The relativistic jerk, snap and all higher-order kinematical D-vectors are defined for the motion of a massive particle in a D-dimensional Minkowski spacetime. We illustrate the formalism with stationary motions, for which we provide a new, Lorentz covariant, classification. We generalize some cases to branes, explaining the relevance to uniform motion in a heat bath. We also consider some non-stationary motions, including motion with constant proper jerk, and free fall into a black hole as viewed from a GEMS perspective.
We discuss kinematical properties of a free relativistic particle with deformed phase space in which momentum space is given by (a submanifold of) de Sitter space. We provide a detailed derivation of the action, Hamiltonian structure and equations of motion for such free particle. We study the action of deformed relativistic symmetries on the phase space and derive explicit formulas for the action of the deformed Poincare' group. Finally we provide a discussion on parametrization of the particle worldlines stressing analogies and differences with ordinary relativistic kinematics.
We present a way to derive a deformation of special relativistic kinematics (possible low energy signal of a quantum theory of gravity) from the geometry of a maximally symmetric curved momentum space. The deformed kinematics is fixed (up to change of coordinates in the momentum variables) by the algebra of isometries of the metric in momentum space. In particular, the well-known example of $κ$-Poincaré kinematics is obtained when one considers an isotropic metric in de Sitter momentum space such that translations are a subgroup of the isometry group, and for a Lorentz covariant algebra one ge…
We show how a deformed composition law of four-momenta can be used to define, at the classical level, a modified notion of spacetime for a system of two particles through the crossing of worldlines in particle interactions. We present a derivation of a generic relativistic isotropic deformed kinematics and discuss the complementarity and relations with other derivations based on $κ$-Poincaré Hopf algebra or on the geometry of a maximally symmetric momentum space.
We propose a method for deforming an extended Galilei algebra that leads to a nonstandard realization of the Poincaré group with the Fock-Lorentz linear fractional transformations. The invariant parameter in these transformations has the dimension of length. Combining this deformation with the standard one (with an invariant velocity $c$) leads to the algebra of the symmetry group of the anti-de Sitter space in Beltrami coordinates. In this case, the action for free point particles contains the dimensional constants $R$ and $c$. The limit transitions lead to the ordinary ($R\to \infty$) or alt…
Doubly special relativity has been studied for the last twenty years as a way to go beyond the special relativistic kinematics, trying to capture residual effects of a quantum gravity theory. In particular, in doubly special relativity the Einstenian relativity principle is generalized, adding to the speed of light another relativistic invariant, the Planck energy. There are several papers in the literature showing a connection between this deformed kinematics and a curved momentum space. Here we review how such kinematics can be derived from geometrical ingredients in a rigorous way, and how …
We study the rest-frame instant form of a new formulation of relativistic perfect fluids in terms of new generalized Eulerian configuration coordinates. After the separation of the relativistic center of mass from the relative variables on the Wigner hyper-planes, we define orientational and shape variables for the fluid, viewed as a relativistic extended deformable body, by introducing dynamical body frames. Finally we define Dixon's multipoles for the fluid.
Fluid mechanics can be formulated on dynamical surfaces of arbitrary co-dimension embedded in a background space-time. This has been the main object of study of the blackfold approach in which the emphasis has primarily been on stationary fluid configurations. Motivated by this approach we show under certain conditions that a given stationary fluid configuration living on a dynamical surface of vanishing thickness and satisfying locally the first law of thermodynamics will behave like an elastic brane when the surface is subject to small deformations. These results, which are independent of th…
Deformed relativistic kinematics have been considered as a way to capture residual effects of quantum gravity. It has been shown that they can be understood geometrically in terms of a curved momentum space on a flat spacetime. In this article we present a systematic analysis under which conditions and how deformed relativistic kinematics, encoded in a momentum space metric on flat spacetime, can be lifted to curved spacetimes in terms of a self-consistent cotangent bundle geometry, which leads to purely geometric, geodesic motion of freely falling point particles. We comment how this construc…
In the context of departures from Special Relativity written as a momentum power expansion in the inverse of an ultraviolet energy scale M, we derive the constraints that the relativity principle imposes between coefficients of a deformed composition law, dispersion relation, and transformation laws, at first order in the power expansion. In particular, we find that, at that order, the consistency of a modification of the energy-momentum composition law fixes the modification in the dispersion relation. We therefore obtain the most generic modification of Special Relativity that preserves the …
Starting from noncommutative generalization of Minkowski space we consider quantum deformed relativistic symmetries which lead to the modification of kinematics of special relativity. The noncommutative field theory framework described by means of the star product formalism is briefly described. We briefly present the quantum modifications of Einstein gravity
There is a vast literature showing the connection between a deformed relativistic kinematics and a curved momentum space, and, in particular, how the former can be obtained from the geometrical properties of the latter. However, there is not any mention about the geometry of a multi-particle system making manifest a possible modification in the metric of one particle due to the presence of others. In this work we explore how a curved momentum metric depending on the particles involved in an interaction arises when considering a process. We also show that the principle of relative locality obta…
Basing on the fundamental symmetry that the space-time inversion is equivalent to particle-antiparticle transformation, a relativistic modification on the stationary Schrodinger equation for many-particle system is made. The eigenvalue in the center of mass system is no longer equal to the negative of binding energy simply. The possible applications in various fields (e.g. the model of quarkonium) are discussed.
A deformation of special relativistic kinematics (possible signal of a theory of quantum gravity at low energies) leads to a modification of the notion of spacetime. At the classical level, this modification is required when one considers a model including single- or multi-interaction processes, for which absolute locality in terms of canonical space-time coordinates is lost. We discuss the different alternatives for observable effects in the propagation of a particle over very large distances that emerge from the new notion of spacetime. A central ingredient in the discussion is the cluster d…
The intrinsic covariant 1-time description (rest-frame instant form) for N relativistic scalar particles is defined. The system of N charged scalar particles plus the electromagnetic field is described in this way: the study of its Dirac observables allows the extraction of the Coulomb potential from field theory and the regularization of the classical self-energy by using Grassmann-valued electric charges. The 1-time covariant relativistic statistical mechanics is defined.
We consider the action principle to derive the classical, relativistic motion of a self-interacting particle in a 4-D Lorentzian spacetime containing a wormhole and which allows the existence of closed time-like curves. In particular, we study the case of a pointlike particle subject to a `hard-sphere' self-interaction potential and which can traverse the wormhole an arbitrary number of times, and show that the only possible trajectories for which the classical action is stationary are those which are globally self-consistent. Generically, the multiplicity of these trajectories (defined as the…
We show that Dirac-Born-Infeld theory possesses a hidden invariance that enhances the local O(1,p) Lorentz symmetry on a Dirichlet p-brane to an O(1,p) x O(1,p) gauge group, encoding both an invariant velocity and acceleration (or length) scale. This enlarged gauge group predicts consequences for the kinematics of observers on Dirichlet branes, with admissible accelerations being bounded from above. An important lesson beyond string theory is that a fundamental length scale can be implemented into the kinematics of general relativity, whilst preserving both space-time as a smooth manifold and …
We apply Lie algebra deformation theory to the problem of identifying the stable form of the quantum relativistic kinematical algebra. As a warm up, given Galileo's conception of spacetime as input, some modest computer code we wrote zeroes in on the Poincare-plus-Heisenberg algebra in about a minute. Further ahead, along the same path, lies a three dimensional deformation space, with an instability double cone through its origin. We give physical as well as geometrical arguments supporting our view that moment, rather than position operators, should enter as generators in the Lie algebra. Wit…
Assuming the validity of the relativity principle, we discuss the implications on relativistic kinematics of a deformation of the Poincaré invariance that preserves the Poincaré algebra, and only modifies its action on phase space in a Lorentz-invariant way. We show that, in contrast to the case where the Poincaré algebra is deformed, the action of boosts on two-particle states is not affected, while the addition law of momenta is to a large extent arbitrary. We give some nontrivial examples of this arising from doubly special relativity and noncommutative geometry and show that Hopf-algebra m…
In the Wigner-covariant rest-frame instant form of dynamics it is possible to develop a relativistic kinematics for the N-body problem. The Wigner hyperplanes define the intrinsic rest frame and realize the separation of the center-of-mass. Three notions of {\it external} relativistic center of mass can be defined only in terms of the {\it external} Poincaré group realization. Inside the Wigner hyperplane, an {\it internal} unfaithful realization of the Poincaré group is defined. The three concepts of {\it internal} center of mass weakly {\it coincide} and are eliminated by the rest-frame cond…
For one dimensional motions, we derive from the Dirac Spinors Equation (DSE) the Quantum Stationary Hamilton-Jacobi Equation for particles with spin 1/2. Then, We give its solution. We demonstrate that the $QSHJES_{1\over2}$ have two explicit forms, which represent the two possible projection of the Spin 1/2.
We study a family of noncommutative spacetimes constructed by one four-vector. The large set of coordinate commutation relations described in this way includes many cases that are widely studied in the literature. The Hopf-algebra symmetries of these noncommutative spacetimes, as well as the structures of star product and twist, are introduced and considered at first order in the deformation, described by four parameters. We also study the deformations to relativistic kinematics implied by this framework, and calculate the most general expression for the momentum dependence of the Lorentz tran…
A deformation of the canonical algebra for kinematical observables of the quantum field theory in Minkowski space-time has been considered under the condition of Lorentz invariance. A relativistic invariant algebra obtained depends on additional fundamental constants M, L and H with the dimensions of mass, length and action, respectively. In some limiting cases the algebra goes over into the well-known Snyder or Yang algebras. In general case the algebra represents a class of Lie algebras, that consists of simple algebras and semidirect sums of simple algebras and integrable ones. Some algebra…
We argue that recently proposed by Amelino-Camelia et all [1,2] so-called doubly special relativity (DSR), with deformed boost transformations identical with the formulae for $κ$-deformed kinematics in bicrossproduct basis is a classical special relativity in nonlinear disguise. The choice of symmetric composition law for deformed fourmomenta as advocated in [1, 2] implies that DSR is obtained by considering nonlinear fourmomenta basis of classical Poincaré algebra and it does not lead to noncommutative space-time. We also show how to construct large two classes of doubly special relativity th…
We analyze the kinematics of electron-positron production in a photon-photon interaction when one has a modification of the special relativistic kinematics as a power expansion in the inverse of a new high-energy scale. We derive the equation for the threshold energy of this reaction to first order in this expansion, including the effects due to a modification of the energy-momentum conservation equation. In contrast with the Lorentz invariance violation case, a scale of the order of a few TeV is found to be compatible with the observations of very high-energy cosmic gamma rays in the case of …
In non-relativistic mechanics the center of mass of an isolated system is easily separated out from the relative variables. For a N-body system these latter are usually described by a set of Jacobi normal coordinates, based on the clustering of the centers of mass of sub-clusters. The Jacobi variables are then the starting point for separating {\it orientational} variables, connected with the angular momentum constants of motion, from {\it shape} (or {\it vibrational}) variables. Jacobi variables, however, cannot be extended to special relativity. We show by group-theoretical methods that two …
Kinematic algebras can be realised on geometric spaces and constrain the physical models that can live on these spaces. Different types of kinematic algebras exist and we consider the interplay of these algebras for non-relativistic limits of a relativistic system, including both the Galilei and the Carroll limit. We develop a framework that captures systematically the corrections to the strict non-relativistic limit by introducing new infinite-dimensional algebras, with emphasis on the Carroll case. One of our results is to highlight a new type of duality between Galilei and Carroll limits th…
Starting from a classical picture of shear viscosity we construct a stationary velocity gradient in a microscopic parton cascade. Employing the Navier-Stokes ansatz we extract the shear viscosity coefficient $η$. For elastic isotropic scatterings we find an excellent agreement with the analytic values. This confirms the applicability of this method. Furthermore for both elastic and inelastic scatterings with pQCD based cross sections we extract the shear viscosity coefficient $η$ for a pure gluonic system and find a good agreement with already published calculations.
This report deals with the basic concepts on deducing transit times for quantum scattering: the stationary phase method and its relation with delay times for relativistic and non-relativistic tunneling particles. We notice that the applicability of this method is constrained by several subtleties in deriving the phase time that describes the localization of scattered wave packets. We investigate the general relation between phase times and dwell times for quantum tunneling/scattering. Considering a symmetrical collision of two identical wave packets with an one-dimensional barrier, we demonstr…
We study the Infra Red (IR) limit of dispersion relations for scalar and fermion fields in a Lifshitz-type Yukawa model, after dressing by quantum fluctuations. Relativistic-like dispersion relations emerge dynamically in the IR regime of the model, after quantum corrections are taken into account. In this regime, dynamical mass generation also takes place, but in such a way that the particle excitations remain massive, even if the bare masses vanish. The group velocities of the corresponding massive particles of course are smaller than the speed of light, in a way consistent with the IR regim…
In kappa-deformed relativistic framework we consider three different definitions of kappa-deformed velocities and introduce corresponding addition laws. We show that one of the velocities has classical relativistic addition law. The relation of velocity formulae with the coproduct for fourmomenta and noncommutative space-time structure is exhibited.
The calculation of an amplitude involving resonance production is presented. This calculation employs for the resonance state a relativistic Gamow vector. It is used for investigating the question of compatibility of the relativistic Gamow vectors kinematics, defined by real 4-velocities and complex mass, with the stable particle kinematics; or in other words, the integration of the Gamow vectors with the conventional Dirac bra-ket formalism. The calculation demonstrates a consistent framework comprising stable and Gamow vectors.
We find a near detailed balance solution to the relativistic Boltzmann equation under the relaxation time approximation with a collision term which differs from the Anderson-Witting model and is dependent on the stationary observer. Using this new solution, we construct an explicit covariant transport equation for the particle flux in response to the generalized temperature and chemical potential gradients in generic stationary spacetimes, with the transport tensors characterized by some integral functions in the chemical potential and the relativistic coldness. To illustrate the application o…
We present a short review describing the use of noncommutative space-time in quantum-deformed dynamical theories: classical and quantum mechanics as well as classical and quantum field theory. We expose the role of Hopf algebras and their realizations (noncommutative modules) as important mathematical tool describing quantum-deformed symmetries: quantum Lie groups and quantum Lie algebras. We consider in some detail the most studied examples of noncommutative space-time geometry: the canonical and $κ$-deformed cases. Finally we briefly describe the modifications of Einstein gravity obtained by…
The Klein-Gordon system describing three scalar particles without interaction is cast into a new form, by transformation of the momenta. Two redundant degrees of freedom are eliminated; we are left with a covariant equation for a reduced wave function with three-dimensional arguments. This new formulation of the mass-shell constraints is equivalent to the original KG system in a sector characterized by positivity of the energies and, if the mass differences are not too large, by a moderately relativistic regime. Then, mutual interaction is introduced and produces a tractable model.
Dynamical symmetries of Born-Infeld theory can be absorbed into the spacetime geometry, giving rise to relativistic kinematics with an additional invariant acceleration scale. The standard Poincare group P is thereby enhanced to its pseudo-complexified version, which is isomorphic to P x P. We construct the irreducible representations of this group, which yields the particle spectrum of a relativistic quantum theory that respects a maximal acceleration. It is found that each standard relativistic particle is associated with a `pseudo'-partner of equal spin but generically different mass. These…
We study a simple microscopic model for the one-dimensional stochastic motion of a (non)relativistic Brownian particle, embedded into a heat bath consisting of (non)relativistic particles. The stationary momentum distributions are identified self-consistently (for both Brownian and heat bath particles) by means of two coupled integral criteria. The latter follow directly from the kinematic conservation laws for the microscopic collision processes, provided one additionally assumes probabilistic independence of the initial momenta. It is shown that, in the nonrelativistic case, the integral cri…
We introduce new Lorentz-symmetry violating kinematics for a four-fermion interaction model, where dynamical mass generation is allowed, irrespectively of the strength of the coupling. In addition, these kinematics lead to a quasi-relativistic dispersion relation, in the sense that it is relativistic in both the infrared and the ultraviolet, but not in an intermediate regime, characterized by the mass $M$. For two fermions, we show that a flavour-mixing mass matrix is generated dynamically, and the Lorentz symmetric limit $M\to\infty$ leads to two free relativistic fermions, with flavour oscil…
