Tuesday, 26 March 2013
Friday, 15 March 2013
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.
Sunday, 3 March 2013
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.
Friday, 22 February 2013
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.
Saturday, 9 February 2013
Saturday, 26 January 2013
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, …
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, …
Wednesday, 16 January 2013
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
"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."
Tuesday, 8 January 2013
Friday, 21 December 2012
Jet manifold formalism (from my Scientific Biography)
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 Italy
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].
Monday, 10 December 2012
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:
[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.
Monday, 3 December 2012
My review “Axiomatic quantum field theory”
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.
Thursday, 22 November 2012
Different citation indices
At present, the following three citation indices are widely accepted:
My ones are: H-index = 31, G-index = 49, I100 = 3.
Thursday, 15 November 2012
My review “Axiomatic classical (prequantum) field theory”
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
Wednesday, 7 November 2012
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.
Monday, 29 October 2012
My Scientific Biography
Gennadi A. SARDANASHVILY, theoretician and
mathematical physicist, principal research
scientist of the Department of Theoretical Physics, Moscow State
University
Was born March 13, 1950, Moscow
In 1967, he graduated from
the Mathematical Superior Secondary School No.2 (Moscow
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:
geometric formulation of
classical field theory, where classical fields are represented by sections of
fibre bundles;
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;
gauge gravitation theory,
where a gravitational field is treated as the Higgs one which is responsible
for spontaneous breaking of space-time symmetries;
covariant (polysymplectic) Hamiltonian field
theory, where momenta correspond to derivatives of fields with respect to all
world coordinates;
geometric formulation of
classical non-relativistic mechanics (in terms of fibre bundles);
geometric formulation of
relativistic mechanics (in terms of one-dimensional submanifolds);
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
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*Y is 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 identities 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
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->X is 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 to h*: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 pseudo-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
associated with LX-> LX/SO(3,1).
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 of Q->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 for completely 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 modeled 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
Student period and the first works
In 1967, I graduated from the Moscow
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