Gennadi A. SARDANASHVILY, theoretician and mathematical physicist, principal research scientist of the Department of Theoretical Physics,
Was born March 13, 1950,
In 1967, he graduated from the Mathematical Superior Secondary School No.2 (
) with a silver award and entered the
Physics Faculty of Moscow State University (MSU). Moscow
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
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:
geometric formulation of classical field theory, where classical fields are represented by sections of fibre bundles;
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;
gauge gravitation theory, where a gravitational field is treated as the Higgs one which is responsible for spontaneous breaking of space-time symmetries;
covariant (polysymplectic) Hamiltonian field theory, where momenta correspond to derivatives of fields with respect to all world coordinates;
geometric formulation of classical non-relativistic mechanics (in terms of fibre bundles);
geometric formulation of relativistic mechanics (in terms of one-dimensional submanifolds);
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 ( ). He is an author of the
course "Modern Methods in Field
Theory" (in Russ.) in five volumes. Italy
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*Y is 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 identities 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
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. Hamilton
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->X is 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 to h*: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 pseudo-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 associated with LX-> LX/SO(3,1).
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 of Q->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 for completely 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 modeled 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
Student period and the first works
In 1967, I graduated from the
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…. # Moscow