scholarly journals NON-COMMUTATIVE SPACE–TIME OF DOUBLY SPECIAL RELATIVITY THEORIES

2003 ◽  
Vol 12 (02) ◽  
pp. 299-315 ◽  
Author(s):  
J. KOWALSKI-GLIKMAN ◽  
S. NOWAK
Keyword(s):  
Phase Space ◽  
Special Relativity ◽  
Space Time ◽  
Kinematic Structure ◽  
Doubly Special Relativity ◽  
Minkowski Space Time ◽  
Space Time Structure ◽  
Commutative Space ◽  
Intrinsic Length ◽  
Heisenberg Double

Doubly Special Relativity (DSR) theory is a recently proposed theory with two observer-independent scales (of velocity and mass), which is to describe a kinematic structure underlining the theory of Quantum Gravity. We observe that there are infinitely many DSR constructions of the energy–momentum sector, each of whose can be promoted to the κ-Poincaré quantum (Hopf) algebra. Then we use the co-product of this algebra and the Heisenberg double construction of κ-deformed phase space in order to derive the non-commutative space–time structure and the description of the whole of DSR phase space. Next we show that contrary to the ambiguous structure of the energy momentum sector, the space–time of the DSR theory is unique and related to the theory with non-commutative space–time proposed long ago by Snyder. This theory provides non-commutative version of Minkowski space–time enjoying ordinary Lorentz symmetry. It turns out that when one builds a natural phase space on this space–time, its intrinsic length parameter ℓ becomes observer-independent.

scholarly journals SCALAR FIELD THEORY ON κ-MINKOWSKI SPACE–TIME AND DOUBLY SPECIAL RELATIVITY

2005 ◽  
Vol 20 (20n21) ◽  
pp. 4925-4940 ◽  
Author(s):  
M. DASZKIEWICZ ◽  
K. IMIŁKOWSKA ◽  
J. KOWALSKI-GLIKMAN ◽  
S. NOWAK
Keyword(s):  
Scalar Field ◽  
Special Relativity ◽  
Minkowski Space ◽  
Space Time ◽  
Doubly Special Relativity ◽  
Minkowski Space Time ◽  
Point Vertex ◽  
Field Action ◽  
Poincaré Algebra ◽  
Energy And Momentum

In this paper we recall the construction of scalar field action on κ-Minkowski space–time and investigate its properties. In particular we show how the coproduct of κ-Poincaré algebra of symmetries arises from the analysis of the symmetries of the action, expressed in terms of Fourier transformed fields. We also derive the action on commuting space–time, equivalent to the original one. Adding the self-interaction Φ4 term we investigate the modified conservation laws. We show that the local interactions on κ-Minkowski space–time give rise to six inequivalent ways in which energy and momentum can be conserved at four-point vertex. We discuss the relevance of these results for Doubly Special Relativity.

scholarly journals On Characterization of Non-commutative Minkowski Space Time

2012 ◽  
Vol 9 (1) ◽  
pp. 123-127
Author(s):  
Dharmendra Kumar ◽  
Sunil KumarYadav
Keyword(s):  
Geodesic Equation ◽  
Space Time ◽  
Gravity Models ◽  
Commutative Case ◽  
Force Equation ◽  
Minkowski Space Time ◽  
Tetrad Formulation ◽  
Commutative Space ◽  
Suitable Technique

The present study aims to derive modified geodesic equation in non-commutative space time. Snyder developed a model for non-commutative space time which provides a suitable technique of quantum structure of the space. We extend Tetrad formulation of general relativity to non-commutative case for complex gravity models. We derive geodesic equation on the k-space time in Non-commutative space, which is a generalization of Feynman’s approach. It has been shown that the homogeneous Maxwell’s equations may be derived by starting with the Newton’s force equation and generalized to relativistic. We show that the geodesic equation in the commutative space time is a suitable for generalization to κ -space time in κ -deformed space time. It shown that the κ-dependent correction to geodesic equation is cubic in velocities.

scholarly journals Approaching space-time through velocity in doubly special relativity

2004 ◽  
Vol 70 (12) ◽  
Author(s):  
R. Aloisio ◽  
A. Galante ◽  
A. F. Grillo ◽  
E. Luzio ◽  
F. Méndez
Keyword(s):  
Special Relativity ◽  
Space Time ◽  
Doubly Special Relativity

scholarly journals DUAL KAPPA POINCAŔE ALGEBRA

2010 ◽  
Vol 25 (09) ◽  
pp. 1881-1890 ◽  
Author(s):  
JOSE A. MAGPANTAY
Keyword(s):  
Phase Space ◽  
Space Time ◽  
Lorentz Algebra ◽  
Poincaré Algebra ◽  
New Space ◽  
Heisenberg Double

We show a different modification of Poincaré algebra that also preserves Lorentz algebra. The change begins with how boosts affect space–time in a way similar to how they affect the momenta in kappa Poincaré algebra; hence the term "dual kappa Poincaré algebra." Since by construction the new space–time commutes, it follows that the momenta cocommute. Proposing a space–time coalgebra that is similar to the momentum coproduct in the bicrossproduct basis of kappa Poincaré algebra, we derive the phase space algebra using the Heisenberg double construction. The phase space variables of the dual kappa Poincaré algebra are then related to SR phase space variables. From these relations, we complete the dual kappa Poincaré algebra by deriving the action of rotations and boosts on the momenta.

scholarly journals Complex Covariance

10.14311/1809 ◽  
2013 ◽  
Vol 53 (3) ◽  
Author(s):  
Frieder Kleefeld
Keyword(s):  
Quantum Theory ◽  
Phase Space ◽  
Special Relativity ◽  
Degrees Of Freedom ◽  
Classical Mechanics ◽  
Space Time ◽  
Complex Phase ◽  
Dirac Equations ◽  
Klein Gordon ◽  
Complex Valued

According to some generalized correspondence principle the classical limit of a non-Hermitian quantum theory describing quantum degrees of freedom is expected to be the well known classical mechanics of classical degrees of freedom in the complex phase space, i.e., some phase space spanned by complex-valued space and momentum coordinates. As special relativity was developed by Einstein merely for real-valued space-time and four-momentum, we will try to understand how special relativity and covariance can be extended to complex-valued space-time and four-momentum. Our considerations will lead us not only to some unconventional derivation of Lorentz transformations for complex-valued velocities, but also to the non-Hermitian Klein-Gordon and Dirac equations, which are to lay the foundations of a non-Hermitian quantum theory.

Computational Space, Time and Quantum Mechanics

2010 ◽  
pp. 253-279 ◽  
Author(s):  
Michael Nicolaidis
Keyword(s):  
Special Relativity ◽  
Quantum Systems ◽  
Space Time ◽  
Quantum Superposition ◽  
Inertial Frames ◽  
Space Time Structure ◽  
The Universe ◽  
Ultimate Limit ◽  
Philosophical Questions ◽  
Non Locality

We start this chapter by introducing an ultimate limit of knowledge: as observers that are part of the universe we have no access on information concerning the fundamental nature of the elementary entities (particles) composing the universe but only on information concerning their behaviour. Then, we use this limit to develop a vision of the universe in which the behaviour of particles is the result of a computation-like process (not in the restricted sense of Turing machine) performed by meta-objects and in which space and time are also engendered by this computation. In this vision, the structure of space-time (e.g. Galilean, Lorentzian, …) is determined by the form of the laws of interactions, important philosophical questions related with the space-time structure of special relativity are resolved, the contradiction between the non-locality of quantum systems and the reversal of the temporal order of events (encountered in special relativity when we change inertial frames) is conciliated, and the “paradoxes” related with the “strange” behaviour of quantum systems (non-determinism, quantum superposition, non-locality) are resolved.

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