On a mathematical hypothesis of the quark confinement

In quantum field theory, the Wick rotation provides the standard technique of computing Feynman diagrams by means of Euclidean propagators.

Let us suppose that quantum fields in an interaction zone are really Euclidean. In contrast with the well-known Euclidean field theory dealing with the Wightman and Schwinger functions of free quantum fields, we address complete Green's functions of interacting fields, i.e., causal forms on the Borchers algebra of quantum fields. They are the Laplace transform of the Euclidean states obeying a certain condition.

If Euclidean states of a quantum field system, e.g., quarks do not satisfy this condition, this system fails to possess Green's functions and, consequently, the S-matrix. One therefore may conclude that it is not observed in the Minkowski space.

References:

G.Sardanashvily, arXiv: hep-th/0511111

The prespinor model (from my Scientific Biography)

My Scientific Biography: ...Nevertheless, the most promising of my nominated ideas was a model of prespinors (which however till now remains only "promising").

It is known that the root diagrams of simple complex Lie algebras admit groups of reflections, which are finite Coxeter groups. Moreover, the classification of simple complex Lie algebras and their real subalgebras is conducted by means of finite Coxeter groups. They are groups of symmetries of the weight diagrams of irreducible representations of these Lie algebras, which the algebras both of internal and space-time symmetries belong to.

There was an idea that Lie algebras and groups of symmetries can be replaced with the corresponding finite Coxeter groups. Generating elements s of these groups have the property ss=1, and the diversity of these groups is due to the fact that different generating elements do not commute between themselves. The simplest Coxeter group consists of two elements (s,1) and serves as a symmetry group of a 2-spinor. Let us suppose that a physical world in its basis is made of such spinors, let us call them the  prespinors, so that, when their interaction, Coxeter groups of their transformations become non-commutative, providing all the known diversity of symmetries of elementary particles. Moreover, we can go even further and identify elements of the simple Coxeter group (s,1) with the simplest logical system of statements ("no", "yes").

D.Ivanenko believed this model to be very promising. He saw in it the prospect of a continuation of Heisenberg’s and his unified nonlinear field theory, which by that time had stepped aside in the light of theory of gauge fields. It became clear that an interaction of elementary particles is described by exchange of mediators, gauge fields, but not nonlinearities, though not everywhere. For example, an interaction of a field of Cooper pairs in the theory of superconductivity is due to non-linearity, and it may happen that an interactions of a Higgs fields and prespinors are of this type.

The  prespinor model is presented in our book  "Gravitation" (1985) (in Russ.) and a few articles,  but no further development has obtained, since it is unclear how to describe the dynamics of systems with finite groups of symmetries.

 

Who is who among universities in 2011

New world ranking of universities "QS Top University Ranking 2011" has been published. It contains 700 universities.

In 2011, my Moscow State University occupies the 112-th place with a coefficient 61.28 of 100, while in the past year - the 93rd place. From Russian universities, in addition to the MSU, there are still 10. The nearest one is Petersburg State University of the 251-th place with a coefficient 41.06.

The top ten positions are occupied by 6 universities of USA and 4 of United Kingdom.

In the top twenty: 13 - USA, 5 - United Kingdom, and one each from Switzerland and Canada.

In the first hundred: 30 – USA; 19 - United Kingdom; 8 - Australia; 6 - Japan; 5 - Canada; 4 - Germany; 3 - Switzerland, China, Hong-Kong, South Korea, the Netherlands; 2 - Singapore, Sweden, France, Denmark; 1 - Ireland, Belgium, New Zealand, Finland and Taiwan.

My Scientific Biography: Student period

In 1967, I graduated from the Moscow mathematical school №2 with a silver medal and entered Physics Faculty of Moscow State University. Besides the standard education program, I began to engage in self-education and went to the circle of theoretical physics, held for students of the junior courses of prof. D.Ivanenko, his staff and post-graduate students. I originally wanted to engage in theoretical physics, but at the faculty there were three theoretical departments. Under the influence of the theoretical circle, his broad topics, I decided to enter to the Department of Theoretical Physics to D.Ivanenko. From time to time, I even attended his scientific seminar.

In the middle of the third year, in spring of 1970, I was assigned to the Department of Theoretical Physics. The best students of the course tried to enter it, as well as on other theoretical departments. Only 12 people could do, and it was necessary to pass the interview. In the course of the interview, I felt that they knowingly take me: I had lost only on ball for all exam sessions and, apparently, D.Ivanenko warned that I am to him.

After entering the Department, I as a future graduate officially joined the group of Ivanenko: went to his scientific seminars, continued self-education, and eyed what anyone in the group is engaged in.

On the fourth course, I began to collaborate with Andrey Bulinski. He graduated from Physics Faculty in 1968, but was not taken in the graduate school and worked at the Department of Higher Mathematics of the Moscow Physical-Technical Institute. He continued to collaborate with D.Ivanenko, and engaged in algebraic quantum theory: algebras of quantum observables, their representations, quantum dynamical systems, etc. All of this was outside the scope of conventional courses of Physics Faculty. Working with him, I got a good experience in this field, which I then is very handy. In one of his articles, published in Journal of Theoretical and Mathematical Physics in 1971, he even thanking me for useful discussion. Although I do not remember that I was really any good. Algebraic quantum theory is rather mathematically sophisticated subject. My level was certainly not enough to get on this topic some original results and prepare a diploma work. Besides, Andrei Bulinski less and less began to come into the band and seminars of Ivanenko, apparently, having lost hope to return to Physics Faculty. Therefore, D.Ivanenko offered me, at least for pragmatic reasons, to change a research subject and a scientific chief (not being Ph.D., A.Bulinski formally could not be a scientific supervisor of my diploma work).

At that time, in science and seminars of Ivanenko, there has been actively discussed conformal field theory on the basis of the 15-parameter conformal group, including the Lorentz and Poincare subgroups. Naturally, the question arose about constructing the spinor representations of this group, as I did. My scientific supervisor was D.Sc. Dmitri Kurdgelaidze, a long-term employee of D.Ivanenko, with whom he developed a nonlinear meson and spinor theory. However, my purely algebraic subject was far away from his interest, and he could not help me. Therefore, I actually worked independently. I obtained a 8-spinor representation of the conformal group, which also implement the CPT transformations, and wrote for them the conformal-invariant Dirac equation. To me, this work still like it. I reported it on the 3-th Soviet gravitational conference in October 1972, and before that submitted an article to  "Vestnik of Moscow State University, Physics and Astronomy". However, for some reason, this article appeared much later, - in March of 1975. In January 1973, I defended my diploma work "Finite-dimensional  representations of the conformal group", with Ivanenko and Kurdgelaidze as supervisors.

To complete this topic, in 1973, I also constructed the nonlinear representation of the conformal group by the method of the so-called "nonlinear realizations". This method shortly before was developed, allowed to build a representation of a group as an extension of a representation of its Cartan subgroup, and was then very popular. This work was presented at the  Symposium "Modern problems of gravitation" in Moscow and went out in its Proceedings. It became my first scientific publication.

After graduating from the Physics Faculty February 1973, I in April was enrolled in the postgraduate school at the Department of Theoretical Physics to D.Ivanenko. My study of the conformal group was completed and, in front of me, there was a wide range of research directions. Interested in very many, D.Ivanenko provided a full freedom of activity of his graduate students. My direct supervisor was he himself, no one was standing between us, and I could do what I will.

First of all, I was interested in out-of-scope of the standard field theory on the basis of new mathematical methods of theoretical physics: algebraic, geometric and topological, because it was clear that the standard field theory had exhausted its possibilities. And I started with the search for and development of such innovative methods. Although the risk was great: could nothing is going to happen, no publications or dissertation. As it turned out, with the publication of problems was not, and that's Ph.D. thesis was delayed.

References:
G. Sardanashvily: Scientific Biography

Illusion of matter

Can a structure be carrier-free? Philosophy says that it is impossible. However, contemporary theoretical physics gives a different answer.

In mathematics, there exist various concepts of a structure: the structure genus of a structure (a rather sophisticated definition of Bourbaki), a lattice (an algebraic notion generalizing a Boolean algebra), a topological structure, a geometric structure, etc. For physical applications, I would propose a mathematical definition of the structure as an n-ary relation on a set defined by some subset of an n-product of this set. This concept correlates with the definition of Bourbaki in some way and absorbs other definitions of a structure. In particular, morphisms of a set are structures in this sense. Nevertheless, in all existent variants, a mathematical structure is introduced on a carrier set.

In physics, however, it appears that a set, carrying a structure, often itself consists of elements of some structure. For example, a classical field, defined as a section of a fibre bundle, is a morphism. i. e., a structure, called the geometric structure. It is obvious, that quantum operators as elements of a certain algebra exemplify an algebraic structure. Moreover, by virtue of the well-known GNS construction in algebraic quantum theory, a Hilbert space of states, which quantum operators act in, consists of equivalence classes of these operators possessing the same average value, and so, it also is a set of elements of an algebraic structure.

Contrary, a point mass in classical mechanics is not part of any structure. However, in modern united models of fundamental interactions, a quantum field acquires a mass as a result of its interaction with a Higgs field. It follows that a mass is a derivative characteristics of two structures. Thus, the massive matter ceases to be a fundamental concept. For example, a particle and an antiparticle, annihilating, are converted into photons.

At present, with theoretical and mathematical viewpoint, all known fundamental physical objects are structures whose carrier consists of elements of some other structure having its carrier another structure, etc. Moreover, a structure can be defined on different carriers or be carrier-free. For instance, morphisms of some vector space are representations of a certain abstract group which is defined for itself and admits other representations.

If all physical objects, e.g., classical and quantum fields are a structure, then what is a carrier of a structure in the physical world? Is there such a carrier? Of course, the matter does not disappear, but is somewhat illusory.