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 (#).

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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)

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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.

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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

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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

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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)

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