Higgs field (classical) explained

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 symmetries of fundamental interactions and provides the Higgs mechanism of generating mass of elementary particles.

G of a principal bundle P\to X to its closed subgroup H . By the well-known theorem, such a reduction takes place if and only if there exists a global section h of the quotient bundle P/H\to X . This section is treated as a classical Higgs field.

A key point is that there exists a composite bundle

P\to P/H\to X where P\to P/H is a principal bundle with the structure group H . Then matter fields, possessing an exact symmetry group H , in the presence of classical Higgs fields are described by sections of some composite bundle E\to P/H\to X , where E\to P/H is some associated bundle to P\to P/H . Herewith, a Lagrangian of these matter fields is gauge invariant only if it factorizes through the vertical covariant differential of some connection on a principal bundle P\to P/H , but not P\to X .X . In the framework of gauge gravitation theory, it is described as a global section of the quotient bundle FX/O( 1 , 3 )\to X where FX is a principal bundle of the tangent frames to X with the structure group GL( 4 , R) .

See also

• Gauge gravitation theory
• Reduction of the structure group
• Spontaneous symmetry breaking

Bibliography

• D. . Ivanenko . Dmitri Ivanenko . G. . Sardanashvily . Gennadi Sardanashvily . 1983 . The gauge treatment of gravity . Phys. Rep. . 94 . 1 . 1. 10.1016/0370-1573(83)90046-7 . 1983PhR....94....1I .
• Book: Trautman , A. . Differential Geometry for Physicists . Bibliopolis . Naples, IT . 1984.
• L. . Nikolova . V. . Rizov . 1984 . Geometrical approach to the reduction of gauge theories with spontaneous broken symmetries . Rep. Math. Phys. . 20 . 287. 10.1016/0034-4877(84)90039-9 .
• M. . Keyl . 1991 . About the geometric structure of symmetry breaking . J. Math. Phys. . 32 . 4 . 1065. 10.1063/1.529385 . 1991JMP....32.1065K .
• Book: G. . Giachetta . L. . Mangiarotti . G. . Sardanashvily . Gennadi Sardanashvily . 2009 . Advanced Classical Field Theory . World Scientific . 978-981-283-895-7.

External links

• G. Sardanashvily, Geometry of classical Higgs fields, Int. J. Geom. Methods Mod. Phys. 3 (2006) 139; .