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

Geometry in quantum theory IV: Modern geometries

Contemporary quantum models appeal to a number of new algebraic structures and the associated geometric techniques. Let us mention the following ones.

(i) Supergeometry of graded manifolds and different types of superrmanifolds (Archive).

(ii) Non-commutative geometry and non-commutative field theory (Archive).

(iii) Hopf algebras, including quantum groups, and, in particular, two types of quantum bundles.

(iv) Formalism of groupoids and Lie groupoids, including quantization via groupoids.

(v) Finally, one of the main point of the formality theorem in deformation quantization is that, for any algebra A over a field of characteristic zero, its Hochschild cochain complex and its Hochschild cohomology are algebras over the same operad. This observation has been the starting point of 'operad renaissance'. Monoidal categories provide numerous examples of algebras for operads. Furthermore, homotopy monoidal categories lead to the notion of a homotopy monoidal algebra for an operad. In a general setting, one considers homotopy algebras and weakened algebraic structures where, e.g., a product operation is associative up to homotopy. At the same time, the formality theorem is also applied to quantization of several algebraic geometric structures such as algebraic varieties.

References:

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

G.Sardanashvily, G.Giachetta, What is geometry in quantum theory, arXiv: hep-th/0401080

Geometry in quantum theory III: Differential geometry of modules and rings

Geometry in quantum systems speaks mainly the algebraic language of rings, modules and sheaves due to the fact that the basic ingredients in the differential calculus and differential geometry on smooth manifolds (except non-linear differential operators) can be restarted in a pure algebraic way.

Any smooth real manifold X is homeomorphic to the real spectrum of the real ring C(X) of smooth real functions on X provided with the Gelfand topology. Furthermore, the sheaf CX of germs of functions from C(X) on this topological space fixes a unique smooth manifold structure on X such that it is the sheaf of smooth functions on X. The pair (X,CX) exemplifies a local-ringed space. One can associate to any commutative ring A the particular local-ringed space, called an affine scheme, on the spectrum Spec(A) of A endowed with the Zariski topology.

Given a connected smooth manifold X and the ring C(X) of smooth real functions on X, the well-known Serre -- Swan theorem states  that a C(X)-module is finitely generated projective iff it is isomorphic to the module of sections of some vector bundle over X. Moreover, this  isomorphism is a categorical. A variant of the Serre -- Swan theorem for Hilbert modules over non-commutative C*-algebras holds.

Let K be a commutative ring, A a commutative K-ring, and P, Q some A-modules. The K-linear Q-valued differential operators on P can be defined. The representative objects of the functor Q->diff(P,Q) are the jet modules JP of P. Using the first order jet module, one also restarts the notion of a connection on an A-module P. For instance, if P is a C(X)-module of sections of a smooth vector bundle Y->X, we come to the familiar notions of a linear differential operator on Y, the jets of sections of Y->X and a linear connection on Y->X. In supergeometry, connections on graded modules  over a graded commutative ring and graded local-ringed spaces are defined.

In non-commutative geometry, different definitions of a differential operator on modules over a non-commutative ring have been suggested. Roughly speaking, the difficulty lies in the fact that, if d is a derivation of a non-commutative ring A, the product ad, where a is from A, need not be so. There are also different definitions of a connection on modules over a non-commutative ring.

Let K be a commutative ring, A a (commutative or non-commutative) K-ring, and Z(A) the center of A. Derivations of A make up a Lie K-algebra d(A). Let us consider the Chevalley -- Eilenberg complex of K-multilinear morphisms of d(A) to A, seen as a d(A)-module. Its subcomplex of Z(A)-multilinear morphisms is a differential graded algebra, called the Chevalley -- Eilenberg differential calculus over A.

If A is the real ring C(X) of smooth real functions on a smooth manifold X, the module d C(X) of its derivations is the Lie algebra of vector fields on X and the Chevalley -- Eilenberg differential calculus over C(X) is exactly the algebra of exterior forms on a manifold X where the Chevalley -- Eilenberg coboundary operator d coincides with the familiar exterior differential. In a general setting, one therefore can think of elements of the Chevalley -- Eilenberg differential calculus over an algebra A as being differential forms over A.

References:

G.Giachetta, L.Mangiarotti, G.Sardanashvily, Geometric and Algebaic Topological Methods in Quantum Mechanics (WS, 2005)
G.Sardanashvily, G.Giachetta, What is a geometry in quantum theory, arXiv: hep-th/0401080

Does Impact Factor show anything?

The Impact Factor 2010 of scientific journals has been recently published. At present, IF is considered as an important characteristic of scientific journals. But does it show anything?

Recall that IF2010 of a journal “A” equals N/M where N is the number of citations in 2010 of articles published in “A” in 2008 – 2009 and M is the number of articles published in “A” in 2008 - 2009. For instance, if N=100 and M=100, then IF=1.000.

I am a mathematical physicist. Therefore, let us look at the Table of IF2010 of journals in mathematical physics. The most authoritative journal in this area is Communications in Mathematical Physics. However, it occupies the 12^th position with IF=2.000 in comparison with the highest IF=3.144.

It is readily observed that thin journals of about 50 articles in a year a priory have advantages over the thick ones. At the same time, thick journals can widen their scope that gives an advantage over the specialized one. For instance, this is a recent practice of Journal in Mathematical Physics (IF=1.291) and Journal of Physics A (IF=1.641).

IF of a journal in mathematical physics is higher than it is closer to applications and theoretical physics. In particular, the above mentioned Journal of Physics A has IF=1.641, whereas the theoretical journal Classical and Quantum Gravity has IF=3.098.

A problem is that only one well-cited article can essentially increase IF of a journal during two years. For instance, in the example above, let a journal “A” in 2007 published an article quoted 101 times every year. Then its IF in 2008 and 2009 becomes equal to 2.000 in comparison with its usual value 1.000

This is the case of our International Journal of Geometric Methods in Modern Physics. One article on gravitation theory published in our Journal in 2007 and cited 491 times made its IF2008=1.464 and IF2009=1.612 in contrast with 0.662 in 2007 and 0.757 in 2010.

Geometry in quantum theory II: Infinite-dimensional fiber bundles

In quantum models, one deals with infinite-dimensional smooth Banach and Hilbert manifolds and (locally trivial) Hilbert and C*-algebra bundles. The definition of smooth Banach (and Hilbert) manifolds follows that of finite-dimensional smooth manifolds in general, but infinite-dimensional Banach manifolds are not locally compact, and they need not be paracompact. In particular, a Banach manifold admits the differentiable partition of unity if and only if its model space does. It is essential that Hilbert manifolds satisfy the inverse function theorem and, therefore, locally trivial Hilbert bundles are defined. However, they need not be bundles with a structure group.

Infinite-dimensional Kahler manifolds provide an important example of Hilbert manifolds. In particular, the projective Hilbert space of complex rays in a Hilbert space E is such a Kahler manifold. This is the space the pure states of a C*-algebra A associated to the same irreducible representation of A in a Hilbert space E. Therefore, it plays a prominent role in many quantum models. For instance, it has been suggested to consider a loop in the projective Hilbert space, instead of a parameter space, in order to describe Berry's phase.

Sections of a Hilbert bundle over a smooth finite-dimensional manifold X make up a particular locally trivial continuous field of Hilbert spaces. Conversely, one can think of any locally trivial continuous field of Hilbert spaces or C*-algebras as being the module of sections of a topological fibre bundle. Given a Hilbert space E, let B be some C*-algebra of bounded operators in E. The following fact reflects the non-equivalence of Schrodinger and Heisenberg quantum pictures. There is the obstruction to the existence of associated (topological) Hilbert and C*-algebra bundles E->X and B->X with the typical fibres E and B, respectively. Firstly, transition functions of E define those of B, but the latter need not be continuous, unless B is the algebra of compact operators in E. Secondly, transition functions of B need not give rise to transition functions of E. This obstruction is characterized by the Dixmier--Douady class of B in the third Cech cohomology of X.

There is a problem of the definition of a connection on C*-algebra bundles which comes from the fact that a C*-algebra need not admit non-zero bounded derivations. An unbounded derivation  of a C*-algebra A obeying certain conditions is an infinitesimal generator of a strongly (but not uniformly) continuous one-parameter group of automorphisms of A. Therefore, one may introduce a connection on a C*-algebra bundle in terms of parallel transport curves and operators, but not their infinitesimal generators. Moreover, a representation of A does not imply necessarily a unitary representation of its strongly (not uniformly) continuous one-parameter group of automorphisms. In contrast, connections on a Hilbert bundle over a smooth manifold can be defined both as particular first order differential operators on the module of its sections.

Instantwise geometric quantization of time-dependent mechanics is phrased in terms of Hilbert bundles over the time axis R. Holonomy operators in a Hilbert bundle with a structure finite-dimensional Lie group are well known to describe the non-Abelian geometric phase phenomena. At present, holonomy operators in Hilbert bundles attract special attention in connection with quantum computation and control theory.

References:

G.Giachetta, L.Mangiarotti, G.Sardanashvily, Geometric and Algebaic Topological Methods in Quantum Mechanics (WS, 2005)
G.Sardanashvily, G.Giachetta, What is a geometry in quantum theory, arXiv: hep-th/0401080
G.Giachetta, L.Mangiarotti, G.Sardanashvily, Geometric Formulation of Classical and Quantum Mechanics (WS, 2009)

Эра пилотируемых космических полетов человечества завершена

Эра пилотируемых космических полетов человечества заканчивается очень надолго, если ни навсегда. Летать человеку в космос незачем и не на чем.

Незачем, потому что пока в Солнечной системе не нашли ничего особенно интересного, важного и полезного, что, тем более, требовало бы присутствия человека.

Не на чем, потому что источники энергии путем химических реакций почти исчерпаны, а гипотетические ядерные или термоядерные двигатели, даже если и будут созданы, ситуацию радикально не изменят. Для свободного передвижения по Солнечной системе нужны скорости порядка 1000 км/сек. Единственным известным источником энергии для достижения таких скоростей космическим кораблем массой несколько тонн является аннигиляция материи и антиматерии, причем нужны десятки килограмм антиматерии. Вряд ли это когда-нибудь станет реальностью.

О полетах человека за пределами Солнечной системы в принципе речь не идет, правда, если не обнаружится что-то, радикально расходящееся с известными физическими законами.