Theory of Classical Higgs Fields. I. Matter Fields

Abstract

Higgs fields are attributes of classical gauge theory on a principal bundle P → X whose structure Lie group G if is reducible to a closed subgroup H. They are represented by sections of the quotient bundle P/H → 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 P → P/H. It is essential that they admit an action of a gauge group G.Higgs fields are attributes of classical gauge theory on a principal bundle P → X if its symmetries are spontaneously broken [2,6,7]. Spontaneous symmetry breaking is a quantum phenomenon, but it is characterized by a classical background Higgs field [8]. Therefore, such a phenomenon also is considered in the framework of classical field theory when a structure Lie group G of a principal bundle P is reduced to a closed subgroup H of exact symmetries.One says that a structure Lie group G of a principal bundle P is reduced to its closed subgroup H if the following equivalent conditions hold:• a principal bundle P admits a bundle atlas with H-valued transition functions,• there exists a principal reduced subbundle of P with a structure group H.A key point is the following.Theorem 1. There is one-to-one correspondence between the reduced H-principal subbundles P h of P and the global sections h of the quotient bundle P/H → X possessing a typical fibre G/H.In classical field theory, global sections of a quotient bundle P/H → X are treated as classical Higgs fields [2,7].In general, there is topological obstruction to reduction of a structure group of a principal bundle to its subgroup. In particular, such a reduction occurs if the quotientGiven a principal bundle P → X whose structure group is reducible to a closed subgroup H, one meets a problem of description of matter fields which admit only an exact symmetry subgroup H. Here, we aim to show that they adequately are represented by sections of the composite bundle Y (5) which is associated to an H-principal bundle P → P/H. A key point is that Y also is a P -associated bundle (Theorem 3), and it admits the action (15) of a gauge group G. In the case of a pseudo-orthogonal group G = SO(1, m) and its maximal compact subgroup H = SO(m), we obtain this action in the explicit form (24).Forthcoming Part II of our work is devoted to Lagrangians of these matter fields and Higgs fields.

Classical higgs fields

“…The following fact is essential when constructing gauge theory with spontaneous symmetry breaking [13,25]. Theorem 14.…”

Theory of Classical Higgs Fields. III. Metric-affine gauge theory

“…of a frame bundle LX such that the corresponding local sections z h ι of LX take their values into a reduced subbundle L g X, and the transition functions of Ψ h (14) between the frames {h a } are SO(m, n − m)-valued. The frames (14):…”

“…Matter fields in metric-affine gauge theory are spinor fields, e.g., Dirac's fermion fields in gravitation theory, we restrict or consideration to gravitation theory where they are spinor fields [2,13]. They do not admit general covariant transformations, and we therefore follow the procedure in our previous works [14,15] in order to describe them. Spinors are conventionally described in the framework of formalism of Clifford algebras [8].…”

“…In this framework, classical gauge theory is theory of principal connections on principal and associated bundles. Following our previous work [14,15], we consider classical gauge theory with spontaneous symmetry breaking on a principal bundle P → X whose structure group G is reducible to a closed subgroup H, and sections of the quotient bundle P/H → X are treated as classical Higgs fields [2,11,16]. In this theory, matter fields with an exact symmetry group H are described by sections of a composite bundle Y → P/H → X.…”

Composite bundles in Clifford algebras. Gravitation theory. Part I

“…These sections are treated as classical Higgs fields [4,16,22]. Matter fields possessing only exact symmetry group H are described in a pair with Higgs fields as sections of composite bundles Y → Σ → X [20,22]. This is just the case of Dirac spinor fields in gravitation theory [4,13,18].…”