GROUP OF TRANSFORMATIONS WITH RESPECT TO THE COUNTERPART OF RAPIDITY AND RELATED FIELD EQUATIONS

The Lorentz-group of transformations usually consists of linear transformations of the coordinates, keeping as invariant the norm of the four-vector in (Minkowski) space-time. Besides those linear transformations, one may construct different forms of nonlinear transformations of the coordinates keeping unchanged that respective invariant. In this paper we explore nonlinear transformations of second-order which have a natural interpretation within the framework of Yamaleev's concept of the counterpart of rapidity (co-rapidity). The purpose of developed concept is to show that the formulae for energy and momentum of the relativistic particle become regular near the zero-mass and speed of light states. Furthermore, in a covariant formulation, the co-rapidity is presented as a four-vector which admits an extension of the Lorentz-group of transformations. In this paper we additionally show, that in the same way as the rapidity is related to the electromagnetic field, the co-rapidity is related to the field of strengths, which are given by a four-vector. The corresponding equations of such a field are also constructed.     

Relativistic Dynamics of Accelerating Particles Derived from Field Equations

In relativistic mechanics the energy-momentum of a free point mass moving without acceleration forms a four-vector. Einstein’s celebrated energy-mass relation E=mc ² is commonly derived from that fact. By contrast, in Newtonian mechanics the mass is introduced for an accelerated motion as a measure of inertia. In this paper we rigorously derive the relativistic point mechanics and Einstein’s energy-mass relation using our recently introduced neoclassical field theory where a charge is not a point but a distribution. We show that both the approaches to the definition of mass are complementary within the framework of our field theory. This theory also predicts a small difference between the electron rest mass relevant to the Penning trap experiments and its mass relevant to spectroscopic measurements.     

Clifford Algebras and the Dirac-Bohm Quantum Hamilton-Jacobi Equation

In this paper we show how the dynamics of the Schr?dinger, Pauli and Dirac particles can be described in a hierarchy of Clifford algebras, C_1,3, C_3,0{\mathcal{C}}_{1,3}, {\mathcal{C}}_{3,0}, and C_0,1{\mathcal{C}}_{0,1}. Information normally carried by the wave function is encoded in elements of a minimal left ideal, so that all the physical information appears within the algebra itself. The state of the quantum process can be completely characterised by algebraic invariants of the first and second kind. The latter enables us to show that the Bohm energy and momentum emerge from the energy-momentum tensor of standard quantum field theory. Our approach provides a new mathematical setting for quantum mechanics that enables us to obtain a complete relativistic version of the Bohm model for the Dirac particle, deriving expressions for the Bohm energy-momentum, the quantum potential and the relativistic time evolution of its spin for the first time.     

The classical limit of the relativistic Vlasov-Maxwell system

Solutions of the relativistic Vlasov-Maxwell system of partial differential equations are considered in three space dimensions. The speed of light,c, appears as a parameter in this system. For smooth Cauchy data, classical solutions are shown to exist on a time interval that is independent ofc. Then, using an integral representation for the electric and magnetic fields due to Glassey and Strauss [6], conditions are given under which solutions of the relativistic Vlasov-Maxwell system converge in pointwise sense to solutions of the non-relativistic Vlasov-Poisson system at the asymptotic rate of 1/c, asc tends to infinity.     

Spacetime quantization,generalized relativistic mechanics,and Mach's principle

The introduction of an elementary lengtha representing the ultimate limit for the smallest measurable distance leads to a generalization of Einstein's energy-momentum relation and of the usual Lorentz transformation. The value ofa is left unspecified, but is found to be equal tohc/2E _u, whereE _u is the total energy content of our universe. Particles of zero rest mass can only move at the velocityc of light in vacuum, while material bodies can move slower or faster than light, whena0, without violating the principle of causality. The laws of relativistic mechanics are actually generalized so that they include Mach's principle, since it is found that the universe as a whole can only be in a state of rest for any particular inertial observer.     

Fractional Dynamics of Relativistic Particle

Fractional dynamics of relativistic particle is discussed. Derivatives of fractional orders with respect to proper time describe long-term memory effects that correspond to intrinsic dissipative processes. Relativistic particle subjected to a non-potential four-force is considered as a nonholonomic system. The nonholonomic constraint in four-dimensional space-time represents the relativistic invariance by the equation for four-velocity u _μ u ^μ +c ²=0, where c is a speed of light in vacuum. In the general case, the fractional dynamics of relativistic particle is described as non-Hamiltonian and dissipative. Conditions for fractional relativistic particle to be a Hamiltonian system are considered.     

Photon wavelength, information transport speed, and mass. An information mechanics perspective

In Newton's mechanics, Maxwell's electromagnetism, Einstein's relativistic mechanics, quantum mechanics, and Kantor's information mechanics (IM), the speed of light in vacuumc has been treated as independent of wavelength. However, in IM, the transport of information by means of an electromagnetic signal appears to offer a perspective for reconsidering photon information transport speed, extending the concept of (rest) mass to treat the photon as having a mass of 1 bit.     

Consequences of the inertial equivalence of energy

The usual macroscopic theory of relativistic mechanics and electromagnetism is formulated so that all assumptions but one are consistent with both special relativity and Newtonian mechanics, the distinguishing assumption being that to any energyE, whatever its form, there corresponds an inertial massE/c ² . The speed of light enters this formulation only as a consequence of the inertial equivalent of energy1/c ² . While, for1/c ² >0 the resulting theory has symmetry under the Poincaré group, including Lorentz transformations, all its physical consequences can be derived and tested in any one inertial frame. In particular, an account is given in one inertial frame for the dynamic causes of relativistic effects for simple accelerated clocks and roads.     

The Lund fragmentation process for a multi-gluon string according to the area law

The Lund area law describes the probability for the production of a set of colourless hadrons from an initial set of partons, in the Lund string fragmentation model. It was derived from classical probability concepts but has later been interpreted as the result of gauge invariance in terms of the Wilson gauge loop integrals. In this paper {we will present a general method to implement the area law for a multi-gluon string state}. In this case the world surface of the massless relativistic string is a geometrically bent (1+1)-dimensional surface embedded in the (1+3)-dimensional Minkowski space. The partonic states are in general given by a perturbative QCD cascade and are consequently defined only down to a cutoff in the energy-momentum fluctuations. We will show that our method defines the states down to the hadronic mass scale inside an analytically calculable scenario. We will then show that there is a differential version of our process which is closely related to the generalized rapidity range , which has been used as a measure on the partonic states. We identify as the area spanned between the directrix curve (the curve given by the parton energy-momentum vectors laid out in colour order, which determines the string surface) and the average curve (to be called the -curve) of the stochastic X-curves (curves obtained when the hadronic energy-momentum vectors are laid out in rank order). Finally {we show that from the X-curve corresponding to a particular stochastic fragmentation situation it is possible to reproduce the directrix curve} (up to one starting vector and a set of sign choices, one for each hadron). This relationship provides an analytical formulation of the notion of parton–hadron duality. The whole effort is made in order to get a new handle to treat the transition region between where we expect perturbative QCD to work and where the hadronic features become noticeable. Received: 3 July 2001 / Published online: 31 August 2001     

Beryllium (boron) clustering quest in relativistic multifragmentation (BECQUEREL Project)

A physical program of irradiation of emulsions in beams of relativistic nuclei named the BECQUEREL Project is reviewed. It is destined to study in detail the processes of relativistic fragmentation of light radioactive and stable nuclei. The expected results would make it possible to answer some topical questions concerning the cluster structure of light nuclei. Owing to the best spatial resolution, the nuclear emulsions would enable one to obtain unique and evident results. The most important irradiations will be performed in the secondary beams of He, Be, B, C, and N radioactive nuclei formed on the basis of JINR Nuclotron beams of stable nuclei. We present results on the charged state topology of relativistic fragmentation of the ¹⁰B nucleus at low energy-momentum transfers as the first step of the research.     

The charged scalar-graviton coupling to order α

The electrodynamical corrections of order α to the charged scalar-graviton vertex are calculated in quantum field theory using the improved energy-momentum tensor. It is found that the deviations from Newton's law are of the r^?2 type, exactly the same as for the spin-$\text{1}\text{2}$ case. Also, the trace of the energy-momentum tensor is found not to vanish, at the one-loop level, in the limit of zero-mass scalar particles.     

Maxwell–Chern–Simons Topologically Massive Gauge Fields in the First-Order Formalism

We find the canonical and Belinfante energy-momentum tensors and their nonzero traces. We note that the dilatation symmetry is broken and the divergence of the dilatation current is proportional to the topological mass of the gauge field. It was demonstrated that the gauge field possesses the ‘scale dimensionality’ d=1/2. Maxwell–Chern–Simons topologically massive gauge field theory in 2+1 dimensions is formulated in the first-order formalism. It is shown that 6×6-matrices of the relativistic wave equation obey the Duffin–Kemmer–Petiau algebra. The Hermitianizing matrix of the relativistic wave equation is given. The projection operators extracting solutions of field equations for states with definite energy-momentum and spin are obtained. The 5×5-matrix Schr?dinger form of the equation is derived after the exclusion of non-dynamical components, and the quantum-mechanical Hamiltonian is obtained. Projection operators extracting physical states in the Schr?dinger picture are found.     

Spectral Analysis of Weakly Coupled Stochastic Lattice Ginzburg–Landau Models

We consider the relaxation to equilibrium of solutions , t>0, , of stochastic dynamical Langevin equations with white noise and weakly coupled Ginzburg–Landau interactions. Using a Feynman–Kac formula, which relates stochastic expectations to correlation functions of a spatially non-local imaginary time quantum field theory, we obtain results on the joint spectrum of H, , where H is the self-adjoint, positive, generator of the semi-group associated with the dynamics, and P ^j , j= 1, …, d are the self-adjoint generators of the group of lattice spatial translations. We show that the low-lying energy-momentum spectrum consists of an isolated one-particle dispersion curve and, for the mass spectrum (energy-momentum at zero-momentum), besides this isolated one-particle mass, we show, using a Bethe–Salpeter equation, the existence of an isolated two-particle bound state if the coefficient of the quartic term in the polynomial of the Ginzburg–Landau interaction is negative and d= 1, 2; otherwise, there is no two-particle bound state. Asymptotic values for the masses are obtained. Received: 27 September 2000 / Accepted: 16 January 2001     

Novel approach to pion and eta production in proton-proton collisions near threshold

We evaluate the threshold matrix–element for the reaction pp→ppπ⁰ in a fully relativistic Feynman diagrammatic approach. We employ a simple effective range approximation to take care of the S–wave pp final–state interaction. The experimental value for the threshold amplitude A = (2.7 −i0.3) fm⁴ can be reproduced by contributions from tree level chiral (long–range) pion exchange and short–range effects related to vector meson exchanges, with ω-exchange giving the largest individual contribution. Pion loop effects appear to be small. We stress that the commonly used heavy baryon formalism is not applicable in the NN–system above the pion production threshold due to the large external momentum, |p|≃ (Mm _π)^−1/2, with M and m _π the nucleon and the pion mass, respectively. We furthermore investigate the reaction pp→pnπ⁺ near threshold within the same approach. We extract from the data the triplet threshold amplitude as B = (2.8 −i1.5) fm⁴. Its real part can be well understood from (relativistic) tree level meson–exchange diagrams. In addition, we investigate the process pp→ppη near threshold. We use a simple factorization ansatz for the ppη final–state interaction and extract from the data the modulus of the threshold amplitude, |C|= 1.32 fm⁴. With g _ηN= 5.3, this value can be reproduced by (relativistic) tree level meson–exchange diagrams and η–rescattering, whose strength is fixed by the ηN scattering length. We also comment on the recent near threshold data for η^′–production. Received: 27 November 1998     

Protons flow in Pb+Pb collisions at 40 <Emphasis Type="Italic">A</Emphasis> GeV energy

For Pb+Pb collisions at 40 A GeV energy, we calculate the side-ward and elliptic differential flow of protons in the microscopic relativistic transport simulation model. We compare our results with the recent data from the NA49 Collaboration as a function of transverse momenta, rapidity and centrality. We find that the side-ward and elliptic flow agree reasonably well with the experimental data with and without momentum-dependent potentials in the simulation model.     

Exact self-consistent plane-symmetric solutions of the equations of two interacting fields (spinor field and scalar field)

A spinor field interacting with a zero-mass neutral scalar field is considered for the case of the simplest type of direct interaction, where the interaction Lagrangian has the formL _int =1/2 ϕ_αϕ_,α F(S) whereF(S) is an arbitrary function of the spinor field invariantS=ψψ. Exact solutions of the corresponding systems of equations that take into account the natural gravitational field in a plane-symmetric metric are obtained. It is proved that the initial system of equations has regular localized soliton-type solutions only if the energy density of the zero-mass scalar field is negative as it “disengages” from interaction with the spinor field. In two-dimensional space-time the system of field equations we are studying describes the configuration of fields with constant energy densityT ₀₀ , i.e., no soliton-like solutions exist in this case. Russian People’s Friendship University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 69–75, July, 1998.     

Might Dark Matter and Energy be Intrinsic Properties of?Space?

It is shown that if a volume element V, of space is assumed to have intrinsic energy E, then basic principles of mechanics, thermodynamics and special relativity lead to the equation of state: E=pV, where p is the pressure. When this equation of state is incorporated in the Einstein equations it leads to the prediction that the orbital speed of matter circling a visible galaxy embedded in a spherical galactic halo should be relativistic, in disagreement with observations. However, it also leads directly to the interesting notion that the inertial mass of such a medium can be understood as a resistance to being compressed via Lorentz contraction. It is then shown that the mathematical structure of thermodynamics suggests another plausible definition of pressure if we allow for the possibility that the intrinsic energy of spacetime may not be described by the same work-energy relationship as ordinary matter. This additional possibility leads to the equation of state: E=−pV. While both of these equations of state describe forms of energy that are quite unlike ordinary energy, neither alone is able to account for observed rotational velocity curves of matter orbiting visible galaxies. Therefore, the possibility that space has two distinct components of energy is investigated. This results in a plausible, two-component equation of state in which the former equation of state is tentatively identified as the “dark matter” (DM) component, the latter as the “dark energy” (DE) component. The effective equation of state of space, accounting for the presence of both components, may then be written in the form: p=w ε, where ε is the total energy density, p the total pressure, and w represents the fractional excess of DM over DE (and therefore satisfies: −1≤w≤+1). Given the wide range of possible spacetime properties implied by this equation it appears to be a viable candidate for explaining observations presently attributed to the presence of both DM and DE. Specifically, the static, spherically symmetric solution of Einstein’s field equations, neglecting effects of ordinary matter, predicts the inverse r ² distribution of intrinsic space energy required to explain observed constant rotational velocity curves for matter in circular orbits around visible galaxies embedded within spherically symmetric galactic halos. The proposed equation of state is also capable of describing regions of space undergoing accelerated expansion as regions where DE is dominant (i.e., w<−1/3).     

What do we really know about cold nuclear matter effects?

The measurement of charmonium suppression in relativistic heavy ion collisions is posited to be an unambiguous probe of the properties of the strongly interacting quark gluon plasma (sQGP). In hot and dense QCD matter Debye color screening prevents charm and anti-charm quark pairs from forming J/ψ mesons if the screening radius is smaller than the binding radius. However, one must have a clear understanding of the expected suppression in normal density QCD matter before interpreting any additional anomalous suppression. The PHENIX experiment has measured J/ψ production from colliding proton + proton and deuteron + gold beams at 200 GeV from the relativistic heavy ion collider (RHIC). The deuteron + gold data can be compared to the proton + proton baseline in order to establish the typical suppression in cold nuclear matter (CNM). For PHENIX, a suppression of J/ψ in cold nuclear matter is observed as one goes forward in rapidity (in the deuteron-going direction), corresponding to a region more sensitive to initial state low-x gluons in the gold nucleus. These results can be convoluted with the nuclear-environment-modified parton distribution functions, extracted from deep inelastic scattering (DIS) and Drell-Yan (DY) data, in order to estimate the J/ψ break up cross section in cold nuclear matter. One can also use a data driven method that does not rely on the assumption of the production mechanism, or PDF parameterization, to extrapolate to the heavy ion collision case. At this time both the predictions for CNM effect suppression in heavy ion collisions are somewhat ambiguous. Future results using the data acquired by the PHENIX experiment in run-6 (p + p) and run-8 (d + Au) will be vital for our understanding. These data, which are in the process of being analyzed, will provide a needed improvement in the statistical and systematic precision of constraints for CNM effects. These constraints must be improved in order to make firm conclusions concerning additional hot nuclear matter charmonium suppression in the sQGP.