Space-Time Grains: Roots of Special and Doubly Special Relativity

We show that the special relativistic dynamics when combined with quantum mechanics and the concept of superstatistics can be interpreted as arising from two interlocked non-relativistic stochastic processes that operate at different energy scales. This interpretation leads to Feynman amplitudes that are in the Euclidean regime identical to transition probability of a Brownian particle propagating through a granular space. Some kind of spacetime granularity could be therefore held responsible for the emergence at larger scales of various symmetries. For illustration we consider also the dynamics and the propagator of a spinless relativistic particle. Implications for doubly special relativity, quantum field theory, quantum gravity and cosmology are discussed.     

Study of propagation of ultraintense electromagnetic wave throughplasma using semi-Lagrangian Vlasov codes

Interaction of relativistically strong laser pulses with underdense and overdense plasmas is investigated by a semi-Lagrangian Vlasov code. These Vlasov simulations revealed a rich variety of phenomena associated with the fast particle dynamics induced by the electromagnetic wave as electron trapping, particle acceleration, and electron plasma wavebreaking. To describe the distribution of accelerated particle momenta and energy will require a very detailed analysis of the kinetic and time history of the plasma wave evolution. The semi-Lagrangian Vlasov code allows us to handle the interaction of ultrashort electromagnetic pulse with plasma at strongly relativistic intensities with a great deal of resolution in phase space     

Spin-1/2 relativistic particle in a magnetic field in NC phase space

This work provides an accurate study of the spin-l/2 relativistic particle in a magnetic field in NC phase space. By detailed calculation we find that the Dirac equation of the relativistic particle in a magnetic field in noncommutative space has similar behaviour to what happens in the Landau problem in commutative space even if an exact map does not exist. By solving the Dirac equation in NC phase space, we not only obtain the energy level of the spin-1/2 relativistic particle in a magnetic field in NC phase space but also explicitly offer some additional terms related to the momentum-momentum non-commutativity.     

Spin-1/2 relativistic particle in a magnetic field in NC phase space

This work provides an accurate study of the spin-1/2 relativistic particle in a magnetic field in NC phase space. By detailed calculation we find that the Dirac equation of the relativistic particle in a magnetic field in noncommutative space has similar behaviour to what happens in the Landau problem in commutative space even if an exact map does not exist. By solving the Dirac equation in NC phase space, we not only obtain the energy level of the spin-1/2 relativistic particle in a magnetic field in NC phase space but also explicitly offer some additional terms related to the momentum-momentum non-commutativity.     

Theory of Stochastic Schrödinger Equation in Complex Vector Space

A generalized Schrödinger equation containing correction terms to classical kinetic energy, has been derived in the complex vector space by considering an extended particle structure in stochastic electrodynamics with spin. The correction terms are obtained by considering the internal complex structure of the particle which is a consequence of stochastic average of particle oscillations in the zeropoint field. Hence, the generalised Schrödinger equation may be called stochastic Schrödinger equation. It is found that the second order correction terms are similar to corresponding relativistic corrections. When higher order correction terms are neglected, the stochastic Schrödinger equation reduces to normal Schrödinger equation. It is found that the Schrödinger equation contains an internal structure in disguise and that can be revealed in the form of internal kinetic energy. The internal kinetic energy is found to be equal to the quantum potential obtained in the Madelung fluid theory or Bohm statistical theory. In the rest frame of the particle, the stochastic Schrödinger equation reduces to a Dirac type equation and its Lorentz boost gives the Dirac equation. Finally, the relativistic Klein–Gordon equation is derived by squaring the stochastic Schrödinger equation. The theory elucidates a logical understanding of classical approach to quantum mechanical foundations.     

Symmetry groups, density-matrix equations and covariant Wigner functions

A representation theory for Lie groups is developed taking the Hilbert space, say , of the w^*-algebra standard representation as the representation space. In this context the states describing physical systems are amplitude wave functions but closely connected with the notion of the density matrix. Then, based on symmetry properties, a general physical interpretation for the dual variables of thermal theories, in particular the thermofield dynamics (TFD) formalism, is introduced. The kinematic symmetries, Galilei and Poincaré, are studied and (density) amplitude matrix equations are derived for both of these cases. In the same context of group theory, the notion of phase space in quantum theory is analysed. Thus, in the non-relativistic situation, the concept of density amplitude is introduced, and as an example, a spin-half system is algebraically studied; Wigner function representations for the amplitude density matrices are derived and the connection of TFD and the usual Wigner-function methods are analysed. For the Poincaré symmetries the relativistic density matrix equations are studied for the scalar and spinorial fields. The relativistic phase space is built following the lines of the non-relativistic case. So, for the scalar field, the kinetic theory is introduced via the Klein–Gordon density-matrix equation, and a derivation of the Jüttiner distribution is presented as an example, thus making it possible to compare with the standard approaches. The analysis of the phase space for the Dirac field is carried out in connection with the dual spinor structure induced by the Dirac-field density-matrix equation, with the physical content relying on the symmetry groups. Gauge invariance is considered and, as a basic result, it is shown that the Heinz density operator (which has been used to develope a gauge covariant kinetic theory) is a particular solution for the (Klein–Gordon and Dirac) density-matrix equation.     

A Classical and Quantum Relativistic Interacting Variable-Mass Model

A classical and quantum relativistic interacting particle formalism is revisited. A Hilbert space is achieved through the use of variable individual particle rest masses, but no c-number mass parameter is required for the relativistic free particle. Boosted center of momentum states feature in both the free and interacting model. The implications of a failure to impose simultaneity conditions at the classical level are explored. The implementation of these conditions at the quantum level leads to a finite uncertainty in interaction times, perhaps more closely modeling the exchange of virtual particles in quantum field theory. This work is compared and contrasted with other variable mass models in the literature.     

Path integral for the relativistic particle and harmonic oscillators

The action for a massive particle in special relativity can be expressed as the invariant proper length between the end points. In principle, one should be able to construct the quantum theory for such a system by the path integral approach using this action. On the other hand, it is well known that the dynamics of a free, relativistic, spinless massive particle is best described by a scalar field which is equivalent to an infinite number of harmonic oscillators. We clarify the connection between these two—apparently dissimilar—approaches by obtaining the Green function for the system of oscillators from that of the relativistic particle. This is achieved through defining the path integral for a relativistic particle rigorously by two separate approaches. This analysis also shows a connection between square root Lagrangians and the system of harmonic oscillators which is likely to be of value in more general context.     

Canonical proper time formulation of relativistic particle dynamics

A canonical (contact) transformation is performed on the time variable (in extended phase space) to reexpress relativistic dynamics in terms of proper time, leaving the generalized coordinates and canonical momentum as functions of this time variable. The form of the energy functional conjugate to this time variable is seen to be similar to that of a nonrelativistic dynamics at all values of particle momenta. The formulation is explored for single- and multiparticle classical systems. The (form) invariance of the theory is determined by a group which results from a similarity action of the contact group on the Poincaré group. One advantage of this approach is that it overcomes the no-interaction difficulties introduced by standard methods.     

Transport coefficients near chiral phase transition

We analyze the transport properties of relativistic fluid composed of constituent quarks at finite temperature and density. We focus on the shear and bulk viscosities and study their behavior near chiral phase transition. We model the constituent quark interactions through the Nambu–Jona-Lasinio Lagrangian. The transport coefficients are calculated within kinetic theory under relaxation time approximation including in-medium modification of quasi-particles dispersion relations. We quantify the influence of the order of chiral phase transition and the critical end point on dissipative phenomena in such a medium.     

The laws of relativistic thermodynamics: II. The density,momentum and energy laws

The conservation laws of particle number, momentum and energy are derived from relativistic kinetic theory. The first law of relativistic thermodynamics is formulated.     

Resolution of the Klein paradox for spin-1/2 particles

The problem of a relativistic spin-1/2 particle scattering from a step potential is solved within the theoretical framework of relativistic dynamics. This treatment avoids the Klein paradox. An experiment for testing the theory is suggested.     

Classical and quantum dynamics of a kicked relativistic particle in a box

We study classical and quantum dynamics of a kicked relativistic particle confined in a one dimensional box. It is found that in classical case for chaotic motion the average kinetic energy grows in time, while for mixed regime the growth is suppressed. However, in case of regular motion energy fluctuates around certain value. Quantum dynamics is treated by solving the time-dependent Dirac equation with delta-kicking potential, whose exact solution is obtained for single kicking period. In quantum case, depending on the values of the kicking parameters, the average kinetic energy can be quasi periodic, or fluctuating around some value. Particle transport is studied by considering spatio-temporal evolution of the Gaussian wave packet and by analyzing the trembling motion.     

A classical analogue of Yukawa coupling: Presymplectic formalism

A unified treatment of Yang-Mills and Higgs fields in classical gauge theory is carried out in a general relativistic context. A presymplectic formalism for a spinless test particle dwelling in this background geometry is described. The mass of this particle is found to depend specifically upon its generalized isospin and the Higgs field. This mass generating process is very much reminiscent of the so-called Yukawa coupling in the (electro-weak) standard model. The space of motions (phase space) is constructed together with a set of generalized Wong equations. Comparison with the Marsden-Weinstein symplectic reduction procedure is achieved.Laboratoire Propre, Centre National de la Recherche Scientifique, LP 7061.     

Passage of a high-current relativistic electron beam through matter

A solution is presented for the problem of passage of a high-current relativistic electron beam through matter in the stationary case with one-dimensional geometry. The system of equations describing the passage of the beam consists of the kinetic equation for fast electrons, which considers the effect of the electric field on the magnitude and direction of particle momentum, the equations for the field produced by the space charge generated by thermalized electrons, and relations connecting the conductivity of the medium to the radiation field. Higher-order perturbation theory is used for the solution. The solutions reveal that the distribution of expended energy, thermalized electrons, and other properties of the flow are highly dependent on the density of the incident flux and the conductivity of the medium. It will be shown that linear transfer theory may be applied to calculation of the passage of high-current beams through matter over a wide range of currents and conductivities, if the barrier thickness does not exceed one-half the path length, but cannot be used for calculation of passage through large-thickness barriers, i.e., with thickness comparable to the electron free path length.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 67–74, September, 1979.The author thanks A. N. Didenko and A. M. Kol'chuzhkin for their evaluation of the study.     

Particle Trajectories for Quantum Field Theory

The formulation of quantum mechanics developed by Bohm, which can generate well-defined trajectories for the underlying particles in the theory, can equally well be applied to relativistic quantum field theories to generate dynamics for the underlying fields. However, it does not produce trajectories for the particles associated with these fields. Bell has shown that an extension of Bohm’s approach can be used to provide dynamics for the fermionic occupation numbers in a relativistic quantum field theory. In the present paper, Bell’s formulation is adopted and elaborated on, with a full account of all technical detail required to apply his approach to a bosonic quantum field theory on a lattice. This allows an explicit computation of (stochastic) trajectories for massive and massless particles in this theory. Also particle creation and annihilation, and their impact on particle propagation, is illustrated using this model.     

Relativistic aberrations in quantum phase space

A phase space treatment of special relativity of quantum systems is developed. In this approach a quantum particle remains localized if subject to inertial transformations, the localization occurring in a finite phase space area. Unlike in the non-relativistic case, relativistic transformations generally distort the phase space distribution function, being equivalent to aberrations in optics. The relativistic aberrations of massive particles are in general different from those of optical images.     

A generalized quantum kinetic equation for models of relativistic nuclear dynamics

Zubarev’s method of non-equilibrium statistical operator is applied to problems of relativistic kinetic theory. Within this method, a generalized relativistic quantum kinetic equation for the relativistic Wigner function is derived with taking into account the drift term of the Vlasov type and the collision integral of the second order in particle interaction. It is shown that this result holds as well for gauge invariant theories in the case of slowly changing fields. An advantage of the developed approach is exemplified by the consideration of relativistic nuclear matter within the Walecka and Nambu-Jona-Lasinio models. Typical relativistic effects like retardation, spin degrees of freedom and antiparticle evolution are taken into consideration.     

Curvature Coupling and Accelerated Expansion of the Universe

A new exactly solvable model for the evolution of a relativistic kinetic system interacting with an internal stochastic reservoir under the influence of a gravitational background expansion is established. This model of self-interaction is based on the relativistic kinetic equation for the distribution function defined in the extended phase space. The supplementary degree of freedom is described by the scalar stochastic variable (Langevin source), which is considered to be the constructive element of the effective one-particle force. The expansion of the Universe is shown to be accelerated for the suitable choice of the non-minimal self-interaction force.