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

Friday, 25 October 2013

Geometry of the composite bundles (from my Scientific Biography)

"The jet formalism, when I first met it, was quite developed in application to theory of differential operators and differential equations, differential geometry, and also, as I have already mentioned, in main aspects to Lagrangian formalism. It seemed that I as a theoretician should only apply it to the particular field models: gauge theory, gravitation theory, etc. However, I had to do develop a number of its basic issues: geometry of composite bundles, Lagrangian theory in formalism of a variational bicomplex, Noether identities and the second Noether theorem.

A composition of fibre bundles Y->S->X is called the composite bundle. They arise in a number of models of field theory and mechanics. In mechanics, these are models with parameters described by sections of a fibre bundle S->X. In field theory, they are systems with a background field and models with spontaneous symmetry breaking, e.g., gravitation theory, when sections of  a fibre bundle S->X are Higgs fields. A key point is that, if h is a section of a fibre bundle S->X, then the restriction of Y->S to a submanifold h(X) of S is a subbundle h*Y->X of a fibre bundle Y->X, describing a system in the presence of a background field (or a parametric function) h(X).

Using a relation between jet manifolds of fibre bundles Y->X, Y->S and S->X, I obtained that between connections on these bundles and, most importantly, the new differential operator on sections of a fibre bundle Y->S, called the vertical covariant differential determined by a connection A on Y->S. The fact is that, being restricted to h(X), this operator coincides with the familiar covariant differential yielded by the restriction of a connection A onto h*Y->X. Thus, this vertical covariant differential should appear in description of the dynamics of field systems on a composite bundle. This result was published in 1991 in the article [64] and was already used in the book [9] for description of spinors in a gravitational field. Subsequently, I have used it in different models of field theory and mechanics. One of them, the key to construct the gauge gravitation theory, is classical field theory with spontaneous symmetry breaking."


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