Wave Propagation and IR/UV Mixing in Noncommutative Spacetimes (original) (raw)

Waves on Noncommutative Space–Time and Gamma-Ray Bursts

International Journal of Modern Physics A, 2000

Quantum group Fourier transform methods are applied to the study of processes on noncommutative Minkowski space–time [xi, t]=ιλxi. A natural wave equation is derived and the associated phenomena of in vacuo dispersion are discussed. Assuming the deformation scale λ is of the order of the Planck length one finds that the dispersion effects are large enough to be tested in experimental investigations of astrophysical phenomena such as gamma-ray bursts. We also outline a new approach to the construction of field theories on the noncommutative space–time, with the noncommutativity equivalent under Fourier transform to non-Abelianness of the "addition law" for momentum in Feynman diagrams. We argue that CPT violation effects of the type testable using the sensitive neutral-kaon system are to be expected in such a theory.

Observables and dispersion relations in κ-Minkowski spacetime

Journal of High Energy Physics

From noncommutative κ-Minkowski to Minkowski space–time

Physics Letters B, 2007

We show that free κ-Minkowski space field theory is equivalent to a relativistically invariant, non local, free field theory on Minkowski space-time. The field theory we obtain has in spectrum a relativistic mode of arbitrary mass m and a Planck mass tachyon. We show that while the energy momentum for the relativistic mode is essentially the standard one, it diverges for the tachyon, so that there are no asymptotic tachyonic states in the theory. It also follows that the dispersion relation is not modified, so that, in particular, in this theory the speed of light is energy-independent.

Observables and Dispersion Relations in k-Minkowski Spacetime

2017

We revisit the notion of quantum Lie algebra of symmetries of a noncommutative spacetime, its elements are shown to be the generators of infinitesimal transformations and are naturally identified with physical observables. Wave equations on noncommutative spaces are derived from a quantum Hodge star operator. This general noncommutative geometry construction is then exemplified in the case of k-Minkowski spacetime. The corresponding quantum Poincare'-Weyl Lie algebra of infinitesimal translations, rotations and dilatations is obtained. The d'Alembert wave operator coincides with the quadratic Casimir of quantum translations and it is deformed as in Deformed Special Relativity theories. Also momenta (infinitesimal quantum translations) are deformed, and correspondingly the Einstein-Planck relation and the de Broglie one. The energy-momentum relations (dispersion relations) are consequently deduced. These results complement those of the phenomenological literature on the subject.

On the IR/UV mixing and experimental limits on the parameters of canonical noncommutative spacetimes

Journal of High Energy Physics, 2004

We investigate some issues that are relevant for the derivation of experimental limits on the parameters of canonical noncommutative spacetimes. By analyzing a simple Wess-Zumino-type model in canonical noncommutative spacetime with soft supersymmetry breaking we explore the implications of ultraviolet supersymmetry on low-energy phenomenology. The fact that new physics in the ultraviolet can modify low-energy predictions affects significantly the derivation of limits on the noncommutativity parameters based on low-energy data. These are, in an appropriate sense here discussed, "conditional limits". We also find that some standard techniques for an effective low-energy description of theories with non-locality at short distance scales are only applicable in a regime where theories in canonical noncommutative spacetime lack any predictivity, because of the strong sensitivity to unknown UV physics. It appears useful to combine high-energy data, from astrophysics, with the more readily available low-energy data.

Particle velocity in noncommutative space-time

Physical Review D, 2002

We investigate a particle velocity in the κ-Minkowski space-time, which is one of the realization of a noncommutative space-time. We emphasize that arrival time analyses by high-energy γ-rays or neutrinos, which have been considered as powerful tools to restrict the violation of Lorentz invariance, are not effective to detect space-time noncommutativity. In contrast with these examples, we point out a possibility that low-energy massive particles play an important role to detect it. 95.85.Pw, 98.70.Sa.

Dispersion relations in κ-noncommutative cosmology

Journal of Cosmology and Astroparticle Physics, 2020

We study noncommutative deformations of the wave equation in curved backgrounds and discuss the modification of the dispersion relations due to noncommutativity combined with curvature of spacetime. Our noncommutative differential geometry approach is based on Drinfeld twist deformation, and can be implemented for any twist and any curved background. We discuss in detail the Jordanian twist — giving κ-Minkowski spacetime in flat space — in the presence of a Friedman-Lemaître-Robertson-Walker (FLRW) cosmological background. We obtain a new expression for the variation of the speed of light, depending linearly on the ratio E ph/E LV (photon energy/Lorentz violation scale), but also linearly on the cosmological time, the Hubble parameter and inversely proportional to the scale factor.

Group velocity in noncommutative spacetime

Journal of Cosmology and Astroparticle Physics, 2003

The realization that forthcoming experimental studies, such as the ones planned for the GLAST space telescope, will be sensitive to Planck-scale deviations from Lorentz symmetry has increased interest in noncommutative spacetimes in which this type of effects is expected. We focus here on κ-Minkowski spacetime, a muchstudied example of Lie-algebra noncommutative spacetime, but our analysis appears to be applicable to a more general class of noncommutative spacetimes. A technical controversy which has significant implications for experimental testability is the one concerning the κ-Minkowski relation between group velocity and momentum. A large majority of studies adopted the relation v = dE(p)/dp, where E(p) is the κ-Minkowski dispersion relation, but recently some authors advocated alternative formulas. While in these previous studies the relation between group velocity and momentum was introduced through ad hoc formulas, we rely on a direct analysis of wave propagation in κ-Minkowski. Our results lead conclusively to the relation v = dE(p)/dp. We also show that the previous proposals of alternative velocity/momentum relations implicitly relied on an inconsistent implementation of functional calculus on κ-Minkowski and/or on an inconsistent description of spacetime translations.

Noether analysis for field theory in κ-Minkowski noncommutative spacetime

The research work reported in this thesis intends to contribute to the understanding of theories constructed in noncommutative spacetimes, spacetimes whose coordinates satisfy commutation relations of the type [xµ, xν]= iΘµν (x), with a noncommutativity matrix Θµν which may be coordinate dependent. Such theories have attracted strong interest in the context of research attempting to apply the principles of quantum mechanics to the fundamental description of spacetime structure.

Birefringence and noncommutative structure of space–time

Physics Letters B, 2011

We analyze the phenomenon of birefringence of the electromagnetic field in the context of noncommutative geometry, using as background a deformed pp-wave solution to noncommutative Einstein's equations. The light-cone structure is determined using a generalized Fresnel equation characterizing the propagation of light in premetric vacuum electrodynamics.

Effective field theory for non-commutative spacetime: a toy model

Physics Letters B, 2003

A novel geometric model of a noncommutative plane has been constructed. We demonstrate that it can be construed as a toy model for describing and explaining the basic features of physics in a noncommutative spacetime from a field theory perspective. The noncommutativity is induced internally through constraints and does not require external interactions. We show that the noncommutative space-time is to be interpreted as having an internal angular momentum throughout. Subsequently, the elementary excitationsi.e. point particles -living on this plane are endowed with a spin. This is explicitly demonstrated for the zero-momentum Fourier mode. The study of these excitations reveals in a natural way various stringy signatures of a noncommutative quantum theory, such as dipolar nature of the basic excitations [7] and momentum dependent shifts in the interaction point . The observation [9] that noncommutative and ordinary field theories are alternative descriptions of the same underlying theory, is corroborated here by showing that they are gauge equivalent.

Quantum Fields on Noncommutative Spacetimes: Theory and Phenomenology

Symmetry, Integrability and Geometry: Methods and Applications, 2010

In the present work we review the twisted field construction of quantum field theory on noncommutative spacetimes based on twisted Poincaré invariance. We present the latest development in the field, in particular the notion of equivalence of such quantum field theories on a noncommutative spacetime, in this regard we work out explicitly the inequivalence between twisted quantum field theories on Moyal and Wick-Voros planes; the duality between deformations of the multiplication map on the algebra of functions on spacetime F (R 4 ) and coproduct deformations of the Poincaré-Hopf algebra HP acting on F (R 4 ); the appearance of a nonassociative product on F (R 4 ) when gauge fields are also included in the picture. The last part of the manuscript is dedicated to the phenomenology of noncommutative quantum field theories in the particular approach adopted in this review. CPT violating processes, modification of two-point temperature correlation function in CMB spectrum analysis and Pauli-forbidden transition in Be 4 are all effects which show up in such a noncommutative setting. We review how they appear and in particular the constraint we can infer from comparison between theoretical computations and experimental bounds on such effects. The best bound we can get, coming from Borexino experiment, is 10 24 TeV for the energy scale of noncommutativity, which corresponds to a length scale 10 −43 m. This bound comes from a different model of spacetime deformation more adapted to applications in atomic physics. It is thus model dependent even though similar bounds are expected for the Moyal spacetime as well as argued elsewhere.

Noncommutative Einstein-Maxwell pp-waves

Phys Rev D, 2006

The field equations coupling a Seiberg-Witten electromagnetic field to noncommutative gravity, as described by a formal power series in the noncommutativity parameters θ αβ , is investigated. A large family of solutions, up to order one in θ αβ , describing Einstein-Maxwell null pp-waves is obtained. The order-one contributions can be viewed as providing noncommutative corrections to pp-waves. In our solutions, noncommutativity enters the spacetime metric through a conformal factor and is responsible for dilating/contracting the separation between points in the same null surface. The noncommutative corrections to the electromagnetic waves, while preserving the wave null character, include constant polarization, higher harmonic generation and inhomogeneous susceptibility. As compared to pure noncommutative gravity, the novelty is that nonzero corrections to the metric already occur at order one in θ αβ .

Lorentzian approach to noncommutative geometry

2011

This thesis concerns the research on a Lorentzian generalization of Alain Connes' noncommutative geometry. In the first chapter, we present an introduction to noncommutative geometry within the context of unification theories. The second chapter is dedicated to the basic elements of noncommutative geometry as the noncommutative integral, the Riemannian distance function and spectral triples. In the last chapter, we investigate the problem of the generalization to Lorentzian manifolds. We present a first step of generalization of the distance function with the use of a global timelike eikonal condition. Then we set the first axioms of a temporal Lorentzian spectral triple as a generalization of a pseudo-Riemannian spectral triple together with a notion of global time in noncommutative geometry. Ph.D. thesis in Mathematics

Unitary quantum field theory on the noncommutative Minkowski space

Fortschritte der Physik, 2003

This is the written version of a talk I gave at the 35th Symposium Ahrenshoop in Berlin, Germany, August 2002. It is an exposition of joint work with S. Doplicher, K. Fredenhagen, and Gh. Piacitelli [1]. The violation of unitarity found in quantum field theory on noncommutative spacetimes in the context of the so-called modified Feynman rules is linked to the notion of time ordering implicitely used in the assumption that perturbation theory may be done in terms of Feynman propagators. Two alternative approaches which do not entail a violation of unitarity are sketched. An outlook upon our more recent work is given.

Noncommutative Spacetime in Very Special Relativity

2010

Very Special Relativity (VSR) framework, proposed by Cohen and Glashow [1], demonstrated that a proper subgroup of the Poincar\'e group, (in particular ISIM(2)), is sufficient to describe the spacetime symmetries of the so far observed physical phenomena. Subsequently a deformation of the latter, DISIMb(2)DISIM_b(2)DISIMb(2), was suggested by Gibbons, Gomis and Pope [2]. In the present work, we introduce a novel Non-Commutative (NC) spacetime structure, underlying the DISIMb(2)DISIM_b(2)DISIMb(2). This allows us to construct explicitly the DISIMb(2)DISIM_b(2)DISIMb(2) generators, consisting of a sector of Lorentz rotation generators and the translation generators. Exploiting the Darboux map technique, we construct a point particle Lagrangian that lives in the NC phase space proposed by us and satisfies the modified dispersion relation proposed by Gibbons et. al. [2]. It is interesting to note that in our formulation the momentum algebra becomes non-commutative.

Towards a first-principles approach to spacetime noncommutativity

Journal of Physics: …, 2007

Our main thesis in this note is that if spacetime noncommutativity is at all relevant in the quantum gravitational regime, there might be a canonical approach to pinning down its form. We start by emphasizing the distinction between an intrinsically noncommuting "manifold", i.e., one with noncommuting coordinate functions, on the one hand, and particles with noncommuting position operators, on the other. Focusing on the latter case, which, we feel, more adequately reflects the experimental nature of our knowledge of spacetime properties, we find that several complementary considerations point to a spin-dependent noncommutativity, which is confirmed in the single-particle sector of Dirac's theory, as well as in Fokker's relativistic "center-of-mass" prescription. Finally, we propose an extension of Jordan and Mukunda's work to gain a glimpse on the effect of curvature on position operator noncommutativity.