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

Tuesday, 26 November 2013

Scientific Biography

"Gennadi A. SARDANASHVILY, theoretician and mathematical physicist, principal research  scientist of the Department of Theoretical Physics, Moscow State University

Was born March 13, 1950, Moscow.

In 1967, he graduated from the Mathematical Superior Secondary School No.2 (Moscow) with a silver award and entered the Physics Faculty of Moscow State University (MSU).

In 1973, he graduated with Honours Diploma from MSU (diploma work: "Finite-dimensional representations of the conformal group").

He was a Ph.D. student of the Department of Theoretical Physics of MSU under the guidance of professor D.D. Ivanenko in 1973–76.

Since 1976 he holds research positions at the Department of Theoretical Physics of MSU: assistant research scientist (1976-86), research scientist (1987-96), senior research scientist (1997-99), principal research scientist (since 1999).

In 1989 - 2004 he also was a visiting professor at the University of Camerino, Italy.

He attained his Ph.D. degree in physics and mathematics from MSU in 1980, with Dmitri Ivanenko as his supervisor (Ph.D. thesis: "Fibre bundle formalism in some models of field theory"), and his D.Sc. degree in physics and mathematics from MSU in 1998 (Doctoral thesis: "Higgs model of a classical gravitational field").

Gennadi Sardanashvily research area is geometric methods in field theory, classical and quantum mechanics; gauge theory; gravitation theory.

His main achievement includes:

(i) comprehensive geometric formulation of classical field theory, where classical fields are represented by sections of fibre bundles, and in that number:

    generalized Noether theorem for reducible degenerate Lagrangian theories (in terms of      cohomology);

    Lagrangian BRST field theory;

    differential geometry of composite bundles;

    classical theory of Higgs fields;

    covariant (polysymplectic) Hamiltonian field theory, where momenta correspond to derivatives of fields with respect to all world coordinates;

(ii) gauge gravitation theory, where a gravitational field is treated as the Higgs one which is responsible for spontaneous breaking  world symmetries;

(iii) geometric formulation of Lagrangian and Hamiltonian time-dependent non-relativistic mechanics (in terms of fibre bundles);

(iv) geometric formulation of relativistic mechanics (in terms of one-dimensional submanifolds);

(v) generalization of the Liouville–Arnold, Nekhoroshev and Mishchenko–Fomenko theorems on completely and partially integrable and superintegrable Hamiltonian systems to the case of non-compact invariant submanifolds.

In 1979 - 2011, he lectures on algebraic and geometrical methods in field theory at the Department of Theoretical Physics of MSU and, In 1989 - 2004, on geometric methods in field theory at University of Camerino (Italy). He is an author of the course "Modern Methods in Field Theory" (in Russ.) in five volumes.

Gennadi Sardanashvily published 20 books and more than 300 scientific articles.

He is the founder and Managing Editor of "International Journal of Geometric Methods in Modern Physics" (World Scientific, Singapore).

Brief exposition of main results

Geometric formulation of classical field theory

In contrast to the classical and quantum mechanics and quantum field theory, classical field theory, the only one that allows for a comprehensive mathematical formulation. It is based on representation of classical fields by sections of smooth fibre bundles.

Lagrangian theory on fibre bundles and graded manifolds

Because classical fields are represented by sections of fibre bundles, Lagrangian field theory is developed as Lagrangian theory on fibre bundles. The standard mathematical technique for the formulation of such a theory are jet manifolds of sections of fibre bundles. As is seen Lagrangian formalism of arbitrary finite order, it is convenient to develop this formalism on the Frechet manifold J*Y of infinite order jets of a fibre bundle Y->X because of operations increasing order. It is formulated in algebraic terms of the variational bicomplex, not by appealing to the variation principle. The jet manifold J*Yis endowed with the algebra of exterior differential forms as a direct limit of algebras exterior differential forms on jet manifolds of finite order. This algebra is split into the so-called variational bicomplex, whose elements include Lagrangians L, and one of its coboundary operator is the variational Euler – Lagrange operator. The kernel of this operator is the Euler - Lagrange equation. Cohomology of the variational bicomplex has been defined that results both in a global solution of the inverse variational problem (what Lagrangians L are variationaly trivial) and the global first variational formula, which the first Noether theorem follows from. Construction of Lagrangian field theory involves consideration of Lagrangian systems of both even, submitted by the sections bundles, and odd Grassmann variables. Therefore, Lagrangian formalism in terms of the variational bicomplex has been generalized to graded manifolds.

Generalized second Noether theorem for reducible degenerate Lagrangian systems

In a general case of a reduced degenerate Lagrangian, the Euler - Lagrange operator obeys nontrivial Noether identities, which are not independent and are subject to nontrivial first-order Noether identities, in turn, satisfying second-order Noether identities, etc. The hierarchy of these Noether identities under a certain cohomology condition is described by the exact cochain complex, called the Kozul - Tate complex. Generalized second Noether theorem associates a certain cochain sequence with this complex. Its ascent operator, called the gauge operator, consists of a gauge symmetry of a Lagrangian and gauge symmetries of first and higher orders, which are parameterized by odd and even ghost fields. This cochain sequence and the Kozul - Tate complex of Noether identitie fully characterize the degeneration of a Lagrangian system, which is necessary for its quantization..

Generalized first Noether theorem for gauge symmetries

In the most general case of a gauge symmetry of a Lagrangian field system, it is shown that the corresponding conserved symmetry current is reduced to a superpotential, i. e., takes the form J=dU +W, where W vanishes on the Euler – Lagrange equations.

Lagrangian BRST field theory

A preliminary step to quantization of a reducible degenerate Lagrangian field system is its so-called BRST extension. Such an extension is proved to be possible if the gauge operator is prolonged to a nilpotent BRST operator, also acting on ghost fields. In this case, the above-mentioned cochain sequence becomes a complex, called the BRST complex, and an original Lagrangian admits the BRST extension, depending on original fields, antifields, indexing the zero and higher order Noether identities, and ghost fields, parameterizing zero and higher order gauge symmetries.

Covariant (polysymplectic) Hamiltonian formalism of classical field theory

Application of symplectic Hamiltonian formalism of conservative classical mechanics to field theory leads to an infinite-dimensional phase space, when canonical variables are values of fields in any given instant. It fails to be a partner of Lagrangian formalism of classical field theory. The Hamilton equations on such a phase space are not familiar differential equations, and they are in no way comparable to the Euler – Lagrange equations of fields. For a field theory with first order Lagrangians, covariant Hamiltonian formalism on polysymplectic manifolds, when canonical momenta are correspondent to derivatives of fields relative to all space-time coordinates, was developed. Lagrangian formalism and covariant Hamiltonian formalism for field models with hyperregular Lagrangians only are equivalent. A comprehensive relation between these formalisms was established in the class of almost regular Lagrangians, which includes all the basic field models.

Differential geometry of composite bundles

In a number of models of field theory and mechanics, one uses composite bundles Y->S->X, when sections of a fibre bundle S->X describe, e.g., a background field, Higgs fields or function of parameters. This is due to the fact that, given a section h of a fibre bundle S->X, the pull-back bundle h*:Y->Xis a subbundle of Y->X. The correlation between connections on bundles Y->X, Y->S, S->X and h*:Y->X were established. As a result, given a connection A on a bundle Y->S, one introduces the so-called vertical covariant differential D on sections of a fibre bundle Y->X, such that its restriction toh*:Y->X coincides with the usual covariant differential for a connection induced on h*:Y->X by a connection A. For applications, it is important that a Lagrangian of a physical model considered on a composition bundle Y->S->X is factorized through a vertical covariant differential D.

Classical theory of Hiigs fields

Although spontaneous symmetry breaking is a quantum effect, it was suggested that, in classical gauge theory on a principal bundle P->X, it is characterized by a reduction of a structure Lie group G of this bundle to some of its closed subgroups Lie H. By virtue to the well-known theorem, such a reduction takes place if and only if the factor-bundle P/H->X admits a global section h, which is interpreted as a classical Higgs field. Let us consider a composite bundle P-> P/H->X and a fibre bundle Y->P/H associated with an H-principal bundle P-> P/H. It is a composite bundle P-> P/H->X whose sections describe a system of matter fields with an exact symmetry group H and Hiigs fields. This is Lagrangian theory on a composite fibre bundle Y->P/H ->X. In particular, a Lagrangian of matter fields depends on  Higgs fields through a vertical covariant differential defined by a connection on a fibre bundle Y->P/H. An example of such a system of matter and Higgs fields are Dirac spinor fields in a gravitational field.

Gauge gravitation theory, where a gravitational field is treated as the Higgs one, responsible for spontaneous breaking of space-time symmetries

Since gauge symmetries of Lagrangians of gravitation theory are general covariant transformations, gravitation theory on a world manifold X is developed as classical field theory in the category of so-called natural bundles over X. Examples of such bundles are tangent TX and cotangent T*X bundles over X, their tensor products and the bundle LX of linear frames in TX. The latter is a principal bundle with the structure group GL(4,R). The equivalence principle in a geometrical formulation sets a reduction of this structure group to the Lorentz SO(1,3) subgroup that stipulates the existence of a global section g of the factor-bundle LX/SO(3,1)->X, which is a pseudo-Riemannian metric, i.e., a gravitational field on X. It enables one to treat a metric gravitation field as the Higgs one. The obtained gravitation theory is the affine-metric one whose dynamic variables are a psudo-Riemannian metric and general linear connections on X. The Higgs field nature of a gravitational field g is characterized the fact that, in different pseudo-Riemannian metrics, the representation of the tangent covectors by Dirac’s matrices  and, consequently, the Dirac operators, acting on spinor fields, are not equivalent. A complete system of spinor fields with the exact Lorentz group of symmetries and gravitational fields is described sections of a composite bundle Z-> LX/SO(3,1)->X where bundle Z-> LX/SO(3,1) is spinor bundle.

Geometric formulation of classical relativistic mechanics in terms of fibre bundles

Hamiltonian formulation of autonomous classical mechanics on symplectic manifolds is not applied to non-autonomous mechanics, subject to time-dependent transformations. that permits depending on the time of conversion. It was suggested to describe non-relativistic mechanics in the complete form, admitting time-dependent transformations, as particular classical field theory on fibre bundles Q->R over the time axis R. However, it differ from classical field theory in that connections on fibre bundles Q->R over R are always flat and, therefore, are not dynamic variables. They characterize reference systems in non-relativistic mechanics. The velocity and phase spaces of non-relativistic mechanics are the first order jet manifold of sections ofQ->R and the vertical cotangent bundle of Q->R. There has been developed a geometric formulation of Hamiltonian and Lagrangian non-relativistic mechanics with respect to an arbitrary reference frame and, in more general setting, of mechanics described by second order dynamic equations.

Geometric formulation of relativistic mechanics in terms of one-dimensional submanifolds

In contrast to non-relativistic mechanics, relativistic mechanics admits transformations of time, depending on spatial coordinates. It is formulated in terms of one-dimensional submanifolds of a configuration manifold Q, when the space of non-relativistic velocities is the first-order jet manifold of one-dimensional submanifolds of a manifold Q, which Lagrangian formalism of relativistic mechanics is based on.

The generalization of the Liouville–Arnold, Nekhoroshev and Mishchenko–Fomenko theorems on the "action-angle" coordinates forcompletely and partially integrable and superintegrable Hamiltonian systems to the case of non-compact invariant submanifolds.

Other published results

Spinor representations of the special conformal group

Topology of stable points of the renormalization group

Homotopy classification of curvature-free gauge fields

Mathematical model of a discrete space-time

Geometric formulation of the equivalence principle

Classification of gravitation singularities as singularities of space-time foliations

The Wheeler-deWitt superspace of spatial geometries with topological transitions

Gauge theory of the “fifth force” as space-time dislocations

Generating functionals in algebraic quantum field theory as true measures in the duals of nuclear spaces

Generalized Komar energy-momentum superpotentials in affine-metric and gauge gravitation theories

Non-holonomic constraints in non-autonomous mechanics

Differential geometry of simple graded manifolds

The geodesic form of second order dynamic equations in non-relativistic mechanics

Classical and quantum mechanics with time-dependent parameters on composite bundles

Geometry of symplectic foliations

Geometric quantization of non-autonomous Hamiltonian mechanics

Bi-Hamiltonian partially integrable systems and the KAM theorem for them

Non-autonomous completely integrable and superintegrable Hamiltonian systems

Geometric quantization of completely integrable and superintegrable Hamiltonian systems in the “action-angle” variables

The covariant Lyapunov tensor and Lyapunov stability with respect to time-dependent Riemannian metrics

Relative and iterated BRST cohomology

Non-equivalent representations of the algebra of canonical commutation relations modelled on an infinite-dimensional nuclear space

Generalization of the Serre – Swan theorem to non-compact and graded manifolds

Definition of higher-order differential operators in non-commutative geometry

Conservation laws in higher-dimensional Chern-Simons models

Classical and quantum Jacobi fields of completely integrable systems

Classical and quantum non-adiabatic holonomy operators for completely integrable systems

Classical and quantum mechanics with respect to different reference frames

Lagrangian and Hamiltonian theory of submanifolds

Geometric quantization of Hamiltonian relativistic mechanics

Supergravity as a supermetric on supermanifolds

Noether identities for differential operators

Differential operators on generalized functions"

Monday, 11 November 2013

On a notion of the mathematical structure

My article “What is a mathematical structure” (2013) came out (#).

"A notion of the mathematical structure was introduced at the beginning of XX century. However, for a long time, mathematical objects were believed to be given always together with some structure, not necessarily unique, but at least natural (canonical). And only a practice, e.g., of functional analysis has led to conclusion that a canonical structure need not exist. For instance, there are different “natural” topologies of a set of rational numbers, different smooth structures of a four-dimensional topological Euclidean space, different measures on a real line, and so on.

In mathematics, different types of structures are considered. These are an algebraic structure, a topological structure, cells whose notion generalizes the Boolean algebras and so on. In the first volume of their course, Bourbaki provide a description of a mathematical structure which enables them to define “espece de structure” and, thus, characterize and compare different structures. However, this is a structure of mathematical theories formulated in terms of logic. We aim to suggest a wider definition of a structure which absorbs the Bourbaki one and the others, but can not characterize different types of structures. This definition is based on a notion of the relation on a set, and it generalizes the definition of a relational system in set theory.

Morphisms and functions are structures in this sense that provides a wide circle of perspective applications of this notion of the structure to mathematical physics.

In particular, let us mention the notions of the universal structure on a set (see Section 2) and the abstract structure on its own elements. One can show that any structure is a constituent of a universal structure, and that any structure admits an exact representation as a constituent of some abstract structure.

Though we follow the von Neumann – Bernays – Gödel set theory, structures on sets only are considered unless otherwise stated. This is sufficient in order to investigate real, e.g., physical systems."

Tuesday, 5 November 2013

Is a metric gravitational field non-quantized?

My conjecture is that, being a classical Higgs field, a metric gravitational field is not quantized, but it is classical in principle (What is gauge gravitation theory about).


G.Sardanashvily, Classical gauge gravitation theory, Int. J. Geom. Methods Mod. Physics, v8 (2011) 1869-1895.
WikipediA: Gauge gravitation theory
WikipediaA: Higgs field (classical)