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

Sunday, 6 November 2011

II. How we developed gauge gravitation theory (from my Scientific Biography)

Proposed by D.Ivanenko and me in the early 80-ies, gauge gravitation theory, where a metric gravity has been described as a Higgs field, had the disadvantage that it is not defined gauge transformations of gravitation theory. This question was discussed [52]. Since gauge gravitation evidently should include Einstein’s General Relativity, its  gauge tsymmetries are general covariant transformations. However, there was no clarity in the definition of general covariant transformations. The answer was found in the framework of fibre bundle formalism, too. These transformations characterize the so-called natural bundles.

Let us restrict ourselves to one-parameter groups of transformations and their infinitesimal generators, which are vector fields. Let Y->X be a fibre bundle. Generators of one-parameter groups of diffeomorphisms of its base X are vector fields on X. Such a vector field can give rise to a vector field on Y in a different way, e.g., by means of connection on Y->X. However, such a lift u->u’, in general, is not functorial, i.e., it is not a homomorphism of a Lie algebra T(X) of vector fields on X to a Lie algebra T(Y) of vector fields on Y since the commutator [u',v'] need not be equal a lift [u,v]' of the commutator of vector felds u and v on X. However, there are fibre bundles which allow a functorial lift Fu of vector fields u on a base X, so that the above-mentioned homomorphism of a Lie algebra T(X) to T(Y) holds. These bundles are called natural. These include tangent TX and cotangent T*X bundles over X, their tensor  products, a linear frame bundle LX and all associated bundles, but not only. A functorial lift Fu on a natural bundle Y of vector fields on its base X, by definition, are generators of one-parameter groups of general covariant transformations of Y. Thus, gravitation theory must be built as classical field theory on natural bundles [73,77,80,98]. In particular, this implies the following.

A group of general covariant transformations is a subgroup of the group of automorphisms Aut(LX) of a linear frame bundle LX. However, Lagrangians of gravitation theory, in particular, a Lagrangian of General Relativity are invariant only under general covariant transformations, but not general frame transformation from Aut(LX). Therefore, a gravitational field (pseudo-Riemannian metric), in contrast to  Higgs fields in gauge theory of internal symmetries, is not brought to the Minkowski metric by gauge transformations and, therefore, it is a dynamic variable.

An energy-momentum current in gravitation theory is a current symmetry along a functorial lift Fu of vector fields u on X. It leads to a generalized Komar energy-momentum superpotential [73,77].

Spinor bundles are not natural, and they do not admit general covariant transformations. Therefore, a question arises about description of Dirac fermion fields in gravitation theory. Because these fields admit only Lorentz transformations, there is a situation of spontaneous symmetry breaking. In this case, a spinor field is described only in a pair with a certain gravitation field g, namely, by sections of a spinor bundle S^g associated with a reduced subbundle L^gX of a linear frame bundle LX. Then, in accordance with a general scheme of description of spontaneous symmetry breaking in classical field theory, all the spinor and gravitational fields are represented by sections of a composite bundle S->LX/SO(1,3)->X, where S->LX/SO(1,3) is a spinor bundle associated with LX->LX/SO(1,3) [80,81]. In particular, a fibre bundle S->X  is natural, and  energy-momentum current of spinor fields can be defined.

The Higgs nature of a gravitational field is clarified by the fact that, for different gravitational fields g and g', spinor bundles S^g and S^g’ are not equivalent, because the representation of tangent covectors by Dirac matrices and, consequently, the Dirac operators are not equivalent.

G.Sardanashvily My Scientific Biography

No comments:

Post a Comment