The God has created a man in order that he creates that the God fails to do

Monday, 5 September 2011

What are classical Higgs fields?

In the 70-s, in field theory, it has already been folklore that spontaneous symmetry breaking is accompanied by Higgs and Goldstone fields, that follows from the theorem of Goldstone in quantum theory, the method of nonlinear realizations of groups (particular case of induced representations), and that provides the Higgs mechanism of generation of masses of particles in united gauge model of fundamental interactions. Spontaneous symmetry breaking is a quantum effect, when a vacuum (or a background state) is fails to be invariant under a whole group of transformations, but only a subgroup of exact symmetries. A problem is how to describe spontaneous symmetry breaking in classical gauge theory. This is necessary because a generating functional for Green functions of quantum fields is expressed through a Lagrangian of classical fields, and it contains classical Higgs fields. Classical gauge theory was described in terms on fibre bundles, and it naturally raised a question what is Higgs field in this formalism.

In classical gauge theory on a principal bundle P->X with a structure Lie group G, spontaneous symmetry breaking is characterized  as a  reduction of a structure group G to its closed (and, consequently, Lie) subgroup H. This means that there is an atlas of a principal bundle P and associate bundles with H-valued transition functions or, equivalently, that there is a principal subbundle P' of P with a structure group H. Then there may exist a fibre bundle Y->X associated with P', whose typical fibre V admits no action of a group G, but only its subgroup H. Section of this fibre bundle describe  matter fields in a situation of a breakdown of symmetries with a group G to a subgroup H of exact symmetries.

A key point is that, by the well-known theorem, reduction of a structure group G to a subgroup H occurs if and only if there exists a global section h of a factor-bundle with a typical fibre G/H. Since this section takes values in a factor-space G/H, one can treat it as a classical Higgs field.

Moreover, there is one-to-one correspondence between such sections h and the H-principal subbundles P[h] of P. Let Y[h]->X be a fibre bundle associated with P[h]. Then its sections s describe matter fields with an exact symmetry group H in the presence of a Higgs field h. A problem, however, is that, for different Higgs fields h, fibre bundles Y[h]->X need not be equivalent. Therefore, matter field s with an exact symmetry group H must be considered only in a pair with a certain Higgs field h. Of course, a question arises, how to describe a totality of matter fields with broken symmetry and Higgs fields.

To do this, one can consider a composite bundle P->P/H->X, where P->P/H is a principal bundle with a structure group H, and a fibre bundle Y->P/H associated with P->P/H, with a typical fibre V. Then section of a composite bundle P->P/H->X describe a desired totality of matter and Higgs fields in a case of spontaneous symmetry breaking. Indeed, in accordance with the above-mentioned properties of composite bundles, the restriction of a fibre bundle Y->P/H to a submanifold h(X) of P/H is exactly a fibre bundle Y[h]->X.

In particular, let X be a 4-dimensional world manifold, and let P=LX be a fibre bundle of linear frames in the tangent bundle TX of X. Its structure group is GL(4,R). By virtue of the geometric equivalence principle (), this structure group is reduced to the Lorentz group H= SO(1,3). Then a global section h of the factor-bundle LX/SO(1,3) is a pseudo-Riemannian metric, i.e., a gravitational field on a manifold X. Thus, a gravitational field exemplifies a classical Hiigs field.


G.Giachetta, L.Mangiarotti, G.Sardanashvily, Advanced Classical Field Theory (WS, 2009)
G.Sardanashvily, Geometry of classical Higgs fields, Int. J. Geom. Methods Mod. Phys. 3 (2006) 139-148.

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