What is Gauge Gravitation Theory about?

Classical field theory admits a comprehensive mathematical formulation in the geometric terms of smooth fibre bundles. For instance, Yang – Mills gauge theory is theory of principal connections on principal bundles.

Gauge gravitation theory as particular classical field theory also is formulated in the terms of fibre bundles.

Studying gauge gravitation theory, one believes reasonable to require that it incorporates Einstein's General Relativity and, therefore, it should be based on Relativity and Equivalence Principles reformulated in the fibre bundle terms.

In these terms, Relativity Principle states that gauge symmetries of classical gravitation theory are general covariant transformations. It should be emphasized that these gauge symmetries differ from gauge symmetries of the above mentioned Yang – Mills gauge theory which constitute a gauge group of vertical automorphisms of a principal bundles. Fibre bundles possessing general covariant transformations constitute the category of so called natural bundles.

Let Y->X be a smooth fibre bundle. Any automorphism of Y, by definition, is projected onto a diffeomorphism of its base X. The converse is not true. A fibre bundle Y->X is called the natural bundle if there exists a monomorphism of the group of diffeomorphisms of X to the group of bundle automorphisms of Y->X, called general covariant transformations of Y. 

The tangent bundle TX of X exemplifies a natural bundle. Any diffeomorphism f of X gives rise to the tangent automorphisms Tf of TX which is a general covariant transformation of TX.The associated principal bundle is a fibre bundle LX of frames in the tangent spaces to X also is a natural bundle. Moreover, all fibre bundles associated with LX are natural bundles. Principal connections on LX yield linear connections on the tangent bundle TX and other associated bundles. They are called the world connections.

Following Relativity Principle, one thus should develop gravitation theory as gauge theory of principal connections on a principal frame bundle LX over a four-dimensional manifold X, called the world manifold. A key point however is that this gauge theory also is characterized by spontaneous symmetry breaking in accordance with geometric Equivalence Principle.

Though spontaneous symmetry breaking is quantum effect, spontaneous symmetry breaking in classical gauge theory on a principal bundle P->X with a structure Lie group G is characterized as a reduction of this structure group to its closed Lie subgroup H. By virtue of the well-known theorem, such a reduction takes place if and only if there exists a global sections h of the quotient bundle P/H->X which are treated as a classical Higgs field.

There are different formulations of Equivalence Principle in gravitation theory. In particular, 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. The weakest Equivalence Principle is restricted to the motion law of a probe point mass in a uniform gravitational field. Its 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.

The above mentioned variants of Equivalence Principle aim to guarantee the transition of General Relativity to Special Relativity in a certain reference frame. However, only the particular weakest and weak Equivalence Principles are true. To overcome this difficulty, Equivalence Principle can be formulated in geometric terms as follows. In the spirit of Felix Klein's Erlanger program, Special Relativity can be characterized as the Klein geometry of Lorentz group invariants. Then geometric Equivalence Principle is formulated to require the existence of Lorentz invariants on a world manifold X. This requirement holds if the tangent bundle of X admits an atlas with Lorentz transition functions, i.e., a structure group of the associated frame bundle LX of linear tangent frames in is reduced to the Lorentz group SO(1,3). By virtue of the above mentioned theorem, this reduction takes place if and only if the quotient bundle LX/SO(1,3) possesses a global section, which is a pseudo-Riemannian metric on X.

Thus geometric Equivalence Principle provides the necessary and sufficient conditions of the existence of a pseudo-Riemannian metric, i.e., a gravitational field on a world manifold. Based on geometric Equivalence Principle, gravitation theory is formulated as gauge theory where a gravitational field is described as a classical Higgs field responsible for spontaneous breakdown of world gauge symmetries which are general covariant transformations.

The character of gravity as a Higgs field responsible for spontaneous breaking of general covariant transformations is displayed as follows. Given different gravitational fields, the representations of holonomic coframes dx by Dirac matrices acting on Dirac spinor fields are nonequivalent. Consequently, Dirac operators in the presence of different gravitational fields fails to be equivalent, too. 

It follows that, since the Dirac operators in the presence of different gravitational fields are nonequivalent, Dirac spinor fields fail to be considered, e.g., in the case of a superposition of different gravitational fields. Therefore, quantization of a metric gravitational field fails to satisfy the superposition principle, and one can suppose that a metric gravitational field as a Higgs field is non-quantized in principle.

References:

WikipediA: Gauge gravitation theory

G.Sardanashvily, Classical gauge gravitation theory, Int. J. Geom. Methods Mod. Phys.8 (2011) 1869-1895.

Facebook: Gravitation Gauge Theory

I. How we developed gauge gravitation theory (from my Scientific Biography)

II. How we developed gauge gravitation theory (from my Scientific Biography)

Introduction to my book “Advanced Differential Geometry for Theoreticians”

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

In contrast with quantum field theory, classical field theory can be formulated in a strict mathematical way by treating classical fields as sections of smooth fibre bundles. This also is the case of time-dependent non-relativistic mechanics on fibre bundles over R. This book aim to compile the relevant material on fibre bundles, jet manifolds, connections, graded manifolds and Lagrangian theory. The book is based on the graduate and post graduate courses of lectures given at the Department of Theoretical Physics of Moscow State University (Russia). It addresses to a wide audience of mathematicians, mathematical physicists and theoreticians. It is tacitly assumed that the reader has some familiarity with the basics of differential geometry.

Contents

1 Geometry of fibre bundles: 1.1 Fibre bundles, 1.2 Vector and affine bundles, 1.3 Vector fields, 1.4 Exterior and tangent-valued forms.

2 Jet manifolds: 2.1 First order jet manifolds, 2.2 Higher order jet manifolds, 2.3 Differential operators and equations, 2.4 Infinite order jet formalism.

3 Connections on fibre bundles: 3.1 Connections as tangent-valued forms, 3.2 Connections as jet bundle sections, 3.3 Curvature and torsion, 3.4 Linear and affine connections, 3.5 Flat connections, 3.6 Connections on composite bundles.

4 Geometry of principal bundles: 4.1 Geometry of Lie groups, 4.2 Bundles with structure groups, 4.3 Principal bundles, 4.4 Principal connections, 4.5 Canonical principal connection, 4.6 Gauge transformations, 4.7 Geometry of associated bundles, 4.8 Reduced structure.

5 Geometry of natural bundles: 5.1 Natural bundles, 5.2 Linear world connections, 5.3 Affine world connections.

6 Geometry of graded manifolds: 6.1 Grassmann-graded algebraic calculus, 6.2 Grassmann-graded differential calculus, 6.3 Graded manifolds, 6.4 Graded differential forms.

7 Lagrangian theory: 7.1 Variational bicomplex, 7.2 Lagrangian theory on fibre bundles, 7.3 Grassmann-graded Lagrangian theory, 7.4 Noether identities, 7.5 Gauge symmetries.

8 Topics on commutative geometry: 8.1 Commutative algebra, 8.2 Differential operators on modules, 8.3 Homology and cohomology of complexes, 8.4 Differential calculus over a commutative ring, 8.5 Sheaf cohomology, 8.6 Local-ringed spaces.

30 Years of “The Gauge Treatment of Gravity”

Thirty years of our pioneer article: D.Ivanenko, G.Sardanashvily, "The Gauge Treatment of Gravity", Physics Reports, 94 (1983) 1-45, where a gravitational field (a pseudo-Riemannian metric) is described as a classical Higgs field responsible for spontaneous breakdown of space-time symmetries in accordance with the geometric Equivalence Principle.

References:

WikipediA: Higgs field (classical)

WikipediA: Gauge gravitation theory

G.Sardanashvily, Classical gauge gravitation theory, Int. J. Geom. Methods Mod. Phys. 8 (2011) 1869-1895.

What is a classical Higgs field

Spontaneous symmetry breaking, a vacuum Higgs field, a Higgs boson are quantum phenomena. A vacuum '''Higgs field''' is responsible for spontaneous symmetry breaking the gauge symmetries of fundamental interactions and provides the Higgs mechanism of generating mass of elementary particles. However, no adequate mathematical model of this Higgs vacuum has been suggested in the framework of quantum gauge theory, though somebody treats it as sui generis a condensate by analogy with that of Cooper pairs in condensed matter physics.

At the same time, classical gauge theory admits comprehensive geometric formulation where gauge fields are represented by connections on principal bundles. In this framework, spontaneous symmetry breaking is characterized as a reduction of the structure group G of a principal bundle P -> X to its closed subgroup H. By the well-known theorem, such a reduction takes place if and only if there exists a global section h of the quotient bundle P/G -> X. This section is treated as a classical Higgs field.

A key point is that there exists a composite bundle P -> P/G -> X where P -> P/G is a principal bundle with the structure group H. Then matter fields, possessing an exact symmetry group H, in the presence of classical Higgs fields are described by sections of some composite bundle E -> P/G -> X, where E -> P/G is some associated bundle to P -> P/G. Herewith, a Lagrangian of these matter fields is gauge invariant only if it factorizes through the vertical covariant differential of some connection on a principal bundle P -> P/G, but not P -> X.

An example of a classical Higgs field is a classical gravitational field identified with a pseudo-Riemannian metric on a world manifold X. In the framework of gauge gravitation theory, it is described as a global section of the quotient bundle FX/O(1,3) ->  X where FX is a principal bundle of the tangent frames to X with the structure group GL(4,R).

WikipediA

Lectures on integrable Hamiltonian systems

G.Sardanashvily, Lectures on integrable Hamiltonian systems, arXiv: 1303.5363 

Abstract. We consider integrable Hamiltonian systems in a general setting of invariant submanifolds which need not be compact. For instance, this is the case a global Kepler system, non-autonomous integrable Hamiltonian systems and integrable systems with

time-dependent parameters.

Introduction

The Liouville -- Arnold theorem for completely integrable systems, the Poincar\'e -- Lyapounov -- Nekhoroshev theorem for partially integrable systems and the Mishchenko -- Fomenko theorem for the superintegrable ones state the existence of action-angle coordinates around a compact invariant submanifold of a Hamiltonian integrable system which is a torus. However, it is well known that global extension of these action-angle coordinates meets a certain topological obstruction.

Note that superintegrable systems sometimes are called non-commutative (or non-Abelian) completely integrable systems.

In these Lectures, we consider integrable Hamiltonian systems in a general setting of invariant submanifolds which need not be compact. These invariant submanifolds are proved to be diffeomorphic to toroidal cylinders. A key point is that a fibred manifold whose fibres are diffeomorphic either to a compact manifold or an Euclidean space is a fibre bundle, but this is not the case of toroidal cylinders.

In particular, this is the case of non-autonomous integrable Hamiltonian systems and Hamiltonian mechanics with time-dependent parameters.

It may happen that a Hamiltonian system on a phase space Z falls into different integrable Hamiltonian systems on different open subsets of Z. For instance, this is the case of the Kepler system. It contains two different globally superintegrable systems on different open subsets of a phase space Z. Their integrals of motion form the Lie algebras so(3) and so(2,1) with compact and non-compact invariant submanifolds, respectively.

Geometric quantization of completely integrable and superintegrable Hamiltonian systems with respect to action-angle variables is considered. The reason is that, since a

Hamiltonian of an integrable system depends only on action variables, it seems natural to provide the Schrodinger representation of action variables by first order differential

operators on functions of angle coordinates.

Throughout the Lectures, all functions and maps are smooth, and manifolds are real smooth and paracompact. We are not concerned with the real-analytic case because a paracompact real-analytic manifold admits the partition of unity by smooth functions. As a consequence, sheaves of modules over real-analytic functions need not be acyclic that is essential for our consideration.

 

My articles in WikipediA

Gauge gravitation theory

Variational bicomplex

Covariant Hamiltonian field theory

Non-autonomous mechanics

Superintegrable Hamiltonian system

Connection (algebraic framework)

Gauge symmetry (mathematics)

Supergeometry

Lagrangian system

Graded manifold

Noether identities

Relativistic system (mathematics)

Free motion equation

Non-autonomous system (mathematics) 

Fibre bundle formulation of time-dependent mechanics

G.Sardanashvily, Fibre bundle formulation of time-dependent mechanics, arXiv: 1303.1735

  

Abstract. We address classical and quantum mechanics in a general setting of arbitrary time-dependent transformations. Classical non-relativistic mechanics is formulated as a particular field theory on smooth fibre bundles over a time axis R. Connections on these bundles describe reference frames. Quantum time-dependent mechanics is phrased in geometric terms of Banach and Hilbert bundles and connections on these bundles. A quantization scheme speaking this language is geometric quantization. 

Introduction

 The technique of symplectic manifolds is well known to provide the adequate Hamiltonian formulation of autonomous mechanics . Its realistic example is a mechanical system whose configuration space is a manifold M and whose phase space is the cotangent bundle T*M of M provided with the canonical symplectic form W. Any autonomous Hamiltonian system locally is of this type.

However, this geometric formulation of autonomous mechanics is not extended to mechanics under time-dependent transformations because the symplectic form W fails to be invariant under these transformations. As a palliative variant, one has developed time-dependent mechanics on a configuration space Q=RxM where R is the time axis. Its phase space RxT*M is provided with the pull-back of the form W. However, this presymplectic form also is broken by time-dependent transformations.

We address non-relativistic mechanics in a case of arbitrary time-dependent transformations. Its configuration space is a fibre bundle Q->R endowed with bundle coordinates (t,q), where t is the standard Cartesian coordinate on the time axis R with transition functions t'=t+const. Its velocity space is the first order jet manifold JQ of sections of Q->R. A phase space is the vertical cotangent bundle V*Q of Q->R.

This formulation of non-relativistic mechanics is similar to that of classical field theory on fibre bundles over a base of dimension >1. A difference between mechanics and field theory however lies in the fact that connections on bundles over R are flat, and they fail to be dynamic variables, but describe reference frames.

Note that relativistic mechanics is adequately formulated as particular classical string theory of one-dimensional submanifolds.

Graded Lagrangian formalism

G.Sardanashvily, Graded Lagrangian formalism, International Journal of Geometric Methods in Modern Physics 10 (2013) N5 1350016 #

Abstract. Graded Lagrangian formalism in terms of a Grassmann-graded variational bicomplex on graded manifolds is developed in a very general setting. This formalism provides the comprehensive description of reducible degenerate Lagrangian systems, characterized by hierarchies of non-trivial higher-order Noether identities and gauge symmetries. This is a general case of classical field theory and Lagrangian non-relativistic mechanics.

Introduction

Conventional Lagrangian formalism on fibre bundles Y->X over a smooth manifold X is formulated in algebraic terms of a variational bicomplex of exterior forms on jet manifolds of sections of Y->X [2, 9, 16, 17, 19, 30, 36, 37]. The cohomology of this bicomplex provides the global first variational formula for Lagrangians and Euler – Lagrange operators, without appealing to the calculus of variations. For instance, this is the case of classical field theory if dimX>1 and non-autonomous mechanics if X=R [19, 20, 35].

However, this formalism is not sufficient in order to describe reducible degenerate Lagrangian systems whose degeneracy is characterized by a hierarchy of higher order Noether identities. They constitute the Kozul--Tate chain complex whose cycles are Grassmann-graded elements of certain graded manifolds [7, 8, 19]. Moreover, many field models also deal with Grassmann-graded fields, e.g., fermion fields, antifields and ghosts [19, 21, 35].

These facts motivate us to develop graded Lagrangian formalism of even and odd variables [8, 17, 19, 34].

Different geometric models of odd variables are described either on graded manifolds or supermanifolds. Both graded manifolds and supermanifolds are phrased in terms of sheaves of graded commutative algebras [5, 19]. However, graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing of sheaves on supervector spaces. Treating odd variables on a smooth manifold X, we follow the Serre – Swan theorem generalized to graded manifolds (Theorem 7). It states that, if a graded commutative C(X)-ring is generated by a projective C(X)-module of finite rank, it is isomorphic to a ring of graded functions on a graded manifold whose body is X. In accordance with this theorem, we describe odd variables in terms of graded manifolds [8, 17, 19, 34].

We consider a generic Lagrangian theory of even and odd variables on an n-dimensional smooth real manifold X. It is phrased in terms of the Grassmann-graded variational bicomplex (28) [4, 7, 8, 17, 19, 34]. Graded Lagrangians L and Euler – Lagrange operators are defined as elements of this bicomplex. Cohomology of the Grassmann-graded variational bicomplex (28) (Theorems 13 - 14) defines a class of variationally trivial graded Lagrangians (Theorem 15) and results in the global decomposition (33) of dL (Theorem 16), the first variational formula (37) and the first Noether Theorem 20.

A problem is that any Euler – Lagrange operator satisfies Noether identities, which therefore must be separated into the trivial and non-trivial ones. These Noether identities obey first-stage Noether identities, which in turn are subject to the second-stage ones, and so on. Thus, there is a hierarchy of higher-stage Noether identities. In accordance with general analysis of Noether identities of differential operators [33], if certain conditions hold, one can associate to a graded Lagrangian system the exact antifield Koszul – Tate complex (62) possessing the boundary operator (60) whose nilpotentness is equivalent to all non-trivial Noether and higher-stage Noether identities [7, 8, 18].

It should be noted that the notion of higher-stage Noether identities has come from that of reducible constraints. The Koszul – Tate complex of Noether identities has been invented similarly to that of constraints under the condition that Noether identities are locally separated into independent and dependent ones [4, 13]. This condition is relevant for constraints, defined by a finite set of functions which the inverse mapping theorem is applied to. However, Noether identities unlike constraints are differential equations. They are given by an infinite set of functions on a Frechet manifold of infinite order jets where the inverse mapping theorem fails to be valid. Therefore, the regularity condition for the Koszul – Tate complex of constraints is replaced with homology regularity Condition 27 in order to construct the Koszul – Tate complex (62) of Noether identities.

The second Noether theorems (Theorems 32 – 34) is formulated in homology terms, and it associates to this Koszul – Tate complex the cochain sequence of ghosts (71) with the ascent operator (72) whose components are non-trivial gauge and higher-stage gauge symmetries of Lagrangian theory.

New Managing Editor of IJGMMP

I was a founder of International Journal of Geometric Methods in Modern Physics (IJGMMP) in 2003 and its Managing Editor this decade. In the middle of 2012, I decided to leave this position because of my age.

New Managing Editor of IJGMMP is Professor Mauro Francaviglia who starts his activity in February 2013 with preparing V 11 (2014) of IJGMMP.

I remains responsible only for V 10 (2013) of the Journal which is already complete. 

Any theory is either incomplete or contradictory

Biographic books of WikipediA: Theoretical Physicists, …

The content of these books primarily consists of articles available from WikipediA or other free sources online.

Theoretical Physicists (Source: Wikipedia , General Books, 2011) - 348 pp.

ISBN: 115104881X, 9781151048813

Contents: Albert Einstein, Isaac Newton, Richard Feynman, Erwin Schr dinger, Niels Bohr, Max Planck, Werner Heisenberg, Enrico Fermi, Christiaan Huygens, Paul Dirac, Wolfgang Pauli, Freeman Dyson, William Rowan Hamilton, J. Robert Oppenheimer, Louis de Broglie, Eugene Wigner, Josiah Willard Gibbs, John Bardeen, Edward Witten, Murray Gell-Mann, Lars Onsager, Max Born, David Deutsch, Satyendra Nath Bose, Lev Landau, Hendrik Lorentz, Wilhelm Wien, John C. Baez, Lee Smolin, John Archibald Wheeler, Rudolf Clausius, Galileo Galilei, Abdus Salam, Pierre-Simon Laplace, Stephen Hawking, James Clerk Maxwell, Riazuddin, Nikolay Bogolyubov, Antony Garrett Lisi, Hans Bethe, Michio Kaku, Hugh Everett III, Leonard Susskind, Gerard 't Hooft, Fayyazuddin, Reinhard Oehme, Masud Ahmad, Steven Weinberg, Frank J. Tipler, Nicolas Rashevsky, Johannes Diderik van der Waals, Julian Schwinger, Faheem Hussain, Alexander Kuzemsky, Alan Guth, Vladimir Korepin, Arthur Compton, Brian Greene, Pieter Zeeman, David Pines, Tsung-Dao Lee, Frank Wilczek, James J. Kay, Rolf Hagedorn, Peter Higgs, Vitaly Ginzburg, Jim Al-Khalili, Chen Ning Yang, Isidor Isaac Rabi, Tasneem Zehra Husain, John D. Barrow, David L. Webster, Francisco Jose Yndurain Mu oz, Johann Rafelski, Philip Warren Anderson, C. R. Hagen, Thomas G. Barnes, Nima Arkani-Hamed, John Clive Ward, Marlan Scully, Xiao-Gang Wen, Feza G rsey, Vladimir Gribov, H. Stanley Allen, Mikhail Shifman, Yoichiro Nambu, Martinus J. G. Veltman, Sean M. Carroll, David A. Frank-Kamenetskii, Sidney Coleman, Sheldon Lee Glashow, Jack Sarfatti, Ferenc Krausz, Fred Alan Wolf, Giorgio Parisi, Gennadi Sardanashvily, A. P. Balachandran, Laurent Nottale, Lisa Randall, Alexei Alexeyevich Abrikosov, Owen Willans Richardson, Kenneth G. Wilson, Fran ois Englert, Kazuhiko Nishijima, Tom W. B. Kibble, Boris Chirikov, Bernard d'Espagnat...

Russian Physicists (Source: Wikipedia , General Books, 2010) - 582 pp

ISBN: 1157385842, 9781157385844

Contents: Leonhard Euler, Mikhail Lomonosov, George Gamow, Igor Dmitriyevich Novikov, Zhores Alferov, Lev Landau, Igor Tamm, A. I. Shlyakhter, Vladimir Steklov, Alexander Prokhorov, Mikhail Lavrentyev, Ilya Prigogine, Nikolay Bogolyubov, Gustav Heinrich Johann Apollon Tammann, Alexei Fridman, Dmitri Z. Garbuzov, Alexander Kuzemsky, Eugene Podkletnov, Victor Veselago, Alexander S. Potupa, Nikolay Neprimerov, Boris Rauschenbach, List of Russian astronomers and astrophysicists, Vitaly Ginzburg, Igor Kurchatov, Yevgeny Zavoisky, Abraham Zelmanov, Aleksandr Danilovich Aleksandrov, List of Russian physicists, Roman Maev, Dmitri Ivanenko, Abram A. Slutskin, Konstantin Novoselov, Roald Sagdeev, Alexander Stepanovich Popov, Rashid Sunyaev, Victor Balykin, Eduard Shpolsky, Akiva Yaglom, Mikhail Shifman, Andrei Monin, Vera Yurasova, Pyotr Valentinovich Trusov, Pyotr Kapitsa, Gennadi Sardanashvily, Alexei Alexeyevich Abrikosov, Vladimir Krivchenkov, Nikolay Semyonov, George M. Zaslavsky, Oleg Losev, Abram Ioffe, Ivan Pulyui, Alexander Friedmann, Evgeny Velikhov, Pavel K. Oshchepkov, Boris Chirikov, Igor Sutyagin, Vladimir Ignatowski, Benjamin Fain, Ludvig Faddeev, Ilya Frank, Evgeny Aramovich Abramyan, Grigory Landsberg, Boris Podolsky, Aleksandr Stoletov, Alexander Markovich Polyakov, Vladimir Teplyakov, Semen Altshuler, Artem Alikhanian, Boris Hessen, Galina Kurlyandskaya, Sergei P. Kurdyumov, Oleg Firsov, Georgy Golitsyn, Anatoly Vlasov, Eugene Levich, Mikhail Rabinovich, Pavel Cherenkov, Yuri Orlov, Vladimir Markovich Entov, Dmitry Zubarev, Nikolay Basov, Lev Bulat, Sergei Kapitsa, Edward Drobyshevski, Grigoriy A. Gamburtsev, Yurii Shirokov, Sergei Tyablikov, Yakov Frenkel, Leonid Isaakovich Mandelstam, Dmitry Shirkov, Sergey Nikitin, Mikhail Ostrogradsky, Viktor Dilman, Serge Timashev, Igor Ternov, Nikolay Umov, Arseny Sokolov, ...

Russian Scientists (Source: Wikipedia , General Books, 2011) - 134 pp

ISBN: 1156961084, 9781156961087

Contents: Dmitri Mendeleev, Ivan Pavlov, Sofia Kovalevskaya, Konstantin Tsiolkovsky, lie Metchnikoff, Ilya Prigogine, List of Russian scientists, Vladimir Kovalevsky, Felix Ziegel, Zaid Orudzhev, Vladimir Shukhov, Jeffrey Yi-Lin Forrest, Victor V. Tikhomirov, Dmitri Z. Garbuzov, Nikolai Vavilov, Vladimir N. Beneshevich, Alexander Kuzemsky, Leibin, Valery Moiseevich, Georgiy B. Shul'pin, Alexander S. Potupa, Anatoly Koryagin, Peter Simon Pallas, Ivan Timokhovich, Dmitry Okhotsimsky, Struve family, Anatoly Fomenko, Askar Akayev, Georg Wilhelm Steller, Mstislav Keldysh, Igor Spassky, Ivan Ostromislensky, Nikolai Kibalchich, Friedrich Georg Wilhelm von Struve, Victor Ovcharenko, Nikolay Ivanovich Pirogov, Boris Dubin, Ivan Vladimirovich Michurin, David A. Frank-Kamenetskii, Grigori Ivanovitch Langsdorff, Gennady Simeonovich Osipov, Vladimir Porfiriev, Gennadi Sardanashvily, Arkhip Mikhailovich Lyulka, Leonid Brekhovskikh, Yuri Izrael, Vladimir Wiese, Sergei M. Plekhanov, Erik Laxmann, Khabibullo Abdusamatov, Aleksei N. Leontiev, Eduard Toll, Vasily Karazin, Nikolai Alexandrovich Morozov, Dmitry Dmitrievich Maksutov, Yuri Levada, Anatoly Vlasov, Alexander von Middendorff, Oleg Ivanovich Mamayev, Dmitry Zubarev, Mikhail Matinsky, Victor Glushkov, Sergey Polyakov, Alexander Abramov, Sergei Tyablikov, Yuri Savenko, Georgy Gause, Igor Ternov, Arseny Sokolov, Alexander Kuchin, Gerhardt Friedrich M ller, Alexei Yuryevich Smirnov, Daniel Gottlieb Messerschmidt, Pyotr Zinchenko, Aleksandr Chudakov, Mikhail Berulava, Anatoli Bugorski, Franz Aepinus, Andrey Alexandrovich Verbitsky, Yuri Nuller, Mikhail Yangel, Igor Smirnov, Simon El'evich Shnoll, Nikolay Mitrofanovich Krylov, Anatoly Sagalevich, Pavel Molchanov, Boris Babkin, Alexander Nadiradze, Afanasiy Ilich Tobonov, Boris Chertok, Alexandr Kapto, …

Moscow State University Faculty: (Source: Wikipedia , General Books, 2010)  - 382 pp

ISBN: 1155911733, 9781155911731

Contents: Andrey Kolmogorov, Mikhail Lomonosov, Vladimir Arnold, Pavel Samuilovich Urysohn, Lev Landau, Igor Tamm, Alexander Prokhorov, Mikhail Suslov, Nikolay Bogolyubov, Zaid Orudzhev, Marina Karaseva, Alexander Luria, Alexei Fridman, Konstantin Pobedonostsev, Vladimir Guerrier, Olga Arsenievna Oleinik, Alexander Piatigorsky, Igor Kurchatov, Nikolai Luzin, Ivan Petrovsky, Anatoly Fomenko, Dmitri Ivanenko, Yakov Borisovich Zel'dovich, Akiva Yaglom, Valentin Afraimovich, Edward George, Baron George, Pyotr Kapitsa, Gennadi Sardanashvily, Alexei Alexeyevich Abrikosov, Vladimir Krivchenkov, Otto Schmidt, Revaz Dogonadze, Sergei Sobolev, Sergei Novikov, Aleksei N. Leontiev, Lev Gudkov, Hasan Shaheed Suhrawardy, Aleksandr Stoletov, Oleg Lupanov, Sergei Adian, Yuri Levada, Boris Hessen, Sergei P. Kurdyumov, Valery Legasov, Anatoly Vlasov, Alexander Gelfond, Pavel Alexandrov, Oleg Ivanovich Mamayev, Dmitri Egorov, Kurt Fabri, Johann Fischer von Waldheim, Iosif Shklovsky, Sergey Alexandrovich Markov, Yurii Shirokov, Pavel Petrovich Parenago, Dmitry Shirkov, Nikolay Burdenko, Igor Ternov, Arseny Sokolov, Yakov G. Sinai, John Naisbitt, Maria Smith-Falkner, Alexei Yuryevich Smirnov, Aleksandr Chudakov, Sergey Ivanovich Vavilov, Nikolai Sergeevich Bakhvalov, D. D. Morduhai-Boltovskoi, Nikolay Zelinsky, Sergei Nikolaevich Trubetskoy, Marina Solodkin, Veniamin Kagan, Vyacheslav Vsevolodovich Ivanov, Valentin Yanin, Khariton Chebotaryov, Aleksandr Khinchin, Nikolay Bogolepov, Lev Schnirelmann, Viktor Sadovnichiy, Tikhon Rabotnov, Albert Shiryaev, Yuri Osipyan, Viktor Buchstaber, Gleb Vladimirovich Nosovsky, Nikolai Brashman, Nikolay Beketov, Boleslav Mlodzeevskii, Pyotr Lebedev, Boris Grakov, Lazar Lyusternik, Haljand Udam, Mikhail Ovsyannikov, Semyon Desnitsky, Irina Antonova, Sergei Fomin, Alexander Spirin, Mikhail Volkenshtein, …

Ambarzumyan, Ivanenko and the inverse Sturm - Liouville problem

I have seen a recent article: arXiv: 1301.3276, devoted to the pioneer work of Victor Ambarzumyan on the Sturm – Liouville inverse problem in 1929 (Zeitschrift f\"ur Physik, 53, 690-695). It is interesting that an idea of this work belongs to Dmitri Ivanenko, as it follows from the letter of Ambarzumyan to Ivanenko in connection with Ivanenko’s 60s jubilee in 1964. The letter is in Russian, and I provide the English translation of this part.

"In one of our discussions, you posed a question on the uniqueness of definition of a mechanical system by the spectrum of its eigenvalues. It motivated me to write my first article of 1929 on the inverse Sturm – Liouvolle problem. After that, many authoritative mathematicians devoted their attempts to this problem. If today some of them mention my work, then I in turn give you thanks for that discussion which occurred in 1928."

My favorite painting in the structuralism style

Nikolai Estis, Composition, 1985

©G.Sardanashvily

Jet manifold formalism (from my Scientific Biography)

My Scientific Biography… Fourth period (1990 - 1999)

In autumn of 1987, in the framework of scientific cooperation between Moscow State University and University of Camerino (Italy) professor Luigi Mangiarotti arrived in Moscow. He made a report at the seminar of Ivanenko. His report was geometric, on the fibre bundle technique, but I understood nothing. And in spring of 1989, I myself went to him for a month in Italy. Since then, our cooperation continues for more than 20 years. I opened new geometric methods for me, which enable me to give an exhaustive mathematical formulation both of classical field theory and classical relativistic mechanics.

Pursuing gauge theory in the language of fibre bundles, I met the fact that the dynamics of this theory is formulated in a traditional form  of an action functional, variations of fields, variational equations and so on, not related to geometrization. At the same time, in mathematics, has long been developed an apparatus of jet manifolds jets for theory of nonlinear differential operators, differential equations and Lagrangian theory. However, it was completely unknown to theoreticians, and now remains little-known to them. It was that Luigi Mangiarotti told at the seminar of Ivanenko.

The essence of formalism of jet manifolds is that sections of a fibre bundle  Y → X are identified by their values and values of their partial derivatives up to some order k at a point  x of a manifold X. The key point is that the set of all such equivalence classes forms a smooth finite-dimensional manifold  J^kY, called the k-order jet manifold of sections of a fibre bundle  Y → X . This enables one, for the analysis of a  k-order differential equation, consider not some infnite-dimensional functional space of smooth sections, but a finite-dimensional jet manifold, and define this differential equation as some its submanifold. Respectively, a differential  operator on sections of  Y → X is defined as a mapping of a jet manifold  J^kY to some vector bundle  E → X , and a k-order Lagrangian L is defined as an n-form (n=dim X) on  J^kY.

Moreover, connections on a fibre bundle  Y → X also are expressed in terms of jet manifolds: they are sections of the jet bundle  J^1Y →Y. Thus, jet manifolds provide the language of differential geometry. The fact is that linear connections as like as linear differential operators can be described in different ways, but the nonlinear ones can be done only in formalism of jet manifolds.

In 1989 - 1990, I was engaged in the study of formalism jet manifolds, and my first works, where it is used, are the articles on classical theory of spontaneous symmetry breaking [63,64], multimomentum Hamiltonian field theory [65,66] and a book on  gauge gravitation theory [9] in 1991 - 92.

At that time, my attention was also attracted to formalism of differential operators on modules over an arbitrary algebra [12]. It also included the machinery of jets of modules, and led to differential geometry (differential forms, connections, etc.) on modules. This formalism, in particular, lies in the basis of non-commutative geometry. Its connection with familiar differential geometry on vector bundles is expressed by the well-known Serre - Swan theorem (generalized by me to non-compact manifolds [15]) that every projective module of finite rank over a ring of smooth functions on a manifold X is a module of sections of some vector bundle over X, and vice versa. Hereinafter, I have repeatedly addressed this formalism for constructing geometry of graded manifolds and for geometric formulation of non-autonomous quantum mechanics [15,16,17].  

D.Ivanenko’s proton-neutron model of atomic nuclei of 1932

In 1932, Soviet physicist Dmitri Ivanenko proposed the proton-neutron model of atomic nuclei. One usually refers to Ivanenko's short letter [1] of April 21, 1932 in Nature, which was quoted by W. Heisenberg in his first work on the model of nuclei submitted to Zs. f. Phys on June 7, 1932 [7].

However, Ivanenko published five works on his model in 1932 [1-5].

In the above-mentioned first one, he proposed that atomic nuclei consist of alpha-particles and neutrons, and assumed the existence of beta-electrons in nuclei as constituents of these alpha-particles and neutrons. In the second and third works [2,3], Ivanenko stated that atomic nuclei contain only protons and neutrons, but electrons are created under beta-decay in accordance with the Ambarzumian - Ivanenko  hypothesis of creation of massive particles of 1930 [6].

In the next articles [4,5], D. Ivanenko and E. Gapon proposed the idea of the shell distribution of protons and neutrons in nuclei.

References:

[1]  Iwanenko D., The neutron hypothesis, Nature, 129, N 3265 (1932) 798.

[2] Iwanenko D., Neutronen und kernelektronen, Physikalische Zeitschrift der Sowjetunion 1 (1932) 820-822.

[3]  Iwanenko D., Sur la constitution des noyaux atomiques, Compt. Rend. Acad Sci. Paris, 195 (1932).439-441.

[4]  Gapon E., Iwanenko D., Zur Bestimmung der isotopenzahl, Die Naturwissenschaften 20 (1932) 792-793.

[5] Gapon E., Iwanenko D., Zur Bestimmung der isotopenzahl, Physikalische Zeitschrift der Sowjetunion 2 (1932) 99-100.

[6] Ambarzumian V., Iwanenko D., Les électrons inobservables et les rayons, Compt. Rend. Acad Sci. Paris 190 (1930) 582.

[7] Heisenberg W., Uber den Bau der Atomkerner I, Zeitschrift für Physik A  77 (1932) 1-11.

My review “Axiomatic quantum field theory”

G. Sardanashvily, Axiomatic quantum field theory. Jet formalism, arXiv: 0707.4257

Jet formalism provides the adequate mathematical formulation of classical field theory, reviewed in hep-th/0612182. A formulation of QFT compatible with this classical one is discussed. We are based on the fact that an algebra of Euclidean quantum fields is graded commutative, and there are homomorphisms of the graded commutative algebra of classical fields to this algebra. As a result, any variational symmetry of a classical Lagrangian yields the identities which Euclidean Green functions of quantum fields satisfy.

Different citation indices

At present, the following three citation indices are widely accepted:

H-index http://en.wikipedia.org/wiki/H-index

G-index http://en.wikipedia.org/wiki/G-index

I-index http://en.wikipedia.org/wiki/I10-index

My ones are: H-index = 31, G-index = 49, I100 = 3.

My review “Axiomatic classical (prequantum) field theory”

G. Sardanashvily, Axiomatic classical (prequantum) field theory. Jet formalism
arXiv:hep-th/0612182

  

Abstract. In contrast with QFT, classical field theory can be formulated in a strict mathematical way if one defines even classical fields as sections of smooth fiber bundles. Formalism of jet manifolds provides the conventional language of dynamic systems (nonlinear differential equations and operators) on fiber bundles. Lagrangian theory on fiber bundles is algebraically formulated in terms of the variational bicomplex of exterior forms on jet manifolds where the Euler–Lagrange operator is present as a coboundary operator. This formulation is generalized to Lagrangian theory of even and odd fields on graded manifolds. Cohomology of the variational bicomplex provides a solution of the global inverse problem of the calculus of variations, states the first variational formula and Noether’s first theorem in a very general setting of  supersymmetries depending on higher-order derivatives of fields. A theorem on the Koszul–Tate complex of reducible Noether identities and Noether’s inverse second theorem extend an original field theory to prequantum field-antifield BRST theory. Particular field models, jet techniques and some quantum outcomes are discussed.

Contents

I. Introduction

II. ACFT. The general framework

1. The main postulate, 2. Jet manifolds, 3. Jets and connections, 4. Lagrangian theory

of even fields, 5. Odd fields, 6. The algebra of even and odd fields, 7. Lagrangian theory

of even and odd fields, 8. Noether’s first theorem in a general setting, 9. The Koszul–Tate complex of Noether identities, 10. Noether’s inverse second theorem, 11. BRST extended field theory, 12. Local BRST cohomology.

III. Particular models

13. Gauge theory of principal connections, 14. Topological Chern–Simons theory, 15.

Topological BF theory, 16. SUSY gauge theory, 17. Field theory on composite bundles,

18. Symmetry breaking and Higgs fields, 19. Dirac spinor fields, 20. Natural and gauge natural bundles. 21. Gauge gravitation theory, 22. Covariant Hamiltonian field theory,

23. Time-dependent mechanics, 24. Jets of submanifolds, 25. Relativistic mechanics, 26. String theory.

IV. Quantum outcomes

27. Quantum master equation, 28. Gauge fixing procedure, 28. Green function identities

Victor Ambartsumian and Dmitri Ivanenko in history of Quantum Field Theory

WikipediA article "History of Quantum Field Theory" says the following.

“… Of great importance are the studies of Soviet physicists, Viktor Ambartsumian and Dmitri Ivanenko, in particular the Ambarzumian - Ivanenko hypothesis of creation of massive particles (published in 1930) which is the cornerstone of the contemporary quantum field theory. The idea is that not only the quanta of the electromagnetic field, photons, but also other particles (including particles having nonzero rest mass) may be born and disappear as a result of their interaction with other particles. This idea of Ambartsumian and Ivanenko formed the basis of modern quantum field theory and theory of elementary particles.”

Reference:

V. Ambarzumian, D. Iwanenko, Les électrons inobservables et les rayons, Compt. Rend. Acad Sci. Paris 190 (1930) 582.