Discrete space-time (from my Scientific Biography)

In 1930. D.Ivanenko and V.Ambarzumian put forward the idea of discreteness of space-time inside an atomic nuclear that, by means of introducing the fundamental length, to solve some of the problems encountered at that time in nuclear physics.

The idea of discreteness then was not constructively developed, but many scientists from time to time turned to this concept from general considerations, but restricted themselves, as a rule, to examination of lattices. A kind of its embodiment is theory of gauge fields on a lattice, which enabled one to do some incentive calculations. In 1965, the historically-review book "Discrete space-time" (M., Science) by A.Vialtcev came out. D.Ivanenko himself repeatedly returned to this idea. In the 1970s, his Ph.D. student G.Gorelik was engaged in this subject, but nothing new has happened. Of course, all of this was discussed at the seminars of Ivanenko, and these discussions have led me to think about the problem.

From the beginning, I refused the discrete space-time as a discrete topological space (which lattices belong to). Then what? I suggested totally disconnected topological spaces that occur in a number of theoretical models (such is, for example, the set of rational numbers). D.Ivanenko again enjoyed it, and we published a few works. Moreover, it was so well-defined mathematical model that my article "Discrete space-time" was taken in very responsible "Mathematical encyclopedia":

“One conceivable hypothesis on the structure of space in the microcosmos, conceived as a collection of disconnected elements in space (points) which cannot be distinguished by observations. An acceptable formalization of discrete space-time can be given in terms of topological spaces Y in which the connected component of its point y is its closure, and, in a Hausdorff space Y, is this point itself (totally disconnected spaces). Examples of Y include a discrete topological space, a rational straight line, analytic manifolds, and Lie groups over fields with ultra-metric absolute values.

The discrete space-time hypothesis was originally developed as a variation of a finite totally-disconnected space, in models of finite geometries on Galois fields V.A. Ambartsumyan and D.D. Ivanenko (1930) were the first to treat it in the framework of field theory (as a cubic lattice in space). In quantum theory the hypothesis of discrete space-time appeared in models in which the coordinate (momentum, etc.) space, like the spectrum of the C*-algebra of corresponding operators, is totally disconnected (e.g. like the spectrum of the C*-algebra of probability measures). It received a serious foundation in the concept of "fundamental lengths" in non-linear generalizations of electrodynamics, mesondynamics and Dirac's spinor theory, in which the constants of field action have the dimension of length, and in quantum field theory, where it is necessary to introduce all kinds of  "cut-off"  factors. These ideas, later in conjunction with non-local models, served as the base for the formulation of the concept of minimal domains in space in which it appears no longer possible to adopt the quantum-theoretical description of micro-objects in terms of their interaction with a macro-instrument. As a result, the space-time continuum is unacceptable for the parameterization of spatial-evolutionary relations in these domains (e.g. the Hamilton formalism in non-local theories), and their points cannot be distinguished by observation (in spaces Y this may be represented as the presence of a non-Hausdorff uniform structure). The discrete space-time hypothesis was developed in the conception of the non-linear vacuum. According to this concept — under extreme conditions inside particles, and possibly also in astrophysical and cosmological singularities — the spatial characteristics may manifest themselves as dynamic characteristics of a physical system, in the models of which the spatial elements are provided with non-commutative binary operations.”

References:

G. Sardanashvily, Discrete space-time, Enciclopaedia of Mathematics (Springer)

D.Ivanenko, G.Sardanashvily, Towards a model of discrete space-time, Russ. Phys. J. 21(1978) 1508.
My Scientific Biography

Archive

What is a mathematical structure?

The notion of a mathematical structure was introduced at the beginning of XX century.

However, for a long time, mathematical objects were believed to be given always together with some structure, not necessarily unique, but at least natural (canonical). And only a practice, e.g., of functional analysis has led to conclusion that a canonical structure need not exist. For instance, there are different “natural” topologies of a set of rational numbers, different smooth structures of a 4-dimensional topological Euclidean space, different measures on a real line, and so on.

In mathematics, different types of structures are considered. These are an algebraic structure, a topological structure, cells whose notion generalizes the Boolean algebras and so on. In the first volume of their course, Bourbaki provide a description of a mathematical structure which enables them to define “espece de structure” and, thus, characterize and compare different structures. However, this is a structure of mathematical theories formulated in terms of logic. Therefore, one can suggest a wider definition of a structure which absorbs the Bourbaki one and the others, but can not characterize different types of structures.

This definition is based on the notion of a relation on a set and generalizes the definition of  a relational system in set theory.

Morphisms and functions are structures in this sense, and this fact provides a wide circle of applications of this notion of a structure to physics. 

The Higgs boson or the Higgs vacuum?

In accordance with the Higgs mechanism of mass generation, quantum particles get a mass due to its interaction with a constant background classical field  responsible for spontaneous symmetry breaking. It is treated as a Higgs vacuum, but not a familiar quantum particle – a Higgs boson. The physical origin of such a Higgs vacuum is unclear. For instance, one thinks of it as being sue generis a condensate by analogy with a condensate of Cooper pairs in superconductivity.  What  then has been discovered?

Impact Factor 2011 of Journals in Mathematical Physics

New Impact Factor 2011 has been announced.

Impact Factor 2011 of some journals closed to our International Journal of Geometric Methods in Modern Physics by subject and style is the following:

Journal Title	Impact Factor 2011	Impact Factor 2010	Impact Factor 2009	Impact Factor 2008	Impact Factor 2007	5-Year Impact Factor
COMMUN MATH PHYS	1.941	2.000	2.067	2.075	2.070	1.998
LETT MATH PHYS	1.819	0.842	0.969	0.916	1.044	1.223
J PHYS A-MATH THEOR	1.564	1.641	1.577	1.540	1.680	1.344
J MATH PHYS	1.291	1.291	1.318	1.085	1.137	1.181
REV MATH PHYS (WS)	1.213	1.290	1.190	1.258	1.386	1.032
IJGMMP (WS)	0.856	0.757	1.612	1.464	0.662	1.102
J GEOM PHYS	0.818	0.652	0.714	0.683	0.986	0.710
REP MATH PHYS	0.643	0.734	0.658	0.576	0.624	0.627

See Total List of journals in mathematical physics.

Of course, IF essentially depends on a research area. Of course, small journals of < 50 articles in a year have the advantage in IF.

Dmitri Ivanenko's archives: Nobel Laureates Letters

The letters of A. Einstein, L. de Broglie, Ch. Raman, W. Heisenberg, P. A. M. Dirac, A. Sommerfeld, P. Jordan, F. Joliot-Curie, W. Pauli, P. Blackett, H. Yukawa, M. Born, E. Segre, O. Chamberlain, E. Wigner, H. Bethe, H. Alfven, A. Bohr, I. Prigogine, and some other famous scientists are kept in Ivanenko's archives

A. Einstein (Nobel prize in 1921) (a)

L. de Broglie (Nobel prize in 1929) (a, b, c)

Ch. Raman (Nobel prize in 1930) (a)

W. Heisenberg (Nobel prize in 1932) (a, b, c, d, e, f, g, h)

P. A. M. Dirac (Nobel prize in 1933) (a, b, c, d)

F. Joliot-Curie (Nobel prize in 1935) (a)

W. Pauli (Nobel prize in 1945) (a)

P. Blackett (Nobel prize in 1948) (a)

H. Yukawa (Nobel prize in 1949) (a)

M. Born (Nobel prize in 1954) (a)

E. Segre (Nobel prize in 1959) (a)

O. Chamberlain (Nobel prize in 1959) (a)

E. Wigner (Nobel prize in 1963) (a)

H. Bethe (Nobel prize in 1967) (a)

H. Alfven (Nobel prize in 1970) (a)

A. Bohr (Nobel prize in 1975) (a)

I. Prigogine (Nobel prize in 1977) (a)

A. Sommerfeld (a)

P. Jordan (a, b, c, d, e)

J. Wheeler (a, b, c, d)

D. Kerst (a)

H. Pollock (a, b)

F. Reines (a)

E. Amaldi (a)

F. Perrin (a)

V. Weisskopf (a)

W. Panofsky (a)

G. Vataghin (a, b, e)

T. Regge (a)

My book: Generalized Hamiltonian Formalism for Field Theory

Generalized Hamiltonian Formalism for Field Theory

(World Scientific, Singapore, 1995 )

G. SARDANASHVILY

(PDF file)

Preface

Classical field theory utilizes traditionally the language of Lagrangian dynamics.

The Hamiltonian approach to field theory was called into play mainly for canonical quantization of fields by analogy with quantum mechanics. The major goal of

this approach has consisted in establishing simultaneous commutation relations of

quantum fields in models with degenerate Lagrangian densities, e.g., gauge theories.

In classical field theory, the conventional Hamiltonian formalism fails to be so

successful. In the straightforward manner, it takes the form of the instantaneous

Hamiltonian formalism when canonical variables are field functions at a given instant of time. The corresponding phase space is infinite-dimensional. Hamiltonian

dynamics played out on this phase space is far from to be a partner of the usual Lagrangian dynamics of field systems. In particular, there are no Hamilton equations

in the bracket form which would be adequate to Euler-Lagrange field equations.

This book presents the covariant finite-dimensional Hamiltonian machinery for

field theory which has been intensively developed from 70th as both the De Donder

Hamiltonian partner of the higher order Lagrangian formalism in the framework of

the calculus of variations and the multisymplectic (or polysimplectic) generalization

of the conventional Hamiltonian formalism in analytical mechanics when canonical

momenta correspond to derivatives of fields with respect to all world coordinates,

not only time. Each approach goes hand-in-hand with the other. They exemplify

the generalized Hamiltonian dynamics which is not merely a time evolution directed

by the Poisson bracket, but it is governed by partial differential equations where

temporal and spatial coordinates enter on equal footing. Maintaining covariance

has the principal advantages of describing field theories, for any preliminary spacetime splitting shades the covariant picture of field constraints.

Contemporary field models are almost always the constraint ones. In field theory,

if a Lagrangian density is degenerate, the Euler-Lagrange equations are underdetermined and need supplementary conditions which however remain elusive in general. They appear automatically as a part of multimomentum Hamilton equations. Thus, the universal procedure is at hand to canonically analize constraint field systems on the covariant finite-dimensional level. This procedure is applied to a number of

contemporary field models including gauge theory, gravitation theory, spontaneous

symmetry breaking and fermion fields.

In the book, we follow the generally accepted geometric formulation of classical

field theory which is phrased in terms of fibred manifolds and jet spaces.

Contents

1 Geometric Preliminary

1.1 Fibred manifolds

1.2 Jet spaces

1.3 General connections

2 Lagrangian Field Theory

2.1 Lagrangian formalism on fibred manifolds

2.2 De Donder Hamiltonian formalism

2.3 Instantaneous Hamiltonian formalism

3 Multimomentum Hamiltonian Formalism

3.1 Multisymplectic Legendre bundles

3.2 Multimomentum Hamiltonian forms

3.3 Hamilton equations

3.4 Analytical mechanics

3.5 Hamiltonian theory of constraint systems

3.6 Cauchy problem

3.7 Isomultisymplectic structure

4 Hamiltonian Field Theory

4.1 Constraint field systems

4.2 Hamiltonian gauge theory

4.3 Electromagnetic fields

4.4 Proca fields

4.5 Matter fields

4.6 Hamilton equations of General Relativity

4.7 Conservation laws

5 Field Systems on Composite Manifolds

5.1 Geometry of composite manifolds

5.2 Hamiltonian systems on composite manifolds

5.3 Classical Berry’s oscillator

5.4 Higgs fields

5.5 Gauge gravitation theory

5.6 Fermion fields

5.7 Fermion-gravitation complex

Our book in Spanish: D.Ivanenko, G.Sardanashvili, Gravitación

En el libro se expone el punto de vista moderno de la teoría de la gravitación, sus éxitos y dificultades, así como las posibilidades de incorporarla en la teoría unificada de las partículas elementales con ayuda de los modelos gauge y generalizados. Se narra la historia de la creación de la teoría de la relatividad y se exponen sus fundamentos. Se analizan los problemas de los sistemas de referencia, la energía del campo gravitatorio, las singularidades gravitatorias y la cuantificación de la gravitación.
Gravitación (PDF) 

Prólogo a la edición en español
Introducción. Historia y problemas de la teoría de la gravitación
1	Teoría relativista de la gravitación
	1.	El espacio-tiempo de Minkowski
	2.	El espacio-tiempo en la teoría de la gravitación de Einstein
	3.	Fundamentos de la geometría de la TGR
	4.	Las ecuaciones de la teoría de la gravitación
	5.	Catálogo de campos gravitatorios
	6.	Confirmación experimental de la TGR
		Ley de gravitación de Newton
		Principio de equivalencia
		Corrimiento gravitatorio al rojo
		Desviación de la luz debido al Sol
		Precesión de las órbitas planetarias
		Localización láser de la Luna
		Precesión de un giroscopio en una órbita próxima a la Tierra
		Radiolocalización de planetas
		Ondas gravitatorias
2	Enfoques modernos en la teoría de la gravitación
	1.	El principio de relatividad y el problema de los sistemas de referencia
	2.	El principio de equivalencia y la partición (3+ 1)
	3.	El problema de la energía del campo gravitatorio
	4.	Singularidades gravitatorias
	5.	Cosmología moderna
		El problema de la singularidad
		El problema de la homogeneidad y la isotropía
		El problema de la planitud
3	Gravitación y partículas elementales
	1.	Elementos de la teoría de grupos y la tabla de las partículas elementales
	2.	Teoría de los campos gauge y el programa de la Gran Unificación
	3.	Teoría gauge de la gravitación
	4.	Generalizaciones de la TGR. Teoría de la gravitación con torsión
	5.	Gravitación cuántica
		Creación de partículas en un espacio con torsión
		Campo de torsión colectivo
	6.	Superunificación de la gravitación y las partículas elementales
Bibliografía
Índice de autores
Índice de materias

El presente libro está dirigido a estudiantes que apenas se inician en el estudio de la teoría de la gravitación. Su objetivo es dar a conocer al lector las ideas y problemas de la teoría de la gravitación, los cuales, generalmente, no encuentran lugar en los textos de estudio para gravitacionistas principiantes. La mayoría de estos textos de estudio se limita a la teoría general de la relatividad de Einstein y a la geometría seudoriemanniana del espacio-tiempo. En la bibliografía al final del libro se indican al inicio tres colecciones de resúmenes de artículos que cubren muchos de los temas tratados aquí.

La concepción einsteiniana de la gravitación como un campo geometrizado se mantiene en el centro de la atención, ya sea como una teoría no sujeta, según la opinión de muchos autores, a variaciones de ningún tipo, ya sea como uno de los modelos de gravitación más elaborados y consistentes con los experimentos, y sobre cuya base se construyen todas las demás generalizaciones.

Al mismo tiempo, la teoría general de la relatividad de Einstein se encontró con todo un conjunto de problemas internos serios, notados ya desde los tiempos de su creación, pero que han sido encubiertos por los éxitos de la teoría einsteiniana, por lo cual la discusión alrededor de ellos renació sólo en los años 60--70. Se trata del problema de los sistemas de referencia, las dificultades que presenta la búsqueda de una expresión para la energía del campo gravitatorio, de las singularidades gravitatorias y del problema de la geometría de fondo, entre otros. Por ejemplo, ni siquiera está claro cuál es la fuente física del espacio de Minkowski y qué determina la geometría y la topología del espacio en las regiones desiertas entre los cúmulos de galaxias. Los intensos esfuerzos por superar estas dificultades no han tenido éxito hasta el momento, pero han estimulado la búsqueda de nuevos métodos en la teoría de la gravitación, así como el surgimiento de diversos enfoques de revisión, ampliación y generalización de la TGR einsteiniana. A esto se debe agregar que la verificación experimental directa (sin hablar de las observaciones astrofísicas y cosmológicas) se limita por ahora, prácticamente, a la primera aproximación postnewtoniana, dejando grandes posibilidades a los modelos alternativos. En la actualidad nos vemos obligados a hablar no de la teoría, sino de muchas teorías de la gravitación, las cuales conforman un catálogo bastante amplio.

Un motivo importante para el desarrollo y la generalización de la teoría de la gravitación fue siempre la tendencia a establecer la conexión de la gravitación con otras interacciones fundamentales. Estimulado por los éxitos de la física de altas energías, este problema salió a un primer plano. La base reconocida de tal unificación es la teoría gauge. Se han propuesto diferentes modelos gauge de gravitación y en todos ellos la gravitación clásica y la cuántica se describen mediante dos campos geométricos independientes. Estos campos son, al igual que en la TGR, la métrica seudoriemanniana (o campo tetrádico) y la conexión lorentziana, la cual desempeña el papel de potencial gauge de la interacción gravitatoria. Así pues, la geometría de la teoría gauge de la gravitación se encuentra lejos de la sencillez de la geometría seudoriemanniana de la TGR de Einstein, es la geometría afinométrica y la geometría de Klein--Chern de invariantes lorentzianos. En el lenguaje de la teoría gauge, se puede decir que la teoría de la gravitación es una teoría con violación espontánea de las simetrías espaciotemporales, donde la simetría exacta es el grupo de Lorentz. Esta violación espontánea de las simetrías se deduce del principio de equivalencia, y su trasfondo físico es la existencia de materia fermiónica, la cual no admite transformaciones general-covariantes de la arena geométrica, sino, únicamente, transformaciones del grupo de Lorentz. El correspondiente campo de Higgs es el campo gravitatorio geométrico de la TGR. Esto aclara, junto con la naturaleza geométrica de la gravitación, la particularidad de la gravitación como campo físico.

La violación espontánea de la simetría es un fenómeno cuántico condicionado por la existencia de un conjunto de vacíos no-equivalentes. Este fenómeno se simula mediante el campo clásico de Higgs, cuyas características son inherentes también al campo gravitatorio. Una confirmación indirecta de la existencia del vacío de Higgs fue proporcionada por los experimentos de búsqueda de los bosones intermedios, responsables de la interacción electrodébil. Sus masas corresponden a los valores pronosticados por la teoría de Weinberg--Salam. Los campos de Higgs están presentes casi en todos los modelos modernos de las interacciones fundamentales. Estos campos aparecen también en la mayoría de escenarios cosmológicos que describen el estadio inflacionario del Universo temprano. Más aún, los datos de las observaciones cosmológicas se convirtieron en un criterio de elección de unas u otras teorías de unificación de las partículas elementales.

La variedad de modelos de gravitación está acompañada de una variedad de métodos matemáticos, utilizados actualmente en la teoría de la gravitación. Entre ellos se cuentan los espacios fibrados, las variedades de chorros (jet manifolds), la geometría espinorial compleja, las supervariedades, las cuerdas y membranas, la geometría no-conmutativa, etcétera. Es de aceptación general que, precisamente, la geometría diferencial es la que proporciona una formulación adecuada de la teoría de campos clásica, cuando los campos clásicos se describen como secciones de fibrados. De esta manera, al nivel de los campos clásicos, la conocida hipótesis de los años 20 de la posibilidad de una geometrización de todas las interacciones se hizo realidad.

Nobel laureates inscriptions on the walls of Ivanenko's office in Moscow State University

Seven Nobel Laureates: P.A.M. Dirac, H. Yukawa, N.Bohr, I.Prigogine, S.Ting, M. Gell-Mann, G. 't Hooft wrote their famous inscriptions with a chalk on the walls of Ivanenko's office in Moscow State University:

"Physical law should have mathematical beauty" (P.A.M. Dirac, 1956)

"Nature is simple in its essence" (H. Yukawa, 1959)

"Contraria non contradictoria sed complementa sunt" (N.Bohr, 1961)

"Time precedes existence" (I.Prigogine, 1987)

"Physics is an experimental science" (S.Ting, 1988)

"Nature Conformable to Herself in Complexity" (M. Gell-Mann, 2007)

"History repeats itself and will continue to do so, but not in a predictable manner" (G. 't Hooft, 2011)

References: Photo 

My lectures on mathematical physics

G.Sardanashvily, Five lectures on the jet manifold methods in field theory, hep-th/ 9411089

G.Sardanashvily, Ten lectures on jet manifolds in classical and quantum field theory, math-ph/ 0203040

G.Sardanashvily, Fibre bundles, jet manifolds and Lagrangian theory. Lectures for theoreticians, arXiv: 0908.1886v1

G.Sardanashvily, Lectures on supergeometry, arXiv: 0910.0092v1

G.Sardanashvily, Lectures on differential geometry of modules and rings, arXiv: 0910.1515v1

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

Lagrangian BRST field theory (from my Scientific biography)

BRST theory emerged in the framework of the quantum theory of gauge fields, where the timing of a degeneration of the Yang - Mills Lagrangian led to its replacement in a generating functional with some modified Lagrangian, depending on ghost fields and invariant under BRST transformations. These BRST transformations resulted from the replacement of parameter functions in gauge transformations with odd ghost fields, and their extension to action on these ghost fields. BRST theory was mainly developed in the framework of Hamiltonian formalism, but its Lagrangian variant also was under consideration. The main works in this direction were the articles of J.Gomis, J.Paris, S.Samuel in 1995 and G.Barnish, F.Brandt, M.Henneaux in 2000 in Physics Reports, as well as preceding works of these authors in the Communication in Mathematical Physics. These works, however, involved the so-called regularity condition which came from BRST theory of Hamiltonian systems with constraints, and which was not appropriate for Noether identities. The latter, in contrast to the algebraic constraint conditions, are the differential identities. Moreover, this BFST theory was developed for fields on .

I was interested in BRST theory, as a kind of prequantum field theory which is a necessary step in the procedure of BV-quantization of fields. Because a BRST operator is nilpotent, I spent the calculation of its relative and iterated cohomology on an arbitrary manifold X [95] in 2000. In 2005, I returned to BRST theory in connection with a consideration of a general type of gauge transformations, depending on the derivatives of fields of arbitrary order [118]. Later, when studying Noether identities, I gave up on the above-mentioned conditions of regularity and introduced a new cohomology condition. In 2008, after constructing a complete description of reducible degenerate Lagrangian systems, I began exploring their BRST extension. Such an extension was proved to be possible if the gauge operator continues to a nilpotent BRST operator acting on ghost fields. In this case, the above-mentioned cochain sequence becomes a complex, called the BRST complex, and an original Lagrangian admits the BRST extension, depending on original fields, antifields, indexing the zero and higher order Noether identities, and ghost fields, parameterizing  zero and higher order gauge symmetries [129]. The BRST extension of some basic field models was built.

References:

 

My Scientific Biography

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

 

Classical mechanics and field theory admit comprehensive geometric formulation

Classical non-relativistic mechanics and classical field theory are adequately formulated in geometric terms of fibre bundles Y->X where dim X>1 in field theory and X=R in mechanics.

Configuration space of a classical non-relativistic mechanics is a fibre bundle Y->R over the time axis R. Its velocity space is a first order jet manifold JY of Y. Its phase space is the vertical tangent bundle VY of Y. Connections on a fibre bundle Y->R characterize non-relativistic reference frames.

Geometric formulation of classical field theory is based on a representation of classical fields by sections of fibre bundles Y->X where dim X>1. Their Lagrangians are densities on finite order jet manifolds of Y.  Connections on Y->X also are fields, e.g., gauge fields which are sections of the first order jet bundle JY->Y. In a very general setting, in order to include odd fields, e.g., fermions  and ghosts, field theory is formulated on a graded manifold whose body is a fibre bundle Y->X.

The formulation of relativistic mechanics generalizes that of non-relativistic mechanics. It is phrased in terms of one-dimensional submanifolds of a configuration space. In a case of two-dimensional submanifolds, we come to classical string theory.

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)
G.Sardanashvily, Fibre bundles, jet manifolds and Lagrangian theory. Lectures for theoreticians, arXiv: 0908.1886
G.Sardanashvily, Advanced mechanics. Mathematical introduction arXiv: 0911.0411