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

Tuesday, 24 December 2013

Dmitri Ivanenko. Scientific Biography

My article: G.Sardanashvily, Dmitri Ivanenko. Scientific Biography, In: The People of Physics Faculty. Selected Papers of the Journal "Soviet Physicist" (1998 - 2006) (#) (see also arXiv: 1607.03828).

"Dmitri Ivanenko (29.07.1904 - 30.12.1994), professor of Moscow State University (since 1943)was one of the great theoreticians of XX century. He made the fundamental contribution to many areas of nuclear physics, field theory and gravitation theory.

His outstanding achievements include:
·                The Fock - Ivanenko coefficients of parallel displacement of spinors in a curved space-time (1929). Nobel laureate Abdus Salam called it the first gauge theory.
·                The Ambartsumian - Ivanenko hypothesis of creation of massive particles which is a corner stone of contemporary quantum field theory (1930).
·                The proton-neutron model of atomic nuclei (1932).
·                The first shell model of nuclei (in collaboration with E. Gapon) (1932).
·                The first model of exchange nuclear forces by means of massive particles (in collaboration with I. Tamm) (1934). Based on this model, Nobel laureate H. Yukawa developed his meson theory.
·                The prediction of synchrotron radiation (in collaboration with I. Pomeranchuk) (1944) and its classical theory (in collaboration with A. Sokolov).
·                Theory of hypernucleus (1956).
·                The hypothesis of quark stars (in collaboration with D. Kurdgelaidze) (1965).

·                The gauge gravitation theory (in collaboration with G. Sardanashvily), where gravity is treated as a Higgs field responsible for spontaneous breaking of space-time symmetries (1983).

Professor D. D. Ivanenko was born on July 29, 1904 in Poltava, where he finished school and began his creative path as a teacher of physics in middle school. In 1923 D. D. Ivanenko entered Petrograd University. In 1926, while still a student, he wrote his first scientific works: with G. A. Gamov on the Kaluza-Klein five-dimensional theory and with L. D. Landau on the problems of relativistic quantum mechanics .... "

Saturday, 14 December 2013

Program for International Student Assessment (PISA 2012). Results

PISA 2012 is program's 5th survey which assessed the competencies of 15-year-olds in reading, mathematics and science (with a focus on mathematics) in 65 countriesAround 510 000 students between the ages of 15 years 3 months and 16 years 2 months participated in the assessment, representing about 28 million 15-year-olds globally.

The students took a paper-based test that lasted 2 hours. The tests were a mixture of open-ended and multiple-choice questions that were organised in groups based on a passage setting out a real-life situation. A total of about 390 minutes of test items were covered.  Students took different combinations of different tests. They and their school principals also answered questionnaires to provide information about the students' backgrounds, schools and learning experiences and about the broader school system and learning environment.

The results are the following (#).

In Russian Federation  (#):
  • The average performance in reading of 15-year-olds is 475 points, compared to an average of 496 points in OECD countries (41-42nd position). Girls perform better than boys with a statistically significant difference of 40 points (OECD average: 38 points higher for girls).
  • On average, 15-year-olds score 482 points in mathematics, the main topic of PISA 2012, compared to an average of 494 points in OECD countries (34-35th position). Girls perform better than boys with a statistically significant difference of 2 points (OECD average: 11 points higher for boys).
  • In science literacy, 15-year-olds in the Russian Federation score 486 points compared to an average of 501 points in OECD countries (37th position). Girls perform better than boys with a non statistically significant difference of 6 points (OECD average: only 1 point higher for boys).

Friday, 6 December 2013

My lectures on Advanced Geometric Methods in Mechanics and Field Theory in arXiv:

The course of my lectures on Advanced Geometric Methods in Mechanics and Field Theory in arXiv:

Fibre bundles, jet manifolds and Lagrangian theory. Lectures for theoreticians arXiv: 0908.1886

Lectures on supergeometry arXiv: 0910.0092

Lectures on differential geometry of modules and rings arXiv: 0910.1515

Advanced mechanics. Mathematical introduction arXiv: 0911.0411

Lectures on integrable Hamiltonian systems arXiv: 1303.5363

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)

Friday, 25 October 2013

Geometry of the composite bundles (from my Scientific Biography)

"The jet formalism, when I first met it, was quite developed in application to theory of differential operators and differential equations, differential geometry, and also, as I have already mentioned, in main aspects to Lagrangian formalism. It seemed that I as a theoretician should only apply it to the particular field models: gauge theory, gravitation theory, etc. However, I had to do develop a number of its basic issues: geometry of composite bundles, Lagrangian theory in formalism of a variational bicomplex, Noether identities and the second Noether theorem.

A composition of fibre bundles Y->S->X is called the composite bundle. They arise in a number of models of field theory and mechanics. In mechanics, these are models with parameters described by sections of a fibre bundle S->X. In field theory, they are systems with a background field and models with spontaneous symmetry breaking, e.g., gravitation theory, when sections of  a fibre bundle S->X are Higgs fields. A key point is that, if h is a section of a fibre bundle S->X, then the restriction of Y->S to a submanifold h(X) of S is a subbundle h*Y->X of a fibre bundle Y->X, describing a system in the presence of a background field (or a parametric function) h(X).

Using a relation between jet manifolds of fibre bundles Y->X, Y->S and S->X, I obtained that between connections on these bundles and, most importantly, the new differential operator on sections of a fibre bundle Y->S, called the vertical covariant differential determined by a connection A on Y->S. The fact is that, being restricted to h(X), this operator coincides with the familiar covariant differential yielded by the restriction of a connection A onto h*Y->X. Thus, this vertical covariant differential should appear in description of the dynamics of field systems on a composite bundle. This result was published in 1991 in the article [64] and was already used in the book [9] for description of spinors in a gravitational field. Subsequently, I have used it in different models of field theory and mechanics. One of them, the key to construct the gauge gravitation theory, is classical field theory with spontaneous symmetry breaking."


Monday, 14 October 2013

What is Nobel Prize in Physics 2013 for?

Nobel Prize in Physics 2013 is awarded to François Englert and Peter Higgs "for the theoretical discovery of a mechanism that contributes to our understanding of the origin of mass of subatomic particles, and which recently was confirmed through the discovery of the predicted fundamental  particle, by the ATLAS and CMS experiments at CERN’s Large Hadron Collider".

A curios is that, in accordance with the Higgs mechanism of mass generation, quantum particles get a mass due to their interaction with a constant background Higgs field  responsible for spontaneous symmetry breaking, but not a quantum Higgs boson. This constant background Higgs field  is treated as a Higgs vacuum, whose physical origin however remains unclear. For instance, one thinks of it as being sue generis a condensate by analogy with a condensate of Cooper pairs in superconductivity.  

Thursday, 3 October 2013

Who is who among universities in 2013 by THE World University Rankings

New world ranking of universities "Times Higher Education World University Rankings 2013-2014" # has been published. It contains 400 universities.

The top ten positions are occupied by 7 universities of USA and the 3 ones of United Kingdom.

In the top twenty: 15 - USA, 3 - United Kingdom, 1 – Switzerland and Canada.

In the first 50 Universities: 29 – USA; 7 - United Kingdom; 3 – Canada; 2 – Switzerland, Australia and China; 1 - Hong-Kong, Japan, Sweden and Korea.

Tuesday, 24 September 2013

Dmitri Ivanenko and Lev Landau - two archival photos

Dmitri Ivanenko and Lev Landau - two soviet genius-physicists.

D. Ivanenko and L. Landau (1927)

D. Ivanenko, L. Landau and Jennie Kannegisser (in future lady Peierls)

Tuesday, 10 September 2013

Who is who among universities in 2013

New world ranking of universities "QS Top University Ranking 2013/2014" has been published. It contains 800 universities.

The top ten positions are occupied by 6 universities of USA and 4 of United Kingdom.

In the top twenty: 11 - USA, 6 - United Kingdom, 2 – Switzerland, 1 -  Canada.

In the first 50 Universities: 19 – USA; 8 - United Kingdom; 5 - China (including 3 of Hong-Kong ); 4 - Australia; 3 – Canada;  2 – Switzerland, France, Japan and Singapore; 1 – Germany, Netherlands  and Korea.

See # for the ranking of Russian universities.

Wednesday, 21 August 2013

My Lectures on Supergeometry

G. Sardanashvily, Lectures on supergeometryarXiv: 0910.0092

Elements of supergeometry are an ingredient in many contemporary classical and quantum field models involving odd fields. For instance, this is the case of SUSY field theory, BRST theory, supergravity. Addressing to theoreticians, these Lectures aim to summarize the relevant material on supergeometry of modules over graded commutative rings, graded manifolds and supermanifolds. 


1. Graded tensor calculus, 2. Graded dierential calculus and connections, 3. Geometry of graded manifolds, 4. Superfunctions, 5. Supermanifolds, 6. DeWitt supermanifolds, 7. Supervector bundles, 8. Superconnections, 9. Principal superconnections, 10. Supermetric, 11. Graded principal bundles. 


Supergeometry is phrased in terms of Z_2-graded modules and sheaves over Z_2-graded commutative algebras. Their algebraic properties naturally generalize those of modules and sheaves over commutative algebras, but supergeometry is not a particular case of noncommutative geometry because of a dierent definition of graded erivations.

In these Lectures, we address supergeometry of modules over graded commutative rings (Lecture 2), graded manifolds (Lectures 3 and 11) and supermanifolds.

It should be emphasized from the beginning that graded manifolds are not supermanifolds, though every graded manifold determines a DeWitt H∞-supermanifold, and vice versa (see Theorem 6.2 below). Both graded manifolds and supermanifolds are phrased in terms of sheaves of graded commutative algebras. However, graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing of sheaves of supervector spaces. Note that there are different types of supermanifolds; these are H∞-, G∞-, GH∞-, G-, and DeWitt supermanifolds. For instance, supervector bundles are defined in the category of G-supermanifolds.

Saturday, 10 August 2013

Is supersymmetry illusive?

“Despite the success of the Large Hadron Collider, evidence for the follow-up theory – supersymmetry – has proved elusive” #

“All would be perfect except that no one has detected any of the many expected supersymmetric particles. “ #

Thus, it seems that supersymmetries, described by generalization of Lie algebras to Lie superalgebras, are illusive. This is also about supergravity based on a super extension of a Poincare Lie algebra.

At the same time, we observe particles both of the even Grassmann parity (photons) and the odd one (fermions). Moreover, gauge symmetries are parameterized by odd ghosts, and BRST theory at present is the generally accepted technique of gauge field quantization. These facts motivate us to develop Grassmann-graded Lagrangian theory of even and odd fields, in general.


G. Giachetta, L. Mangiarotti, G. Sardanashvily, Advanced Classical Field Theory (2009)

Sunday, 4 August 2013

Thursday, 25 July 2013

Who is who in modern cosmology history

V.Mukhanov and A.Starobinsky were awarded the 2013 Gruber Cosmology Prize (#) . The following Alexei Starobinsky Laureate Profile (#) provides a brilliant sketch of modern cosmology history.

In 1979, the universe was in trouble – at least from a cosmologist’s point of view.  Compelling evidence for the Big Bang theory – an interpretation of the universe as expanding over time – dated only to the mid-1960s.  But already theorists found themselves confronting a problem that threatened to undermine that theory:  Why is the universe so uniform, or homogeneous, on scales much greater than the size of its largest structures – the web of superclusters of galaxies that span hundreds of millions of light-years (a light year being the distance light travels in a year, or about 6 trillion miles)? 

According to the Big Bang theory, galaxies on the whole are being carried away from one another on the expansion of space itself, so that no matter where you are in the universe, the rest of the universe seems to be receding from you.  Yet if you look at the most distant part of the universe in one direction, and the most distant part of the universe in the opposite direction, they will be remarkably similar.  They’re billions of light years apart, double the distance that light or any other kind of information could have traveled since the Big Bang, so how could they “know” to be alike?

Alexei Starobinsky, then a senior researcher at the Landau Institute for Theoretical Physics in Moscow, wasn’t working on that problem, but he helped to solve it anyway. 

He had been trying, instead, to figure out how the origin of a Big Bang universe might have worked, a task that took him down the rabbit hole of quantum gravity – the attempt to combine quantum mechanics and the general theory of relativity.  In 1979, he discovered that the universe could have gone through an extraordinarily rapid exponential expansion in the first moments of its existence. In the same year he calculated the generation of gravitational waves during this exponential expansion.

Shortly after it, the American physicist Alan Guth proposed a brilliant idea that the stage of the exponential expansion of the early universe, which he called “inflation,” could explain the incredible uniformity of our universe and resolve many other outstanding problems of the Big Bang cosmology. This clarified a potential significance of the regime of the exponential expansion. However, Guth immediately recognized that his proposal had a flaw: the world described by his scenario would become either empty or very non-uniform at the end of inflation. This problem was solved by Andrei Linde, who introduced several major modifications of inflationary theory, such as “new inflation” (later also developed by Albrecht and Steinhardt), “chaotic inflation”, and “eternal chaotic inflation.” A new cosmological paradigm was born.

Starobinsky’s work inspired two of his fellow theoreticians in Moscow, Viatcheslav Mukhanov and G. V. Chibisov (now deceased).  While an exponential expansion of a newborn universe would explain the large-scale homogeneity we see today, Mukhanov and Chibisov also realized that Heisenberg’s uncertainty principle prohibits absolute homogeneity. 
“There would always remain small wiggles, or small inhomogeneities, in the distribution of the matter,” Mukhanov explains.  “But normally these kinds of inhomogeneities are extremely small.”  What would have happened, Mukhanov and Chibisov wondered, to the inhomogeneities that were present during the exponential expansion?  In 1981, Mukhanov and Chibisov concluded that the exponentially rapid expansion would stretch tiny quantum fluctuations to an enormously large size. After that, these fluctuations would grow in amplitude and become the seeds for the galaxy formation.

“We were thinking we could take these small inhomogeneities and amplify them in the expanding universe,” Mukhanov says.  He and Chibisov concluded that in a certain sense these primordial wiggles would be the universe today:  the things that make the universe inhomogeneous on smaller scales; the structures that make the universe more than empty space. 
In 1982, several scientists, including Starobinsky, outlined a theory of quantum fluctuations generated in new inflation. This theory was very similar to the theory developed by Mukhanov and Chibisov in the context of the Starobinsky model. Investigation of inflationary fluctuations culminated in 1985 in the work by Mukhanov, who developed a rigorous theory of these fluctuations applicable to a broad class of inflationary models, including new and chaotic inflation.

Later, cosmologists calculated how those inhomogeneities would appear in the cosmic microwave background (CMB), the relic radiation dating to the moment when the universe was 380,000 years old. At that time, hydrogen atoms and photons (packets of light) decoupled, leaving a kind of “flashbulb” image that pervades the universe to this day.  Since then, numerous observations of the CMB have found an exquisite match with Mukhanov and Chibisov’s theoretical predictions, most recently in the release of data from the European Space Agency’s Planck observatory.”

Monday, 1 July 2013

Impact Factor 2012 of Journals in mathematical physics

New Impact Factor 2012 has been announced.

Impact Factor 2012 of some journals closed to our International Journal of Geometric Methods in Modern Physics (IJGMMP) by subject and style is the following:

Journal Title

Impact Factor 2012
Impact Factor 2011
Impact Factor 2010
Impact Factor 2009
Impact Factor 2008

See also Total List of journals in mathematical physics.