Over the last decade there were significant advances in the understanding of quantum gravity coupled to point particles in 3D (2+1-dimensional) spacetime. Most notably it is emerging that the theory can be effectively described as a theory of free particles on a momentum space with anti-deSitter geometry and with noncommutative spacetime coordinates of the type $[x^μ,x^ν]=i \hbar \ell \varepsilon^{μν}_{\phantom{μν}ρ} x^ρ$. We here show that the recently proposed relative-locality curved-momentum-space framework is ideally suited for accommodating these structures characteristic of 3D quantum gravity. Through this we obtain an intuitive characterization of the DSR-deformed Poincaré symmetries of 3D quantum gravity, and find that the associated relative spacetime locality is of the type producing dual-gravity lensing.
We argue that a (slightly) curved space-time probed with a finite resolution, equivalently a finite minimal length, is effectively described by a flat non-commutative space-time. More precisely, a small cosmological constant (so a constant curvature) leads the kappa-deformed Poincaré flat space-time of deformed special relativity (DSR) theories. This point of view eventually helps understanding some puzzling features of DSR. It also explains how DSR can be considered as an effective flat (low energy) limit of a (true) quantum gravity theory. This point of view leads us to consider a possible g…
By studying the notion of a fundamentally minimal length scale in asymptotically safe gravity we find that a specific version of deformed special relativity (DSR) naturally arises in this approach. We then consider two thought experiments to examine the interpretation of the scenario and discuss similarities and differences to other approaches to DSR.
We investigate the properties of multi-particle states in Deformed Special Relativity (DSR). Starting from the Lagrangian formalism with an energy dependent metric, the conserved Noether current can be derived which is additive in the usual way. The integrated Noether current had previously been discarded as a conserved quantity, because it was correctly realized that it does no longer obey the DSR transformations. We identify the reason for this mismatch in the fact that DSR depends only on the extensive quantity of total four-momentum instead of the energy-momentum densities as would be appr…
In many different ways, Deformed Special Relativity (DSR) has been argued to provide an effective limit of quantum gravity in almost-flat regime. Unfortunately DSR is up to now plagued by many conceptual problems (in particular how it describes macroscopic objects) which forbids a definitive physical interpretation and clear predictions. Here we propose a consistent framework to interpret DSR. We extend the principle of relativity: the same way that Special Relativity showed us that the definition of a reference frame requires to specify its speed, we show that DSR implies that we must also ta…
Deformed Special Relativity (DSR) is obtained by imposing a maximal energy to Special Relativity and deforming the Lorentz symmetry (more exactly the Poincaré symmetry) to accommodate this requirement. One can apply the same procedure deforming the Galilean symmetry in order to impose a maximal speed (the speed of light). This leads to a non-commutative space structure, to the expected deformations of composition of speed and conservation of energy-momentum. In doing so, one runs into most of the ambiguities that one stumbles onto in the DSR context. However, this time, Special Relativity is t…
We show that the existence of a fundamental length, introduced in Deformed Special Relativity (DSR) inspired minisuper (phase-) space, causes the behavior of the scale factor of the universe to change from that of a universe filled with dust to an accelerating universe driven by a cosmological constant.
Deformed Special Relativity (DSR) is a generalization of Special Relativity based on a deformed Minkowski space, i.e. a four-dimensional space-time with metric coefficients depending on the energy. We show that, in the DSR framework, it is possible to derive the value of the electron mass from the space-time geometry via the experimental knowledge of the parameter of local Lorentz invariance breakdown, and of the Minkowskian threshold energy $E_{0,em}$ for the electromagnetic interaction.
The phase space of a classical particle in DSR contains de Sitter space as the space of momenta. We start from the standard relativistic particle in five dimensions with an extra constraint and reduce it to four dimensional DSR by imposing appropriate gauge fixing. We analyze some physical properties of the resulting theories like the equations of motion, the form of Lorentz transformations and the issue of velocity. We also address the problem of the origin and interpretation of different bases in DSR.
We study a de-Sitter model in the framework of a Deformed Special Relativity (DSR) inspired structure. The effects of this framework appear as the existence of a fundamental length which influences the behavior of the scale factor. We show that such a deformation can either be used to control the unbounded growth of the scale factor in the present accelerating phase or account for the inflationary era in the early evolution of the universe.
Some classes of Deformed Special Relativity (DSR) theories are reconsidered within the Hopf algebraic formulation. For this purpose we shall explore a minimal framework of deformed Weyl-Heisenberg algebras provided by a smash product construction of DSR algebra. It is proved that this DSR algebra, which uniquely unifies $κ$-Minkowski spacetime coordinates with Poincaré generators, can be obtained by nonlinear change of generators from undeformed one. Its various realizations in terms of the standard (undeformed) Weyl-Heisenberg algebra opens the way for quantum mechanical interpretation of DSR…
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 analyse the quantum geometry of 3-dimensional deformed special relativity (DSR) and the notion of spacetime points in such a context, identified with coherent states that minimize the uncertainty relations among spacetime coordinates operators. We construct this system of coherent states in both the Riemannian and Lorentzian case, and study their properties and their geometric interpretation.
The relation between Conformal generators and Magueijo Smolin Deformed Special Relativity term, added to Lorentz boosts, is achieved. The same is performed for Fock Lorentz transformations. Through a dimensional reduction procedure, it is demonstrated that a massless relativistic particle living in a $d$ dimensional space, is isomorphic to one living in a $d+2$ space with pure Lorentz invariance and to a particle living in a $AdS_{d+1}$ space. To accomplish these identifications, the Conformal Group is extended and a nonlinear algebra arises. Finally, because the relation between momenta and v…
The doubly special relativity (DSR) theories are investigated in order to take into account an observer-independent length scale in special relativity framework. It is widely believed that any quantum theory of gravity would reduce to a DSR model at the flat limit when purely gravitational and quantum mechanical effects are negligible. Gravity's rainbow is a simple generalization of DSR theories to incorporate gravity. In this paper, we show that the effective Friedmann equations that are suggested by loop quantum cosmology (LQC) can be exactly reobtained in rainbow cosmology setup. The deform…
The dissertation presents possibilities of applying noncommutative spacetimes description, particularly kappa-deformed Minkowski spacetime and Drinfeld's deformation theory, as a mathematical formalism for Doubly Special Relativity theories (DSR), which are thought as phenomenological limit of quantum gravity theory. Deformed relativistic symmetries are described within Hopf algebra language. In the case of (quantum) kappa-Minkowski spacetime the symmetry group is described by the (quantum) kappa-Poincare Hopf algebra. Deformed relativistic symmetries were used to construct the DSR algebra, wh…
Theories with a curved momentum space, which became recently of interest in the quantum-gravity literature, can in general violate many apparently robust aspects of our current description of the laws of physics, including relativistic invariance, locality, causality and global momentum conservation. We here explore some aspects of the particularly severe pathologies arising in generic theories with curved momentum space for what concerns causality and momentum conservation. However, we also report results suggesting that when momentum space is maximally symmetric, and the theory is formulated…
The doubly special relativity (DSR) theories are suggested in order to incorporate an observer-independent length scale in special theory of relativity. The Magueijo-Smolin proposal of DSR is realizable through a particular form of the noncommutative (NC) spacetime (known as $κ$-Minkowski spacetime) in which the Lorentz symmetry is preserved. In this framework, the NC parameter $κ$ provides the origin of natural cutoff energy scale. Using a nonlinear deformed relativistic dispersion relation along with the Lorentz transformations, we investigate some phenomenological facets of two-body collisi…
DSR-relativistic spacetime picture and the phenomenology of Planck-scale-modified time dilation
Università di Napoli “Federico II”（那不勒斯费德里科二世大学）；INFN, Sezione di Napoli（意大利国家核物理研究所那不勒斯分部）；Università di Cagliari（卡利亚里大学）；INFN, Sezione di Cagliari（意大利国家核物理研究所卡利亚里分部）
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…
The Poincaré sector of a recently deformed conformal algebra is proposed to describe, after the identification of the deformation parameter with the Planck length, the symmetries of a new relativistic theory with two observer-independent scales (or DSR theory). Also a new non-commutative space-time is proposed. It is found that momentum space exhibits the same features of the DSR proposals preserving Lorentz invariance in a deformed way. The space-time sector is a generalization of the well known non-commutative $κ$-Minkowski space-time which however does not preserve Lorentz invariance, not e…
The interest of part of the quantum-gravity community in the possibility of Planck-scale-deformed Lorentz symmetry is also fueled by the opportunities for testing the relevant scenarios with analyses, from a signal-propagation perspective, of observations of bursts of particles from cosmological distances. In this respect the fact that so far the implications of deformed Lorentz symmetry have been investigated only for flat (Minkowskian) spacetimes represents a very significant limitation, since for propagation over cosmological distances the curvature/expansion of spacetime is evidently tangi…
In this paper, we analyze a possible formalization of the deformed special relativity as a five-dimensional theory. This is not the first attempt to do so, but we feel that either these previous treatments are too arbitrary in the choice of the new enlarged space, or they lack a satisfactory physical interpretation. In this work, we propose an algorithm which fixes the shape of the enlarged space. Afterwards, we focus our attention on the consequences of our formalism, proposing a physical interpretation.
In deformed or doubly special relativity (DSR) the action of the lorentz group on momentum eigenstates is deformed to preserve a maximal momenta or minimal length, supposed equal to the Planck length. The classical and quantum dynamics of a particle propagating in kappa-Minkowski spacetime is discussed in order to examine an apparent paradox of locality which arises in the classical dynamics. This is due to the fact that the Lorentz transformations of spacetime positions of particles depend on their energies, so whether or not a local event, defined by the coincidence of two or more particles,…
Much attention has been recently devoted to the possibility that quantum gravity effects could lead to departures from Special Relativity in the form of a deformed Poincarè algebra. These proposals go generically under the name of Doubly or Deformed Special Relativity (DSR). In this article we further explore a recently proposed class of quantum field theories, involving noncanonically commuting complex scalar fields, which have been shown to entail a DSR-like symmetry. An open issue for such theories is whether the DSR-like symmetry has to be taken as a physically relevant symmetry, or if in …
A mass-like quantum Weyl-Poincare algebra is proposed to describe, after the identification of the deformation parameter with the Planck length, a new relativistic theory with two observer-independent scales (or DSR theory). Deformed momentum representation, finite boost transformations, range of rapidity, energy and momentum, as well as position and velocity operators are explicitly studied and compared with those of previous DSR theories based on kappa-Poincare algebra. The main novelties of the DSR theory here presented are the new features of momentum saturation and a new type of deformed …
In this article we elaborate on a recently proposed interpretation of DSR as an effective measurement theory in the presence of non-negligible (albeit small) quantum gravitational fluctuations. We provide several heuristic arguments to explain how such a new theory can emerge and discuss the possible observational consequences of this framework.
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 question the notion of line element in some quantum spaces that are expected to play a role in quantum gravity, namely non-commutative deformations of Minkowski spaces. We recall how the implementation of the Leibniz rule forbids to see some of the infinitesimal deformed Poincare transformations as good candidates for Noether symmetries. Then we recall the more fundamental view on the line element proposed in noncommutative geometry, and re-interprete at this light some previous results on Connes' distance formula.
We study the effects of relative locality dynamics in the case of the Snyder model. Several properties of this model differ from those of the widely studied $κ$-Poincaré models: for example, in the Snyder case the action of the Lorentz group is preserved, and the composition law of momenta is deformed by terms quadratic in the inverse Planck energy. From the investigation of time delay and dual curvature lensing we deduce that, because of these differences, in the Snyder case the properties of the detector are essential for the observation of relative locality effects. The deviations from spec…
We describe firstly the basic features of quantum $κ$-Poincaré symmetries with their Hopf algebra structure. The quantum $κ$-Poincaré framework in any basis relates rigidly the quantum $κ$-Poincaré algebra with quantum $κ$-Poincaré group, noncommutative space-time and $κ$-deformed phase space. Further we present the approach of Doubly Special Relativity (DSR) theories, which introduce(in the version DSR1) kinematically the frame - independent fundamental mass parameter as described by maximal three-momentum $|\overrightarrow{p}|=κc$. We argue why the DSR theories in one-particle sector can be …
We argue that the natural framework for embedding the ideas of deformed, or doubly, special relativity (DSR) into a curved spacetime is a generalisation of Einstein-Cartan theory, considered by Stelle and West. Instead of interpreting the noncommuting "spacetime coordinates" of the Snyder algebra as endowing spacetime with a fundamentally noncommutative structure, we are led to consider a connection with torsion in this framework. This may lead to the usual ambiguities in minimal coupling. We note that observable violations of charge conservation induced by torsion should happen on a time scal…
We establish the correspondence between two apparently unrelated but in fact complementary approaches of a relativistic deformed kinematics: the geometric properties of momentum space and the loss of absolute locality in canonical spacetime, which can be restored with the introduction of a generalized spacetime. This correspondence is made explicit for the case of $κ$-Poincaré kinematics and compared with its properties in the Hopf algebra framework.
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 space-time and the phase space algebra. Next we investigate the relation between our geometric formulation and the one based on quantum $κ$-deformations of the Poincaré…
We study the nonrelativistic limit of the motion of a classical particle in a model of deformed special relativity and of the corresponding generalized Klein-Gordon and Dirac equations, and show that they reproduce nonrelativistic classical and quantum mechanics, respectively, although the rest mass of a particle no longer coincides with its inertial mass. This fact clarifies the meaning of the different definitions of velocity of a particle available in DSR literature. Moreover, the rest mass of particles and antiparticles differ, breaking the CPT invariance. This effect is close to observati…
We study the interplay of non-locality and Lorentz invariance in a version of deformed or doubly special relativity (DSR) based on kappa-Minkowski spacetime. We find that Einstein's procedure for an inertial observer to assign coordinates to distant events becomes ambiguous for sufficiently distant events. The accuracy to which two clocks can be synchronized turns out to depend on the distance between them. These are consequences of the non-commutativity of space and time coordinates or a dependence of the speed of light on energy in relativistic theories. These ambiguities grow with distance …
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 is 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 known construction of $κ$-deformed phase space via Heisenberg double in order to derive the non-commutative space-time structure and description of the whole of the DSR …
In this paper, we study twist deformed quantum field theories obtained by combining the Wightman axiomatic approach with the idea of spacetime noncommutativity. We prove that the deformed fields with deformation parameters of opposite sign satisfy the condition of mutual asymptotic commutativity, which was used earlier in nonlocal quantum field theory as a substitute for relative locality. We also present an improved proof of the wedge localization property discovered for the deformed fields by Grosse and Lechner, and we show that the deformation leaves the asymptotic behavior of the vacuum ex…
Deformed special relativity is embedded in deformed general relativity using the methods of canonical relativity and loop quantum gravity. Phase-space dependent deformations of symmetry algebras then appear, which in some regimes can be rewritten as non-linear Poincare algebras with momentum-dependent deformations of commutators between boosts and time translations. In contrast to deformed special relativity, the deformations are derived for generators with an unambiguous physical role, following from the relationship between canonical constraints of gravity with stress-energy components. The …
The most studied doubly-special-relativity scenarios, theories with both the speed-of-light scale and a length/inverse-momentum scale as non-trivial relativistic invariants, have concerned the possibility of enforcing relativistically some nonlinear laws on momentum space. For the "relative-locality framework" recently proposed in arXiv:1101.0931 a central role is played by nonlinear laws on momentum space, with the guiding principle that they should provide a characterization of the geometry of momentum space. Building on previous doubly-special-relativity results I here identify a criterion …
A common misconception is that Lorentz invariance is inconsistent with a discrete spacetime structure and a minimal length: under Lorentz contraction, a Planck length ruler would be seen as smaller by a boosted observer. We argue that in the context of quantum gravity, the distance between two points becomes an operator and show through a toy model, inspired by Loop Quantum Gravity, that the notion of a quantum of geometry and of discrete spectra of geometric operators, is not inconsistent with Lorentz invariance. The main feature of the model is that a state of definite length for a given obs…
It is explained why the argument in arXiv:1104.2019 does not answer the question how to describe multi-particle states in models with a deformed Lorentz-symmetry in momentum space.
This is my PhD thesis. In this thesis we study the gauge theories on noncommutative Moyal space. We find new static solitons and instantons in terms of the so called generalized Bose operators. Generalized Bose operators are constructed to describe reducible representation of the oscillator algebra. They create/annihilate $k$-quanta, $k$ being a positive integer. We start with giving an alternative description to the already found static magnetic flux tube solutions of the noncommutative gauge theories in terms of generalized Bose operators. The Nielsen-Olesen vortex solutions found in terms o…
The symmetry properties of a proposal to go beyond relativistic quantum field theory based on a modification of the commutation relations of fields are identified. Poincaré invariance in an auxiliary spacetime is found in the Lagrangian version of the path integral formulation. This invariance is contrasted with the idea of Doubly (or Deformed) Special Relativity (DSR). This analysis is then used to go from the free theory of a complex field to an interacting field theory.
The hypersurface deformation algebra consists in a fruitful approach to derive deformed solutions of general relativity based on symmetry considerations with quantum gravity effects, whose linearization has been recently demonstrated to be connected to the DSR program by the $κ$-Poincaré symmetry. Based on this approach, we analyzed the solution derived for the interior of a black hole and we found similarities with the, so called, rainbow metrics, like a momentum-dependence of the metric functions. Moreover, we derived an effective, time-dependent Planck length and compared different regulari…
I shall recall in historical perspective some results from nineties and show further how $κ$-deformed symmetries and $κ$-Minkowski space inspired DSR (Doubly of Deformed Special Relativity) approach proposed after 2000. As very recent development I shall show how to describe quantum-covariant $κ$-deformed phase spaces by passing from Hopf algebras to Hopf algebroids (arXiv:1507.02612) and I will briefly describe the $κ$-deformations of $AdS_5 \times S^5$ superstring target spaces (arXiv:1510.030.83).
We show that the $κ$-Poincaré Hopf algebra can be interpreted in the framework of curved momentum space leading to the relativity of locality \cite{AFKS}. We study the geometric properties of the momentum space described by $κ$-Poincaré, and derive the consequences for particles propagation and energy-momentum conservation laws in interaction vertices, obtaining for the first time a coherent and fully workable model of the deformed relativistic kinematics implied by $κ$-Poincaré. We describe the action of boost transformations on multi-particles systems, showing that in order to keep covariant…
We analyze the algebra of Dirac observables of the relativistic particle in four space-time dimensions. We show that the position observables become non-commutative and the commutation relations lead to a structure very similar to the non-commutative geometry of Deformed Special Relativity (DSR). In this framework, it appears natural to consider the 4d relativistic particle as a five dimensional massless particle. We study its quantization in terms of wave functions on the 5d light cone. We introduce the corresponding five-dimensional action principle and analyze how it reproduces the physics …
We construct recently introduced palatial NC twistors by considering the pair of conjugated (Born-dual) twist-deformed $D=4$ quantum inhomegeneous conformal Hopf algebras $\mathcal{U}_{θ}(su(2,2)\ltimes T^{4}$) and $\mathcal{U}_{\barθ}(su(2,2)\ltimes\bar{T}^{4}$), where $T^{4}$ describe complex twistor coordinatesand $\bar{T}^{4}$ the conjugated dual twistor momenta. The palatial twistors are suitably chosen as the quantum-covariant modules (NC representations) of the introduced Born-dual Hopf algebras. Subsequently we introduce the quantum deformations of $D=4$ Heisenberg-conformal algebra (H…
When applied to some models of noncommutative geometry, the formalism of relative locality predicts the occurrence of a delay in the time of arrival of massless particle of different energies emitted by a distant observer. In this letter, we show that this is not the case with Snyder spacetime, essentially because the Lorentz invariance is not deformed in this case. This conclusion is in accordance with the findings of doubly special relativity. Distant observers may however measure different times of flight for massive particle.
We review the main features of models where relativistic symmetries are deformed at the Planck scale. We cover the motivations and links to other quantum gravity approaches. We describe in some detail the most studied theoretical frameworks, including Hopf algebras, relative locality, and other scenarios with deformed momentum space geometry. We discuss possible phenomenological consequences, and point out current open questions.
Several arguments suggest that the Planck scale could be the characteristic scale of curvature of momentum space. As other recent studies we assume that the metric of momentum space determines the condition of on-shellness while the momentum-space affine connection governs the form of the law of composition of momenta. We show that the possible choices of laws of composition of momenta are more numerous than the possible choices of affine connection on a momentum space. This motivates us to propose a new prescription for associating an affine connection to momentum composition, which we compar…
We demonstrate that the covariance of the algebra of quantum NC fields under quantum-deformed Poincare symmetries implies the appearence of braided algebra of fields and the notion of braided locality in NC QFT. We briefly recall the historical development of NC QFT which was firstly formulated in the framework using classical relativistic symmetries but further it was described as generated by the quantum-deformed symmetries. We argue that consistent covariant quantum-deformed formalism requires "braiding all the way", in particular braided commutator of deformed field oscillators as well as …
We present a version of relative locality based on the geometry of twistor space. This can also be thought of as a new kind of deformation of twistor theory based on the construction of a bundle of twistor spaces over momentum space. Locality in space-time is emergent and is deformed in a precise way when a connection on that bundle is non-flat. This gives a precise and controlled meaning to Penrose's hypothesis that quantum gravity effects will deform twistor space in such a way as to maintain causality and relativistic invariance while weakening the notion that interactions take place at poi…
