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

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