"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"
