Spontaneous symmetry breaking Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or ...

, a vacuum

Higgs field The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the excited state, quantum excitation of the Higgs field, one of the field (physics), fields in particl ...

, and its associated

fundamental particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions (quarks, leptons, antiqu ...

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 In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other be ...

of generating mass of elementary particles. At the same time, classical gauge theory admits comprehensive geometric formulation where gauge fields are represented by connections on

principal bundle In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...

s. In this framework, spontaneous symmetry breaking is characterized as a

reduction of the structure group In differential geometry, a ''G''-structure on an ''n''-manifold ''M'', for a given structure group ''G'', is a principal ''G''- subbundle of the tangent frame bundle F''M'' (or GL(''M'')) of ''M''. The notion of ''G''-structures includes vario ...

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/G\to X

. This section is treated as a classical Higgs field. A key point is that there exists a composite bundle

P\to P/G\to X

where

P\to P/G

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/G\to X

, where

E\to P/G

is some

associated bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces wit ...

P\to P/G

. Herewith, a

Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...

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/G

, but not

P\to X

. An example of a classical Higgs field is a classical gravitational field identified with a

pseudo-Riemannian metric In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...

on a

world manifold In gravitation , gravitation theory, a world manifold endowed with some Lorentzian manifold, Lorentzian pseudo-Riemannian manifold, pseudo-Riemannian metric and an associated space-time structure is a spacetime, space-time. Gravitation theory is f ...

X

. In the framework of

gauge gravitation theory In quantum field theory, gauge gravitation theory is the effort to extend Yang–Mills theory, which provides a universal description of the fundamental interactions, to describe gravity. ''Gauge gravitation theory'' should not be confused with th ...

, 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,\mathbb R)

Bibliography

* * * * *

External links

* G. Sardanashvily, Geometry of classical Higgs fields, Int. J. Geom. Methods Mod. Phys. 3 (2006) 139; . Theoretical physics Gauge theories Symmetry {{quantum-stub

See also

Bibliography

External links