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
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:
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 (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 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 Moscow
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