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Register a new userTheory of Classical Higgs Fields. II. Lagrangians
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Added by
Gennady Sardanashvily
Publication date
2014
fields
Physics
and research's language is
English
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We consider classical gauge theory with spontaneous symmetry breaking on a principal bundle $Pto X$ whose structure group $G$ is reducible to a closed subgroup $H$, and sections of the quotient bundle $P/Hto X$ are treated as classical Higgs fields. In this theory, matter fields with an exact symmetry group $H$ are described by sections of a composite bundle $Yto P/Hto X$. We show that their gauge $G$-invariant Lagrangian necessarily factorizes through a vertical covariant differential on $Y$ defined by a principal connection on an $H$-principal bundle $Pto P/H$.
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Higgs fields are attributes of classical gauge theory on a principal bundle $Pto X$ whose structure Lie group $G$ if is reducible to a closed subgroup $H$. They are represented by sections of the quotient bundle $P/Hto X$. A problem lies in description of matter fields with an exact symmetry group $H$. They are represented by sections of a composite bundle which is associated to an $H$-principal bundle $Pto P/H$. It is essential that they admit an action of a gauge group $G$.
We consider classical gauge theory on a principal bundle P->X in a case of spontaneous symmetry breaking characterized by the reduction of a structure group G of P->X to its closed subgroup H. This reduction is ensured by the existence of global sections of the quotient bundle P/H->X treated as classical Higgs fields. Matter fields with an exact symmetry group H in such gauge theory are considered in the pairs with Higgs fields, and they are represented by sections of a composite bundle Y->P/H->X, where Y->P/H is a fiber bundle associated to a principal bundle P->P/H with a structure group H. A key point is that a composite bundle Y->X is proved to be associated to a principal G-bundle P->X. Therefore, though matter fields possess an exact symmetry group H, their gauge G-invariant theory in the presence of Higgs fields can be developed. Its gauge invariant Lagrangian factorizes through the vertical covariant differential determined by a connection on a principal H-bundle P->P/H. In a case of the Cartan decomposition of a Lie algebra of G, this connection can be expressed in terms of a connection on a principal bundle P->X, i.e., gauge potentials for a group of broken symmetries G.
Higgs fields on gauge-natural prolongations of principal bundles are defined by invariant variational problems and related canonical conservation laws along the kernel of a gauge-natural Jacobi morphism.
We consider classical gauge theory with spontaneous symmetry breaking on a principal bundle $Pto X$ whose structure group $G$ is reducible to a closed subgroup $H$, and sections of the quotient bundle $P/Hto X$ are treated as classical Higgs fields. Its most comprehensive example is metric-affine gauge theory on the category of natural bundles where gauge fields are general linear connections on a manifold $X$, classical Higgs fields are arbitrary pseudo-Riemannian metrics on $X$, and matter fields are spinor fields. In particular, this is the case of gauge gravitation theory.
We consider the variation of the surface spanned by closed strings in a spacetime manifold. Using the Nambu-Goto string action, we induce the geodesic surface equation, the geodesic surface deviation equation which yields a Jacobi field, and we define the index form of a geodesic surface as in the case of point particles to discuss conjugate strings on the geodesic surface.
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