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

Wednesday, 2 November 2011

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

… Enrolling in a graduate school in 1973, I among other things addressed to gauge gravitation theory. This direction was developed in Ivanenko’s group in the early 1960s, but then subsided with the departure of G. Sokolik, though continued to be discussed at the seminar of Ivanenko because it led to theory of gravity with torsion that Ivanenko engaged in.

By that time it became clear that gauge theory was adequately formulated in the formalism of fibre bundles, although a comprehensive formulation appeared later in the two articles: M. Daniel and C. Viallet in Reviews of Modern Physics and T. Eguchi, P. Gilkey and A. Hanson in Physics Reports in 1980. I therefore actively engaged in the study of differential geometry with the help of the translation of the book R.Sulanke and P.Wintgen, "Differential geometry and Fibre Bundles" which was released in 1975. The well-known two volumes of S.Kobayashi and K.Nomizu in the Russian translation appeared only in 1981. Simultaneously, I learned general topology on the books of Bourbaki and K.Kuratowski.

My first article on gauge gravitation theory [18] was released in September 1974. It was the author of I, but D.Ivanenko, to be sure, brought in as my co-author B.Frolov, who previously was engaged in gauge theory of gravity. In the article already mentions fibre bundles. After three months, it was published my second article [19], where I was a sole author.

By the time, when I turned to gauge gravitation theory, the problem was already almost 20 years. In 1954, C. Yang and R. Mills proposed first gauge model for a symmetry group SU(2). And already in 1956, R. Utiyama generalized this theory for an arbitrary Lie groups of internal symmetries G, including theory of gravity as a gauge theory of the Lorentz group. It is natural to assume that gauge gravitation theory should contain Einstein’s General Relativity. In General Relativity, a gravitational field is identified with a pseudo-Riemannian metric, and its symmetries are general covariant transformations. However, the difficulty was with the status of pseudo-Riemannian metrics and general covariant transformations, which have no analogue in the Yang – Mills gauge scheme because gauge fields are connections on a fibre bundle Y->X with a structure group G, and gauge transformations are vertical automorphisms of Y projected onto the identity map of X. General covariant transformations are not so. To overcome these difficulties in the work of Utiyama, in the beginning of 60-s T. Kibbl, D. Sciama et al. have proposed to treat gravity, represented by a tetrad field, as a gauge field for a translation group. All the same, it is beyond the scope of Yang - Mills – Utiyama gauge theory for internal symmetries, as evidenced not identical morphism of a base X of tensor bundles. I, too, began with this model, but soon withdrew from it, because it did not fit into fibre bundle formalism. Almost four years I was ineffectual, fiddling with the other options, until I came to interpretation of gravitation as a Higgs field, which was first described in my article [22] in 1978 .

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) 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.

One of sections of the above mentioned book "Differential geometry and Fibre Bundles" by R.Sulanke and P.Wintgen was devoted to the so-called G-structures, when a structure group of a principal frame bundle LX over a manifold X is reduced to its closed subgroup H. In a general case of an arbitrary principal bundles P with a structure group Lie G, a construction of the structure group reduction was described in the book "The Topology of Fibre Bundles" by N.Steenrod in 1953, which I found in the library of the Mathematical Faculty. The well-known theorem states that such reduction takes place if and only if there is a global section h of a factor bundle P/H->X. Since this section takes values in a factor-space G/H, one can treat it as a classical Higgs (or Goldstone) field. If P=LX and H is the Lorentz group SO(1,3), then h is a global section of  LX/SO(1,3) which is a pseudo-Riemannian metric on a manifold X. Therefore, I concluded that a pseudo-Riemannian metric, i.e., a gravitational field has the status of a Higgs field in gauge gravitation theory. This result was published in my report on the 8-th International gravitational conference in Canada in 1977, and the article [26].

D.Ivanenko liked such interpretation of gravity because even in the middle of the 60's he supposed that a gravitational field can be the Goldstone one by its physical nature due to breakdown of space-time symmetries caused by a curvature. However, such a symmetry breaking (and, consequently, the Higgs nature of a gravitational field) did not result from the gauge principle, and it should be lead from a principled basis. And I found such a principle. It is the equivalence principle, but reformulated in geometric terms.

In the above mentioned book by R.Sulanke and P.Wintgen, the G-structures were considered as a type of the Klein - Chern geometry of invariants, namely: if a structure group G is reduced to its subgroup H, then there is a bundle atlas of this fibre bundle with H-valued transition functions and, therefore, H-invariants on this fibre bundle are defined. At that time, the equivalence principle in gravitation theory, its different variants (weakest, weak, middle-strong, strong, etc.) were not once discussed o the seminar of Ivanenko. All of these variants were too physical for its language, to become as a basis for mathematical formulation of gauge gravitation theory. They characterize the possibility of transition to Special Relativity with respect to some reference frame. Describing Special Relativity as geometry of invariants of the Lorentz group, I came to an idea to formulate the equivalence principle in the spirit of geometry of invariants as a requirement of the existence of Lorentz invariants in some reference frame. This in turn implies a reduction of a structure group of the frame bundle LX over a manifold X to the Lorentz group, and, consequently, the existence of a gravitational field on X [28,29,31]. This geometric equivalence principle has summed up the foundation under our interpretation of gravity as a Higgs field in gauge gravitation theory. Gauge theory of gravitation was as a whole formed. It was a affine-metric theory whose dynamic variables were a pseudo-Riemannian metric as a Higgs field and general linear connections as a gauge field. D.Ivanenko and I published the review [35] in Physics Reports in 1983, which is traditionally quoted among the fundamental works on gauge gravitation theory. Our proposed gauge model of gravity also was present in the books [2,8].

Our version of the gauge theory of gravity was seen, nobody denied it, but it did not became widely recognized. Theoreticians do not hurry to refuse the treatment of a gravitational field as a gauge field of translations. Although still in 1982, I published an article [34] which specifically argued in bundle formalism that identification of tetrad fields with the so-called soldering form (a translational part of a general affine connection) is a mathematical mistake.

Therefore, I began investigating a possible physical interpretation of translation components of an affine connection. I knew the book “A Gauge Theory of Dislocations and Disclinations” by A. Kadic and D. Edelen published in 1983 (its Russian translation appeared in 1987), where gauge fields of translations on a 3-dimensional manifold described dislocations in continuum medium theory. Based on this result, I developed a model where a translational part of an affine connection on a 4-dimensional manifold described a new hypothetical structure: a kind of deformations of a world manifold [45,47]. In particular, they could be responsible for an additional Yukawa term to the Newton gravitation potential: the so-called "fifth force" [58]. At that time, such an amendment was actively investigated, but as a result, at least at laboratory distances nothing was found.

Geometric equivalence principle determines not only the existence of a gravitational field on a manifold, but a space-time structure on it. The point is that, if a structure group of a frame bundle LX is reduced to a Lorentz group (let g be the corresponding gravitational field), it is always reduced to its maximal compact subgroup SO(3). The associated Higgs field is a 3-dimensional space-like subbundle F of the tangent bundle TX of a manifold X, which defines a space-time decomposition of TX, i.e. a space-time structure on X. If a subbundle F is involutive, we have a space-time foliation of X associated with a gravitational field g. Hence, I had an idea to describe gravitational singularities as those of space-time foliations because the most recognized criterion of gravitational singularities by the so-called b-incompleteness of geodetics had a number of disadvantages [32,33]. It was given a classification of the singularities of space-time foliations, including a violation of causality, topological transitions through critical points, caustics of foliations [39,43,48]. However, this way of describing gravitational singularities also is not ideal. For example, caustic of space-time foliation can take place in the case of a regular gravitational field.
One of the most actively developed generalizations of gravitation theory of gravity is supergravitation. However, it largely built as a generalization of gauge theory of the Poincare group by extending its Lie algebra to some superalgebra. Obtained in this approach, Higgs superfields treated as a supergravity field do not have a geometrical nature. Therefore, I suggested that on should develop theory of supergravity as a supermetric on a supermanifold, introducing it from the condition of reduction of a structure supergroup of an appropriate superbundle. This was done in the framework of existed then formalism of supermanifolds [41,42]. I returned to this subject almost ten years later, already on the other mathematical level.

And, after all, gauge gravitation theory was not completed. Firstly, it remained unclear physical background of the geometric equivalence principle which looked formal. Secondly, it was unclear what are gauge transformations in gauge gravitation theory. In Einstein's General Relativity, which gauge gravitation theory should include, they are general covariant transformations. However, what are these transformations in fibre bundle formalism?

It took almost another 10 years and more advanced mathematical apparatus to all fell into place.

G.Sardanashvily, My Scientific Biography

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