Jerzy Kowalski-Glikman

The principle of relative locality

We propose a deepening of the relativity principle according to which the invariant arena for nonquantum physics is a phase space rather than spacetime. Descriptions of particles propagating and interacting in spacetimes are constructed by observers, but different observers, separated from each other by translations, construct different spacetime projections from the invariant phase space. Nonetheless, all observers agree that interactions are local in the spacetime coordinates constructed by observers local to them. This framework, in which absolute locality is replaced by relative locality, results from deforming energy-momentum space, just as the passage from absolute to relative simultaneity results from deforming the linear addition of velocities. Different aspects of energy-momentum space geometry, such as its curvature, torsion and nonmetricity, are reflected in different kinds of deformations of the energy-momentum conservation laws. These are in principle all measurable by appropriate experiments. We also discuss a natural set of physical hypotheses which singles out the cases of energy-momentum space with a metric compatible connection and constant curvature.

Doubly special relativity theories as different bases of κ-Poincaré algebra

Abstract Doubly Special Relativity (DSR) theory is a theory with two observer-independent scales, of velocity and mass (or length). Such a theory has been proposed by Amelino-Camelia as a kinematic structure which may underline quantum theory of relativity. Recently another theory of this kind has been proposed by Magueijo and Smolin. In this Letter we show that both these theories can be understood as particular bases of the κ -Poincare theory based on quantum (Hopf) algebra. This observation makes it possible to construct the space–time sector of Magueijo and Smolin DSR. We also show how this construction can be extended to the whole class of DSRs. It turns out that for all such theories the structure of space–time commutators is the same. This results lead us to the claim that physical predictions of properly defined DSR theory should be independent of the choice of basis.

Deformed boost transformations that saturate at the Planck scale

Abstract We derive finite boost transformations based on the Lorentz sector of the bicross-product-basis κ -Poincare Hopf algebra. We emphasize the role of these boost transformations in a recently-proposed new relativistic theory, and their relevance for experimental studies presently being planned. We find that when the (dimensionful) deformation parameter is identified with the Planck length, which together with the speed-of-light constant has the status of observer-independent scale in the new relativistic theory, the deformed boosts saturate at the value of momentum that corresponds to the inverse of the Planck length.

NON-COMMUTATIVE SPACE–TIME OF DOUBLY SPECIAL RELATIVITY THEORIES

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-Poincare 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 par...

2+1 gravity and doubly special relativity

It is shown that gravity in 2+1 dimensions coupled to point particles provides a nontrivial example of doubly special relativity (DSR). This result is obtained by interpretation of previous results in the field and by exhibiting an explicit transformation between the phase space algebra for one particle in 2+1 gravity found by Matschull and Welling and the corresponding DSR algebra. The identification of 2+1 gravity as a DSR system answers a number of questions concerning the latter, and resolves the ambiguity of the basis of the algebra of observables. Based on this observation a heuristic argument is made that the algebra of symmetries of ultra high energy particle kinematics in 3+1 dimensions is described by some DSR theory.

De Sitter space as an arena for doubly special relativity

Abstract We show that Doubly Special Relativity (DSR) can be viewed as a theory with energy–momentum space being the four-dimensional de Sitter space. Different formulations (bases) of the DSR theory considered so far can be therefore understood as different coordinate systems on this space. The emerging geometrical picture makes it possible to understand the universality of the non-commutative structure of space–time of doubly special relativity. Moreover, it suggests how to construct the most natural DSR basis, which turns out to be the bicrossproduct basis.

Doubly special relativity and de Sitter space

In this paper we recall the construction of doubly special relativity (DSR) as a theory with energy–momentum space being the four-dimensional de Sitter space. Then the bases of the DSR theory can be understood as different coordinate systems on this space. We investigate the emerging geometrical picture of doubly special relativity by presenting the basis independent features of DSR that include the non-commutative structure of spacetime and the phase-space algebra. Next we investigate the relation between our geometric formulation and the one based on quantum κ-deformations of the Poincare algebra. Finally we re-derive the five-dimensional differential calculus using the geometric method, and use it to write the deformed Klein–Gordon equation and to analyse its plane-wave solutions.

Relative locality: A deepening of the relativity principle

We describe a recently introduced principle of relative locality which we propose governs a regime of quantum gravitational phenomena accessible to experimental investigation. This regime comprises phenomena in which

From noncommutative κ-Minkowski to Minkowski space–time

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