The quantum structure of spacetime at the Planck scale and quantum fields

Abstract

We propose uncertainty relations for the different coordinates of spacetime events, motivated by Heisenberg's principle and by Einstein's theory of classical gravity. A model of Quantum Spacetime is then discussed where the commutation relations exactly implement our uncertainty relations. We outline the definition of free fields and interactions over QST and take the first steps to adapting the usual perturbation theory. The quantum nature of the underlying spacetime replaces a local interaction by a specific nonlocal effective interaction in the ordinary Minkowski space. A detailed study of interacting QFT and of the smoothing of ultraviolet divergences is deferred to a subsequent paper. In the classical limit where the Planck length goes to zero, our Quantum Spacetime reduces to the ordinary Minkowski space times a two component space whose components are homeomorphic to the tangent bundle TS^2 of the 2-sphere. The relations with Connes' theory of the standard model will be studied elsewhere.

TeX, 37 pages. Since recent and forthcoming articles (hep-th/0105251, hep-th/0201222, hep-th/0301100) are based on this paper, we thought it would be convenient for the readers to have it available on the web

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• And what if gravity is intrinsically quantic ?
• The Weyl realizations of Lie algebras and left-right duality
• Building a linear equation of state for trapped gravitons from finite size effects and the Schwarzschild black hole case
• A Review of Noncommutative Field Theories
• Non-commutative Geometry from Perturbative Quantum Gravity in de Sitter spacetime
• Noncommutative magnetic moment, fundamental length and lepton size
• Born--Oppenheimer decomposition for quantum fields on quantum spacetimes
• Space-time: Commutative or noncommutative ?
• Equivalence principle in context of large uniform acceleration - a quantum mechanical perspective
• Deformation of the quintom cosmological model and its consequences
• Noncommutative deformations of quantum field theories, locality, and causality
• Generalized Non-Commutative Inflation
• On Poisson geometries related to noncommutative emergent gravity
• Magnetic Monopole in Noncommutative Space-Time and Wu-Yang Singularity-Free Gauge Transformations
• Photonic realization of the kappa-deformed Dirac equation
• Two-photon annihilation of singlet cold dark matters due to noncommutative space-time
• Quantum space and quantum completeness
• Dimensional flow in the kappa-deformed space-time
• Wedge locality and asymptotic commutativity
• Field Theory on a Non-commutative Plane: a Non-Perturbative Study
• Noncommutative of space-time and the relativistic Hydrogen atom
• Generalized twist deformations of Poincare and Galilei Hopf algebras
• Quantum instability of gauge theories on $\kappa$-Minkowski space
• Classical dynamics on three dimensional fuzzy space: Connecting the short and long length scales
• From Snyder space-times to doubly $\kappa$-dependent Yang quantum phase spaces and their generalizations
• Symmetric ordering and Weyl realizations for quantum Minkowski spaces
• Bound state energies and phase shifts of a non-commutative well
• Low-Energy Lorentz Invariance in Lifshitz Nonlinear Sigma Models
• Quantum field theories on noncommutative R^4 versus theta-expanded quantum field theories
• Closedness of orbits in a space with SU(2) Poisson structure
• Aspects of perturbative quantum field theory on non-commutative spaces
• Conjugate variables in quantum field theory: the basic case
• Magnetic monopoles in noncommutative quantum mechanics
• Two roads to noncommutative causality
• Noncommutative quantum mechanics of simple matter systems interacting with circularly polarized gravitational waves
• The entropy of dense non-commutative fermion gases
• Analytical and numerical analysis of a rotational invariant D=2 harmonic oscillator in the light of different noncommutative phase-space configurations
• Non-Commutative Correction to Thin Shell Collapse in Reissner Nordstr$\ddot{o}$m Geometry
• Virtual Black Holes from Generalized Uncertainty Principle and Proton Decay
• A note on broken dilatation symmetry in planar noncommutative theory
• N-Particle Scattering in Relativistic Wedge-Local Quantum Field Theory
• Twist for Snyder space
• Particle Dynamics And Emergent Gravity
• Gravitational measurements in higher dimensions
• Faddeev-Jackiw approach of the noncommutative spacetime Podolsky electromagnetic theory
• Twisted submanifolds of R^n
• Causal Fermion Systems and the ETH Approach to Quantum Theory
• Noncommutative Brownian motion
• Central tetrads and quantum spacetimes
• Complete solution of the LSZ Model via Topological Recursion
• Quantum Aspects of the Noncommutative Sine-Gordon Model
• One loop radiative corrections to the translation-invariant noncommutative Yukawa Theory
• Gravitational probe of quantum spacetime
• Information Theoretical Approach to Detecting Quantum Gravitational Corrections
• ESA Voyage 2050 white paper -- GrailQuest: hunting for Atoms of Space and Time hidden in the wrinkle of Space-Time
• Statistical description of massless excitations within a sphere with a linear equation of state and the dark energy case
• Landau levels in a 2D noncommutative space: matrix and quaternionic vector coherent states
• Extended noncommutative Minkowski spacetimes and hybrid gauge symmetries
• On the consequences of twisted Poincare' symmetry upon QFT on Moyal noncommutative spaces
• Thermodynamic phase transition and Joule-Thomson expansion of a quantum corrected black hole in AdS spacetime
• Two arguments for more fundamental building blocks
• Comment on "Can we measure structures to a precision better than the Planck length?"[arXiv:1205.3636], by Sabine Hossenfelder
• Cornwall-Jackiw-Tomboulis effective potential for canonical noncommutative field theories
• Length and distance on a quantum space
• Propagators and Matrix Basis on Noncommutative Minkowski Space
• Calculating the jet quenching parameter in the plasma of NCYM theory from gauge/gravity duality
• Star Product Geometries
• Phase space quantization, noncommutativity and the gravitational field
• Twist Deformations of the Supersymmetric Quantum Mechanics
• TeV-photon paradox and space with SU(2) fuzziness
• Learning a Deep Model for Human Action Recognition from Novel Viewpoints
• Maximal acceleration in a Lorentz invariant non-commutative space-time
• Preconjugate variables in quantum field theory and their use
• Emergence of maximal acceleration from non-commutativity of space-time
• Consistency of an alternative CPT-odd and Lorentz-violating extension of QED
• Noncommutative (supersymmetric) electrodynamics in the Yang-Feldman formalism
• Quantization of the open string on plane-wave limits of dS_n x S^n and non-commutativity outside branes
• Holographic Bound in Quantum Field Energy Density and Cosmological Constant
• Realization of bicovariant differential calculus on the Lie algebra type noncommutative spaces
• Matrix Field Theory
• Noncommutative quasinormal modes of Schwarzschild black hole
• Energy corrections due to the noncommutative phase-space of the charged isotropic harmonic oscillator in a uniform magnetic field in 3D
• The ultraviolet-finite Hamiltonian approach on the noncommutative Minkowski space
• Conformal quantum mechanics and holography in noncommutative space-time
• Noncommutative supersymmetric Chern-Simons-matter model
• Uniformly Accelerated Observer in Moyal Spacetime
• Computing the spectral action for fuzzy geometries: from random noncommutative geometry to bi-tracial multimatrix models
• Unitarity of noncommutative field theories from string theory
• Symplectic Deformations, Non Commutative Scalar Fields and Fractional Quantum Hall Effect
• Spectral distances on doubled Moyal plane using Dirac eigen-spinors
• Multi-particle systems in quantum spacetime and a novel challenge for center-of-mass motion
• Quantum Entanglement of Fermions- Antifermions Pair Creation modes in Non-commutative Bianchi I Space-time
• Possible quantum kinematics. II. Non-minimal case
• Poincaré Covariant k-Minkowski Spacetime
• Quantized Kähler Geometry and Quantum Gravity
• Noncommutative Vortices and Instantons from Generalized Bose Operators
• Basic quantizations of $D=4$ Euclidean, Lorentz, Kleinian and quaternionic ${\mathfrak{o}}^{\star }(4)$ symmetries
• Phase structure of fuzzy black holes
• The cosmological constant from Planckian fluctuations and the averaging procedure
• Noncommutative Time in Quantum Field Theory
• Twisted Covariance and Weyl Quantisation
• Spin and Statistics and First Principles
• Quantum Kaluza-Klein theory with $M_2(\mathbb{C})$
• Twisted Fermionic Oscillator Algebra in $\kappa$-Minkowski space-time
• Effects of wave propagation in canonical Poisson gauge theory under an external magnetic field
• UV Finite Field Theories on Noncommutative Spacetimes: the Quantum Wick Product and Time Independent Perturbation Theory
• Notes on several phenomenological laws of quantum gravity
• On the Quantization of Length in Noncommutative Spaces
• Reply to "Comment on 'Noncommutative gauge theories and Lorentz symmetry'"
• Linear hyperbolic PDEs with non-commutative time
• Position Operators and Center of Mass: New Perspectives
• Spontaneous symmetry breaking and masses numerical results in DFR noncommutative space-time
• Quantum Riemannian geometry of the discrete interval and q-deformation
• Poisson electrodynamics on $\kappa$-Minkowski space-time
• Space-time noncommutativity with a bifermionic parameter
• Two-particle system in Coulomb potential for twist-deformed space-time
• QED Representation for the Net of Causal Loops
• Harmonic oscillator in twisted Moyal plane: eigenvalue problem and relevant properties
• A proposal for Heisenberg uncertainty principle and STUR for curved backgrounds: an application to white dwarf, neutron stars and black holes
• Asymptotic commutativity of quantized spaces: the case of ${\mathbb{CP}}^{p,q}$
• Central force problem in space with SU(2) Poisson structure
• On Charge Conservation and The Equivalence Principle in the Noncommutative Spacetime
• Exponential Formulas, Normal Ordering and the Weyl-Heisenberg Algebra
• Trace of phase-space noncommutativity in the response of a free particle to linearized gravitational waves
• Spectral dimension of Horava-Snyder spacetime and the $Ad{S}_2\times S^2$ momentum space
• Random versus holographic fluctuations of the background metric. I. (Cosmological) running of space-time dimension
• Conserved symmetries in noncommutative quantum mechanics
• The physical origin of the cosmological constant: continuum limit and the analogy with the Casimir effect
• Dominance of a single topological sector in gauge theory on non-commutative geometry
• The instability of intersecting fuzzy spheres
• Noncommutative Spherically Symmetric Spaces
• Noncommutative field theories from a deformation point of view
• Newtonian cosmology and Evolution of kappa-deformed universe
• Revisiting Connes' Finite Spectral Distance on Non-commutative Spaces : Moyal Plane and Fuzzy Sphere
• $U(N)$ Yang-Mills in non-commutative space time
• The Hierarchy Problem: From the Fundamentals to the Frontiers
• Deformation Quantization of Surjective Submersions and Principal Fibre Bundles
• Covariant Fields of C*-Algebras under Rieffel Deformation
• Interacting Relativistic Particle: Time-Space Noncommutativity And Symmetries
• Fermi acceleration under Lorentz invariance violation
• Fuzzy hyperspheres via confining potentials and energy cutoffs
• Exact solutions for scalars and spinors on quantized (Euclidean) $Ad{S}_2$ and the correspondence principle
• A Stochastic Origin of Spacetime Non-Commutativity
• Thermodynamical features of Verlinde's approach for a non-commutative Schwarzschild-anti-deSitter black hole in a broad range of scales
• Scalar fields in a non-commutative space
• Novel Ambiguities in the Seiberg-Witten Map and the Emergent Gravity
• Propagation of spinors on a noncommutative spacetime: equivalence of the formal and the effective approach
• One-loop unitarity of scalar field theories on Poincare invariant commutative nonassociative spacetimes
• Non-Commutative space-time and Hausdorff dimension
• Noncommutative Lagrange Mechanics
• Localization at threshold in noncommutative space
• An Arena for Model Building in the Cohen-Glashow Very Special Relativity
• Two approaches to quantum gravity and M-(atrix) theory at large number of dimensions
• Strict deformations of quantum field theory in de Sitter spacetime
• The physical nature of the cosmological constant and the decoherence scale in a renormalization-group approach
• Residual gauge-invariance in a massive Lorentz-violating extension of QED
• Quantum Klein Space and Superspace
• Quantum groups and noncommutative spacetimes with cosmological constant
• Generalized Heisenberg algebra, realizations of the $\mathfrak{gl}(n)$ algebra and applications
• The Strong Interaction Model in DFR noncommutative space-time
• Beyond second-moment approximation in fuzzy-field-theory-like matrix models
• A Deformation Quantization for Non-Flat Spacetimes and Applications to QFT
• Regularization of Kepler Problem in $\kappa$-spacetime
• Quantization of Spacetime Based on Spacetime Interval Operator
• Covariant dynamics on the energy-momentum space: scalar field theory
• Pinhole interference in three-dimensional fuzzy space
• Non-commutative coordinates from quantum gravity
• $\kappa$-deformed power spectrum and modified Unruh temperature
• Twisting loops and global momentum non-conservation in Relative Locality
• A scale-covariant quantum space-time
• Neutron Star in Quantized-space-time
• Yang-Mills field from fuzzy sphere quantum Kaluza-Klein model
• Braided Field Quantization from Quantum Poincare Covariance
• Towards new relativistic doubly $\kappa$-deformed D=4 quantum phase spaces
• Estimating the strength of Lorentzian distribution in non-commutative geometry by solar system tests
• Noncommutative geometry of the Moyal plane: translation isometries, Connes' distance on coherent states, Pythagoras equality
• Light Propagation in $\kappa$-Minkowski Space-Time: Gauge Ambiguities and Invariance
• Noncommutativity and Duality through the Symplectic Embedding Formalism
• Families of vector-like deformed relativistic quantum phase spaces, twists and symmetries
• All-Loop Renormalizable Wess-Zumino Model on Bosonic-Fermionic Noncommutative Superspace
• Bicrossproduct construction versus Weyl-Heisenberg algebra
• kappa-Deformed Oscillators: Deformed Multiplication Versus Deformed Flip Operator and Multiparticle Clusters
• Could the photon dispersion relation be non-linear ?
• Covariant formulation of Noether's Theorem for translations on kappa-Minkowski spacetime
• Is life a thermal horizon ?
• Causality in Non-Commutative Quantum Field Theories
• Morita Equivalence, Picard Groupoids and Noncommutative Field Theories
• Perturbative Algebraic Quantum Field Theory on Quantum Spacetime: Adiabatic and Ultraviolet Convergence
• Deformations of the Canonical Commutation Relations and Metric Structures
• Search for the Footprints of New Physics with Laboratory and Cosmic Neutrinos
• A Poincaré Covariant Noncommutative Spacetime?
• Spinor calculus for q-deformed quantum spaces II
• Commutative deformations of general relativity: nonlocality, causality, and dark matter
• Projective module description of embedded noncommutative spaces
• Radiative corrections to the Chern-Simons term at finite temperature in the noncommutative Chern-Simons-Higgs model
• Twisted Supersymmetry in a Deformed Wess-Zumino Model in (2+1) Dimensions
• Some considerations on duality concerning kappa-Minkowski spacetime theories
• Global Space-Time Symmetries of Quantized Euclidean and Minkowski Superspaces
• Pale Glares of Dark Matter in Quantum Spacetime
• Total momentum and other Noether charges for particles interacting in a quantum spacetime
• Quantized Equations of Motion and Currents in Noncommutative Theories
• Consistent Construction of Perturbation Theory on Noncommutative Spaces
• The coordinate coherent states approach revisited
• Ultraviolet-infrared mixing on the noncommutative Minkowski space in the Yang-Feldman formalism
• Noncommutative Geometry: Fuzzy Spaces, the Groenewold-Moyal Plane
• DFR Perturbative Quantum Field theory on Quantum Space Time, and Wick Reduction
• Quantum-gravitational running/reduction of space-time dimension
• General Covariance in Algebraic Quantum Field Theory
• Dirac Operators on Noncommutative Curved Spacetimes
• On localized and coherent states on some new fuzzy spheres
• Quantum harmonic oscillator in a time dependent noncommutative background
• T-Minkowski noncommutative spacetimes I: Poincaré groups, differential calculi and braiding
• Wigner functions for gauge equivalence classes of unitary irreducible representations of noncommutative quantum mechanics
• Braided Tensor Products and the Covariance of Quantum Noncommutative Free Fields
• A brief introduction to the noncommutative geometry description of particle physics standard model
• Generalized twist deformations of Poincare and Galilei quantum groups
• Curving Flat Space-Time by Deformation Quantization?
• Wave Propagation and IR/UV Mixing in Noncommutative Spacetimes
• Separation of noncommutative differential calculus on quantum Minkowski space
• Some Features of Scattering Problem in a $\kappa$-Deformed Minkowski Spacetime
• Generalization of Weyl realization to a class of Lie superalgebras
• Relativistic Corrections to the Moyal-Weyl Spacetime
• Energy momentum tensor for translation invariant renormalizable noncommutative field theory
• Pair production of Dirac particles in a d+1-dimensional noncommutative space-time
• Quantum Cosmological Backreactions III: Deparametrised Quantum Cosmological Perturbation Theory
• Quantisation of Lorentz invariant scalar field theory in non-commutative space-time and its consequence
• To the Memory of Alexander Grothendieck: a Great and Mysterious Genius of Mathematics
• A Spectral Triple for a Solenoid Based on the Sierpinski Gasket
• Supersymmetry breaking in noncommutative quantum mechanics
• Fuzzy field theories and related matrix models
• Symplectic Fluctuations for Electromagnetic Excitations of Hall Droplets
• Twisting kappa-deformed phase space
• Quantum Gravity models - brief conceptual summary
• Study on Noncommutative Representations of Galilean Generators
• Yang model revisited
• Time dependent transitions with time-space noncommutativity & its implications in Quantum Optics
• Quantum speed limit as a sensitive probe of Planck-scale effects
• Fermions Coupled to Emergent Noncommutative Gravity
• Random versus holographic fluctuations of the background metric. II. Note on the dark energies arising due to microstructure of space-time
• Thermostatistical analysis and negative heat capacities of Yukawa and Lee-Wick potentials in noncommutative phase spaces
• A noncommutative geometric approach to the quantum structure of spacetime
• Quantum theory, noncommutative gravity, and the cosmological constant problem
• The 1/N Expansion in Noncommutative Quantum Mechanics
• Effects of quantum geometry on the decoherence induced by black holes
• Two ${\theta }_{\mu \nu }$ -deformed covariant relativistic quantum phase spaces as Poincare-Hopf algebroids
• Horizon quantum geometries and decoherence
• Entropic force and bouncing behaviour in $\kappa$-Minkowski space-time
• The DFR-Algebra for Poisson Vector Bundles
• Entropy of black holes, charged probes and noncommutative generalization
• A note on the implementation of Poincaré symmetry in noncommutative field theory
• Non-commutative correction of ideal gas thermodynamics
• The Slavnov-Taylor Identities for the 2+1 Dimensional Noncommutative CP${}^{N-1}$ Model
• On the UV/IR mixing of Lie algebra-type noncommutatitive ${\varphi }^4$-theories
• The Majid-Ruegg model and the Planck scales
• Local Space-Time Curvature Effects on Quantum Orbital Angular Momentum
• Thermal effective potential in two- and three- dimensional non-commutative spaces
• Eigenvalue problem for radial potentials in space with SU(2) fuzziness
• One-Loop Renormalizable Wess-Zumino Model on Bosonic-Fermionic Noncommutative Superspace
• Vulcanized Vortex
• A general relativity formulation for the Doplicher-Fredenhagen-Roberts noncommutative space-time
• Noncommutative $Ad{S}_2$ I: Exact solutions
• Retarded Functions in Noncommutative Theories
• Waves on Noncommutative Spacetimes
• q-Deformed Superalgebras
• Signatures of quantum geometry from exponential corrections to the black hole entropy
• Green's Functions for Translation Invariant Star Products
• Testing Lorentz Violation Using Propagating UHECRs
• The standard electroweak model in the noncommutative $DFR$ space-time
• Yang-Mills action from minimally coupled bosons on R^4 and on the 4D Moyal plane
• Conjugate variables in quantum field theory and a refinement of Paulis theorem
• Spherically symmetric potential in noncommutative spacetime with a compactified extra dimensions
• Deformation of Noncommutative Quantum Mechanics
• Spinorial Snyder and Yang Models From Superalgebras And Noncommutative Quantum Superspaces
• Space-time noncommutativity and (1+1) Higgs Model
• Einstein Equations for a Noncommutative Spacetime of Lie-Algebraic Type
• Effective Potential and Spontaneous Symmetry Breaking in the Noncommutative phi^6 Model
• The Structure of Spacetime and Noncommutative Geometry
• Exploring the gravity sector of emergent higher-spin gravity: effective action and a solution
• The $q$-linked complex Minkowski space, its real forms and deformed isometry groups
• Quantum Spacetime and the Universe at the Big Bang, vanishing interactions and fading degrees of freedom
• Chiral Spin Noncommutative Space and Anomalous Dipole Moments
• Braided algebras and the kappa-deformed oscillators
• On Finite Noncommutativity in Quantum Field Theory
• From quantum deformations of relativistic symmetries to modified kinematics and dynamics
• Towards an Axiomatic Formulation of Noncommutative Quantum Field Theory. II
• Twisted geometry for submanifolds of ${\mathbb{R}}^n$
• Aspects of noncommutativity in field theory, strings and membranes
• The G-wavefront set and the twisted convolution product
• Review of Twisted Poincare Symmetry
• Corrections to the Black hole entropy from a Bose Einstein condensate: a semi-classical phenomenological approach
• Quantum Space-Time and Noncommutative Gauge Field Theories
• Quantum kinematics on q-deformed quantum spaces II, Wave functions on position and momentum space
• The Photon Dispersion as an Indicator for New Physics ?
• Quantum kinematics on q-deformed quantum spaces I, Mathematical Framework
• Einstein's lifts and topologies: topological investigations on the Principle of Equivalence
• Noncommutative spaces with twisted symmetries and second quantization
• Compactification, T-Duality and Quantum Erasers
• Non-relativistic Schroedinger theory on q-deformed quantum spaces III, Scattering theory
• Non-relativistic Schroedinger theory on q-deformed quantum spaces II, The free non-relativistic particle and its interactions
• Wick rotation for quantum field theories on degenerate Moyal space(-time)
• The 3D perturbed Schrödinger Hamiltonian in a Friedmann flat spacetime testing the primordial universe in a non commutative spacetime
• Statistical interparticle potential on noncommutative space
• A Quantum Energy Inequality for a Non-commutative QFT
• Energy corrections due to the Non-commutative Phase-Space of the Charged Harmonic Oscillator in a constant magnetic field in 3D
• Propagation features of Lorentz-violating electrodynamics
• Quantum Spacetime: Echoes of basho
• Deriving Ontological Statements from the Unnatural Higgs Mass
• Quantization of Length in Spaces with Position-Dependent Noncommutativity
• Statistical physics on Euclidean Snyder space: connections with the GUP and cosmological implications
• Quantum geometry from commutators: a Heisenberg-picture framework and a toy application to early structure
• Algebraic structures in $\kappa$-Poincaré invariant gauge theories
• A relation between deformed superspace and Lee-Wick higher-derivative theories
• Unruh effect using Doppler shift method in DSR framework
• Realizations of the Extended Snyder Model
• On the nonlinear electrodynamics in a Lorentz-breaking scenario
• Vacuum Pair Creation in Spin Noncommutative of Coordinates: Volkov Background and Constant Electric Field
• A Twisted ${\mathcal{C}}^{\star }$-algebra formulation of Quantum Cosmology with application to the Bianchi I model
• The universality question for noncommutative quantum field theory
• A note on the fuzzy sphere area spectrum, black hole luminosity, and the quantum nature of spacetime
• Noncommutative $\mathcal{N}=2$ Chern-Simons-matter model
• Theories on noncommutative spaces and deformed symmetries
• Quantum Spacetimes from General Relativity?
• Radiation Reaction in Non-commutative Electrodynamics
• Twist-Deformed Lorentzian Heisenberg-Algebras
• Electromagnetic Excitations of Hall Systems on Four Dimensional Space
• Fubctional integral approach to quantum gauge theories on a noncommutative space-time
• N=1/2 Global SUSY: R-Matrix Approach
• Renormalization on the DFR Quantum Spacetime
• Comments on Noncommutative Field Theories
• On 2d gravity and parallel transport field equation on quantum sphere
• Dynamic and static properties of Quantum Hall and Harmonic Oscillator systems on the non-commutative plane
• The flux of noncommutative U(1) instanton through the fuzzy spheres
• Noncommutativity in Configuration Space Induced by A Conjugate Magnetic Field in Phase Space
• General $O(D)$-equivariant fuzzy hyperspheres via confining potentials and energy cutoffs
• Braided Statistics from Abelian Twist in $\kappa$-Minkowski Spacetime
• Lattice oscillator model on noncommutative space: eigenvalues problem for the perturbation theory
• $\hslash$-perturbative solutions of quantum Snyder and Yang models with parameters describing spontaneous symmetry breaking
• On the Non-Flatness Nature of Noncommutative Minkowski Spacetime and the Singular Behavior of Probes
• Noncommutative scalar fields from symplectic deformation