Review on our book "Geometric Formulation of Classical and Quantum Mechanics" in Mathematical Reviews

MR2761736
G.Giachetta, L.Mangiarotti, G.Sardanashvily
Geometric formulation of classical and quantum mechanics. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. xii+392 pp. ISBN: 978-981-4313-72-8; 981-4313-72-6

Whereas most textbooks on the differential geometrical approach to classical and quantum mechanics are concerned with the case of autonomous (i.e., time-independent) systems, the present book addresses the case of time-dependent mechanical systems. Except for chapter 10, which explicitly deals with the relativistic case, the treatment is confined to non-relativistic mechanics. The extended configuration space of a time-dependent system is taken to be a fibre bundle Q over R, the time axis, and the corresponding velocity space is the first jet bundle JQ. The resulting description of non-relativistic mechanics becomes covariant, but not invariant under bundle transformations, i.e., time-dependent coordinate and reference frame transformations.
   The first chapter starts with some general preliminaries about fibre bundles, jet bundles, connections and the notions of first- and second-order dynamic equations. After the definition of a reference frame in terms of a connection on the configuration bundle, attention is paid, among other things, to the Newtonian formulation of time-dependent mechanics. Chapters 2 and 3 then deal with the Lagrangian and Hamiltonian description of a time-dependent non-relativistic system, respectively. The Lagrangian formulation is based on the variational bicomplex and the first variational formula and, besides the classical Lagrange equations of motion, the Cartan equations and the Hamilton-De Donder equations are also considered within this framework. A further topic that is discussed is the connection between the conservation laws of Lagrangian systems and variational symmetries, according to Noether's theorem. The Hamiltonian formulation of non-relativistic mechanics is developed on the vertical cotangent bundle V*Q of the configuration bundle Q->R, and it is shown that to any Hamiltonian system on V*Q there corresponds an equivalent autonomous symplectic Hamiltonian system on T*Q. The connections between the Lagrangian and Hamiltonian formulations of time-dependent mechanics are also investigated.
   Chapters 4 to 6 are devoted to the quantization of time-dependent mechanical systems. In chapter 4, a geometric framework for non-relativistic quantum mechanics is presented in terms of Banach and Hilbert manifolds and locally trivial Hilbert and C*-algebra bundles. A quantization scheme in the spirit of geometric quantization is then developed in chapter 5. Chapter 6 studies the geometric quantization of Hamiltonian systems with time-dependent constraints.
   In chapter 7, completely integrable, partially integrable and superintegrable Hamiltonian systems are treated in a general setting of invariant submanifolds which need not be compact. Using appropriate action-angle coordinates, the geometric quantization of completely integrable and superintegrable Hamiltonian systems is discussed. In chapter 8, the vertical extension of a mechanical system is considered from the configuration bundle Q->R to the vertical tangent bundle VQ->R, and the Jacobi fields of the Lagrange and the Hamilton equations of the system are investigated. It is shown, for instance, that the Jacobi fields of a completely integrable Hamiltonian system make up a completely integrable system in twice the number of degrees of freedom, whereby the additional first integrals characterize the relative motion. Chapter 9 deals with mechanical systems with time-dependent parameters. The Lagrangian and Hamiltonian description is analysed, treating the parameters at the same level of the dynamical variables. Next, the geometric quantization of these systems is studied.
   Leaving the non-relativistic setting, chapter 10 is concerned with the description of relativistic mechanics, both Lagrangian and Hamiltonian, and the geometric quantization of a relativistic mechanical system is discussed. Finally, chapter 11 contains several appendices, devoted to various mathematical topics which complement the main treatment, making it somewhat more self-contained (e.g., commutative algebras, geometry of fiber bundles, jet manifolds, connections, differential operators on modules, etc.).
   Although this book is addressed to a wide audience of mathematicians and theoretical physicists, even at an (advanced) undergraduate level, in my opinion it will primarily be appreciated by more experienced researchers who already have some acquaintance with the geometric approach to classical and quantum mechanics.

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

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

Proposed by D.Ivanenko and me in the early 80-ies, gauge gravitation theory, where a metric gravity has been described as a Higgs field, had the disadvantage that it is not defined gauge transformations of gravitation theory. This question was discussed [52]. Since gauge gravitation evidently should include Einstein’s General Relativity, its  gauge tsymmetries are general covariant transformations. However, there was no clarity in the definition of general covariant transformations. The answer was found in the framework of fibre bundle formalism, too. These transformations characterize the so-called natural bundles.

Let us restrict ourselves to one-parameter groups of transformations and their infinitesimal generators, which are vector fields. Let Y->X be a fibre bundle. Generators of one-parameter groups of diffeomorphisms of its base X are vector fields on X. Such a vector field can give rise to a vector field on Y in a different way, e.g., by means of connection on Y->X. However, such a lift u->u’, in general, is not functorial, i.e., it is not a homomorphism of a Lie algebra T(X) of vector fields on X to a Lie algebra T(Y) of vector fields on Y since the commutator [u',v'] need not be equal a lift [u,v]' of the commutator of vector felds u and v on X. However, there are fibre bundles which allow a functorial lift Fu of vector fields u on a base X, so that the above-mentioned homomorphism of a Lie algebra T(X) to T(Y) holds. These bundles are called natural. These include tangent TX and cotangent T*X bundles over X, their tensor  products, a linear frame bundle LX and all associated bundles, but not only. A functorial lift Fu on a natural bundle Y of vector fields on its base X, by definition, are generators of one-parameter groups of general covariant transformations of Y. Thus, gravitation theory must be built as classical field theory on natural bundles [73,77,80,98]. In particular, this implies the following.

A group of general covariant transformations is a subgroup of the group of automorphisms Aut(LX) of a linear frame bundle LX. However, Lagrangians of gravitation theory, in particular, a Lagrangian of General Relativity are invariant only under general covariant transformations, but not general frame transformation from Aut(LX). Therefore, a gravitational field (pseudo-Riemannian metric), in contrast to  Higgs fields in gauge theory of internal symmetries, is not brought to the Minkowski metric by gauge transformations and, therefore, it is a dynamic variable.

An energy-momentum current in gravitation theory is a current symmetry along a functorial lift Fu of vector fields u on X. It leads to a generalized Komar energy-momentum superpotential [73,77].

Spinor bundles are not natural, and they do not admit general covariant transformations. Therefore, a question arises about description of Dirac fermion fields in gravitation theory. Because these fields admit only Lorentz transformations, there is a situation of spontaneous symmetry breaking. In this case, a spinor field is described only in a pair with a certain gravitation field g, namely, by sections of a spinor bundle S^g associated with a reduced subbundle L^gX of a linear frame bundle LX. Then, in accordance with a general scheme of description of spontaneous symmetry breaking in classical field theory, all the spinor and gravitational fields are represented by sections of a composite bundle S->LX/SO(1,3)->X, where S->LX/SO(1,3) is a spinor bundle associated with LX->LX/SO(1,3) [80,81]. In particular, a fibre bundle S->X  is natural, and  energy-momentum current of spinor fields can be defined.

The Higgs nature of a gravitational field is clarified by the fact that, for different gravitational fields g and g', spinor bundles S^g and S^g’ are not equivalent, because the representation of tangent covectors by Dirac matrices and, consequently, the Dirac operators are not equivalent.

Reference:
G.Sardanashvily My Scientific Biography

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

… Enrolling in a graduate school in 1973, I among other things addressed to gauge gravitation theory. This direction was developed in Ivanenko’s group in the early 1960s, but then subsided with the departure of G. Sokolik, though continued to be discussed at the seminar of Ivanenko because it led to theory of gravity with torsion that Ivanenko engaged in.

By that time it became clear that gauge theory was adequately formulated in the formalism of fibre bundles, although a comprehensive formulation appeared later in the two articles: M. Daniel and C. Viallet in Reviews of Modern Physics and T. Eguchi, P. Gilkey and A. Hanson in Physics Reports in 1980. I therefore actively engaged in the study of differential geometry with the help of the translation of the book R.Sulanke and P.Wintgen, "Differential geometry and Fibre Bundles" which was released in 1975. The well-known two volumes of S.Kobayashi and K.Nomizu in the Russian translation appeared only in 1981. Simultaneously, I learned general topology on the books of Bourbaki and K.Kuratowski.

My first article on gauge gravitation theory [18] was released in September 1974. It was the author of I, but D.Ivanenko, to be sure, brought in as my co-author B.Frolov, who previously was engaged in gauge theory of gravity. In the article already mentions fibre bundles. After three months, it was published my second article [19], where I was a sole author.

By the time, when I turned to gauge gravitation theory, the problem was already almost 20 years. In 1954, C. Yang and R. Mills proposed first gauge model for a symmetry group SU(2). And already in 1956, R. Utiyama generalized this theory for an arbitrary Lie groups of internal symmetries G, including theory of gravity as a gauge theory of the Lorentz group. It is natural to assume that gauge gravitation theory should contain Einstein’s General Relativity. In General Relativity, a gravitational field is identified with a pseudo-Riemannian metric, and its symmetries are general covariant transformations. However, the difficulty was with the status of pseudo-Riemannian metrics and general covariant transformations, which have no analogue in the Yang – Mills gauge scheme because gauge fields are connections on a fibre bundle Y->X with a structure group G, and gauge transformations are vertical automorphisms of Y projected onto the identity map of X. General covariant transformations are not so. To overcome these difficulties in the work of Utiyama, in the beginning of 60-s T. Kibbl, D. Sciama et al. have proposed to treat gravity, represented by a tetrad field, as a gauge field for a translation group. All the same, it is beyond the scope of Yang - Mills – Utiyama gauge theory for internal symmetries, as evidenced not identical morphism of a base X of tensor bundles. I, too, began with this model, but soon withdrew from it, because it did not fit into fibre bundle formalism. Almost four years I was ineffectual, fiddling with the other options, until I came to interpretation of gravitation as a Higgs field, which was first described in my article [22] in 1978 .

In the 70-s, in field theory, it has already been folklore that spontaneous symmetry breaking is accompanied by Higgs and Goldstone fields, that follows from the theorem of Goldstone in quantum theory, the method of nonlinear realizations of groups (particular case of induced representations), and that provides the Higgs mechanism of generation of masses of particles in united gauge model of fundamental interactions. Spontaneous symmetry breaking is a quantum effect, when a vacuum (or a background state) fails to be invariant under a whole group of transformations, but only a subgroup of exact symmetries. A problem is how to describe spontaneous symmetry breaking in classical gauge theory. This is necessary because a generating functional for Green functions of quantum fields is expressed through a Lagrangian of classical fields, and it contains classical Higgs fields. Classical gauge theory was described in terms on fibre bundles, and it naturally raised a question what is Higgs field in this formalism.

One of sections of the above mentioned book "Differential geometry and Fibre Bundles" by R.Sulanke and P.Wintgen was devoted to the so-called G-structures, when a structure group of a principal frame bundle LX over a manifold X is reduced to its closed subgroup H. In a general case of an arbitrary principal bundles P with a structure group Lie G, a construction of the structure group reduction was described in the book "The Topology of Fibre Bundles" by N.Steenrod in 1953, which I found in the library of the Mathematical Faculty. The well-known theorem states that such reduction takes place if and only if there is a global section h of a factor bundle P/H->X. Since this section takes values in a factor-space G/H, one can treat it as a classical Higgs (or Goldstone) field. If P=LX and H is the Lorentz group SO(1,3), then h is a global section of  LX/SO(1,3) which is a pseudo-Riemannian metric on a manifold X. Therefore, I concluded that a pseudo-Riemannian metric, i.e., a gravitational field has the status of a Higgs field in gauge gravitation theory. This result was published in my report on the 8-th International gravitational conference in Canada in 1977, and the article [26].

D.Ivanenko liked such interpretation of gravity because even in the middle of the 60's he supposed that a gravitational field can be the Goldstone one by its physical nature due to breakdown of space-time symmetries caused by a curvature. However, such a symmetry breaking (and, consequently, the Higgs nature of a gravitational field) did not result from the gauge principle, and it should be lead from a principled basis. And I found such a principle. It is the equivalence principle, but reformulated in geometric terms.

In the above mentioned book by R.Sulanke and P.Wintgen, the G-structures were considered as a type of the Klein - Chern geometry of invariants, namely: if a structure group G is reduced to its subgroup H, then there is a bundle atlas of this fibre bundle with H-valued transition functions and, therefore, H-invariants on this fibre bundle are defined. At that time, the equivalence principle in gravitation theory, its different variants (weakest, weak, middle-strong, strong, etc.) were not once discussed o the seminar of Ivanenko. All of these variants were too physical for its language, to become as a basis for mathematical formulation of gauge gravitation theory. They characterize the possibility of transition to Special Relativity with respect to some reference frame. Describing Special Relativity as geometry of invariants of the Lorentz group, I came to an idea to formulate the equivalence principle in the spirit of geometry of invariants as a requirement of the existence of Lorentz invariants in some reference frame. This in turn implies a reduction of a structure group of the frame bundle LX over a manifold X to the Lorentz group, and, consequently, the existence of a gravitational field on X [28,29,31]. This geometric equivalence principle has summed up the foundation under our interpretation of gravity as a Higgs field in gauge gravitation theory. Gauge theory of gravitation was as a whole formed. It was a affine-metric theory whose dynamic variables were a pseudo-Riemannian metric as a Higgs field and general linear connections as a gauge field. D.Ivanenko and I published the review [35] in Physics Reports in 1983, which is traditionally quoted among the fundamental works on gauge gravitation theory. Our proposed gauge model of gravity also was present in the books [2,8].

Our version of the gauge theory of gravity was seen, nobody denied it, but it did not became widely recognized. Theoreticians do not hurry to refuse the treatment of a gravitational field as a gauge field of translations. Although still in 1982, I published an article [34] which specifically argued in bundle formalism that identification of tetrad fields with the so-called soldering form (a translational part of a general affine connection) is a mathematical mistake.

Therefore, I began investigating a possible physical interpretation of translation components of an affine connection. I knew the book “A Gauge Theory of Dislocations and Disclinations” by A. Kadic and D. Edelen published in 1983 (its Russian translation appeared in 1987), where gauge fields of translations on a 3-dimensional manifold described dislocations in continuum medium theory. Based on this result, I developed a model where a translational part of an affine connection on a 4-dimensional manifold described a new hypothetical structure: a kind of deformations of a world manifold [45,47]. In particular, they could be responsible for an additional Yukawa term to the Newton gravitation potential: the so-called "fifth force" [58]. At that time, such an amendment was actively investigated, but as a result, at least at laboratory distances nothing was found.

Geometric equivalence principle determines not only the existence of a gravitational field on a manifold, but a space-time structure on it. The point is that, if a structure group of a frame bundle LX is reduced to a Lorentz group (let g be the corresponding gravitational field), it is always reduced to its maximal compact subgroup SO(3). The associated Higgs field is a 3-dimensional space-like subbundle F of the tangent bundle TX of a manifold X, which defines a space-time decomposition of TX, i.e. a space-time structure on X. If a subbundle F is involutive, we have a space-time foliation of X associated with a gravitational field g. Hence, I had an idea to describe gravitational singularities as those of space-time foliations because the most recognized criterion of gravitational singularities by the so-called b-incompleteness of geodetics had a number of disadvantages [32,33]. It was given a classification of the singularities of space-time foliations, including a violation of causality, topological transitions through critical points, caustics of foliations [39,43,48]. However, this way of describing gravitational singularities also is not ideal. For example, caustic of space-time foliation can take place in the case of a regular gravitational field.

One of the most actively developed generalizations of gravitation theory of gravity is supergravitation. However, it largely built as a generalization of gauge theory of the Poincare group by extending its Lie algebra to some superalgebra. Obtained in this approach, Higgs superfields treated as a supergravity field do not have a geometrical nature. Therefore, I suggested that on should develop theory of supergravity as a supermetric on a supermanifold, introducing it from the condition of reduction of a structure supergroup of an appropriate superbundle. This was done in the framework of existed then formalism of supermanifolds [41,42]. I returned to this subject almost ten years later, already on the other mathematical level.

And, after all, gauge gravitation theory was not completed. Firstly, it remained unclear physical background of the geometric equivalence principle which looked formal. Secondly, it was unclear what are gauge transformations in gauge gravitation theory. In Einstein's General Relativity, which gauge gravitation theory should include, they are general covariant transformations. However, what are these transformations in fibre bundle formalism?

It took almost another 10 years and more advanced mathematical apparatus to all fell into place.

Reference:

G.Sardanashvily, My Scientific Biography

On a mathematical hypothesis of the quark confinement

In quantum field theory, the Wick rotation provides the standard technique of computing Feynman diagrams by means of Euclidean propagators.

Let us suppose that quantum fields in an interaction zone are really Euclidean. In contrast with the well-known Euclidean field theory dealing with the Wightman and Schwinger functions of free quantum fields, we address complete Green's functions of interacting fields, i.e., causal forms on the Borchers algebra of quantum fields. They are the Laplace transform of the Euclidean states obeying a certain condition.

If Euclidean states of a quantum field system, e.g., quarks do not satisfy this condition, this system fails to possess Green's functions and, consequently, the S-matrix. One therefore may conclude that it is not observed in the Minkowski space.

References:

G.Sardanashvily, arXiv: hep-th/0511111

The prespinor model (from my Scientific Biography)

My Scientific Biography: ...Nevertheless, the most promising of my nominated ideas was a model of prespinors (which however till now remains only "promising").

It is known that the root diagrams of simple complex Lie algebras admit groups of reflections, which are finite Coxeter groups. Moreover, the classification of simple complex Lie algebras and their real subalgebras is conducted by means of finite Coxeter groups. They are groups of symmetries of the weight diagrams of irreducible representations of these Lie algebras, which the algebras both of internal and space-time symmetries belong to.

There was an idea that Lie algebras and groups of symmetries can be replaced with the corresponding finite Coxeter groups. Generating elements s of these groups have the property ss=1, and the diversity of these groups is due to the fact that different generating elements do not commute between themselves. The simplest Coxeter group consists of two elements (s,1) and serves as a symmetry group of a 2-spinor. Let us suppose that a physical world in its basis is made of such spinors, let us call them the  prespinors, so that, when their interaction, Coxeter groups of their transformations become non-commutative, providing all the known diversity of symmetries of elementary particles. Moreover, we can go even further and identify elements of the simple Coxeter group (s,1) with the simplest logical system of statements ("no", "yes").

D.Ivanenko believed this model to be very promising. He saw in it the prospect of a continuation of Heisenberg’s and his unified nonlinear field theory, which by that time had stepped aside in the light of theory of gauge fields. It became clear that an interaction of elementary particles is described by exchange of mediators, gauge fields, but not nonlinearities, though not everywhere. For example, an interaction of a field of Cooper pairs in the theory of superconductivity is due to non-linearity, and it may happen that an interactions of a Higgs fields and prespinors are of this type.

The  prespinor model is presented in our book  "Gravitation" (1985) (in Russ.) and a few articles,  but no further development has obtained, since it is unclear how to describe the dynamics of systems with finite groups of symmetries.

 

Who is who among universities in 2011

New world ranking of universities "QS Top University Ranking 2011" has been published. It contains 700 universities.

In 2011, my Moscow State University occupies the 112-th place with a coefficient 61.28 of 100, while in the past year - the 93rd place. From Russian universities, in addition to the MSU, there are still 10. The nearest one is Petersburg State University of the 251-th place with a coefficient 41.06.

The top ten positions are occupied by 6 universities of USA and 4 of United Kingdom.

In the top twenty: 13 - USA, 5 - United Kingdom, and one each from Switzerland and Canada.

In the first hundred: 30 – USA; 19 - United Kingdom; 8 - Australia; 6 - Japan; 5 - Canada; 4 - Germany; 3 - Switzerland, China, Hong-Kong, South Korea, the Netherlands; 2 - Singapore, Sweden, France, Denmark; 1 - Ireland, Belgium, New Zealand, Finland and Taiwan.

My Scientific Biography: Student period

In 1967, I graduated from the Moscow mathematical school №2 with a silver medal and entered Physics Faculty of Moscow State University. Besides the standard education program, I began to engage in self-education and went to the circle of theoretical physics, held for students of the junior courses of prof. D.Ivanenko, his staff and post-graduate students. I originally wanted to engage in theoretical physics, but at the faculty there were three theoretical departments. Under the influence of the theoretical circle, his broad topics, I decided to enter to the Department of Theoretical Physics to D.Ivanenko. From time to time, I even attended his scientific seminar.

In the middle of the third year, in spring of 1970, I was assigned to the Department of Theoretical Physics. The best students of the course tried to enter it, as well as on other theoretical departments. Only 12 people could do, and it was necessary to pass the interview. In the course of the interview, I felt that they knowingly take me: I had lost only on ball for all exam sessions and, apparently, D.Ivanenko warned that I am to him.

After entering the Department, I as a future graduate officially joined the group of Ivanenko: went to his scientific seminars, continued self-education, and eyed what anyone in the group is engaged in.

On the fourth course, I began to collaborate with Andrey Bulinski. He graduated from Physics Faculty in 1968, but was not taken in the graduate school and worked at the Department of Higher Mathematics of the Moscow Physical-Technical Institute. He continued to collaborate with D.Ivanenko, and engaged in algebraic quantum theory: algebras of quantum observables, their representations, quantum dynamical systems, etc. All of this was outside the scope of conventional courses of Physics Faculty. Working with him, I got a good experience in this field, which I then is very handy. In one of his articles, published in Journal of Theoretical and Mathematical Physics in 1971, he even thanking me for useful discussion. Although I do not remember that I was really any good. Algebraic quantum theory is rather mathematically sophisticated subject. My level was certainly not enough to get on this topic some original results and prepare a diploma work. Besides, Andrei Bulinski less and less began to come into the band and seminars of Ivanenko, apparently, having lost hope to return to Physics Faculty. Therefore, D.Ivanenko offered me, at least for pragmatic reasons, to change a research subject and a scientific chief (not being Ph.D., A.Bulinski formally could not be a scientific supervisor of my diploma work).

At that time, in science and seminars of Ivanenko, there has been actively discussed conformal field theory on the basis of the 15-parameter conformal group, including the Lorentz and Poincare subgroups. Naturally, the question arose about constructing the spinor representations of this group, as I did. My scientific supervisor was D.Sc. Dmitri Kurdgelaidze, a long-term employee of D.Ivanenko, with whom he developed a nonlinear meson and spinor theory. However, my purely algebraic subject was far away from his interest, and he could not help me. Therefore, I actually worked independently. I obtained a 8-spinor representation of the conformal group, which also implement the CPT transformations, and wrote for them the conformal-invariant Dirac equation. To me, this work still like it. I reported it on the 3-th Soviet gravitational conference in October 1972, and before that submitted an article to  "Vestnik of Moscow State University, Physics and Astronomy". However, for some reason, this article appeared much later, - in March of 1975. In January 1973, I defended my diploma work "Finite-dimensional  representations of the conformal group", with Ivanenko and Kurdgelaidze as supervisors.

To complete this topic, in 1973, I also constructed the nonlinear representation of the conformal group by the method of the so-called "nonlinear realizations". This method shortly before was developed, allowed to build a representation of a group as an extension of a representation of its Cartan subgroup, and was then very popular. This work was presented at the  Symposium "Modern problems of gravitation" in Moscow and went out in its Proceedings. It became my first scientific publication.

After graduating from the Physics Faculty February 1973, I in April was enrolled in the postgraduate school at the Department of Theoretical Physics to D.Ivanenko. My study of the conformal group was completed and, in front of me, there was a wide range of research directions. Interested in very many, D.Ivanenko provided a full freedom of activity of his graduate students. My direct supervisor was he himself, no one was standing between us, and I could do what I will.

First of all, I was interested in out-of-scope of the standard field theory on the basis of new mathematical methods of theoretical physics: algebraic, geometric and topological, because it was clear that the standard field theory had exhausted its possibilities. And I started with the search for and development of such innovative methods. Although the risk was great: could nothing is going to happen, no publications or dissertation. As it turned out, with the publication of problems was not, and that's Ph.D. thesis was delayed.

References:
G. Sardanashvily: Scientific Biography

Illusion of matter

Can a structure be carrier-free? Philosophy says that it is impossible. However, contemporary theoretical physics gives a different answer.

In mathematics, there exist various concepts of a structure: the structure genus of a structure (a rather sophisticated definition of Bourbaki), a lattice (an algebraic notion generalizing a Boolean algebra), a topological structure, a geometric structure, etc. For physical applications, I would propose a mathematical definition of the structure as an n-ary relation on a set defined by some subset of an n-product of this set. This concept correlates with the definition of Bourbaki in some way and absorbs other definitions of a structure. In particular, morphisms of a set are structures in this sense. Nevertheless, in all existent variants, a mathematical structure is introduced on a carrier set.

In physics, however, it appears that a set, carrying a structure, often itself consists of elements of some structure. For example, a classical field, defined as a section of a fibre bundle, is a morphism. i. e., a structure, called the geometric structure. It is obvious, that quantum operators as elements of a certain algebra exemplify an algebraic structure. Moreover, by virtue of the well-known GNS construction in algebraic quantum theory, a Hilbert space of states, which quantum operators act in, consists of equivalence classes of these operators possessing the same average value, and so, it also is a set of elements of an algebraic structure.

Contrary, a point mass in classical mechanics is not part of any structure. However, in modern united models of fundamental interactions, a quantum field acquires a mass as a result of its interaction with a Higgs field. It follows that a mass is a derivative characteristics of two structures. Thus, the massive matter ceases to be a fundamental concept. For example, a particle and an antiparticle, annihilating, are converted into photons.

At present, with theoretical and mathematical viewpoint, all known fundamental physical objects are structures whose carrier consists of elements of some other structure having its carrier another structure, etc. Moreover, a structure can be defined on different carriers or be carrier-free. For instance, morphisms of some vector space are representations of a certain abstract group which is defined for itself and admits other representations.

If all physical objects, e.g., classical and quantum fields are a structure, then what is a carrier of a structure in the physical world? Is there such a carrier? Of course, the matter does not disappear, but is somewhat illusory.

On the strangeness of relativistic mechanics

From the mathematical viewpoint, relativistic mechanics fails to be a step between non-relativistic mechanics and classical field theory. Classical field theory is formulated in terms of fibre bundles Y->X (Archive). Non-relativistic mechanics can be treated as a particular field theory in terms of fibre bundles over the time axis X=R (Archive). Relativistic mechanics is formulated in terms of one-dimensional submanifolds of a its configuration space Q (Archive). In a sense, this is a generalization of non-relativistic mechanics because sections of a fibre bundle Q->R are particular one-dimensional submanifolds of Q. However, this is a generalization towards string theory, but not field theory. Indeed, one can develop theory whose dynamic variables are submanifolds and, if they are two-dimensional submanifolds, we are in the case of classical string theory.

From the physical viewpoint, we do not observe classical relativistic masses of velocities more than 0.0001 of the light one.

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)

“Quantum” causality of the ancient Greeks

In ancient Greek philosophy and art, the following problem was carefully developed. According to Greek religion, the destiny of a man (or gods) was predetermined: great Moirae span its thread. But a man is not deprived of the freedom of will and action. Just, whatever the ways he does not choose, they all lead to a predetermined result. Troy had to be destroyed, and no matter how events developed, it fell. Nobody specially arranged this, just it always happened by itself. Thus, knowledge of the final (the fall of Troy) could not prevent this final: in this sense, the principle of causality was not violated.

So in quantum mechanics, a quantum system is transformed from one fixed state into another fixed state, but the way of transition is not pre-ordained. For example, an electron in a hydrogen atom passes from one energy level to another and radiates,  that is well-described, but the way that it transits is unknown. Let us call this the "quantum" principle of causality. Knowledge of the future does not violate it.

Having found out his future in some way, a man can change his behaviour, but nothing, that he could make, can not change the predetermined final. Indeed, everyone knows that he is mortal, but none that he does, he dies - the "quantum" causality principle in action.

Seers and travels to the future do not violate the "quantum" principle of causality.

Why a citation list for a theoretician?

At present, different administrations, from universities till WikipediA, become to request a list of citations of a scientist. Moreover, they often require of him to follow one or another certain database.

Certainly, a citation list is an important characteristic of a scientist, unless he is a genius. A genius needs no citation list. I keep my citation list because, time by time, somebody says me that my works are very abstract and mathematically sophisticated, and nobody reads them.

There are different citation databases. A problem lies in the following: (i) none of them is complete, (ii) they do not separate self and non-self citations, (iii) they treat a work published in different issues (e.g., in a journal and arXiv) as different publications and, thus, double a number of citations in this work. To obtain a real picture of citations, one therefore should use several databases.

Let me restrict my consideration to publications in theoretical and mathematical physics.

The ISI Web of Knowledge database (http://www.isiwebofknowledge.com/) certainly is most recognized. However, it is not free, but only by subscription. Therefore, I use it only on an occasion. It does not separate self and non-self citations. The main disadvantage of this database is that it takes into account only references in journals of ISI Journal List, but not in other issues, e.g., books, papers in arXiv and others.

The citation search in AMS database (http://www.ams.org/mathscinet/) covers a wider cycle of issues in mathematical physics, but it is rather young, and is by subscription. I use it on an occasion, too.

In comparison with ISI Web of Knowledge, Google Scholar (http://code.google.com/p/citations-gadget/) takes into account any issue, including the electronic ones. However, it possesses all three above-mentioned disadvantages. I complement it by search in Google Books (http://books.google.com/books) and, directly, in Google.

Some years ago, I followed the Hep Search (High-Energy Physics Literature Database) (http://www.slac.stanford.edu/spires/hep/search/), but it mainly is concerned with references to papers in arXiv. At present, these references can be found in “Experimental full text search” of arXiv itself (http://xxx.lanl.gov/find/), however this search fails to be complete.

I also would recommend SAO/NASA Astrophysics Data System (ADS Harvard) (http://adsabs.harvard.edu/) and SCIRUS (http://www.scirus.com/srsapp/).

For search of citations in journals of IOP Science (http://iopscience.iop.org/journals), Springer  ( http://www.springerlink.com/ ) and AIP (http://scitation.aip.org/search_scitation), one can use their own databases, which are rather complete.

As my experience shows, by use of all these database, I however can collect only about 80% citations of my works.

What are classical Higgs fields?

In the 70-s, in field theory, it has already been folklore that spontaneous symmetry breaking is accompanied by Higgs and Goldstone fields, that follows from the theorem of Goldstone in quantum theory, the method of nonlinear realizations of groups (particular case of induced representations), and that provides the Higgs mechanism of generation of masses of particles in united gauge model of fundamental interactions. Spontaneous symmetry breaking is a quantum effect, when a vacuum (or a background state) is fails to be invariant under a whole group of transformations, but only a subgroup of exact symmetries. A problem is how to describe spontaneous symmetry breaking in classical gauge theory. This is necessary because a generating functional for Green functions of quantum fields is expressed through a Lagrangian of classical fields, and it contains classical Higgs fields. Classical gauge theory was described in terms on fibre bundles, and it naturally raised a question what is Higgs field in this formalism.

In classical gauge theory on a principal bundle P->X with a structure Lie group G, spontaneous symmetry breaking is characterized  as a  reduction of a structure group G to its closed (and, consequently, Lie) subgroup H. This means that there is an atlas of a principal bundle P and associate bundles with H-valued transition functions or, equivalently, that there is a principal subbundle P' of P with a structure group H. Then there may exist a fibre bundle Y->X associated with P', whose typical fibre V admits no action of a group G, but only its subgroup H. Section of this fibre bundle describe  matter fields in a situation of a breakdown of symmetries with a group G to a subgroup H of exact symmetries.

A key point is that, by the well-known theorem, reduction of a structure group G to a subgroup H occurs if and only if there exists a global section h of a factor-bundle with a typical fibre G/H. Since this section takes values in a factor-space G/H, one can treat it as a classical Higgs field.

Moreover, there is one-to-one correspondence between such sections h and the H-principal subbundles P[h] of P. Let Y[h]->X be a fibre bundle associated with P[h]. Then its sections s describe matter fields with an exact symmetry group H in the presence of a Higgs field h. A problem, however, is that, for different Higgs fields h, fibre bundles Y[h]->X need not be equivalent. Therefore, matter field s with an exact symmetry group H must be considered only in a pair with a certain Higgs field h. Of course, a question arises, how to describe a totality of matter fields with broken symmetry and Higgs fields.

To do this, one can consider a composite bundle P->P/H->X, where P->P/H is a principal bundle with a structure group H, and a fibre bundle Y->P/H associated with P->P/H, with a typical fibre V. Then section of a composite bundle P->P/H->X describe a desired totality of matter and Higgs fields in a case of spontaneous symmetry breaking. Indeed, in accordance with the above-mentioned properties of composite bundles, the restriction of a fibre bundle Y->P/H to a submanifold h(X) of P/H is exactly a fibre bundle Y[h]->X.

In particular, let X be a 4-dimensional world manifold, and let P=LX be a fibre bundle of linear frames in the tangent bundle TX of X. Its structure group is GL(4,R). By virtue of the geometric equivalence principle (), this structure group is reduced to the Lorentz group H= SO(1,3). Then a global section h of the factor-bundle LX/SO(1,3) is a pseudo-Riemannian metric, i.e., a gravitational field on a manifold X. Thus, a gravitational field exemplifies a classical Hiigs field.

References:

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

G.Sardanashvily, Geometry of classical Higgs fields, Int. J. Geom. Methods Mod. Phys. 3 (2006) 139-148.

The generalized Serre – Swan theorem is a cornerstone of classical field theory

Classical field theory admits the adequate geometric formulation in the terms of fiber bundles and graded manifolds (Archive). Why? A cornerstone of classical field theory is the generalized Serre – Swan theorem.

Let X be a compact smooth manifold and C(X) the ring of smooth functions on X. The original Serre – Swan theorem states that a C(X)-module is a projective module of finite rank if and only if it is isomorphic to a module of sections of some vector bundle over X.  This theorem has been extended to an arbitrary smooth manifold due to the fact that any smooth manifold admits a finite manifold atlas.

It follows from the Serre – Swan theorem that, if classical fields are assumed to constitute a projective C(X)-module of finite rank, they are represented by sections of a vector bundle.

In a general setting, theory of Grassman-graded even and odd classical fields is considered. There are different models of odd classical fields in the terms of graded manifolds and supermanifolds. Combination of the well-known Batchelor theorem and the above mentioned Serre – Swan theorem results in a generalization of the Serre – Swan theorem to graded manifolds as follows.

Given a smooth manifold X, a graded commutative C(X)-algebra is isomorphic to the structure ring of a graded manifold with a body X if and only if it is the exterior algebra of some projective C(X)-module of finite rank.

References:

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

G.Sardanashvily,  Classical field theory. Advanced mathematical formulation, arXiv: 0811.0331

Metric gravity as a non-quantized Higgs field

If gravity is a pseudo-Riemannian metric, it is a Higgs field. Being a Higgs field, a metric gravitational field is non-quantized.

Classical field theory is adequately described in terms of fibre bundles (Archiv). Classical gravitation theory formulated in these terms is metric-affine theory whose dynamic variables are a pseudo-Riemannian metric, treated as a metric gravitational field, and a general linear connection on a world manifold X.

We concentrate our attention to a metric gravitation field.  A pseudo-Riemannian metric on a world manifold X is defined as a global section of the quotient LX/SO(1,3) of the linear frame bundle LX by the Lorentz group SO(1,3). Therefore, it exemplifies a Higgs field in classical field theory on fibre bundles.

Its Higgs character is displayed as follows. Given different pseudo-Riemannian metrics g and g', the representations of the holonomic coframes dx by the Dirac matrices acting on Dirac spinor fields are not equivalent.

It follows that a Dirac spinor field can not be considered in the case of a superposition of different metric gravitational fields. Therefore, quantization of a metric gravitational field fails to satisfy the superposition principle, and we think that it is non-quantized.

References:

G. Sardanashvily, Metric gravity as a non-quantized Higgs field (2010).

What are gauge symmetries?

In mathematics, any Lagrangian system generally admits gauge symmetries, though it may happen that they are trivial.

In theoretical physics, the notion of gauge symmetries depending on parameter functions is a cornerstone of contemporary field theory. It comes from gauge theory on principal bundles whose vertical automorphisms, called the gauge transformations, are gauge symmetries of the Yang – Mills Lagrangian of gauge fields. Gauge symmetries of gravitation theory are general covariant transformations.

A gauge symmetry of a Lagrangian L is defined as a differential operator on some vector bundle E taking its values in the linear space of (variational or exact) symmetries of L. Therefore, a gauge symmetry of L depends on sections of E and their partial derivatives. For instance, this is the case of gauge symmetries in classical field theory.

Gauge symmetries possess the following two peculiarities.

(i) Being Lagrangian symmetries, gauge symmetries of a Lagrangian satisfy first Noether’s theorem, but the corresponding conserved current J^μ takes a particular superpotential form J^μ = W^μ + d_νU^νμ where the first term W^μ vanishes on solutions of the Euler – Lagrange equations and the second one is a boundary term, where U^νμ is called a superpotential.

(ii) In accordance with second Noether’s theorem there is one-to-one correspondence between the gauge symmetries of a Lagrangian and the Noether identities which the Euler–Lagrange operator satisfies. Consequently, gauge symmetries characterize the degeneracy of a Lagrangian system.

Note that, in quantum field theory, a generating functional fail to be invariant under gauge transformations, and gauge symmetries are replaced with the BRST symmetries, depending on ghosts and acting both on fields and ghosts.

References:

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

G.Gaichetta, L.Mangiarotti, G.Sardanashvily, On the notion of gauge symmetries of generic Lagrangian field theory, arXiv: 0807.3003

Why connections in classical mechanics?

The main reasons why connections play a prominent role in many theoretical models field models lie in the fact that they enable us to deal with well (globally, invariantly) defined objects.

Connections in classical field theory have been discussed (Why connections in classical field theory?).

Classical non-relativistic mechanics is formulated as a particular field theory on smooth fibre bundles Q->R over the time axis R (Mechanics as particular classical field theory). Its velocity phase space is the first order jet bundle JQ->R. Its momentum phase space is the vertical cotangent bundle V*Q of Q. The concept of a connection is the central ingredient in this geometric formulation as follows.

(i) An essential difference between classical mechanics and field theory lies in the fact that connections on a fibre bundle Q->R are flat and, therefore, they fail to be dynamic variables. They describe non-relativistic reference frames. This fact enables us to define relative velocities and accelerations, and describe non-relativistic mechanics with respect to different reference frames.

In particular, one can define a free motion equation and the geodesic reference frame for it which is called the inertial reference frame. However, an absolute inertial frame fails to be defined.

(ii) Equations of motion of non-relativistic mechanics almost always are of second order. Second order dynamic equations on a fiber bundle Q->R are conventionally defined as the holonomic connections on the jet bundle JQ->R. These equations also are represented by connections on the jet bundle JQ->Q and, due to the canonical imbedding of JQ to the tangent bundle TQ, they are proved to be equivalent to non-relativistic geodesic equations on TQ.

(iii) In Hamiltonian non-relativistic mechanics on the momentum phase space V*Q,

Hamiltonian connections on V*Q->R define the Hamilton equations.

References:

L.Mangiarotti, G.Sardanashvily, Gauge Mechanics (WS, 1998)

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

G.Giachetta, L.Mangiarotti, G.Sardanashvily, Advanced mechanics. Mathematical introduction, arXiv: 0911.0411