Modification of special relativity and formulation of convergent and invariant quantum field theory

Besides two fundamental postulates, (i) the principle of relativity and (ii) the constancy of the speed of light in all inertial frames of reference, the special theory of relativity uses another assumption. This other assumption concerns the Euclidean structure of gravity-free space and the homogeneity of gravity-free time in the usual inertial coordinate system. Introducing the primed inertial coordinate system, in addition to the usual inertial coordinate system, for each inertial frame of reference, we assume the Euclidean structures of gravity-free space and time in the primed inertial coordinate system and their generalized Finslerian structures in the usual inertial coordinate system. We combine the alternative assumption with the two postulates (i) and (ii) to modify the special theory of relativity. The modified special relativity theory involves two versions of the light speed, infinite c′ in the primed inertial coordinate system and finite c in the usual inertial coordinate system. It also involves the c′-type Galilean transformation between any two primed inertial coordinate systems and the localized Lorentz transformation between two corresponding usual inertial coordinate systems. Since all our experimental data are collected and expressed in the usual inertial coordinate system, the physical principle is: the c′-type Galilean invariance in the primed inertial coordinate system plus the transformation from the primed inertial coordinate system to the usual inertial coordinate system. This principle is applied to a reformulation of mechanics, field theory and quantum field theory. Relativistic mechanics in the usual inertial coordinate system is unchanged, while field theory is developed and divergence-free. Any c′-type Galilean-invariant field system can be quantized by using the canonical quantization method in the primed inertial coordinate system. We establish a transformation law for quantized field systems as they are transformed from the primed to the usual inertial coordinate system. It is shown that the modified special relativity theory, together with quantum mechanics, leads to a convergent and invariant quantum field theory, in full agreement with experimental facts. The formulation of this quantum field theory does not demand departures from the concepts such as local Lorentz invariance in the usual inertial coordinate system, locality of interactions, and local or global gauge symmetries.     

Equivalence of Two Gyrogroup Structures on Unit Balls

In special relativity all possible velocities form an open ball, R_c ³, of radius c in R³ with a binary operation, ⊕, given by Einstein’s velocity composition law. The concept of gyrogroups arose in the study of the algebraic structure underlying the resulting groupoid, (R_c ³, ⊕), giving rise to a generalization of the usual abelian group theory. Two different gyrogroup structures were defined on the unit ball of the Euclidean space resulting in the so-called standard and nonstandard Lorentzian groups [14]. The aim of this paper is (i) to define gyrogroup isomorphism and hence (ii) to show that the two structures are isomorphic in the gyrogroup sense; and (iii) to show that the resulting two Lorentzian groups are isomorphic, in the ordinary group sense, as pointed out by Urbantke [23]. Our results also include determining the group of all continuous automorphisms of the unit ball, when the latter is considered as a gyrogroup.     

Weakly Associative Groups

The space ? _c ³ of 3-dimensional relativistically admissible velocities possesses (i) a binary operation which represents the relativistic velocity composition law; and (ii) a mapping from the cartesian product ? _c ³ ×? _c ³ into a subgroup of its automorphism group, Aut(? _c ³ ), representing the Thomas precession of special relativity. These binary operation and mapping are studied in special relativity as two isolated phenomena. It was recently discovered, however, that they are linked by an algebraic structure which gives rise to a theory of weakly associative and weakly associative-commutative groups. The axioms of these groups are presented in this paper and employed to obtain various interesting results. The algebraic structure underlying these nonstandard groups has been discovered and studied in a totally different context by Karzel (1965), Kerby and Wefelscheid.     

The most intensive fluctuation in chaotic time series and relativity principle

Relativity principle in mechanics and principle of invariant speed of light lead to Einstein theory. The exponent of p-order momentum, derived from a piece of multi-scale chaotic time series, varies with the order p and cannot exceeds a maximum, so there exists the principle of scale relativity. Its special case is the same one as Lorenz transformation from Einstein theory.     

Special relativity and theory of gravity via maximum symmetry and localization

Like Euclid,Riemann and Lobachevski geometries are on an almost equal footing,based on the principle of relativity of maximum symmetry proposed by Professor Lu Qikeng and the postulate on invariant universal constants c and R,the de Sitter/anti-de Sitter(dS/AdS)special relativity on dS/AdS-space with radius R can be set up on an almost equal footing with Einstein's special relativity on the Minkowski-space in the case of R→∞. Thus the dS-space is coin-like:a law of inertia in Beltrami atlas with Beltrami time simultaneity for the principle of relativity on one side,and the proper-time simultaneity and a Robertson-Walker-like dS-space with entropy and an accelerated expanding S~3 fitting the cosmological principle on another side. If our universe is asymptotic to the Robertson-Walker-like dS-space of R(?)(3/Λ)~(1/2),it should be slightly closed in O(A)with entropy bound S(?)3πc~3k_B/ΛGh.Contrarily,via its asymptotic behavior, it can fix on Beltrami inertial frames without‘an argument in a circle’and acts as the origin of inertia. There is a triality of conformal extensions of three kinds of special relativity and their null physics on the projective boundary of a 5-d AdS-space,a null cone modulo projective equivalence[N](?)_p(AdS~5). Thus there should be a dS-space on the boundary of S~5×AdS~5 as a vacuum of supergravity. In the light of Einstein's‘Galilean regions’,gravity should be based on the localized principle of relativity of full maximum symmetry with a gauge-like dynamics.Thus,this may lead to the theory of gravity of corresponding local symmetry.A simple model of dS-gravity characterized by a dimensionless constant g(?)(AGh/3c~3)~(1/2)～10~(-61)shows the features on umbilical manifolds of local dS-invariance. Some gravitational effects out of general relativity may play a role as dark matter. The dark universe and its asymptotic behavior may already indicate that the dS special relativity and dS-gravity be the foundation of large scale physics.     

Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka–Volterra competition system with diffusion

We study the existence, uniqueness, and asymptotic stability of time periodic traveling wave solutions to a periodic diffusive Lotka–Volterra competition system. Under certain conditions, we prove that there exists a maximal wave speed c^? such that for each wave speed c?c^?, there is a time periodic traveling wave connecting two semi-trivial periodic solutions of the corresponding kinetic system. It is shown that such a traveling wave is unique modulo translation and is monotone with respect to its co-moving frame coordinate. We also show that the traveling wave solutions with wave speed c<c^? are asymptotically stable in certain sense. In addition, we establish the nonexistence of time periodic traveling waves for nonzero speed c>c^?.     

Conservation laws of shallow water theory and the Galilean relativity principle

By the example of a model of shallow water theory, it is shown that the compatibility analysis of the Hugoniot conditions for various basic systems of conservation laws in the coordinate system moving together with a strong discontinuity can lead to erroneous results. It is connected with the hierarchy of conservation laws in shallow water theory with respect to the Galilean transformation, according to which the conservation law for total energy is unconditionally noninvariant with respect to this transformation, which leads to the dependence of the corresponding Hugoniot condition on the velocity of the inertial reference frame. It is shown that the specified shortcoming of the classical shallow water theory is absent in the model of vortex shallow water suggested by V. M. Teshukov.     

Non-linear determinacy of minimum wave speed for a Lotka-Volterra competition model

For a reaction-diffusion system that serves as a 2-species Lotka-Volterra diffusive competition model, suppose that the corresponding reaction system has one stable boundary equilibrium and one unstable boundary equilibrium. Then it is well known that there exists a positive number c^?, called the minimum wave speed, such that, for each c larger than or equal to c^?, the reaction-diffusion system has a positive traveling wave solution of wave speed c connecting these two equilibria if and only if c?c^?. It has been shown that the minimum wave speed for this system is identical to another important quantity - the asymptotical speed of population spread towards the stable equilibrium. Hence to find the minimum wave speed c^? not only is of the interest in mathematics but is of the importance in application. It has been conjectured that the minimum wave speed can be determined by studying the eigenvalues of the unstable equilibrium, called the linear determinacy. In this paper we will show that the conjecture on the linear determinacy is not true in general.     

Spatial dynamics of a nonlocal and time-delayed reaction-diffusion system

In this paper, we investigate the spatial dynamics of a nonlocal and time-delayed reaction-diffusion system, which is motivated by an age-structured population model with distributed maturation delay. The spreading speed c^*, the existence of traveling waves with the wave speed c?c^*, and the nonexistence of traveling waves with c<c^* are obtained. It turns out that the spreading speed coincides with the minimal wave speed for monotone traveling waves.     

Spreading speed and traveling waves for a multi-type SIS epidemic model

The theory of asymptotic speeds of spread and monotone traveling waves for monotone semiflows is applied to a multi-type SIS epidemic model to obtain the spreading speed c^∗, and the nonexistence of traveling waves with wave speed c<c^∗. Then the method of upper and lower solutions is used to establish the existence of monotone traveling waves connecting the disease-free and endemic equilibria for c?c^∗. This shows that the spreading speed coincides with the minimum wave speed for monotone traveling waves. We also give an affirmative answer to an open problem presented by Rass and Radcliffe [L. Rass, J. Radcliffe, Spatial Deterministic Epidemics, Math. Surveys Monogr. 102, Amer. Math. Soc., Providence, RI, 2003].     

Fiber Bundles, Connections, General Relativity, and the Einstein-Cartan Theory – Part II

The formalism of tetrads and spin connections is presented, and with it, the Einstein-Cartan (E-C) equations are derived from the usual actions. It is shown how torsion vanishes in the case of pure gravity. Then, the equations in the presence of the Dirac and the Maxwell fields are derived. A possibility of choosing a locally inertial coordinate system in the presence of a totally antisymmetric torsion is proved. We discuss a conflict between torsion and the local gauge invariance of the electromagnetic field. Finally, we study Lorentz and Poincaré gauge invariance of general relativity and the E-C theory.     

SLEs as boundaries of clusters of Brownian loops

In this research announcement, we show that SLE curves can in fact be viewed as boundaries of certain clusters of Brownian loops (of the clusters in a Brownian loop soup). For small densities c of loops, we show that the outer boundaries of the clusters created by the Brownian loop soup are SLE_κ-type curves where κ∈(8/3,4] and c related by the usual relation c=(3κ?8)(6?κ)/2κ (i.e., c corresponds to the central charge of the model). This gives (for any Riemann surface) a simple construction of a natural countable family of random disjoint SLE_κ loops, that behaves “nicely” under perturbation of the surface and is related to various aspects of conformal field theory and representation theory. To cite this article: W. Werner, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Local existence and non-relativistic limits of shock solutions to a multidimensional piston problem for the relativistic Euler equations

The multidimensional piston problem is a special initial-boundary value problem. The boundary conditions are given in two conical surfaces: one is the boundary of the piston, and the other is the shock whose location is to be determined later. In this paper, we are concerned with spherically symmetric piston problem for the relativistic Euler equations. A local shock front solution with the state equation p = a ² ρ, a is a constant and has been established by the Newton iteration. To overcome the difficulty caused by the free boundary, we introduce a coordinate transformation to fix it and employ the linear iteration scheme to establish a sequence of approximate solutions to the auxiliary problems by iteration. In each step, the value of the solution of the previous problem is taken as the data to determine the solution of the next problem. We obtain the existence of the original problem by establishing the convergence of these sequences. Meanwhile, we establish the convergence of the local solution as c → ∞ to the corresponding solution of the classical non-relativistic Euler equations.     

Existence and uniqueness of traveling waves for non-monotone integral equations with applications

A class of integral equations without monotonicity is investigated. It is shown that there is a spreading speed c^∗>0 for such an integral equation, and that its limiting integral equation admits a unique traveling wave (up to translation) with speed c?c^∗ and no traveling wave with c<c^∗. These results are also applied to some nonlocal reaction-diffusion population models.     

An LP estimate for Maxwell's equations with source term

In this Note we establish an interior L^p-type estimate for the solutions of Maxwell's equations with source term in a domain filled with two different materials separated by a $\text{C}{}^2$ interface. Due to the singularity of the dielectric permittivity, the usual elliptic estimates cannot be applied directly. A special curl–div decomposition is introduced for the electric field to reduce the problem to an elliptic equation in divergence form with discontinuous coefficients. The potential theory analysis and the jump condition lead to the L^p estimates which are superior to the straightforward Nash–Moser estimates. The reduction procedure is expected to be useful for numerical simulation. Such an estimate is crucial for solving nonlinear Maxwell's equations that arise for example in the modeling of nonlinear optics. To cite this article: G. Bao et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

The nonbifurcation of periodic solutions when the variational matrix has a zero eigenvalue

It is assumed that the variational matrix of the 2-dimensional system x′ = F(x, ?) has at least one zero eigenvalue rather than the usual Hopf assumption of two conjugate pure imaginary eigenvalues. It is then shown that genetically, although one may expect a bifurcation of stationary solutions, a bifurcation of periodic solutions will not occur.     

Spreading speed and traveling waves for a nonmonotone reaction–diffusion model with distributed delay and nonlocal effect

In this paper, spreading speed and traveling waves for reaction–diffusion model with distributed delay and nonlocal effect without monotonicity are investigated. It is shown that there exists the spreading speed c^∗ which coincides with the minimal wave speed, and its limiting integral equation has an unique traveling wave with speed c > c^∗, and no traveling wave with c < c^∗. Moreover, the dependence of the spreading speed on the delay and the nonlocal effect is considered.     

Dielectric properties of polyacrylamide and its utilization as a hydrogel

Some of the dielectric properties of polyacrylamide are given such as the variation of the dielectric constant, ε^′, and the dielectric loss, ε^′′, and the power factor tan δ with the applied frequency from 10⁵ to 10⁷ Hz within the temperature range 20–50°C. The dielectric dispersion and absorption bands appear within these ranges of frequencies and temperatures. The relaxation process is explained and discussed, also the relaxation times are calculated and the activation energy and entropy change of dielectric relaxation are calculated from the usual rate equation and discussed. The process of polymerization of polyacrylamide is explained through comparing the dielectric properties of the monomer (acrylamide) and the polymer. Out of two approaches to obtain a hydrogel from polyacrylamide, the physical approach is thoroughly investigated and discussed.     

Periodic behavior of a nonlinear third order vibrating system

A theorem is proved to show that the third order differential equation x^‴+f(t,x,x^′,x^″)=0 has nontrivial solutions characterized by x^′(0)=x^′(τ)=0 when x,x^′,x^″ and f(t,x,x^′,x^″) are bounded. A second condition is introduced to prove the existence of periodic solution for this equation. It is shown that the equation has a τ-periodic solution if f(t,x,x^′,x^″) is an even function with respect to x^′. The existence and periodicity conditions would be applied to third order systems such as viscoelastic mechanical vibration isolator system. The concepts of Green’s function and the Schauder’s fixed-point theorem have been used for proving the third-order-existence theorem.