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

Sunday, 25 September 2011

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.

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)

Sunday, 18 September 2011

“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.

Sunday, 11 September 2011

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 ( 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 ( 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 ( 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 ( and, directly, in Google.

Some years ago, I followed the Hep Search (High-Energy Physics Literature Database) (, 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 (, however this search fails to be complete.

I also would recommend SAO/NASA Astrophysics Data System (ADS Harvard) ( and SCIRUS (

For search of citations in journals of IOP Science (, Springer  ( ) and AIP (, 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.

Monday, 5 September 2011

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.


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.