Principal bundle formulation of the Higgs field
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.
At the same time, classical gauge theory admits comprehensive geometric formulation where gauge fields are represented by connections on principal bundles. In this framework, spontaneous symmetry breaking is characterized as a reduction of the structure group formula_1 of a principal bundle formula_2 to its closed subgroup formula_3. By the well-known theorem, such a reduction takes place if and only if there exists a global section formula_4 of the quotient bundle formula_5. This section is treated as a classical Higgs field.
A key point is that there exists a composite bundle formula_6 where formula_7 is a principal bundle with the structure group formula_3. Then matter fields, possessing an exact symmetry group formula_3, in the presence of classical Higgs fields are described by sections of some composite bundle formula_10, where formula_11 is some associated bundle to formula_7. 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 formula_7, but not formula_2.
An example of a classical Higgs field is a classical gravitational field identified with a pseudo-Riemannian metric on a world manifold formula_15. In the framework of gauge gravitation theory, it is described as a global section of the quotient bundle formula_16 where formula_17 is a principal bundle of the tangent frames to formula_18 with the structure group formula_19.