Noncommutative Spacetime Research Articles

A deformation quantization for non-flat spacetimes and applications to QFT

We provide a deformation quantization, in the sense of Rieffel, for all globally hyperbolic spacetimes with a Poisson structure. The Poisson structures have to satisfy Fedosov type requirements in order for the deformed product to be associative. We apply the novel deformation to quantum field theories and their respective states and we prove that the deformed state (i.e. a state in non-commutative spacetime) has a singularity structure resembling Minkowski, i.e. is Hadamard, if the undeformed state is Hadamard. This proves that the Hadamard condition, and hence the quantum field theoretical implementation of the equivalence principle is a general concept that holds in spacetimes with quantum features (i.e. a non-commutative spacetime).

How does Casimir energy fall in κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}-deformed space-time?

We investigate the response of Casimir energies to fluctuations in a scalar field in the background of a weak gravitational field in κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}-deformed space-time. We model the Casimir plates in a gravitational field by κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}-deformed Rindler coordinates and calculate the Casimir energy using the κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}-deformed scalar field. We show that the Casimir energy accelerates in a weak gravitational field like a mass. Thus, our calculations show that the mass–energy equivalence principle holds in κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}-deformed space-time even though a length scale is introduced through space-time non-commutativity.

Quaternionic generalization of telegraph equations

Using non-commutative spacetime quaternion algebra, we represent the generalization of one-dimensional and three-dimensional telegraph equations, which are widely applied to consider the propagation of an electromagnetic signal in communication lines, as well as to describe particle diffusion and heat transfer. It is shown that the system of telegraph equations can be represented in compact form as a single quaternion equation taking into account the spacetime properties of physical quantities. The distinctive features of the one-dimensional and three-dimensional telegraph equations are discussed.

T-Minkowski Noncommutative Spacetimes I: Poincaré Groups, Differential Calculi, and Braiding

Abstract This paper introduces and investigates a class of noncommutative spacetimes that I will call “T-Minkowski,” whose quantum Poincaré group of isometries exhibits unique and physically motivated characteristics. Notably, the coordinates on the Lorentz subgroup remain commutative, while the deformation is confined to the translations (hence the T in the name), which act like an integrable set of vector fields on the Lorentz group. This is similar to Majid’s bicrossproduct construction, although my approach allows the description of spacetimes with commutators that include a constant matrix as well as terms that are linear in the coordinates (the resulting structure is that of a centrally extended Lie algebra). Moreover, I require that one can define a covariant braided tensor product representation of the quantum Poincaré group, describing the algebra of N-points. This also implies that a 4D bicovariant differential calculus exists on the noncommutative spacetime. The resulting models can all be described in terms of a numerical triangular R-matrix through RTT relations (as well as RXX, RXY, and RXdX relations for the homogeneous spacetime, the braiding, and the differential calculus). The R-matrices I find are in one-to-one correspondence with the triangular r-matrices on the Poincaré group without quadratic terms in the Lorentz generators. These have been classified, up to automorphisms, by Zakrzewski, and amount to 16 inequivalent models. This paper is the first of a series, focusing on the identification of all the quantum Poincaré groups that are allowed by my assumptions, as well as the associated quantum homogeneous spacetimes, differential calculi, and braiding constructions.

Holography on the quantum disk

Motivated by recent study of DSSYK and the non-commutative nature of its bulk dual, we review and analyze an example of a non-commutative spacetime known as the quantum disk proposed by L. Vaksman. The quantum disk is defined as the space whose isometries are generated by the quantum algebra Uqsu1,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {U}_q\\left(\\mathfrak{s}{\\mathfrak{u}}_{1,1}\\right) $$\\end{document}. We review how this algebra is defined and its associated group SUq(1, 1) that it generates, highlighting its non-trivial coproduct that sources bulk non-commutativity. We analyze the structure of holography on the quantum disk and study the imprint of non-commutativity on the putative boundary dual.

Exploring non-commutativity as a perturbation in the Schwarzschild black hole: quasinormal modes, scattering, and shadows

In this work, by a novel approach to studying the scattering of a Schwarzschild black hole, the non-commutativity is introduced as perturbation. We begin by reformulating the Klein–Gordon equation for the scalar field in a new form that takes into account the deformed non-commutative spacetime. Using this formulation, an effective potential for the scattering process is derived. To calculate the quasinormal modes, we employ the WKB method and also utilize fitting techniques to investigate the impact of non-commutativity on the scalar quasinormal modes. We thoroughly analyze the results obtained from these different methods. Moreover, the greybody factor and absorption cross section are investigated. Additionally, we explore the behavior of null geodesics in the presence of non-commutativity. Specifically, we examine the photonic, and shadow radius as well as the light trajectories for different non-commutative parameters. Therefore, by addressing these various aspects, we aim to provide a comprehensive understanding of the influence of non-commutativity on the scattering of a Schwarzschild-like black hole and its implications for the behavior of scalar fields and light trajectories.

Testing space-time non-commutativity with TianQin

The direct detection of gravitational waves offers a powerful tool to explore the nature of gravity and the structure of space-time. This paper focuses on the capabilities of space-based gravitational wave detectors in testing space-time non-commutativity. Our findings indicate that TianQin has the potential to impose constraints on the non-commutative scale at a sub-Planckian level using massive black hole binaries. Additionally, we have developed a pipeline tailored to this specific topic.

Schwarzschild black hole surrounded by a cavity and phase transition in the non-commutative gauge theory of gravity

In this work, we investigate the phase transition of the Schwarzschild black hole (SBH) inside an isothermal spherical cavity in the context of the non-commutative (NC) gauge theory of gravity, by using the Seiberg–Witten (SW) map and the star product. Firstly, we compute the NC correction to the Hawking temperature and derive the logarithmic correction to the entropy, then we derive the local temperature and local energy of NC SBH in isothermal cavity. Our results show that the non-commutativity removes the commutative divergence behavior of temperature, and prevents the SBH from the complete evaporation, which leads to a remnant black hole, and this geometry has predicted a minimal length in the order of Planck scale Θ∼lplanck. Therefore, the thermodynamic stability and phase transition is studied by analyzing the behavior of the local heat capacity and the Helmholtz free energy in the NC spacetime, where the results show that, the NC SBH has a two second-order phase transition and one first-order phase transition, with two Hawking–Page phase transition in the NC gauge theory.

Photon scattering by an electric field in noncommutative spacetime

As is known, the existence of a small noncommutativity between coordinates would generate nonlocal self-interactions in the electromagnetic theory. To explore some consequences of this effect on the propagation of photons we consider Moyal space half-filled with a static and homogeneous electric field and analyze electromagnetic fluctuations on top of this step-like background. Both the localization of photons and the possibility of photon production by strong electric fields are addressed. Several aspects of the Klein paradox in this setup are discussed as well.

Noncommutative Schwarzschild black hole surrounded by quintessence: Thermodynamics, Shadows and Quasinormal modes

In Campos et al. (2022), Campos et al studied the quasinormal modes and shadows of noncommutative Schwarzschild black holes. Since we know that the quintessence matter surrounding black holes has significant effects on the black hole quantities, in this study, we aimed to show this influence by revisiting the same problem in the presence of quintessence matter field. To this end, we first examined the thermodynamics of noncommutative Schwarzschild black holes embedded in quintessence matter using Hawking temperature, entropy, and specific heat functions. After that we discussed phase transition and stability features. We then investigated the shadow images in the presence of plasma. After visualizing these results qualitatively, we calculated the quasinormal modes in WKB and Mashhoon approximations and we demonstrated the impacts of quintessence matter and noncommutative spacetime on the whole quantities.

Symplectic groupoids and Poisson electrodynamics

We develop a geometric approach to Poisson electrodynamics, that is, the semi-classical limit of noncommutative U(1) gauge theory. Our framework is based on an integrating symplectic groupoid for the underlying Poisson brackets, which we interpret as the classical phase space of a point particle on noncommutative spacetime. In this picture gauge fields arise as bisections of the symplectic groupoid while gauge transformations are parameterized by Lagrangian bisections. We provide a geometric construction of a gauge invariant action functional which minimally couples a dynamical charged particle to a background electromagnetic field. Our constructions are elucidated by several explicit examples, demonstrating the appearances of curved and even compact momentum spaces, the interplay between gauge transformations and spacetime diffeomorphisms, as well as emergent gravity phenomena.

On the UV/IR mixing of Lie algebra-type noncommutatitive ϕ4-theories

We show that a UV divergence of the propagator integral implies the divergences of the UV/IR mixing in the two-point function at one-loop for a ϕ4-theory on a generic Lie algebra-type noncommutative space-time. The UV/IR mixing is defined as a UV divergence of the planar contribution and an IR singularity of the non-planar contribution, the latter being due to the former UV divergence, and the UV finiteness of the non-planar contribution. Some properties of this general treatment are discussed. The UV finiteness of the non-planar contribution and the renormalizability of the theory are not treated but commented. Applications are performed for the Moyal space, having a UV/IR mixing, and the κ-Minkowski space for which the two-point function at one-loop is finite.

Neutron Star in Quantized Space-Time

We construct and analyze a model of a neutron star in a κ-deformed space-time. This is conducted by first deriving the κ-deformed generalization of the Einstein tensor, starting from the non-commutative generalization of the metric tensor. By generalizing the energy-momentum tensor to the non-commutative space-time and exploiting the κ-deformed dispersion relation, we then set up Einstein’s field equations in the κ-deformed space-time. As we adopt a realization of the non-commutative coordinates in terms of the commutative coordinates and their derivatives, our model is constructed in terms of commutative variables. Using this, we derive the κ-deformed generalization of the Tolman–Oppenheimer–Volkoff equation. Now, by treating the interior of the star as a perfect fluid as in the commutative space-time, we investigate the modification of the neutron star’s mass due to the non-commutativity of space-time, valid up to first order in the deformation parameter. We show that the non-commutativity of space-time enhances the mass limit of the neutron star. We show that the radius and maximum mass of the neutron star depend on the deformation parameter. Further, our study shows that the mass increases as the radius increases for fixed values of the deformation parameter. We show that maximum mass and radius increase as the deformation parameter increases. We find that the mass varies from 0.26M⊙ to 3.68M⊙ as the radius changes from 8.45 km to 18.66 km. Using the recent observational limits on the upper bound of the mass of a neutron star, we find the deformation parameter to be |a|∼10−44 m. We also show that the compactness and surface redshift of the neutron star increase with its mass.

Singlet scalar dark matter in the noncommutative space–time: A viable hypothesis to explain the gamma-ray excess in the galactic center

In this paper, we explore the noncommutative space–time to revive the idea that gamma-ray excess in the galactic center may stem from the annihilation of particle dark matter. In the noncommutative theory, the photon spectrum is produced by direct emission during this annihilation wherein a photon can be embed in the final state together with other direct products in new vertices. In the various configurations of dark matter phenomenology, we pursue the most prevalent model known as singlet scalar. Calculating the relevant aspects of the model and determining the parameters phase space, we derive the photon flux in the galactic center. This region, known for its high density and occasional existence of robust magnetic fields, serves as an ideal location for investigating theories that encompass the concept of Lorentz symmetry breaking. Upon comparing our numerical achievements with experimental data, it becomes evident that noncommutative space–time can be a reliable framework to explain the gamma-ray excess.

The [formula omitted]-deformation of the generalized DKP oscillator

We consider the Duffin–Kemmer–Petiau (DKP) oscillator for spin-zero bosons, coupled to a pseudoscalar Coulomb potential, in a (3+1)-dimensional noncommutative space–time characterized by κ-deformation. We use the method of Dirac derivatives to build the κ-deformed DKP equation. We work out the solutions of the generalized oscillator so obtained using a perturbation method: This allowed to examine first-order effects of the non-commutativity on its eigenenergies and eigenfunctions. Notably, we find that, due to the deformation, the energy levels of s-states become discontinuous at a certain critical value of the coupling constant of the Coulomb term. We also note an asymmetry between particle and antiparticle energies revealing the breaking of the charge-conjugation symmetry for deformed DKP equation. In addition, we discuss the κ-deformed DKP oscillator as a particular case of the considered model.

A coherent state approach to the Casimir effect for a massive scalar field in a noncommutative spacetime

In this paper we investigate the Casimir effect in the classical geometry of two parallel conducting plates, separated by a distance L, for a massive scalar field in a noncommutative spacetime within a coherent state approach. Noncommutative geometry is implemented by means of modified commutation relations for the spacetime coordinates, thus introducing a fundamental length scale l in the theory. Following a coordinate coherent states approach, we define mean values over the noncommutative spacetime and introduce the massive scalar field theory. We derive an expression for the zero-point energy which we evaluate by means of a dimensional regularization process. We express the Casimir energy in terms of powers of the ratios (l/L)2n in the massless case and (ml2/L)2n in the massive case. Our results are discussed both, theoretically and numerically. We finally estimate boundary effects by introducing an specific model for the plates.

Noncommutative spacetimes versus noncommutative spaces of geodesics

The aim of this contribution is twofold. First, we show that when two (or more) different quantum groups share the same noncommutative spacetime, such an ‘ambiguity’ can be resolved by considering together their corresponding noncommutative spaces of geodesics. In any case, the latter play a mathematical/physical role by themselves and, in some cases, they can be interpreted as deformed phase spaces. Second, we explicitly show that noncommutative spacetimes can be reproduced from ‘extended’ noncommutative spaces of geodesics which are those enlarged by the time translation generator. These general ideas are described in detail for the κ-Poincaré and κ-Galilei algebras.

Formal developments in curved momentum space: the quantum field theory roadmap

We advocate that the dual picture of spacetime noncommutativity, i.e. the existence of a curved momentum space, could be a way out to solve some of the open conceptual problems in the field, such as the basis dependence of observables. In this framework, we show how to build deformed Klein–Gordon and Dirac equations. In addition, we give an outlook of how one could define quantum field theories, both free and interacting ones.

Lie-Poisson gauge theories and κ-Minkowski electrodynamics

We consider gauge theories on Poisson manifolds emerging as semiclassical approximations of noncommutative spacetime with Lie algebra type noncommutativity. We prove an important identity, which allows to obtain simple and manifestly gauge-covariant expressions for the Euler-Lagrange equations of motion, the Bianchi and the Noether identities. We discuss the non-Lagrangian equations of motion, and apply our findings to the κ-Minkowski case. We construct a family of exact solutions of the deformed Maxwell equations in the vacuum. In the classical limit, these solutions recover plane waves with left-handed and right-handed circular polarization, being classical counterparts of photons. The deformed dispersion relation appears to be nontrivial.

Quantum tunneling from Schwarzschild black hole in non-commutative gauge theory of gravity

In this letter, we present the first study of Hawking radiation as a tunneling process within the framework of non-commutative (NC) gauge theory of gravity. First, we reconstruct the non-commutative Schwarzschild black hole (NC SBH) within the gauge theory of gravity, employing the Seiberg-Witten (SW) map and the star product. Then, we compute the emission spectrum of outgoing massless particles using the quantum tunneling mechanism. In the first scenario, we calculate the tunneling rate of massless particles crossing the event horizon of the NC SBH with lower frequencies. Our results reveal pure thermal radiation. Notably, we find that the Hawking temperature remains consistent in both the classical thermodynamics and the quantum tunneling approach, suggesting equivalence between these two approaches in NC spacetime. However, in the case of massless particle emission with higher frequencies, we account for energy conservation resulting in the tunneling rate to deviate from pure thermal radiation. This tunneling rate remains consistent with an underlying unitary quantum theory. We establish a relationship between this deviation and the change in the black hole entropy, revealing a logarithmic correction to the entropy within this geometry. Furthermore, we demonstrate that non-commutativity enhances the correlations between two successively emitted particles. Additionally, we determine the NC density number of particle emission and conclude by discussing the implications of our findings.