Revisit of the Faddeev Model in Dimension Two

The Faddeev model is a fundamental model in relativistic quantum ?eld theory used to model elementary particles. The Faddeev model can be regarded as a system of non-linear wave equations with both quasi-linear and semi-linear non-linearities, which is particularly challenging in two space dimensions. A key feature of the system is that there exist undi?erentiated wave components in the non-linearities, which somehow causes extra di?culties. Nevertheless, the Cauchy problem in two space dimenions was tackled by Lei-Lin-Zhou (2011) with small, regular, and compactly supported initial data, using Klainerman’s vector ?eld method enhanced by a novel angular-radial anisotropic technique. In the present paper, the authors revisit the Faddeev model and remove the compactness assumptions on the initial data by Lei-Lin-Zhou (2011). The proof relies on an improved L2 norm estimate of the wave components in Theorem 3.1 and a decomposition technique for non-linearities of divergence form.     

The <Emphasis Type="Italic">k</Emphasis>-Essence in the Relativistic Theory of Gravitation and General Relativity

We consider a model of a scalar field with a nontrivial kinetic part (k-essence) on the background of a flat homogeneous isotropic universe in the framework of the relativistic theory of gravitation and general relativity. Such a scalar field simulates the substance of an ideal fluid and serves as a model of dark energy because it leads to cosmological acceleration at later times. For finding a suitable cosmological scenario, it is more convenient to determine the dependence of the energy density of such a field on the scale factor and only then find the corresponding Lagrangian. Based on the solution of such an inverse problem, we show that in the relativistic theory of gravitation, either any scalar field of this type leads to instabilities, or the compression stage ends at an unacceptably early stage. We note that a consistent model of dark energy in the relativistic theory of gravitation can be a scalar field with a negative potential (ekpyrosis) of Steinhardt–Turok. In general relativity, the k-essence model is viable and can represent both dark energy and dark matter. We consider several specific k-essence models.     

Generalized o-minimality for partial orders

We consider partially ordered models. We introduce the notions of a weakly (quasi-)p.o.-minimal model and a weakly (quasi-)p.o.-minimal theory. We prove that weakly quasi-p.o.-minimal theories of finite width lack the independence property, weakly p.o.-minimal directed groups are Abelian and divisible, weakly quasi-p.o.-minimal directed groups with unique roots are Abelian, and the direct product of a finite family of weakly p.o.-minimal models is a weakly p.o.-minimal model. We obtain results on existence of small extensions of models of weakly quasi-p.o.-minimal atomic theories. In particular, for such a theory of finite length, we find an upper estimate of the Hanf number for omitting a family of pure types. We also find an upper bound for the cardinalities of weakly quasi-p.o.-minimal absolutely homogeneous models of moderate width.     

The resummation <Emphasis Type="Italic">L</Emphasis>-factor in the relativistic quasipotential approach

We obtain a new relativistic Coulomb-like resummation factor (L-factor) for an arbitrary orbital moment ℓ ≥ 0. We work within the fully covariant quasipotential approach in quantum field theory formulated in the relativistic configuration representation in the case of two interacting relativistic particles with arbitrary masses.     

Breaking of two-dimensional relativistic electron oscillations under small deviations from axial symmetry

A numerical asymptotic model for the breaking of two-dimensional plane relativistic electron oscillations under a small deviation from axial symmetry is developed. The asymptotic theory makes use of the construction of time-uniformly applicable solutions to weakly nonlinear equations. A special finite-difference algorithm on staggered grids is used for numerical simulation. The numerical solutions of axially symmetric one-dimensional relativistic problems yield two-sided estimates for the breaking time. Some of the computations were performed on the “Chebyshev” supercomputer (Moscow State University).     

Lieb–Robinson Bounds,Arveson Spectrum and Haag–Ruelle Scattering Theory for Gapped Quantum Spin Systems

We consider translation invariant gapped quantum spin systems satisfying the Lieb–Robinson bound and containing single-particle states in a ground state representation. Following the Haag–Ruelle approach from relativistic quantum field theory, we construct states describing collisions of several particles, and define the corresponding S-matrix. We also obtain some general restrictions on the shape of the energy–momentum spectrum. For the purpose of our analysis, we adapt the concepts of almost local observables and energy–momentum transfer (or Arveson spectrum) from relativistic QFT to the lattice setting. The Lieb–Robinson bound, which is the crucial substitute of strict locality from relativistic QFT, underlies all our constructions. Our results hold, in particular, in the Ising model in strong transverse magnetic fields.     

Analysis of the properties of solutions of the relativistic theory of gravity in the vicinity of a singular sphere

The properties of the static, spherically symmetric metric tensor of the relativistic theory of gravity are analyzed in the vicinity of a singular sphere. It is shown that a massive particle with a nongeodesic radial motion may reach this sphere and remain there at rest. Based on this property, it is inferred that a sphere formed by massive particles can serve as a source of singular metrics in the relativistic theory of gravity. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 111, No. 1, pp. 144–148, April, 1997.     

Quantum integrable systems of particles as gauge theories

We study quantum integrable systems of interacting particles from the point of view proposed by A. Gorsky and N. Nekrasov. We obtain the Sutherland system by a Hamiltonian reduction of an integrable system on the cotangent bundles to an affine algebra and show that it coincides with the Yang-Mills theory on a cylinder.We point out that there exists a tower of 2d quantum field theories. The top of this tower is the gauged G/G WZW model on a cylinder with an inserted Wilson line in an appropriate representation, which in our approach corresponds to Ruijsenaars' relativistic Calogero model. Its degeneration yields the 2d Yang-Mills theory, whose small radius limit is the Calogero model itself. We make some comments about the spectra and eigenstates of the models, which one can get from their equivalence with the field theories. Also we point out some possibilities of elliptic deformations of these constructions.Institute of Theoretical and Experimental Physics, 117259, Bol. Cheremushkinskaya 25, Moscow, Russia. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 1, pp. 97–103, July, 1994.     

Localizing the axioms

We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by Loc(ZFC), says that every set belongs to a transitive model of ZFC. LZFC consists of Loc(ZFC) plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All ${\Pi_2}We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by Loc(ZFC), says that every set belongs to a transitive model of ZFC. LZFC consists of Loc(ZFC) plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All P₂{\Pi_2} consequences of ZFC are provable in LZFC. LZFC strongly extends Kripke-Platek (KP) set theory minus Δ₀-Collection and minus ?{\in} -induction scheme. ZFC+ “there is an inaccessible cardinal” proves the consistency of LZFC. In LZFC we focus on models rather than cardinals, a transitive model being considered as the analogue of an inaccessible cardinal. Pushing this analogy further we define α-Mahlo models and P₁¹{\Pi_1^1} -indescribable models, the latter being the analogues of weakly compact cardinals. Also localization axioms of the form Loc(ZFC+f){Loc({\rm ZFC}+\phi)} are considered and their global consequences are examined. Finally we introduce the concept of standard compact cardinal (in ZFC) and some standard compactness results are proved.     

The Newton–Wigner Problem in the Relativistic Quantum Mechanics of Free Particles

We discuss the old Newton–Wigner problem, which is understood as the problem of a correct coordinate interpretation of the relativistic quantum mechanics of free particles. This problem is still relevant for quantum field theory because the S-matrix approach assumes that asymptotic fields describe relativistic free quantum-mechanical particles. From the modern standpoint, the original solution of this problem by Newton and Wigner already cannot be considered sufficient because it admits the smearing of wave packets with a superlight velocity. We discuss a possibility of overcoming this difficulty. This possibility is connected with relativistic deformations of the standard Heisenberg algebra. We describe situations in which a sort of desingularization of the effective free Hamiltonian occurs for some special deformations, which possibly allows preserving sublight velocity in the theory.     

The Theory of Kairons

In relativistic quantum mechanics wave functions of particles satisfy field equations that have initial data on a space-like hypersurface. We propose a dual field theory of “wavicles” that have their initial data on a time-like worldline. Propagation of such fields is superluminal, even though the Hilbert space of the solutions carries a unitary representation of the Poincaré group of mass zero. We call the objects described by these field equations “Kairons”. The paper builds the field equations in a general relativistic framework, allowing for a torsion. Kairon fields are section of a vector bundle over space-time. The bundle has infinite-dimensional fibres.     

Methods of statistical modeling in the process of sieve separation of heterogeneous particles

In the article we undertook the attempt of using the theory of random fields in the analysis of the separation process of particles in non-contiguous systems. The presentation of the complete model is a complicated problem. We can say that this article is start point of the theory, which will be supplemented gradually in the future. The separation process of particles on sieves is a phenomenon with random character. It is a result of random character of cleaned mixture and the nature of the separation process. Undoubtedly, an important part is played by the theory of random fields in which system-events, inherently not determined, are described in undetermined-probabilistic way. However, we can perceive one paradox in such actions – correctly constructed probabilistic models serve to impose deterministic frames on the given random process. We can say that stochastic models generate the matrix of the causal-consecutive dependence, which will give the ability of controlling a given process.     

Sudden-perturbation approximation for the dirac equation

We solve the Dirac equation describing the behavior of a hydrogen-like atom interacting with a spatially inhomogeneous ultrashort electromagnetic field pulse in the sudden-perturbation approximation. We express the corresponding transition probabilities through the known inelastic atomic form factors widely used in the theory of relativistic collisions of charged particles with atoms.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 1, pp. 58–63, January, 2005.     

On the formulation of initial-value problems for systems consisting of relativistic particles

We discuss questions related to the well-posedness of problems on the motion of relativistic many-body systems. For one-dimensional relativistic motion of N similar charges, we prove that an ordinary Cauchy problem usual in Newton mechanics can be stated; this is done in the framework of microscopic Maxwell-Lorentz electrodynamics (including a model with self-action) or Wheeler-Feynman theory. __________ Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 11, No. 1, Geometry, 2005.     

Group-Like Structure Underlying the Unit Ball in Real Inner Product Spaces

Abstraction of the relativistic velocity addition law and of the Thomas rotation of the special theory of relativity yields a means of endowing the unit ball in any real inner product space with a group- like structure, in which the standard associative- commutative laws are relaxed by means of the Thomas rotation. The resulting group- like object is called a complete weakly associative- commutative groupoid. Any complete WACG can be extended to a group analogous to the Lorentz group of the special theory of relativity.     

Small extensions of models of o-minimal theories and absolute homogeneity

We obtain some results on existence of small extensions of models of weakly o-minimal atomic theories. In particular, we find a sharp upper estimate for the Hanf number of such a theory for omitting an arbitrary family of pure types. We also find a sharp upper estimate for cardinalities of weakly o-minimal absolutely homogeneous models and a sufficient condition for absolute homogeneity.     

Smoothing effects for classical solutions of the relativistic Landau-Maxwell system

The relativistic Landau-Maxwell system is one of the most fundamental and complete models for describing the dynamics of a dilute hot plasma in which particles interact through Coulomb collisions and their self-consistent electromagnetic field. In this work, we prove that the classical solutions obtained by Strain and Guo become immediately smooth with respect to all variable under the extra assumption of the electromagnetic field. As a by-product, we also prove that the classical solutions to the relativistic Landau-Poisson system and the relativistic Landau equation have the same property without any extra assumption.