Doubly Special Relativity in Position Space Starting from the Conformal Group

Abstract

We propose version of doubly special relativity theory starting from position space. The version is based on deformation of ordinary Lorentz transformations due to the special conformal transformation. There is unique deformation which does not modify rotations. In contrast to the Fock-Lorentz realization (as well as to recent position-space proposals), maximum signal velocity is position (and observer) independent scale in our formulation by construction. The formulation admits one more invariant scale identified with radius of three-dimensional space-like hypersection of space-time. We present and discuss the Lagrangian action for geodesic motion of a particle on the DSR space. For the present formulation, one needs to distinguish the canonical (conjugated to $x^{\mu }$) momentum $p^{\mu }$ from the conserved energy-momentum. Deformed Lorentz transformations for $x^{\mu }$ induce complicated transformation law in space of canonical momentum. $p^{\mu }$ is not a conserved quantity and obeys to deformed dispersion relation. The conserved energy-momentum $P^{\mu }$ turns out to be different from the canonical one, in particular, $P^{\mu }$-space is equipped with nontrivial commutator. The nonlinear transformations for $x^{\mu }$ induce the standard Lorentz transformations in space of $P^{\mu }$. It means, in particular, that composite rule for $P^{\mu }$ is ordinary sum. There is no problem of total momentum in the theory. $P^{\mu }$ obeys the standard energy-momentum relation (while has nonstandard dependence on velocity).

Latex twice, 12 pages, final version

References in corpus (35)

• Lorentz invariance with an invariant energy scale
• Relativity in space-times with short-distance structure governed by an observer-independent (Planckian) length scale
• Testable scenario for Relativity with minimum-length
• Bicrossproduct structure of $\kappa$-Poincare group and non-commutative geometry
• Generalized Lorentz invariance with an invariant energy scale
• Applied Conformal Field Theory
• Classical and Quantum Mechanics of Free \k Relativistic Systems
• Non-commutative space-time of Doubly Special Relativity theories
• Deformed Boost Transformations That Saturate at the Planck Scale
• Observer-independent quanta of mass and length
• De Sitter space as an arena for Doubly Special Relativity
• Doubly Special Relativity and de Sitter space
• Non-Linear Relativity in Position Space
• Triply Special Relativity
• Harmonic oscillator with minimal length uncertainty relations and ladder operators
• Monopole and Berry Phase in Momentum Space in Noncommutative Quantum Mechanics
• Newton's Second Law in a Noncommutative Space
• Velocity of particles in Doubly Special Relativity
• Phases in noncommutative quantum mechanics on (pseudo)sphere
• Poincare covariant mechanics on noncommutative space
• Noncommutative relativistic particle on the electromagnetic background
• Large-scale non-locality in "doubly special relativity" with an energy-dependent speed of light
• On the definition of velocity in doubly special relativity theories
• Quantum mechanics on noncommutative plane and sphere from constrained systems
• Noncommutative Planar Particle Dynamics with Gauge Interactions
• Group velocity in noncommutative spacetime
• A time-space varying speed of light and the Hubble Law in static Universe
• A note about the quantum of area in a non-commutative space
• Fock-Lorentz transformations and time-varying speed of light
• Constant magnetic field and 2d non-commutative inverted oscillator
• Position space versions of Magueijo-Smolin doubly special relativity proposal and the problem of total momentum
• Classical O(N) nonlinear sigma model on the half line: a study on consistent Hamiltonian description
• Position space of Doubly Special Relativity
• Propagation of Light in Doubly Special Relativity
• Conformal Generators and Doubly Special Relativity Theories

Cited by in corpus (21)

• Self-consistency in Theories with a Minimal Length
• Deformed Special Relativity as an effective theory of measurements on quantum gravitational backgrounds
• Conversion of second class constraints by deformation of Lagrangian local symmetries
• Inertial frames without the relativity principle
• Modified Special Relativity on a fluctuating spacetime
• Gravity's rainbow: a bridge between LQC and DSR
• Approaching Space Time Through Velocity in Doubly Special Relativity
• Position space versions of Magueijo-Smolin doubly special relativity proposal and the problem of total momentum
• Lorentz invariant mass and length scales
• Generalized commutation relations and Non linear momenta theories, a close relationship
• Time delay of light signals in an energy-dependent spacetime metric
• A Projective Interpretation of Some Doubly Special Relativity Theories
• Generalization of the Extended Lagrangian Formalism on a Field Theory and Applications
• Variational problem for Hamiltonian system on so(k, m) Lie-Poisson manifold and dynamics of semiclassical spin
• Equilibrium properties of blackbody radiation with an ultraviolet energy cut-off
• Planck scale effect on the thermodynamics of photon gas
• Five-Dimensional Mechanics as the Starting Point for the Magueijo-Smolin Doubly Special Relativity
• From Noncommutative Sphere to Nonrelativistic Spin
• 5-Dimensional $SO(1,4)$-Invariant Action as an Origin to the Magueijo-Smolin Doubly Special Relativity Proposal
• Towards an Entropic Geometrodynamics of Quantum Particles: Mouths of Casimir Wormholes as Planckeons
• Note About String Theory with Deformed Dispersion Relations