Higgs field (classical)

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

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