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."

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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).

References:

G.Sardanashvily, Classical gauge gravitation theory, Int. J. Geom. Methods Mod. Physics, v8 (2011) 1869-1895.
WikipediA: Gauge gravitation theory
WikipediaA: Higgs field (classical)

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Geometry of the composite bundles (from my Scientific Biography)

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."

References:

WikipediA: Connection (composite bundle)

G. Sardanashvily, Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory (2013).

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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.  

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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.

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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)

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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.

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Sardanashvily Internet addresses

WikipediA: http://en.wikipedia.org/wiki/Gennadi_Sardanashvily

Personal site: http://www.g-sardanashvily.ru/

Site at Google: https://sites.google.com/site/sardanashvily/home

This Blogger: http://sardanashvily.blogspot.ru/

Livejournal: http://sardanashvily.livejournal.com/

Facebook: https://www.facebook.com/pages/Gennadi-Sardanashvily/140633719336824

Twitter: https://twitter.com/sardanashvily

Linkedin: http://www.linkedin.com/pub/gennadi-sardanashvily/38/b40/69b

Researchgate: http://www.researchgate.net/profile/G_Sardanashvily/?ev=hdr_xprf

Mendeley: http://www.mendeley.com/profiles/gennadi-sardanashvily/

Google  Scholar: http://scholar.google.com/citations?user=frcpTQEAAAAJ&hl=en

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My Lectures on Supergeometry

G. Sardanashvily, Lectures on supergeometry, arXiv: 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. 

Contents

1. Graded tensor calculus, 2. Graded differential 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. 

Introduction

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 different 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.

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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.

References

G. Sardanashvily, Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory (2013)

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

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Solvay conference 1927: They created contemporary physics

This is the well-known picture of 5th Solvay Conference (1927).  But look at them once more – they created contemporary physics.

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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.”

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How we are small in the Universe

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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	5-Year Impact Factor
LETT MATH PHYS	2.415	1.819	0.842	0.969	0.916	1.367
COMMUN MATH PHYS	1.971	1.941	2.000	2.067	2.075	2.012
J PHYS A-MATH THEOR	1.766	1.564	1.641	1.577	1.540	1.514
J MATH PHYS	1.296	1.291	1.291	1.318	1.085	1.284
REV MATH PHYS (WS)	1.092	1.213	1.290	1.190	1.258	1.102
J GEOM PHYS	1.055	0.818	0.652	0.714	0.683	0.911
IJGMMP (WS)	0.951	0.856	0.757	1.612	1.464	1.265
REP MATH PHYS	0.756	0.643	0.734	0.658	0.576	0.626

See also Total List of journals in mathematical physics.

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"Albert Einstein" of Vadim Sidur

"Albert Einstein" of Vadim Sidur, Princeton, USA

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«Теорминимум-XXI». Современный курс теоретической физики.

Этот курс был задуман как современный «Теорминимум-XXI» в качестве альтернативы известному "Курсу теоретической физики" Ландау и Лифшица, который отражает уровень теоретической физики середины прошлого века. Но уже тогда в 70-е годы зародилась совсем другая теоретическая физика, основанная на математическом аппарате дифференциальной геометрии и алгебраической топологии. Она была стимулирована успехами теории калибровочных полей как универсального механизма описания фундаментальных взаимодействий и ее строгой математической формулировкой в терминах геометрии расслоенных пространств.

Расслоения, связности и многообразия струй, суперсимметрии, супергеометрия и некоммутативная геометрия, гомологии и когомологии, солитоны, инстантоны и топологические заряды, многомерные модели, топологическая теория поля, аномалии, квантовые группы и алгебры Хопфа, геометрическое и деформационное квантования, группоиды, алгеброиды и т. д. составляют стандартный контент современных квантовых и полевых моделей. Ничего этого нет ни у Ландау - Лифшица, ни в подавляющем большинстве отечественных университетских учебников и курсов.

Представляемый курс теоретической физики «Современные методы теории поля» включает 5 томов:

Г.А. Сарданашвили, «Современные методы теории поля. 1. Геометрия и классические поля» (УРСС, 1996) (2-е изд. 2011)

Г.А. Сарданашвили, «Современные методы теории поля. 2. Геометрия и классическая механика» (УРСС, 1998)

Г.А. Сарданашвили, «Современные методы теории поля. 3. Алгебраическая квантовая теория» (УРСС, 1999) (2-е изд. 2011)

Г.А. Сарданашвили, «Современные методы теории поля. 4. Геометрия и квантовые поля» (УРСС, 2000)

Г.А. Сарданашвили, «Современные методы теории поля. 5. Гравитация» (УРСС, 1996) (2-е изд. 2011).

Это своего рода адаптированный «Теорминимум-XXI» для тех, кто собирается начать заниматься современной теоретической и математической физикой. Но для профессиональной работы он недостаточен. Экспертное изложение необходимых математических методов и теоретических моделей дано в монографиях:

L. Mangiarotti, G. Sardanashvily, Connections in Classical and Quantum Field Theory (World Scientific, 2000),

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

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

G. Giachetta, L. Mangiarotti, G. Sardanashvily, Geometric Methods in Classical and Quantum Mechanics (World Scientific, 2010),

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

G.Sardanashvily, Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory (Lambert Academic Publishing, Saarbrucken, 2013),

которые доступны на странице Monographs моего сайта, его дубликата в Google, а также в MendeleY.

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