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

Friday, 21 December 2012

Jet manifold formalism (from my Scientific Biography)

My Scientific BiographyFourth period (1990 - 1999)

In autumn of 1987, in the framework of scientific cooperation between Moscow State University and University of Camerino (Italy) professor Luigi Mangiarotti arrived in Moscow. He made a report at the seminar of Ivanenko. His report was geometric, on the fibre bundle technique, but I understood nothing. And in spring of 1989, I myself went to him for a month in Italy. Since then, our cooperation continues for more than 20 years. I opened new geometric methods for me, which enable me to give an exhaustive mathematical formulation both of classical field theory and classical relativistic mechanics.

Pursuing gauge theory in the language of fibre bundles, I met the fact that the dynamics of this theory is formulated in a traditional form  of an action functional, variations of fields, variational equations and so on, not related to geometrization. At the same time, in mathematics, has long been developed an apparatus of jet manifolds jets for theory of nonlinear differential operators, differential equations and Lagrangian theory. However, it was completely unknown to theoreticians, and now remains little-known to them. It was that Luigi Mangiarotti told at the seminar of Ivanenko.

The essence of formalism of jet manifolds is that sections of a fibre bundle  Y → X are identified by their values and values of their partial derivatives up to some order k at a point  x of a manifold X. The key point is that the set of all such equivalence classes forms a smooth finite-dimensional manifold  J^kY, called the k-order jet manifold of sections of a fibre bundle  Y → X . This enables one, for the analysis of a  k-order differential equation, consider not some infnite-dimensional functional space of smooth sections, but a finite-dimensional jet manifold, and define this differential equation as some its submanifold. Respectively, a differential  operator on sections of  Y → X is defined as a mapping of a jet manifold  J^kY to some vector bundle  E → X , and a k-order Lagrangian L is defined as an n-form (n=dim X) on  J^kY.

Moreover, connections on a fibre bundle  Y → X also are expressed in terms of jet manifolds: they are sections of the jet bundle  J^1Y →Y. Thus, jet manifolds provide the language of differential geometry. The fact is that linear connections as like as linear differential operators can be described in different ways, but the nonlinear ones can be done only in formalism of jet manifolds.

In 1989 - 1990, I was engaged in the study of formalism jet manifolds, and my first works, where it is used, are the articles on classical theory of spontaneous symmetry breaking [63,64], multimomentum Hamiltonian field theory [65,66] and a book on  gauge gravitation theory [9] in 1991 - 92.

At that time, my attention was also attracted to formalism of differential operators on modules over an arbitrary algebra [12]. It also included the machinery of jets of modules, and led to differential geometry (differential forms, connections, etc.) on modules. This formalism, in particular, lies in the basis of non-commutative geometry. Its connection with familiar differential geometry on vector bundles is expressed by the well-known Serre - Swan theorem (generalized by me to non-compact manifolds [15]) that every projective module of finite rank over a ring of smooth functions on a manifold X is a module of sections of some vector bundle over X, and vice versa. Hereinafter, I have repeatedly addressed this formalism for constructing geometry of graded manifolds and for geometric formulation of non-autonomous quantum mechanics [15,16,17].  

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