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

Monday, 29 October 2012

My Scientific Biography

Scientific  Biography (up to 2011) (in English #, in Russian #)
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:

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). 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*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 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->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 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…. #

Sunday, 21 October 2012

Humanity will never leave the Solar system

If the known physical laws are not violated, humanity will forever remains in the Solar system. But in the Solar system, apparently, there is nothing principally new and interesting for fundamental science. Perhaps there is only some primitive form of life in atmospheres of the gas giants.

However, to move freely even in the Solar system, we need a velocity at least of about 1000 km/sec. Annihilation of matter and antimatter only is the known source of energy for that a spacecraft of mass of a few hundred tons could reach this velocity. And one needs tens of kilograms of antimatter. This is unlikely to ever become a reality.

Saturday, 13 October 2012

Introduction to my book “Lectures on Differential Geometry of Modules and Rings”

G. Sardanashvily, “Lectures on Differential Geometry of Modules and Rings. Application to Quantum Theory" (Lambert Academic Publishing, Saarbrucken, 2012) #

Geometry of classical mechanics and field theory mainly is differential geometry of finite-dimensional smooth manifolds and fibre bundles.

At the same time, the standard mathematical language of quantum mechanics and field theory has been long far from geometry. In the last twenty years, the incremental development of new physical ideas in quantum theory has called into play advanced geometric techniques, based on the deep interplay between algebra, geometry and topology.

Geometry in quantum systems speaks mainly the algebraic language of rings, modules and sheaves due to the fact that basic ingredients in the differential calculus and differential geometry of smooth manifolds (except non-linear differential operators) can be restarted in a pure algebraic way.

Let X be a smooth manifold and C(X) a ring of smooth real functions on X. A key point is that, by virtue of the well-known Serre--Swan theorem, a C(X)-module is finitely generated and projective if and only if it is isomorphic to a module of sections of some smooth vector bundle over X. Moreover, this isomorphism is a categorial equivalence. Therefore, differential geometry of smooth vector bundles can be adequately formulated in algebraic terms of a ring C(X), its derivations and the Koszul connections.

In a general setting, let K be a commutative ring, A an arbitrary commutative K-ring, and P, Q some A-modules. The K-linear Q-valued differential operators on P can be defined. The representative objects of functors Q-> Dif(P,Q) are jet modules JP of P. Using the first order jet module J^1P, one also restarts the notion of a connection on an A-module P.

As was mentioned above, if P is a C(X)-module of sections of a smooth vector bundle Y-> X, we come to the familiar notions of a linear differential operator on Y, the jets of sections of Y-> X and a linear connection on Y-> X.

In quantum theory, Banach and Hilbert manifolds, Hilbert bundles and bundles of C*-algebras over smooth manifolds are considered. Their differential geometry also is formulated as geometry of modules, in particular, C(X)-modules.

Let K be a commutative ring, A a (commutative or non-commutative) K-ring, and Z(A) the center of A. Derivations of A constitute a Lie K-algebra DA. Let us consider the Chevalley-Eilenberg complex of K-multilinear morphisms of DA to A, seen as a DA-module. Its subcomplex O*(A) of Z(A)-multilinear morphisms is a differential graded algebra, called the Chevalley-Eilenberg differential calculus over A. If A is an R-ring C(X) of smooth real functions on a smooth manifold X, the module DC(X) of its derivations is a Lie algebra of vector fields on X, and the Chevalley-Eilenberg differential calculus over C(X) is exactly an algebra of exterior forms on a manifold $X$ so that the Chevalley-Eilenberg coboundary operator d coincides with an exterior differential, i.e., O*(A) is the familiar de Rham complex. In a general setting, one therefore can think of elements of the Chevalley-Eilenberg differential calculus over an algebra A as being differential forms over A.

Similarly, the differential calculus over a Grassmann-graded commutative ring is constructed. This is the case of supergeometry. In supergeometry, connections on graded manifolds and supervector bundles are defined as those on graded modules over a graded commutative ring and graded local-ringed spaces.

Note that a Grassmann-graded commutative ring is a particular non-commutative ring. However, the definition of its derivations differs from the non-commutative Leibniz rule. Therefore, supergeometry is not particular non-commutative geometry.

Non-commutative geometry also is developed as a generalization of the calculus in commutative rings of smooth functions. In a general setting, any non-nommutative K-ring A over a commutative ring K can be called into play. One can consider the above mentioned Chevalley-Eilenberg differential calculus over A. However, the definition of differential operators on modules over commutative rings fails to be straightforwardly extended to the non-commutative ones. A key point is that A-module morphisms fail to be zero order differential operators if A is non-commutative. In this case, several nonequivalent definitions of differential operators have been suggested. Accordingly, there are also different definitions of a connection on modules over a non-commutative ring.

The present book aims to summarize the relevant material on the differential calculus and differential geometry on modules and rings. Some basic applications to quantum models are considered.

For the sake of convenience of the reader, several topics on cohomology are compiled in Appendix.

The book is based on the graduate and post graduate courses of lectures given at the Department of Theoretical Physics of Moscow State University (Russia) and the Department of Mathematics and Physics of University of Camerino (Italy). It addresses to a wide audience of mathematicians, mathematical physicists and theoreticians.


1 Commutative geometry: 1.1 Commutative algebra, 1.2 Dierential operators on modules and rings, 1.3 Connections on modules and rings, 1.4 Dierential calculus over a commutative ring, 1.5 Local-ringed spaces, 1.6 Dierential geometry of C(X)-modules, 1.7 Connections on local-ringed spaces.

2 Geometry of quantum systems: 2.1 Geometry of Banach manifolds, 2.2 Geometry of Hilbert manifolds, 2.3 Hilbert and C*-algebra bundles, 2.4 Connections on Hilbert and C*--algebra bundles, 2.5 Instantwise quantization, 2.6 Berry connection.

3 Supergeometry: 3.1 Graded tensor calculus, 3.2 Graded dierential calculus and connections, 3.3 Geometry of graded manifolds, 3.4 Supermanifolds, 3.5 Supervector bundles, 3.6 Superconnections.

4 Non-commutative geometry: 4.1 Modules over C*-algebras, 4.2 Non-commutative dierential calculus, 4.3 Dierential operators in non-commutative geometry, 4.4 Connections in non-commutative geometry, 4.5 Matrix geometry, 4.6 Connes’ non-commutative geometry, 4.7 Dierential calculus over Hopf algebras.

5 Appendix. Cohomology: 5.1 Cohomology of complexes, 5.2 Cohomology of Lie algebras, 5.3 Sheaf cohomology.

Wednesday, 3 October 2012

Infinite-dimensional differential geometry

In quantum models, one deals with infinite-dimensional smooth Banach and Hilbert manifolds and (locally trivial) Hilbert and C*-algebra bundles. The definition of smooth Banach (and Hilbert) manifolds follows that of finite-dimensional smooth manifolds in general, but infinite-dimensional Banach manifolds are not locally compact, and they need not be paracompact. In particular, a Banach manifold admits the differentiable partition of unity if and only if its model space does. It is essential that Hilbert manifolds satisfy the inverse function theorem and, therefore, locally trivial Hilbert bundles are defined. However, they need not be bundles with a structure group.

Infinite-dimensional Kahler manifolds provide an important example of Hilbert manifolds. In particular, the projective Hilbert space of complex rays in a Hilbert space E is such a Kahler manifold.

Note that differential geometry of Banach and Hilbert bundles over a finite-dimensional smooth manifold X can be formulated in terms of differential geometry of modules over a ring of smooth functions on XIn particular, connections on a Hilbert bundle over a smooth manifold can be defined both as particular first order differential operators on the module of its sections. This is the case of time-dependent quantum mechanics where X=R.


G.Giachetta, L. Mangiarotti, G. Sardanashvily, Geometric and Algebraic Topological Methods in Quantum Mechanics (WS, 2005) #

G.Sardanashvily, Differential Geometry of Module and Rings. Application to Quantum Theory. (LAP, 2012)  #