. 
 |volume=32 |issue=4
 |page=1065
|doi=10.1063/1.529385
 |bibcode=1991JMP....32.1065K
 }}

* {{cite book
 |first1=G. |last1=Giachetta
 |first2=L. |last2=Mangiarotti 
 |first3=G. |last3=Sardanashvily |author3-link=Gennadi Sardanashvily
 |year=2009
 |title=Advanced Classical Field Theory
 |publisher=World Scientific
 |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; {{arXiv|hep-th/0510168}}.

{{theoretical-physics-stub}}
{{quantum-stub}}

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