At present, the following three citation indices are widely accepted:
My ones are: H-index = 31, G-index = 49, I100 = 3.
Thursday, 22 November 2012
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
Sunday, 21 October 2012
Humanity will never leave the Solar system
If the known physical laws are not violated, humanity will forever
remains in the Solar system. But in the Solar system, apparently, there is
nothing principally new and interesting for fundamental science. Perhaps there
is only some primitive form of life in atmospheres of the gas giants.
However, to move freely even in the Solar system, we need a velocity at least of
about 1000 km/sec. Annihilation of matter and antimatter only is the known
source of energy for that a spacecraft of mass of a few hundred tons could
reach this velocity. And one needs tens of kilograms of antimatter. This is
unlikely to ever become a reality.
Saturday, 13 October 2012
Introduction to my book “Lectures on Differential Geometry of Modules and Rings”
G. Sardanashvily, “Lectures on Differential Geometry of Modules
and Rings. Application to Quantum Theory" (Lambert Academic Publishing, Saarbrucken
Geometry of classical mechanics and field theory mainly is differential
geometry of finite-dimensional smooth manifolds and fibre bundles.
At the same time, the standard mathematical language of quantum
mechanics and field theory has been long far from geometry. In the last twenty
years, the incremental development of new physical ideas in quantum theory has
called into play advanced geometric techniques, based on the deep interplay
between algebra, geometry and topology.
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 (except
non-linear differential operators) 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 if and only if 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.
In quantum theory, Banach and Hilbert manifolds, Hilbert bundles and
bundles of C*-algebras over smooth
manifolds are considered. Their differential geometry also is formulated as
geometry of modules, in particular, C(X)-modules.
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 A over 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.
The present book aims to summarize the relevant material on the
differential calculus and differential geometry on modules and rings. Some
basic applications to quantum models are considered.
For the sake of convenience of the reader, several topics on cohomology
are compiled in Appendix.
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 ) and the Department of Mathematics and
Physics of University of Camerino (Italy
Contents
1
Commutative geometry: 1.1 Commutative algebra, 1.2 Differential operators
on modules and rings, 1.3 Connections on modules and rings, 1.4 Differential calculus
over a commutative ring, 1.5 Local-ringed spaces, 1.6 Differential geometry
of C(X)-modules, 1.7 Connections on local-ringed spaces.
2
Geometry of quantum systems: 2.1 Geometry of Banach manifolds, 2.2 Geometry of
Hilbert manifolds, 2.3 Hilbert and C*-algebra bundles, 2.4 Connections on
Hilbert and C*--algebra bundles, 2.5 Instantwise quantization, 2.6 Berry
3
Supergeometry: 3.1 Graded tensor calculus, 3.2 Graded differential calculus
and connections, 3.3 Geometry of graded manifolds, 3.4 Supermanifolds, 3.5
Supervector bundles, 3.6 Superconnections.
4
Non-commutative geometry: 4.1 Modules over C*-algebras, 4.2 Non-commutative differential calculus, 4.3 Differential operators in non-commutative geometry, 4.4
Connections in non-commutative geometry, 4.5 Matrix geometry, 4.6 Connes’
non-commutative geometry, 4.7 Differential calculus over Hopf algebras.
5
Appendix. Cohomology: 5.1 Cohomology of complexes, 5.2 Cohomology of Lie
algebras, 5.3 Sheaf cohomology.
Wednesday, 3 October 2012
Infinite-dimensional differential geometry
In quantum models, one deals
with infinite-dimensional smooth Banach and Hilbert manifolds and (locally
trivial) Hilbert and C*-algebra bundles. The definition of smooth Banach (and Hilbert)
manifolds follows that of finite-dimensional smooth manifolds in general, but
infinite-dimensional Banach manifolds are not locally compact, and they need
not be paracompact. In particular, a Banach manifold admits the differentiable
partition of unity if and only if its model space does. It is essential that Hilbert
manifolds satisfy the inverse function theorem and, therefore, locally trivial
Hilbert bundles are defined. However, they need not be bundles with a structure
group.
Infinite-dimensional Kahler
manifolds provide an important example of Hilbert manifolds. In particular, the
projective Hilbert space of complex rays in a Hilbert space E is such a Kahler manifold.
Note that differential geometry of Banach and Hilbert bundles over a finite-dimensional smooth manifold X can be formulated in terms of differential geometry of modules over a ring of smooth functions on X. In particular, connections on a
Hilbert bundle over a smooth manifold can be defined both as particular first
order differential operators on the module of its sections. This is the case of time-dependent quantum mechanics where X=R.
References:
G.Giachetta, L. Mangiarotti, G. Sardanashvily, Geometric and Algebraic Topological Methods in Quantum Mechanics (WS, 2005) #
G.Sardanashvily, Differential Geometry of Module and Rings. Application to Quantum Theory. (LAP, 2012) #
Thursday, 27 September 2012
My new book on differential geometry of modules and rings
G. Sardanashvily, “Lectures on Differential Geometry of Modules
and Rings. Application to Quantum Theory" (Lambert Academic Publishing, Saarbrucken
Differential geometry of
smooth vector bundles can be formulated in algebraic terms of modules over
rings of smooth function. Generalizing this construction, one can define the
differential calculus, differential operators and connections on modules over
arbitrary commutative, graded commutative and non-commutative rings. For
instance, this is the case of quantum theory, supergeometry and non-commutative
geometry, respectively. The book aims to summarize the relevant material on
this subject. Some basic applications to quantum theory are considered.
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 )
and the Department of Mathematics and Physics of University of Camerino (Italy
Contents
1
Commutative geometry: 1.1 Commutative algebra, 1.2 Differential operators
on modules and rings, 1.3 Connections on modules and rings, 1.4 Differential calculus
over a commutative ring, 1.5 Local-ringed spaces, 1.6 Differential geometry
of C(X)-modules, 1.7 Connections on local-ringed spaces.
2
Geometry of quantum systems: 2.1 Geometry of Banach manifolds, 2.2 Geometry of
Hilbert manifolds, 2.3 Hilbert and C*-algebra bundles, 2.4 Connections on
Hilbert and C*--algebra bundles, 2.5 Instantwise quantization, 2.6 Berry
3
Supergeometry: 3.1 Graded tensor calculus, 3.2 Graded differential calculus
and connections, 3.3 Geometry of graded manifolds, 3.4 Supermanifolds, 3.5
Supervector bundles, 3.6 Superconnections.
4
Non-commutative geometry: 4.1 Modules over C*-algebras, 4.2 Non-commutative differential calculus, 4.3 Differential operators in non-commutative geometry, 4.4 Connections in
non-commutative geometry, 4.5 Matrix geometry, 4.6 Connes’ non-commutative
geometry, 4.7 Differential calculus over Hopf
algebras.
5
Appendix. Cohomology: 5.1 Cohomology of complexes, 5.2 Cohomology of Lie
algebras, 5.3 Sheaf cohomology.
Wednesday, 19 September 2012
New Managing Editor of IJGMMP
A letter to
the Editors of International Journal of
Geometric Methods in Modern Physics (IJGMMP):
Dear
Editors of IJGMMP,
I am going
to leave the Journal as a Managing Editor. I prepare v10 (2013) of IJGMMP, but would prefer that new
Managing Editor produces v11 (2014). Then he should start in March-April 2013.
Please,
think if you could suggest candidates for Managing Editor. Of course, the
Publisher decides, but you can recommend one or a few persons.
The candidates
can contact me for information they need.
Gennadi
Sardanashvily
Saturday, 8 September 2012
The list of my posts in 2011 - 2012
Why to gauge
gravity?
Discrete
space-time (from my Scientific Biography)
What is a
mathematical structure?
The Higgs boson
or the Higgs vacuum?
Impact Factor
2011 of Journals in Mathematical Physics
Dmitri Ivanenko’s
archive: Nobel Laureates Letters
My book:
Generalized Hamiltonian Formalism for Field Theory
Nobel laureates
inscriptions on the walls of Ivanenko's office in Moscow State University
My lectures on
mathematical physics
Lagrangian BRST
theory (from my Scientific biography)
Classical
mechanics and field theory admit comprehensive geometric formulation
My Library: Completely
integrable and superintegrable Hamiltonian systems with noncompact invariant
submanifolds
Lagrangian
dynamics of higher-dimensional submanifolds
A problem of an
inertial reference frame in classical mechanics
My Library: General
Noether theorems
On a gauge model
of the fifth force
My Library: Jet
Manifold Formalism
Freedom is an
immanent property of living nature
Review on our
book "Geometric and Algebraic Topological Methods in Quantum Mechanics"
in Mathematical Reviews
An
energy-momentum is not uniquely defined
My Library:
Time-dependent mechanics
Review on our
book "Advanced Classical Field Theory" in Mathematical Reviews
My Library:
Advanced Classical Field Theory
My Library: Gauge
gravitation theory
Is a momentum
space of quantum fields Euclidean?
Hierarchy of
Noether identities (from my Scientific Biography)
Can contemporary
mathematics describe quantum physics?
Why only
electromagnetic and gravitational interactions are in classical physics?
“Antropomorphic”
mathematics and the crisis of science
What is a reference
frame in field theory and mechanics
Covariant
(polysymplectic) Hamiltonian
field theory (from my Scientific Biography)
Why a classical
system admits different non-equivalent quantization
Five fundamental
problems of contemporary physics
Integrable
Hamiltonian systems: generalization to a case of non-compact invariant
submanifolds (from my Scientific Biography)
On a mathematical
hypothesis of quantum space-time
Review on our
book “Geometric Formulation of Classical and Quantum mechanics” in Mathematical
Reviews
II. How we
developed gauge gravitation theory (from my Scientific Biography)
I. How we
developed gauge gravitation theory (from my Scientific Biography)
On a mathematical
hypothesis of the quark confinement
The prespinor
model (from my Scientific Biography)
Who is who among Universities
in 2011
My Scientific
Biography: Student period
Illusion of
matter
On the
strangeness of relativistic mechanics
”Quantum”
causality of ancient Greeks
Why a citation
list for a theoretician?
What are
classical Higgs fields?
The generalized
Serre – Swan theorem is a cornerstone of classical field theory
Metric gravity as
a non-quantized Higgs field
What are gauge
symmetries?
Why connections
in classical field theory?
Geometry in
quantum theory IV: Modern geometries
Geometry in
quantum theory III: Differential geometry of modules and rings
Does Impact
Factor show anything?
Geometry in
quantum theory II: Infinite-dimensional fiber bundles
Geometry in
quantum theory I: Why familiar differential geometry contributes to quantum
theory
Problems of
gravitation theory: What is a criterion of gravitational singularities?
On geometric
formulation of mechanics
Why connections
in field theory
What are general
covariant transformations?
What is classical
field theory really about?
Non-commutative
geometry meets a serious problem
What is meant by
supergeometry
Quantum field
theory: Functional integrals are not true integrals?
What is a
discrete space-time?
What is true
Equivalence principle?
On relativistic
mechanics in a very general setting
Mechanics as particular
classical field theory
What is a
fundamental science?
Well-known
mathematics that theoreticians do not know
What is true
Hamiltonian field theory?
Classical field
theory is complete: the strict geometric formulation
My teacher Dmitri
Ivanenko, a great theoretician of XX century
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