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

Contents

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)

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.

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.

Reference:

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

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.

References:

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)

My Library: Time-Dependent Mechanics

Classical and quantum non-autonomous mechanics with respect to different reference frames is formulated in terms of fibre bundles over the time axis R.

The file Library2.pdf (7 Mb) contains the attached PDF files of my main works in classical and quantum time-dependent (non-autonomous) mechanics.

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Geometric and Algebraic Topological Methods in Quantum Mechanics (World Scientific, Singapore, 2005)

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Geometric Formulation of Classical and Quantum Mechanics (World Scientific, Singapore, 2010)

G.Sardanashvily, Hamilton time-dependent mechanics, J. Math. Phys. 39 (1998) 2714-2729

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Non-holonomic constraints in time-dependent mechanics, J. Math. Phys. 40 (1999) 1376-1390

L.Mangiarotti and G.Sardanashvily, On the geodesic form of second order dynamic equations, J. Math. Phys. 41 (2000) 835-844

L.Mangiarotti and G.Sardanashvily, Constraints in Hamiltonian time-dependent mechanics,

J. Math. Phys. 41 (2000) 2858-2876

G.Sardanashvily, Classical and quantum mechanics with time-dependent parameters, J. Math. Phys. 41 (2000) 5245-5255

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Covariant geometric quantization of nonrelativistic time-dependent mechanics, J. Math. Phys. 43 (2002) 56-68

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Geometric quantization of mechanical systems with time-dependent parameters, J. Math. Phys. 43 (2002) 2882-2894

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Geometric quantization of completely integrable Hamiltonian systems in action-angle coordinates, Phys. Lett. A 301 (2002) 53-57

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Action-angle coordinates for time-dependent completely integrable Hamiltonian systems, J. Phys. A 35 (2002) L439-L445

E.Fiorani, G.Giachetta and G.Sardanashvily, Geometric quantization of time-dependent completely integrable Hamiltonian systems, J. Math. Phys. 43 (2002) 5013-5025

E.Fiorani, G.Giachetta and G.Sardanashvily, The Liouville -- Arnold -- Nekhoroshev theorem for non-compact invariant manifolds, J. Phys. A 36 (2003) L101-L107

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Jacobi fields of completely integrable systems, Phys. Lett. A 309 (2003) 382-386

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Bi-Hamiltonian partially integrable systems, J. Math. Phys. 44 (2003) 1984-1987

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Nonadiabatic holonomy operators in classical and quantum completely integrable systems, J. Math. Phys. 45 (2004) 76-86

E.Fiorani and G.Sardanashvily, Noncommutative integrability on noncompact invariant manifolds, J. Phys. A 39 (2006) 14035-14042

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Quantization of noncommutative completely integrable systems, Phys. Lett. A 362 (2007) 138-142

E.Fiorani and G.Sardanashvily, Global action-angle coordinates for completely integrable systems with noncompact invariant submanifolds, J. Math. Phys. 48 (2007) 032901

L.Mangiarotti and G.Sardanashvily, Quantum mechanics with respect to different reference frames, J. Math. Phys. 48 (2007) 082104

G.Sardanashvily, Superintegrable Hamiltonian systems with noncompact invariant submanifolds. Kepler system, Int. J. Geom. Methods Mod. Phys. 6 (2009) 1391-1420

G.Sardanashvily, Relativistic mechanics in a general setting, Int. J. Geom. Methods Mod. Phys. 7 (2010) 1307-1319

Review on our book "Advanced Classical Field Theory" in Mathematical Reviews

MR2527556 (2010h:70028)
Giachetta, Giovanni; Mangiarotti, Luigi; Sardanashvily, Gennadi
Advanced classical field theory. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2009. x+382 pp. ISBN: 978-981-283-895-7; 981-283-895-3

Unlike quantum field theory, classical field theory is a theory that can be dealt with in a purely mathematical way. This book aims at providing a complete mathematical basis of classical Lagrangian field theory and its BRST extension as a preliminary step towards quantum field theory. Lagrangian field theory is thereby treated in a very general framework relying, among other things, on a geometric approach to the theory of nonlinear differential operators.
  

The first chapter is devoted to an overview of the basic facts about the geometry of fibre bundles, the Frölicher-Nijenhuis calculus of vector-valued forms, jet manifolds (of finite and infinite order), connections on fibre bundles and differential operators. Chapter 2 deals with Lagrangian field theory on fibre bundles, with a discussion of the variational bicomplex and of Lagrangian symmetries and gauge symmetries. Special attention is thereby paid to the case of first-order Lagrangian field theories. Chapter 3 passes to the Lagrangian theory of even and odd fields by means of the Grassmann-graded variational bicomplex. First, an introduction to Grassmann-graded algebraic and differential calculus and to the geometry of graded manifolds is given.
  

Chapter 4 deals with Lagrangian BRST theory. Degenerate Lagrangians are characterised by a family of nontrivial Noether identities. These form a hierarchy which, under certain conditions, can be described by the exact Koszul-Tate complex. By means of a formulation of the inverse second Noether theorem in homology terms, this complex is associated to a cochain sequence of ghosts with an ascent operator, called a gauge operator. The components of this operator represent nontrivial gauge and higher-stage gauge symmetries. Whereas the gauge operator itself in general is not nilpotent, in some cases it may admit a nilpotent extension, which is called the BRST operator and which turns the cochain sequence of ghosts into the BRST complex.
  

Gauge theory on principal bundles is the topic of Chapter 5 with, among others, a study of Yang-Mills gauge theory and supergauge theory, and a discussion of matter fields and Higgs fields. Chapters 6, 7 and 8 are devoted to a complete treatment of gravitation theory on natural bundles, the theory of spinor fields (Dirac spinor and universal spinor structure) and topological field theories (Chern-Simons topological field theory), respectively. Finally, in Chapter 9, some aspects of covariant Hamiltonian field theory are described.
  

To make the exposition as self-contained as possible, the book ends with ten appendices devoted to several mathematical topics, such as differential operators on modules, homology and cohomology theory, sheaf cohomology, local-ringed spaces, leafwise and fibrewise cohomology. In addition, almost every chapter ends with an appendix in which some specific mathematical concept, relevant to the chapter under consideration, is further elucidated.
  

In conclusion, this is a very interesting book which contains a wealth of information regarding the mathematics underlying classical field theories. It is primarily oriented towards a mathematical audience: although the treatment is fairly self-contained, the reader is nevertheless supposed to have a solid background in differential geometry. In the beginning one gets a bit overwhelmed by the rapid succession of definitions, properties and notational conventions, but the effort of struggling through it is definitely rewarding.

Reference:

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

My Library: Advanced Classical Field Theory

Fibre bundles and jet manifolds provide the adequate mathematical formulation of classical field theory and its prequantum BRST extension.

The file Library1.pdf (12 Mb) contains the attached PDF files of my main works on geometric formulation of classical field theory

Contents

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

L.Mangiarotti and G.Sardanashvily, Connections in Classical and Quantum Field Theory (World Scientific, Singapore, 2000)

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

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Covariant Hamilton equations for field theory, J. Phys. A 32 (1999) 6629-6642

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Iterated BRST cohomology, Lett. Math. Phys. 53 (2000) 143-156

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

D.Bashkirov, G.Giachetta, L.Mangiarotti and G.Sardanashvily, The antifield Koszul-Tate complex of reducible Noether identities, J. Math. Phys. 46 (2005) 103513

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

D.Bashkirov, G.Giachetta, L.Mangiarotti and G.Sardanashvily, The KT-BRST complex of a degenerate Lagrangian system, Lett. Math. Phys. 83 (2008) 237-252

G.Sardanashvily, Classical field theory. Advanced mathematical formulation, Int. J. Geom. Methods Mod. Phys. 5 (2008) 1163-1189

G.Giachetta, L.Mangiarotti, G.Sardanashvily, On the notion of gauge symmetries of generic Lagrangian field theory, J. Math. Phys. 50 (2009) 012903

G.Sardanashvily, Gauge conservation laws in a general setting: Superpotential, Int. J. Geom. Methods Mod. Phys. 6 (2009) 1047-1056

My Library: Gauge Gravitation Theory

Gravitation theory is formulated as gauge theory on natural bundles where a gravitational field is a Higgs field responsible for spontaneous breaking of space-time symmetries.

The file Library0.pdf (14 Mb) contains the attached PDF files of my main works on gauge gravitation theory.

Contents

G.Sardanashvily, Gravity as a Goldstone field in the Lorentz gauge theory, Phys. Lett. A 75 (1980) 257-258

D.Ivanenko and G.Sardanashvily, The gauge treatment of gravity, Physics Reports 94 (1983) 1-45

G.Sardanashvily, The gauge model of the fifth force, Acta Phys. Polon. B 21 (1990) 583-587

G.Sardanashvily and O. Zakharov, Gauge Gravitation Theory (World Scientific, Singapore, 1992)

G.Sardanashvily, Stress-energy-momentum conservation law in gauge gravitation theory, Class. Quant. Grav. 14 (1997) 1371-1386

G.Sardanashvily, Covariant spin structure, J. Math. Phys. 39 (1998) 4874-4890

G.Sardanashvily, Classical gauge theory of gravity, Theor. Math. Phys. 132 (2002) 1163-1171

G.Sardanashvily, Gauge gravitation theory from the geometric viewpoint, Int. J. Geom. Methods Mod. Phys. 3 (2006) N1, v-xx

G.Sardanashvily, Supermetrics on supermanifolds, Int. J. Geom. Methods Mod. Phys. 5 (2008) 271-286

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

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

　

Is a momentum space of quantum fields Euclidean?

The Wick rotation provides the standard technique of computing Feynman diagrams by means of Euclidean propagators. It is the Laplace transform from a Minkowski coordinate space to the Euclidean momentum one.

Then one can suppose that a momentum space of quantum fields in an interaction zone really is Euclidean.

Reference:

G.Sardanashvily, hep-th/ 0511111

Hierarchy of Noether identities (from my Scientific Biography)

Every Euler - Lagrange operator obeys Noether identities which, however, can be trivial. If they are not trivial, a Lagrange system is called degenerate. Nontrivial Noether identities always satisfy first-order Noether identity, which also may be trivial, and etc. If first-order Noether identities are not-trivial, a degenerate Lagrangian system is called reducible. A problem was to separate trivial and non-trivial Noether identities of zero and higher orders.

This separation was effected in terms of cohomology. As a result, we described a generic reducible degenerate Lagrangian system, whose Euler - Lagrange operator satisfies non-trivial Noether identities which are not independent, but are subject to non-trivial first-order Noether identities, satisfying, in turn, non-trivila second-order Noether identities, etc. Under a certain cohomology condition, the hierarchy of these Noether identities is described by the exact cochain complex of odd and even antifields, called the Kozul - Tate complex [119,127].

This description was extended to Noether identities for an arbitrary differential operator [120].

Having received the hierarchy of Noether identities for reducible degenerate Lagrangian systems, I believed natural to generalize to it the second Noether theorem, linking the Noether identities with the gauge transformations of zero and higher order. This has been done. Generalized second Noether theorem corresponds to the Koszul – Tate complex some cochain sequence. 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 [129,133]. This cochain sequence and the Kozul - Tate complex of Noether identitie fully characterize the degeneration of a Lagrangian  system, which is necessary for its quantization.

References:

My Scientific Biography

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

Can contemporary mathematics describe quantum physics?

Created by humans, the existent mathematics is anthropomorphic, but not universal. Even in the basics of mathematical logic and set theory, it emanates from the everyday experience of people dealing with classical macroscopic objects. This mathematics meets fundamental challenges when trying to describe, for example, quantum systems.

Indeed, it is the mathematics of sets. Based on this mathematics, theoretical physics treats any physical system as a set. This treatment, a posteriori, seems to be adequate in the case of macroscopic classical systems.

However, what is about quantum systems? Is any quantum system a set? Does it consist of elements? For instance, is a photon a set? The concepts of an element, a subset, the complement of a subset, an empty set, the union and intersection of sets etc are not so evident in a quantum world. In particular, if a quantum system (e.g., a hydrogen atom) is made up by two quantum systems A and B, it does not contains neither A nor B as a subsystem. This is the well known entanglement problem in quantum theory.

Reference:

G.Sardanashvily's site: Frontier problems

Why only electromagnetic and gravitational interactions are in classical physics?

In fact, any system of classical physics (classical mechanics, classical field theory) is a system of fermions interacting with classical electromagnetic field and gravitational field. Why nothing more?

It is surprising that the strict mathematical formulation of relativistic mechanics (in the terms of one-dimensional submanifolds of a world manifold) describes relativistic particles interacting only with electromagnetic and gravitational fields, and nothing more, too.  An equation of motion of such a particle is a geodesic equation in a pseudo-Riemannian space with a torsion, which is a strength of an electromagnetic field.

Thus, one can conclude that space-time transformations are compatible only with gravitational and electromagnetic interactions, and no others, e.g., a scalar field.

References:

G.Giachetta, L.Mangiarotti, G.Sardanashvily, Geometric Formulation of Classical and Quantum Mechanics (WS, 2010)

G.Sardanashvily, Relativistic mechanics in a general setting, arXiv: 1005.1212

“Antropomorphic” mathematics and the crisis of science

Created by humans, our science is anthropomorphic, but not universal. Even in the basics of mathematical logic and axioms of set theory, it emanates from the everyday experience of people. This science meets fundamental challenges when trying to describe, for example, quantum systems,

One of the main achievements of mathematics of XX century are Godel’s incompleteness theorems which state that any formal system in mathematical logic, capable of expressing elementary arithmetic, can not be both consistent and complete. Namely, there are statements expressible in its language that are unprovable. Godel’s theorems developed an axiomatic theory of natural numbers of R. Dedekind and G. Peano. Published in 1931, they showed the failure of Hilbert's program to formalize mathematics. At present, Godel’s incompleteness theorems provide the main principle of methodology of modern science.

Indeed, contemporary theoretical physics forces us to conclude that any complicated physical system is not described by a unique theoretical model, but one needs several models, each of them has its own area of application and describes only a part or a certain aspect of a physical system. Moreover, these models at the intersection of their application areas fail to be consistent in principal.

In particular, until recently, theoreticians followed famous Dirac’s thesis: "A physical law should have mathematical beauty", written by him on the wall of D.D. Ivanenko’s office in Moscow State University. However, almost none of existent realistic theories satisfy this thesis. For example, the unified Standard Model of electroweak interaction is far from to be mathematically elegant. At present, only classical field theory admits the comprehensive mathematical formulation in terms of fibre bundles. Fundamental problems remain in classical mechanics: for instance, there is no intrinsic definition of inertial reference frames. In quantum mechanics, we have different non-consistent quantization techniques, e.g., algebraic quantization (the GNS construction) and canonical quantization.

However, the main "headache" of contemporary theoretical physics is quantum field theory. Some its parts (algebraic quantum theory, perturbative quantum theory, quantum electrodynamics) themselves look rather satisfactory. However, an integrated mathematical formulation of quantum field theory fails to exist yet. Moreover, there are doubts whether such a formulation within the existent mathematics is possible at all.

This mathematic is based on the mathematical logic which formalizes the logic of human thinking. It results from evolution of mental processes of a human mind, and it is the logic of statements in a language of words. This logic is not universal, it is "anthropomorphic". For example, an intelligent ocean in "Solaris" of Stanislaw Lem exemplifies a different logic, not the logic of statements.

In addition to the mathematical logic, the foundation of contemporary mathematics also contains the axiomatic set theory. In the initial period of its development at the fall XIX century (e.g., by G. Cantor), set theory was based on the intuitive notion of a set. However, soon it turned out that the uncertainty of this notion led to contradictions. The most famous of them are antinomies of Russell (1902) and Cantor (1899). Unfolded around antinomies debate has stimulated the development of axiomatic set theory, although its axioms are based on intuitive ideas, too. First axioms of set theory were suggested by Zermelo in 1908. At the present, there are several axiomatic systems of set theory, which are divided into four groups. Let us mention the Zermelo - Fraenkel system and the Von Neumann – Bernays – Godel one. The latter mainly is used in mathematical physics since it is a base of theory of categories. In the framework of this axiomatics, in addition to sets, another basic concept of the class is introduced in order not to consider too "big" sets that leads to contradictions. For example, all of the sets form a class, but not a set. Classes, unlike sets, can not be elements of classes and sets. With all the variety of axiomatic systems of set theory, all of them include some basic concepts and axioms, e.g., the notions of that a set consists of elements, the subset, the complement of a subset, the empty set, and axioms of the existence of the union and intersection of sets. All of these concepts came from the everyday experience of people dealing with classical macroscopic objects. However, they are not so evident, for example, in a quantum world. In particular, a quantum system may not consist of elements, or not admit a subsystem, or a subsystem has no a complement, etc.

Thus, our mathematics based on the logic of statements and set theory fails to be adequate in order to study the inanimate nature, where there are no “statements” and "words”. Therefore, our science fails to be universal, and it is both limited in its subject and incomplete in the image.

About twenty years ago, the idea was put forward to develop a new "quantum" logic and new "quantum" mathematics. However, the problem is not in that a new system of axioms must be offered, but in the fact that such a system could lead to “rich” mathematical theory. It is not possible yet. At the same time, the existent mathematics is meaningful because it simply follows an observable reality. Figuratively speaking, it solves a problem which has a solution a posteriori, and this solution needs to be recorded only. Developing one or another "quantum" mathematics, we do not know whether the problem has a solution in principle. Unfortunately, we can not put ourselves in the place of quarks and, therefore, we do not understand something important in a quantum world.

References:

Sardanashvily's blog post  Archive

What is a reference frame in field theory and mechanics

Non-relativistic mechanics as like as classical field theory is formulated in terms of fibre bundles.

In classical field theory on a fibre bundle Y->X, a reference frame is defined as an atlas of this fibre bundle, i.e. a system of its local trivializations.

If it is a gauge theory, Y->X is a fibre bundle with a structure Lie group G, and a reference frame equivalently defined as a system of local sections of an associated principal bundle P->X.

In particular, in gravitation theory on a world manifold X, a reference frame is a system of local frame fields, i.e. local sections of a fibre bundle P->X of linear frames in the tangent bundle TX of X.  This also is the case of relativistic mechanics. Therefore, one can treat the components of a tangent reference frame as relativistic velocities of some observers.

There are some reasons to assume that a world manifold X is parallelizable, i.e. its tangent bundle is trivial, and there exists a global section of a frame bundle P->X, i.e. a global reference frame. In this case, by virtue of the well-known theorem, there exists a flat connection K on a world manifold X which is trivial (K=0) with respect to this reference frame, and vice versa. Consequently, a reference frame on a parallelizable world manifold can be defined as a flat connection. The corresponding covariant differential provides relative velocities with respect to this reference frame.

Lagrangian and Hamiltonian non-relativistic mechanics is formulated as Lagrangian and Hamiltonian theory on fibre bundles Q->R over the time axis R. Such fibre bundle is always trivial. Therefore, a reference frame in non-relativistic mechanics can be defined both as a trivialization of a fibre bundle Q->R and a connection K on this fibre bundle. Absolute velocities are represented by elements v of the jet bundle JQ of Q, and their covariant differential v-K are relative velocities with respect to a reference frame K.

This description of a reference frame as a connection enables us to formulate non-relativistic mechanics with respect to any reference frame and arbitrary reference frame transformations.

References:

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

G.Giachetta, L.Mangiarotti, G.Sardanashvily,  Geometric Formulation of Classical and Quantum Mechanics (WS, 2010)

Sardanashvily's blog post Archive

 

Covariant (polysymplectic) Hamiltonian field theory (from my Scientific Biography)

Classical field theory is formulated as Lagrangian theory. All fundamental field equations are Euler – Lagrange equations derived from some Lagrangian. At the same time, classical autonomous (conservative) mechanics admits both Lagrangian and Hamiltonian formulations, which, however, are not equivalent. A Hamiltonian formulation of mechanics is based on symplectic geometry which a phase space is provided with. Naturally, a long time ago there arose a question about a Hamiltonian formulation of field theory. However, a straightforward application of symplectic Hamiltonian formalism to field theory, when canonical momenta are correspondent to derivatives of field functions only with respect to time, leads to an infinite-dimensional phase space, where  canonical variables are functions in any given instant. The Hamilton equations on such a phase space are not familiar differential equations and, in no way, comparable to the Euler - Lagrange equations of field theory. Such a symplectic Hamiltonian construction is utilized exclusively in quantum field theory to obtain the commutation relations of quantum field operators.

At the same time, a finite-dimensional phase space can be obtained if one considers canonical momenta correspondent to derivatives of field functions relative to all space-time coordinates. Such an approach is called the covariant Hamiltonian field theory. Its different variants are considered. These are polysymplectic, multisymplectic, k-symplectic Hamiltonian theories in accordance with a choice of a phase space and entered structure on it, generalizing symplectic geometry. In 1990, this question attracted attention of my student and collaborator Oleg Zakharov, who, in 1992, published an article in Journal of Mathematical Physics. However, he met a problem of constructing Hamilton equations of fields similar to the Euler – Lagrange ones. I built these equations, and then close interested in this topic.

We restricted our consideration to first order field theory, and developed polysymplectic Hamiltonian formalism on fibre bundles which, in the case of fibre bundles over the temporal axis X=R, led to non-autonomous Hamiltonian mechanics with the usual canonical variables. We constructed a globally defined polysymplectic form on a phase space, developed polysymplectic Hamiltonian formalism and, given a Lagrangian, built the associated Hamiltonians. The main results were presented in [65,66] in 1992 and, in 1993, we published already quite detailed theory [67]. The main problem was that Lagrangian and Hamiltonian formalisms on fibre bundles are not equivalent, unless only a Lagrangian is hyperregular, i.e., when the Legendre map of a configuration space to a phase space is a diffeomorphism. In a general case, one and the same Lagrangian is associated to different Hamiltonians, or no one. The comprehensive relationship between Lagrangian and polysymplectic Hamiltonian formalisms can be given in the case of the so-called semiregular and almost regular Lagrangians. The basic theorems are presented in papers [67,69] and books [10,11], and the final theory was published in the book [12] in 1997 and in the  article [88] in 1999.

Polysymplectic Hamiltonian formalism was considered in application to the basic field models, all of which are almost regular. We studied a possibility of quantization of fields in covariant canonical variables [70,114]. However, a question remains still open because additional gauge symmetries, arising in field models in covariant canonical variables, are not studied till now.

Reference:

G.Sardanashvily, My Scientific Biography 

Blog post Archive

Why a classical system admits different non-equivalent quantization

A classical mechanical system admits equivalent description in different variables whose transformation law need not be linear.

In particular, a Hamiltonian classical system is equivalently described by variables related by arbitrary canonical transformations. 

If we have a completely integrable Hamiltonian system, its descriptions in original variables and the action-angle ones also are equivalent, though the transformation law between these variables is neither linear nor canonical in general.

In contrast with classical variables, quantum operators are linear operators in Hilbert spaces of quantum states and, therefore, they admit only linear transformations.  For instance, let a classical system be described in an equivalent way with respect to different variables (q,p) and (q’,p’) which possess some non-linear transformation law q’=Q(q,p), p’=P(q,p). Let (q, p) and (q’,p’) be quantization of these variables by operators in Hilbert spaces E and E’, respectively. Then the quantum systems characterized by quantum operators (q, p) and (q’,p’)  fail to be equivalent because there is no Hilbert space morphism E->E’ which transform (q, p)->(q’,p’).

In particular, there is no quantum partner of classical canonical transformations ubless they are linear.

Quantization of a completely integrable Hamiltonian system with respect to original variables and the action-angle ones is not equivalent and leads to different energy spectrums. For instance, this is the case of a Kepler system, whose familiar Schrodinger quantization provides the well-known energy spectrum of a hydrogen atom, but its quantization with respect to action-angle variables leads to a different energy spectrum.

Thus, a classical system can admit non-equivalent quantization. A problem is that nobody generally knows what its quantization is true.

References:

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

G.Giachetta, L.Mangiarotti, G.Sardanashvily, Geometric quantization of completely integrable Hamiltonian systems in the action-angle variables, Phys. Lett. A 301 (2002) 53-57; arXiv: quant-ph/0112083

Five fundamental problems of contemporary physics

Classical mechanics: There is no intrinsic definition of an inertial reference frame.

Relativistic mechanics: What is a physical origin of a Minkowski space-time?

Quantum mechanics: Why are quantum operators represented by the differential ones?

Classical field theory: Why is classical field theory the Lagrangian one?

Quantum field theory: What are quantum fields?

Integrable Hamiltonian systems: generalization to a case of non-compact invariant submanifolds (from my Scientific Biography)

The study of integrable Hamiltonian systems in conservative mechanics did not lay in the mainstream of my research, and they were in my field of vision by accident. Moreover, it was difficult to imagine a possibility of generalization of the fundamental Liouville - Arnold theorem on "action-angle" coordinates in a neighborhood of an invariant submanifolds of a completely integrable Hamiltonian system.

This theorem was proved for a case of compact invariant submanifolds. First, it is proved that a compact invariant submanifold is a multi-dimensional torus, and then, this fact is used in a simple way  that every function on a torus is cyclic.

It seemed to me that this condition can be avoided. Not assuming initially a compactness of an invariant submanifold of an integrable Hamiltonian system, we proved that it is a multi-dimensional cylinder, and then managed to build generalized  "action-angle" coordinates in its neighborhood [102,103]. It all took less than a month. Bring the chronology of events related to that.

It was further naturally to go to partially integrable Hamiltonian systems and, in 2003, we generalized the Nekhoroshev theorem to the case of noncompact  invariant submanifolds [106,108]. In connection with them, we considered bi-Hamiltonian systems, and described a class of Poisson structures with respect to which a Hamiltonian system is partially integrable [108].

As integrable Hamiltonian systems are still, not my subject, at that time I did not suspect about existence of superintegrable Hamiltonian systems. They caught me in the eyes in 2006, and we have generalized the Mishchenko - Fomenko theorem to the case of non-compact invariant submanifolds [123]. We used the fact that such a submanifold in fact is an invariant submanifold of a partially integrable Hamiltonian system, and referred  to our generalization of the Nekhoroshev theorem.

Our results touched generalized "action-angle" coordinates in some neighborhood of non-compact invariant submanifolds. There were known topological obstructions to the existence of global "action-angle" coordinates for completely integrable Hamiltonian systems with compact invariant submanifolds. We generalized these results to a non-compact case [125]. Moreover, it turned out that, in a general case, a phase space of a superintegrable system system is decomposed into open areas, where a system is different, i.e., its integrals of motion form different Lie algebra [135]. An example is the Kepler system whose phase space is split into two areas. In one of them, invariant submanifolds are ellipses, and integrals of motion form the Lie algebra so(3), but in the other, they are hyperboles, and the Lie algebra of integrals of motion is so(2,1).

An example of integrable Hamiltonian systems with non-compact invariant submanifolds are non-autonomous integrable Hamiltonian systems whose invariant submanifolds obviously contain the time axis R. The theory of such integrable Hamiltonian systems has been developed [17,103].

Using the method of geometrical quantization, we have implemented quantization of completely integrable and superintegrable Hamiltonian systems in "action-angle" variables [104,124], including non-autonomous completely integrable systems [102]. It should be noted that, since transformations between original variables and "action-angle" variables are non-linear, quantization in those and other variables are not equivalent. However, as already noted, in "action-angle" variables, we can build non-adiabatic classical and quantum holonomy operators for completely integrable Hamiltonian system [15,17,112].

Reference:

G.Sardanashvily, My Scientific Biography

On a mathematical hypothesis of quantum space-time

A space-time in field theory, except noncommutative field theory, is traditionally described as a finite-dimensional smooth manifold, locally homeomorphic to an Euclidean topological space E. The following fact enables us to think that a space-time might be a wider space of Schwartz distributions on E.

Let E be an Euclidean topological space. Let D(E) be a space of smooth complex functions F of compact support on E. The space of continuous forms on D(E)  is the space D'(E) of Schwartz distributions on E, which includes the subspace T(E) of Dirac’s delta-functions dl_x such that, for any function F on E, we have dl_x(F)=F(x).

A key point is that there exists a homeomorphism x->dl_x of E onto the subset T(E) of delta-functions of D'(E). Moreover, the injection E-> T(E)-> D'(E) is smooth. Therefore, we can identify E with a topological subspace E=T(E) of the space of Schwartz distributions. Herewith, any smooth function F of compact support on E= T(E) is extended to a continuous form

F’(dl_x+w)=F(x) + F’(w)

on the space of Schwartz distributions D'(E). One can think of this extension F’ as being a quantum deformation of F.

In quantum models, one therefore should replace integration of functions over E with that over D'(E).

Reference:
G.Sardanashvily, On the mathematical origin of quantum space-time, arXiv: 0709.3475