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].  

Monday, 10 December 2012

D.Ivanenko’s proton-neutron model of atomic nuclei of 1932

In 1932, Soviet physicist Dmitri Ivanenko proposed the proton-neutron model of atomic nuclei. One usually refers to Ivanenko's short letter [1] of April 21, 1932 in Nature, which was quoted by W. Heisenberg in his first work on the model of nuclei submitted to Zs. f. Phys on June 7, 1932 [7].

However, Ivanenko published five works on his model in 1932 [1-5].

In the above-mentioned first one, he proposed that atomic nuclei consist of alpha-particles and neutrons, and assumed the existence of beta-electrons in nuclei as constituents of these alpha-particles and neutrons. In the second and third works [2,3], Ivanenko stated that atomic nuclei contain only protons and neutrons, but electrons are created under beta-decay in accordance with the Ambarzumian - Ivanenko  hypothesis of creation of massive particles of 1930 [6].

In the next articles [4,5], D. Ivanenko and E. Gapon proposed the idea of the shell distribution of protons and neutrons in nuclei.


[1]  Iwanenko D., The neutron hypothesis, Nature, 129, N 3265 (1932) 798.

[2] Iwanenko D., Neutronen und kernelektronen, Physikalische Zeitschrift der Sowjetunion 1 (1932) 820-822.

[3]  Iwanenko D., Sur la constitution des noyaux atomiques, Compt. Rend. Acad Sci. Paris, 195 (1932).439-441.

[4]  Gapon E., Iwanenko D., Zur Bestimmung der isotopenzahl, Die Naturwissenschaften 20 (1932) 792-793.

[5] Gapon E., Iwanenko D., Zur Bestimmung der isotopenzahl, Physikalische Zeitschrift der Sowjetunion 2 (1932) 99-100.

[6] Ambarzumian V., Iwanenko D., Les électrons inobservables et les rayons, Compt. Rend. Acad Sci. Paris 190 (1930) 582.

[7] Heisenberg W., Uber den Bau der Atomkerner I, Zeitschrift für Physik A  77 (1932) 1-11.

Monday, 3 December 2012

My review “Axiomatic quantum field theory”

G. Sardanashvily, Axiomatic quantum field theory. Jet formalism, arXiv: 0707.4257

Jet formalism provides the adequate mathematical formulation of classical field theory, reviewed in hep-th/0612182. A formulation of QFT compatible with this classical one is discussed. We are based on the fact that an algebra of Euclidean quantum fields is graded commutative, and there are homomorphisms of the graded commutative algebra of classical fields to this algebra. As a result, any variational symmetry of a classical Lagrangian yields the identities which Euclidean Green functions of quantum fields satisfy.