Canonical Noncommutative Spacetime Research Articles

Extended noncommutative Minkowski spacetimes and hybrid gauge symmetries

We study the Lie bialgebra structures that can be built on the one-dimensional central extension of the Poincaré and (A)dS algebras in (1 + 1) dimensions. These central extensions admit more than one interpretation, but the simplest one is that they describe the symmetries of (the noncommutative deformation of) an Abelian gauge theory, U(1) or SO(2) on Minkowski or (A)dS spacetime. We show that this highlights the possibility that the algebra of functions on the gauge bundle becomes noncommutative. This is a new way in which the Coleman–Mandula theorem could be circumvented by noncommutative structures, and it is related to a mixing of spacetime and gauge symmetry generators when they act on tensor-product states. We obtain all Lie bialgebra structures on centrally-extended Poincaré and (A)dS which are coisotropic w.r.t. the Lorentz algebra, and therefore admit the construction of a noncommutative principal gauge bundle on a quantum homogeneous spacetime. It is shown that several different types of hybrid noncommutativity between the spacetime and gauge coordinates are allowed. In one of these cases, an alternative interpretation of the central extension leads to a new description of the well-known canonical noncommutative spacetime as the quantum homogeneous space of a quantum Poincaré algebra.

Noncommutative SO(2,3)⋆ gravity: Noncommutativity as a source of curvature and torsion

Noncommutative (NC) gravity is constructed on the canonical noncommutative (Moyal-Weyl) space-time as a noncommutative $SO(2,3)_\star$ gauge theory. The NC gravity action consists of three different terms: the first term is of Mac-Dowell Mansouri type, while the other two are generalizations of the Einstein-Hilbert action and the cosmological constant term. The expanded NC gravity action is then calculated using the Seiberg-Witten (SW) map and the expansion is done up second order in the deformation parameter. We analyze in details the low energy sector of the full model. We calculate the equations of motion, discuss their general properties and present one solution: the NC correction to Minkowski space-time. Using this solution, we explain breaking of the diffeomorphism symmetry as a consequence of working in a particular coordinate system given by the Fermi normal coordinates.

Upper bound of the time-space non-commutative parameter from gravitational quantum well experiment

Starting from a field theoretic description of the gravitational well problem in a canonical non-commutative spacetime, we have studied the effect of time-space non-commutativity in the gravitational well scenario. The corresponding first quantized theory reveals a spectrum with leading order perturbation term of non-commutative origin. GRANIT experimental data are used to estimate an upper bound on the time-space non-commutative parameter.

MINIMAL LENGTH IN QUANTUM SPACE AND INTEGRATIONS OF THE LINE ELEMENT IN NONCOMMUTATIVE GEOMETRY

We question the emergence of a minimal length in quantum spacetime, comparing two notions that appeared at various points in the literature: on the one side, the quantum length as the spectrum of an operator L in the Doplicher Fredenhagen Roberts (DFR) quantum spacetime, as well as in the canonical noncommutative spacetime (θ-Minkowski); on the other side, Connes' spectral distance in noncommutative geometry. Although in the Euclidean space the two notions merge into the one of geodesic distance, they yield distinct results in the noncommutative framework. In particular, in the Moyal plane, the quantum length is bounded above from zero while the spectral distance can take any real positive value, including infinity. We show how to solve this discrepancy by doubling the spectral triple. This leads us to introduce a modified quantum length d′L, which coincides exactly with the spectral distance dDon the set of states of optimal localization. On the set of eigenstates of the quantum harmonic oscillator — together with their translations — d′Land dDcoincide asymptotically, both in the high energy and large translation limits. At small energy, we interpret the discrepancy between d′Land dDas two distinct ways of integrating the line element on a quantum space. This leads us to propose an equation for a geodesic on the Moyal plane.

Gauging the twisted Poincaré symmetry as a noncommutative theory of gravitation

Einstein's theory of general relativity was formulated as a gauge theory of Lorentz symmetry by Utiyama in 1956, while the Einstein-Cartan gravitational theory was formulated by Kibble in 1961 as the gauge theory of Poincare transformations. In this framework, we propose a formulation of the gravitational theory on canonical noncommutative space-time by covariantly gauging the twisted Poincare symmetry, in order to fulfil the requirement of covariance under the general coordinate transformations, an essential ingredient of the theory of general relativity. It appears that the twisted Poincare symmetry cannot be gauged by generalizing the Abelian twist to a covariant non-Abelian twist, nor by introducing a more general covariant twist element. The advantages of such a formulation as well as the related problems are discussed and possible ways out are outlined.

Θ-TWISTED GRAVITY

We describe a theory of gravitation on canonical noncommutative space–times. The construction is based on θ-twisted general coordinate transformations and local Lorentz invariance.

Noether analysis of the twisted Hopf symmetries of canonical noncommutative spacetimes

We study the twisted Hopf-algebra symmetries of observer-independent canonical spacetime noncommutativity, for which the commutators of the spacetime coordinates take the form [x{sup {mu}},x{sup {nu}}]=i{theta}{sup {mu}}{sup {nu}} with observer-independent (and coordinate-independent) {theta}{sup {mu}}{sup {nu}}. We find that it is necessary to introduce nontrivial commutators between transformation parameters and spacetime coordinates, and that the form of these commutators implies that all symmetry transformations must include a translation component. We show that with our noncommutative transformation parameters the Noether analysis of the symmetries is straightforward, and we compare our canonical-noncommutativity results with the structure of the conserved charges and the ''no-pure-boost'' requirement derived in a previous study of {kappa}-Minkowski noncommutativity. We also verify that, while at intermediate stages of the analysis we do find terms that depend on the ordering convention adopted in setting up the Weyl map, the final result for the conserved charges is reassuringly independent of the choice of Weyl map and (the corresponding choice of) star product.

κ-deformed spacetime from twist

We twist the Hopf algebra of igl(n,R) to obtain the κ-deformed spacetime coordinates. Coproducts of the twisted Hopf algebras are explicitly given. The κ-deformed spacetime obtained this way satisfies the same commutation relation as that of the conventional κ-Minkowski spacetime, but its Hopf algebra structure is different from the well-known κ-deformed Poincaré algebra in that it has larger symmetry algebra than the κ-Minkowski case. There are some physical models which consider this symmetry [R. Percacci, Phys. Lett. B 144 (1984) 37; R. Percacci, Geometry of Nonlinear Field Theories, World Scientific, Singapore, 1986; L. Smolin, Nucl. Phys. B 132 (1978) 138; R. Floreanini, R. Percacci, Class. Quantum Grav. 7 (1990) 975]. Incidentally, we obtain the canonical (θ-deformed) non-commutative spacetime from canonically twisted igl(n,R) Hopf algebra.

Charge conservation and the equivalence principle in noncommutative spacetime

We investigate one of the consequences of the twisted Poincare symmetry. We derive the charge conservation law and show that the equivalence principle is satisfied in the canonical noncommutative spacetime. We applied the twisted Poincare symmetry to Weinberg's analysis [S. Weinberg, Phys. Rev. 135, B1049 (1964).]. To this end, we generalize our earlier construction of the twisted S matrix [J. G. Bu, H.C. Kim, Y. Lee, C.H. Vac, and J.H. Yee, Phys. Rev. D 73, 125001 (2006).], which applies the noncommutativity to the Fourier modes, to the massless fields of integer spins. The transformation formula for the twisted S matrix for the massless fields of integer spin has been obtained. For massless fields of spin 1, we obtain the conservation of charge, and the universality of the coupling constant for massless fields of spin 2, which can be interpreted as the equality of gravitational mass and inertial mass, i.e., the equivalence principle.

Quantum electrodynamics on noncommutative spacetime

We propose a new method to quantize gauge theories formulated on a canonical noncommutative spacetime with fields and gauge transformations taken in the enveloping algebra. We show that the theory is renormalizable at one loop and compute the beta function and show that the spin dependent contribution to the anomalous magnetic moment of the fermion at one loop has the same value as in the commutative quantum electrodynamics case.

Second order noncommutative corrections to gravity

In this work, we calculate the leading order corrections to general relativity formulated on a canonical noncommutative spacetime. These corrections appear in the second order of the expansion in theta. First order corrections can only appear in the gravity-matter interactions. Some implications are briefly discussed.

Quantum mechanics on noncommutative spacetime

We consider electrodynamics on a noncommutative spacetime using the enveloping algebra approach and perform a nonrelativistic expansion of the effective action. We obtain the Hamiltonian for quantum mechanics formulated on a canonical noncommutative spacetime. An interesting new feature of quantum mechanics formulated on a noncommutative spacetime is an intrinsic electric dipole moment. We note, however, that noncommutative intrinsic dipole moments are not observable in present experiments searching for an electric dipole moment of leptons or nuclei such as the neutron since they are spin independent. These experiments are sensitive to the energy difference between two states and the noncommutative effect thus cancels out. Bounds on the noncommutative scale found in the literature relying on such intrinsic electric dipole moments are thus incorrect.

Noncommutative general relativity

We define a theory of noncommutative general relativity for canonical noncommutative spaces. We find a subclass of general coordinate transformations acting on canonical noncommutative spacetimes to be volume-preserving transformations. Local Lorentz invariance is treated as a gauge theory with the spin connection field taken in the so(3,1) enveloping algebra. The resulting theory appears to be a noncommutative extension of the unimodular theory of gravitation. We compute the leading order noncommutative correction to the action and derive the noncommutative correction to the equations of motion of the weak gravitation field.

On duality of the noncommutative extension of the Maxwell–Chern–Simons model

We study issues of duality in 3D field theory models over a canonical noncommutative spacetime and obtain the noncommutative extension of the self-dual model induced by the Seiberg–Witten map. We apply the dual projection technique to uncover some properties of the noncommutative Maxwell–Chern–Simons theory up to first-order in the noncommutative parameter. A duality between this theory and a model similar to the ordinary self-dual model is established. The correspondence of the basic fields is obtained and the equivalence of algebras and equations of motion are directly verified. We also comment on previous results in this subject.

CORNWALL–JACKIW–TOMBOULIS EFFECTIVE POTENTIAL FOR CANONICAL NONCOMMUTATIVE FIELD THEORIES

We apply the Cornwall–Jackiw–Tomboulis (CJT) formalism to the scalar λφ4 theory in canonical-noncommutative space–time. We construct the CJT effective potential and the gap equation for general values of the noncommutative parameter θμν. We observe that under the hypothesis of translational invariance, which is assumed in the effective potential construction, differently from the commutative case (θμν=0), the renormalizability of the gap equation is incompatible with the renormalizability of the effective potential. We argue that our result, is consistent with previous studies, suggesting that a uniform ordered phase would be inconsistent with the infrared structure of canonical noncommutative theories.

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

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.

Phenomenology of particle production and propagation in string-motivated canonical noncommutative spacetime

We outline a phenomenological programme for the search of effects induced by (string-motivated) canonical noncommutative spacetime. The tests we propose are based, in analogy with a corresponding programme developed over the last few years for the study of Lie-algebra noncommutative spacetimes, on the role of the noncommutativity parameters in the $E(p)$ dispersion relation. We focus on the role of deformed dispersion relations in particle-production collision processes, where the noncommutativity parameters would affect the threshold equation, and in the dispersion of gamma rays observed from distant astrophysical sources. We emphasize that the studies here proposed have the advantage of involving particles of relatively high energies, and may therefore be less sensitive to "contamination" (through IR/UV mixing) from the UV sector of the theory. We also explore the possibility that the relevant deformation of the dispersion relations could be responsible for the experimentally-observed violations of the GZK cutoff for cosmic rays and could have a role in the observation of hard photons from distant astrophysical sources.

Coproduct and star product in field theories on Lie-algebra noncommutative space-times

We propose a new approach to field theory on \ensuremath{\kappa}-Minkowski noncommutative space-time, a popular example of Lie-algebra space-time. Our proposal is essentially based on the introduction of a star product, a technique which is proving to be very fruitful in analogous studies of canonical noncommutative space-times, such as the ones recently found to play a role in the description of certain string-theory backgrounds. We find to be incorrect the expectation, previously reported in the literature, that the lack of symmetry of the \ensuremath{\kappa}-Poincar\'e coproduct should lead to interaction vertices that are not symmetric under exchanges of the momenta of identical particles entering the relevant processes. We show that in \ensuremath{\kappa}-Minkowski the coproduct and the star product must indeed treat momenta in a nonsymmetric way, but the overall structure of interaction vertices is symmetric under exchange of identical particles. We also show that in \ensuremath{\kappa}-Minkowski field theories it is convenient to introduce the concepts of ``planar'' and ``nonplanar'' Feynman loop diagrams, again in close analogy with the corresponding concepts previously introduced in the study of field theories in canonical noncommutative space-times.