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 Italy
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].
