The God has created a man in order that he creates that the God fails to do

Friday, 23 March 2012

My Library: Jet Manifold Formalism

Jet manifold formalism is the conventional technique of theory of (nonlinear) differential operators and differential equations, Lagrangian theory and differential geometry of connections on fibre bundles.  It also provides the adequate geometric formulation of classical field theory and Lagrangian and Hamiltonian time-dependent mechanics.

The file Library4.pdf (11 Mb) contains the attached PDF files of my main works on jet manifold formalism


G.Giachetta, L.Mangiarotti and G.Sardanashvily,  New Lagrangian and Hamiltonian Methods in Field Theory  (World Scientific, Singapore, 1997)

G.Giachetta, L.Mangiarotti and G.Sardanashvily,  Cohomology of the infinite-order jet space and the inverse  problem, J. Math. Phys. 42 (2001) 4272-4282

G. Sardanashvily, Cohomology of the variational complex in the class of exterior forms of finite jet order, Int. J. Math. and Math. Sci. 30 (2002) 39-48

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Lagrangian supersymmetries depending on derivatives. Global analysis and cohomology, Commun. Math. Phys. 259 (2005) 103-128

G.Sardanashvily, Graded infinite order jet manifolds, Int. J. Geom. Methods Mod. Phys. 4 (2007) 1335-1362

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Advanced Classical Field Theory (World Scientific, Singapore, 2009)

Sunday, 18 March 2012

Freedom is an immanent property of living nature

In biology, there are no exhaustive criteria of living nature, and living organisms are characterized by a number of phenomenological features such as the ability to move, irritability, the ability to reproduction, adaptation to changing external environment, etc. However, these only are characteristics of a particular protein form of living nature that exists on Earth.

Therefore, I would suggest the following general definition of living nature.

Life is a structure participating in the emergence of a similar structure which cannot appear if an original structure has not existed.

Thus, the thesis - The meaning of life is in its very existence – is the main methodological principle of the study of living organisms.

Since a living structure is involved in the emergence of a similar structure, it should be inherently active, and the possibility of implementation of this activity reveals itself as freedom.

Therefore, freedom is an immanent property of living nature.

Saturday, 10 March 2012

Review on our book "Geometric and Algebraic Topological Methods in Quantum Mechanics" in Mathematical Reviews

MR2218620 (2007m:58001) Giachetta, Giovanni (I-CAM); Mangiarotti, Luigi (I-CAM); Sardanashvily, Gennadi (RS-MOSC)
Geometric and algebraic topological methods in quantum mechanics. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. x+703 pp. ISBN: 981-256-129-3
58-02 (37J05 53D55 81-02 81R10 81R60 81S10)

Representation theory, functional analysis, differential geometry, and other classical mathematical concepts have proven their relevance to the formulation and understanding of models in theoretical and mathematical physics. These theories might nowadays be common knowledge for physicists working in these fields. Within the last 20 years in quantum theory new ideas have been developed, e.g. super- and BRST symmetries, geometric and deformation quantization, topological field theories, quantization of conformal field theories, non-commutativity, strings, branes, etc. These developments triggered the use and sometimes even the development of more advanced mathematics related to geometry, to algebraic geometry, and to algebra. All these techniques have a certain algebraic flavor. It is the passage from the commutative world to the noncommutative world (either by the quantization itself or by considering field theory over noncommutative space) which forces us to replace the category of "usual'' geometric objects by its dual category, the category of function algebras with certain additional structures. The dual category might admit an extension (e.g. a deformation) into the noncommutative world.
It is the goal of the authors of the book under review to introduce the mathematical definitions of these (mainly algebraic) objects, to collect some of the most relevant facts, and to give a guide to the literature. The book has the following chapters. 1. Commutative geometry, including homology of complexes, groups and algebras, algebraic varieties. 2. Classical Hamiltonian systems, including the cohomology of Kähler manifolds, Poisson manifolds, groupoids. 3. Algebraic quantization, including GNS construction. 4. Geometry of algebraic quantization, including Berezin's quantization, Banach and Hilbert manifolds. 5. Geometric quantization. 6. Supergeometry, including graded manifolds, BRST complex of constrained systems, superconnections. 7. Deformation quantization, including the relevant cohomology, Fedosov's and Kontsevich's construction. 8. Non-commutative geometry, including C* algebras, noncommutative differential calculus, Connes' noncommutative geometry, Morita equivalence, K- and KK-theory. 9. Geometry of quantum groups, including the differential calculus for Hopf algebras, quantum principal bundles. After a general appendix recalling some more basic material assumed in the book, such as categories, Hopf algebras, groupoids, algebroids, measures on non-compact spaces, and more information on fibre bundles, the book closes with a very useful extensive bibliography (452 items). Note that, in the above, this is only a selection of the topics of subsections.
With respect to a prospective reader having a reasonably good background in mathematics, the notions, concepts, etc. are introduced in a self-contained but condensed manner. In most cases, proofs are not supplied for the results presented. But in any case reference to the literature is given. The book is not a mathematical "textbook'' in the usual sense. Also, because of the number of covered subjects and hence necessarily condensed style, there is not enough space for internal mathematical motivation for the introduced concepts.   

The book gives a very helpful supply of mathematical tools needed by a theoretical or mathematical physicist to effect entry into some of the new directions in theoretical physics. Also, a mathematician might appreciate the condensed presentation of definitions and results in one of the modern fields of mathematics for which one may be seeking an overview. Clearly, as the main goal of the book is to present the more algebraic background of modern geometry, certain other geometric methods of importance in modern theoretical physics are beyond the intended scope of the book. One example is the notion of moduli spaces of geometric structures and their generalisation.

G.Giachett, L.Mangiarotti, G.Sardanashvily Geometric and Algebraic Topological Methods in Quantum Mechanics (WS, 2005)

Saturday, 3 March 2012

An energy-momentum is not uniquely defined

The fact is that, in classical Lagrangian field theory on a fibre bundle Y->X, an energy-momentum current, by definition, is a symmetry current along a vector field u on Y, which has a certain non-zero projection s on X. Such a lift u of s is not unique, and therefore an energy-momentum current fails to be unique. The difference of two different lifts u and u' is a vertical vector field v=u-u' on Y possessing a zero projection on X Symmetry currents along vertical vector fields are Noether currents of internal symmetries of a Lagrangian. Therefore, different energy-momentum currents differ from each other in Noether symmetry currents. Moreover, any conserved energy-momentum current in general contains a Noether component, determined by internal symmetries of a Lagrangian.


G.Sardanashvily, Energy-momentum conservation laws in gauge theory with broken gauge aymmetries, arXiv: hep-th/0203275 

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