Gennadi Sardanashvily

At present the hypothesis of quark stars is discussed from different viewpoints, and there are some candidates for quark stars (Wikipedia: Quark star ).

The idea of quark stars has been suggested by D. Ivanenko and D. Kurdgelaidze in 1964. They were motivated both by the hypothesis of a neutron star and the quark model just announced in 1964. At first, Ivanenko and Kurdgelaidze published their quark star model in Soviet journal “Astrophysics” 1 (1965) 479-481 (in Russian) (#) and then in Lettere al Nuovo Cimento 2 (1969) 13-16 (#).

Archive

Tuesday, 8 April 2014

Foundations of Modern Physics 4: Equivalence principle

The Equivalence principle is treated as one of the corner-stones of gravitation theory. However, there exist its different formulations. One separates “weakest”, “weak”, “middle-strong” and “strong” Equivalence principles. All of them are based on the empirical equality of inertial mass, gravitational active and passive charges, and they establish the existence of a certain reference frame, where physical laws would take the known special relativistic form, i. e., a gravitational field effectively disappears.

The “weakest” Equivalence principle is restricted to the motion law of a probe point mass in a uniform gravitational field.

Its sui generis localization is the “weak” Equivalence principle that states the existence of a desired local inertial frame at a given world point. This is the case of equations depending on a gravitational field and its first order derivatives, e. g., the equations of mechanics of probe point masses, and the equations of electromagnetic and Dirac fermion fields.

The “middle-strong” Equivalence principle is concerned with any matter, except a gravitational field, while the “strong” one is applied to all physical laws.

Apparently, only the “weakest” and “weak” Equivalence principles are true. It is the “weak” Equivalence principle that the identification of a gravitational field to a pseudo-Riemannian metric satisfies to. However, the “weak” Equivalence principle provides a necessary, but not sufficient condition of such identification. Moreover, it does not explain the existence of a gravitational field itself.

To overcome these difficulties, we have reformulated the Equivalence principle as follows.

In geometric terms Special Relativity can be characterized as the geometry of Lorentz invariants. Then the Equivalence principle can be formulated to require the existence of Lorentz invariants on a world manifold X. We agree to call it the geometric Equivalence principle. Its requirement holds if and only if the tangent bundle TX of X admits an atlas with Lorentz transition functions, i. e., a structure group of the associated principal bundle LX of frames in TX is reduced to the Lorentz group SO(1,3). By virtue of the well known theorem, this reduction takes place if and only if the quotient bundle LX/SO(1,3)->X admits a global section, which is a pseudo-Riemannian metric on X.

Thus the geometric Equivalence principle provides the necessary and sufficient conditions of the existence of a pseudo-Riemannian metric on a world manifold that we observe as a gravitational field.

Moreover, if a structure group of the frame bundle LX is reduced to the Lorentz group, it always is reduced to the spatial rotation group SO(3). In accordance with the above mentioned theorem, this reduction defines a space-time decomposition of the tangent bundle TX and, thus, makes a world manifold X into a space-time.

The geometric Equivalence principle also provides the necessary condition of the existence of Dirac’s spinor fields, possessing Lorentz symmetries, on a world manifold.

Thus, one can think of an observable Dirac fermion matter as being the underlying physical reason of the geometric Equivalence principle and, consequently, the existence of a pseudo-Riemannian gravitational field.

In gravitation theory, the geometric Equivalence principle characterizes spontaneous symmetry breaking of space-time symmetries and, thus, clarifies the physical nature of a gravitational field as a Higgs field responsible for this symmetry breakdown.

References:

D. Ivanenko, G. Sardanashvily, The gauge treatment of gravity, Physics Reports 94 (1983) 1-45.

G. Sardanashvily, Gauge gravitation theory from the geometric viewpoint, Int. J. Geom. Methods Mod. Phys. 3 (2006) N1 v-xx; arXiv: gr-qc/0512115

WikipediA: Equivalence principle (geometric)

Archive

Friday, 21 March 2014

Einstein's letter in Dmitri Ivanenko's Archive

In honour of Einstein's birthday:

In 1929, Albert Einstein was invited to participate to 1^st Soviet Conference on Theoretical Physics in Kharkiv, USSR. This is his answer that, unfortunately, he cannot do, and where he discusses some scientific questions.

This letter is in the Archive of Dmitri Ivanenko, an organizer of that Conference.

Archive

Friday, 14 March 2014

Foundations of Modern Physics 3: Classical field theory

Classical field theory admits a comprehensive formulation in terms of fibre bundles and graded manifolds.

Observable classical fields are even (commutative) electromagnetic and gravitational fields and odd (anti-commutative) Dirac spinor fields. One also considers classical non-Abelian gauge fields and Higgs fields. Classical gauge and gravitation field theories are conventionally formulated in the geometric terms of fibre bundles. Generalizing this geometric formulation, one comes to the following.

Axiom I. Even classical fields are sections of smooth fibre bundles over smooth manifolds.

As a consequence, it is essential that classical fields on a smooth manifold X represented by sections of a fibre bundle over X form a projective module of finite rank over the ring C(X) of smooth real functions on X in accordance with the well-known Serre – Swan theorem.

There are different descriptions of odd fields either on graded manifolds or supermanifolds. Note that both graded manifolds and supermanifolds are described in terms of sheaves of graded commutative algebras, but graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing of sheaves on supervector spaces. In order to treat odd and even fields on the same level, one can follow the above mentioned Serre – Swan theorem, which states that, if an anti-commutative algebra is generated by a projective C(X)-module of finite rank, it is isomorphic to the algebra of graded functions on a graded manifold whose body is X.

Axiom II. Odd classical fields on a smooth manifold X are elements of the structure algebra of a graded manifold whose body is X.

Dynamic equations of all observable classical fields including electromagnetic, spinor and gravitational fields are Euler – Lagrange equations derived from a Lagrangian. This fact leads us to the following.

Axiom III. Classical field theory is a Lagrangian theory.

Lagrangian theory on fibre bundles and graded manifolds is adequately formulated in algebraic terms of the variational bicomplex of exterior forms on jet manifolds. The Euler – Lagrange operator is a coboundary operator of this bicomplex and its cohomology provides the first variational formula, the first Noether theorem and conservation laws. Thus classical field theory is formulated in a complete way, but we obtain something more, namely, its prequantization.

Quantization of Lagrangian field theory essentially depends on its degeneracy characterized by a family of non-trivial reducible Noether identities. These Noether identities can obey first-stage Noether identities, which in turn are subject to the second-stage ones, and so on. This hierarchy of Noether identities is described by the exact Koszul - Tate chain complex of antifields. The second Noether theorem associates to this Koszul--Tate complex the cochain sequence of ghosts with the ascent gauge operator, whose components are gauge and higher-stage gauge symmetries of Lagrangian field theory. If gauge symmetries are algebraically closed, this gauge operator admits a nilpotent BRST prolongation. Thus we come to the BRST extension of original Lagrangian field theory which is a first step towards its quantization.

References:

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

G.Sardanashvily, Classical field theory. Advanced mathematical formulation, Int. J. Geom. Methods Modern Physics, 5 (2008) 1163 (#)

WikipediA: Covariant classical field theory

Archive

Tuesday, 28 January 2014

Foundations of Modern Physics: 2. The notion of an inertial frame

The key problem of classical mechanics is that there is no intrinsic definition of an inertial reference frame.

Classical non-relativistic mechanics admits the adequate mathematical formulation in terns of fibre bundle Q->R over the time axis R. In this framework, a reference frame is defined as a trivialization of this fibre bundle or, equivalently, as a connection on Q->R.

A second order dynamic equation is called a free motion equation if it can be brought into the form of a zero acceleration ddq/dtdt=0 with respect to some reference frame, and this reference frame is said to be inertial for this equation. Thus a definition of an inertial frame depends on the choice of a free motion equation.

A problem is that, given a different free motion equation ddq’/dtdt=0, an inertial reference frame for it fails to be so the first free motion equation ddq/dtdt=0, and their relative velocity is not constant.

In view of this problem, one should write dynamic equations of non-relativistic mechanics in terms of relative velocities and accelerations with respect to an arbitrary reference frame. However, in this case the strict mathematical notions of a relative acceleration and a non-inertial force are rather sophisticated.

References:

G.Sardanashvily, Relative non-relativistic mechanics, arXiv: 0708.2998

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Geometric Formulation of Classical and Quantum Mechanics (WS, 2010)

WikipediA: Free motion equation

Archive

Thursday, 23 January 2014

Божественное на физфаке МГУ

На Физическом ф-те Московского государственного университета, вслед за МИФИ, расскажут о божественном http://lenta.ru/news/2014/01/22/religia/

Я считаю это позором моего факультета - следствием беспринципности его декана и ректора МГУ

Да и формально, кто бы во что бы ни сподобился уверовать, при всем уважении к его вере, но физфак – факультет Московского ГОСУДАРСТВЕННОГО университета, а церковь отделена от государства, и религиозная пропаганда от его имени, а тем более в его стенах представляется неуместной.

Вот программа этой конференции, проводимой как секция XXII МЕЖДУНАРОДНЫХ РОЖДЕСТВЕНСКИХ ОБРАЗОВАТЕЛЬНЫХ ЧТЕНИЙ .

11.8. Конференция «Христианство и наука».

Сопредседатели: протоиерей Кирилл Копейкин, секретарь Санкт-Петербургской
Духовной академии и семинарии, кандидат богословия, к. ф.-м. н.;
Владимиров Юрий Сергеевич, профессор физического факультета МГУ им.
М. В. Ломоносова, д. ф.-м. н.
Куратор: Белинский Александр Витальевич, д. ф.-м. н., ст. н. с. физического факультета
МГУ им. М. В. Ломоносова.

Время проведения: 28 января, 10.00.

Место проведения: Физический факультет МГУ им. М.В. Ломоносова, аудитория 5-19.

Проезд: м. «Университет», далее одна остановка транспортом или пешком в сторону
Главного здания МГУ (ост. «Ул. Лебедева»).

1. Владимиров Юрий Сергеевич, профессор физического факультета МГУ имени
М. В. Ломоносова, д. ф.-м. н. «Фундаментальная физика и религия».

2. Протоиерей Кирилл Копейкин, секретарь Санкт-Петербургской Духовной
академии и семинарии, кандидат богословия, к. ф.-м. н. «Мир – физический
или психический?»

3. Катасонов Владимир Николаевич, доктор богословия, д. филос. н., профессор,
зав. кафедрой философии Общецерковной аспирантуры и докторантуры имени
свв. равноапп. Кирилла и Мефодия. «Парадоксы Галилео Галилея».

4. Протоиерей Димитрий Кирьянов. Тобольская духовная семинария.
«Эволюционное объяснение религии: выход за границы возможного».

5. Поройков Сергей Юрьевич, член Российского философского общества, к. ф.-
м. н. «Общие научные основания мировых религий».

6. Ефремов Александр Петрович, профессор Российского университета дружбы
народов, д. ф.-м. н. «О “частице Бога” и физической мифологии».

7. Векшенов Сергей Александрович, профессор Российской академии образования,
д. ф.-м. н. «Наука и вера в русской математической школе».

8. Постовалова Валентина Ильинична, профессор Института языкознания РАН,
д. филол. н. «Наука, философия, религия: богословские воззрения
протоиерея Александра Геронимуса».

9. Кречет Владимир Георгиевич, профессор Ярославского государственного
педагогического университета, д. ф.-м. н. «О метафизике разума и
человеческого сознания».

10. Протоиерей Василий Павлов, Симферопольская и Крымская епархия.
«Эволюция ноосферы».

11. Родионов Борис Устинович, профессор Российской академии образования, д. ф.-
м. н. «Божественные параллели аэронавтики».

12. Захаров Валерий Дмитриевич, ст. н. с. Всероссийского института научной и
технической информации, к. ф.-м. н. «Этнос и космос в свете религиозного
сознания (к религиозной интерпретации этнологии Льва Гумилева)».

13.Соркин Эдуард Исаевич, руководитель отделения Российского философского
общества «Основания и конструкции знания» школы Ю.П. Трусова. «Научная и
семейная реализация “вечной женственности”».

14. Игумен Владимир (Маслов), Свято-Екатерининский монастырь. «Не видеть
очевидного».

15. Белинский Александр Витальевич, ст. н. с. физического факультета МГУ им.
М.В. Ломоносова, д. ф.-м. н. «Мир сей во зле лежит».

16.Владимирова Татьяна Евгеньевна, профессор ЦМО МГУ им. М.В. Ломоносова,
д. филол. н. «Человек как духовное самосозидание».

17. Ерохин Владимир Петрович, редактор издательства «Волшебный фонарь».
«Недвойственные постижения: перекличка культур».

18. Муравник Галина Леонидовна, генетик, преподаватель Свято-Филаретовского
православного института. «Генетически модифицированные организмы.
Развенчание мифов и преодоление стереотипов».

И вот что получилось: http://www.gazeta.ru/science/2014/01/29_a_5872289.shtml

Архив

Wednesday, 15 January 2014

Foundations of Modern Physics: 1. The differential calculus on modules and rings

Differential geometry of smooth fiber bundles provides the comprehensive formulation of classical field theory and mechanics.

At the same time, 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 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 iff 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.

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 Aover 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.

References:

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

G. Sardanashvily, Lectures on differential geometry of modules and rings, arXiv: 0910.1515

WikipediA: Differential calculus over commutative algebras

WikipediA: Connection (algebraic framework)

Archive

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 .... "

Archive

Saturday, 14 December 2013

Program for International Student Assessment (PISA 2012). Results

PISA 2012 is program's 5^th survey which assessed the competencies of 15-year-olds in reading, mathematics and science (with a focus on mathematics) in 65 countries. Around 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).

Archive

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

Archive

Tuesday, 26 November 2013

Scientific Biography

My "Gennadi Sardanashvily: Scientific Biography" came out.

"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"

Archive

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."

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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).

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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