Noncommutative Field Research Articles

Noncommutative field with constant background fields and neutral fermions

Introducing constant background fields into the noncommutative gauge theory, we first obtain a Hermitian fermion Lagrangian which involves a Lorentz violation term, then we generalize it to a new deformed canonical noncommutation relations for fermion field. Massless neutrino oscillation in the deformed canonical noncommutation relations is analyzed. The restriction of the noncommutative coefficients is also discussed. By comparing with the existing experimental data of conventional neutrino oscillations, the order of noncommutative deformed coefficients is given from different ways.

Approximate analytical solutions of a pair of coupled anharmonic oscillators

The Hamiltonian and the corresponding equations of motion involving the field operators of two quartic anharmonic oscillators indirectly coupled via a linear oscillator are constructed. The approximate analytical solutions of the coupled differential equations involving the non-commuting field operators are solved up to the second order in the anharmonic coupling. In the absence of nonlinearity these solutions are used to calculate the second order variances and hence the squeezing in pure and in mixed modes. The higher order quadrature squeezing and the amplitude squared squeezing of various field modes are also investigated where the squeezing in pure and in mixed modes are found to be suppressed. Moreover, the absence of a nonlinearity prohibits the higher order quadrature and higher ordered amplitude squeezing of the input coherent states. It is established that the mere coupling of two oscillators through a third one is unable to produce any squeezing effects of input coherent light, but the presence of a nonlinear interaction may provide squeezed states and other nonclassical phenomena.

SO(2, 3) noncommutative gravity model

In this paper the noncommutative gravity is treated as a gauge theory of the non-commutative SO(2, 3)★ group, while the noncommutativity is canonical. The Seiberg-Witten (SW) map is used to express noncommutative fields in terms of the corresponding commutative fields. The commutative limit of the model is the Einstein-Hilbert action plus the cosmological term and the topological Gauss-Bonnet term. We calculate the second order correction to this model and obtain terms that are zeroth, first, ... and fourth power of the curvature tensor. Finally, we discuss physical consequences of those correction terms in the limit of big cosmological constant.

Pair production of Dirac particles in a $$d+1$$ d + 1 -dimensional noncommutative space–time

This work addresses the computation of the probability of fermionic particle pair production in \((d+1)\)-dimensional noncommutative Moyal space. Using Seiberg–Witten maps, which establish relations between noncommutative and commutative field variables, up to the first order in the noncommutative parameter \(\theta \), we derive the probability density of vacuum–vacuum pair production of Dirac particles. The cases of constant electromagnetic, alternating time-dependent, and space-dependent electric fields are considered and discussed.

Highly effective action from largeNgauge fields

Recently John H. Schwarz put forward a conjecture that the world-volume action of a probe D3-brane in an AdS5 x S5 background of type IIB superstring theory can be reinterpreted as the highly effective action (HEA) of four-dimensional N=4 superconformal field theory on the Coulomb branch. We argue that the HEA can be derived from the noncommutative (NC) field theory representation of the AdS/CFT correspondence and the Seiberg-Witten (SW) map defining a spacetime field redefinition between ordinary and NC gauge fields. It is based only on the well-known facts that the master fields of large N matrices are higher-dimensional NC U(1) gauge fields and the SW map is a local coordinate transformation eliminating U(1) gauge fields known as the Darboux theorem in symplectic geometry.

Volume independence for Yang–Mills fields on the twisted torus

We review some recent results related to the notion of volume independence in SU (N) Yang–Mills theories. The topic is discussed in the context of gauge theories living on a d-dimensional torus with twisted boundary conditions. After a brief introduction reviewing the formalism for introducing gauge fields on a torus, we discuss how volume independence arises in perturbation theory. We show how, for appropriately chosen twist tensors, perturbative results to all orders in the 't Hooft coupling depend on a specific combination of the rank of the gauge group (N) and the periods of the torus (l), given by lN2/d, for d even. We discuss the well-known relation to noncommutative field theories and address certain threats to volume independence associated to the occurrence of tachyonic instabilities at one-loop order. We end by presenting some numerical results in 2+1 dimensions that extend these ideas to the nonperturbative domain.

Preface

Fortschritte der PhysikVolume 62, Issue 9-10 p. 706-708 PrefaceFree Access Preface Konstantinos Anagnostopoulos, Konstantinos AnagnostopoulosSearch for more papers by this authorOlaf Lechtenfeld, Olaf LechtenfeldSearch for more papers by this authorDieter Lüst, Dieter LüstSearch for more papers by this authorHikaru Kawai, Hikaru KawaiSearch for more papers by this authorJun Nishimura, Jun NishimuraSearch for more papers by this authorHarold Steinacker, Harold SteinackerSearch for more papers by this authorRichard Szabo, Richard SzaboSearch for more papers by this authorSatoshi Watamura, Satoshi WatamuraSearch for more papers by this authorGeorge Zoupanos, George ZoupanosSearch for more papers by this author Konstantinos Anagnostopoulos, Konstantinos AnagnostopoulosSearch for more papers by this authorOlaf Lechtenfeld, Olaf LechtenfeldSearch for more papers by this authorDieter Lüst, Dieter LüstSearch for more papers by this authorHikaru Kawai, Hikaru KawaiSearch for more papers by this authorJun Nishimura, Jun NishimuraSearch for more papers by this authorHarold Steinacker, Harold SteinackerSearch for more papers by this authorRichard Szabo, Richard SzaboSearch for more papers by this authorSatoshi Watamura, Satoshi WatamuraSearch for more papers by this authorGeorge Zoupanos, George ZoupanosSearch for more papers by this author First published: 12 September 2014 https://doi.org/10.1002/prop.201406210AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Volume62, Issue9-10Special Issue: Proceedings of the “Workshop on Noncommutative Field Theory and Gravity” (CORFU2013), 08–15 September 2013, Corfu (Greece)September 2014Pages 706-708 RelatedInformation

Universal κ-Poincaré covariant differential calculus over κ-Minkowski space

Unified graded differential algebra, generated by κ-Minkowski noncommutative (NC) coordinates, Lorentz generators and anticommuting one-forms, is constructed. It is compatible with κ-Poincaré–Hopf algebra. For time- and space-like deformations, the super-Jacobi identities are not satisfied. By introducing additional generator, interpreted as exterior derivative, we find a new unique algebra that satisfies all super-Jacobi identities. It is universal and valid for all type of deformations (time-, space-, and light-like). For time-like deformations this algebra coincides with the one in A. Sitarz, Phys. Lett. B349, 42 (1995), arXiv:hep-th/9409014. Different realizations of our algebra in terms of super-Heisenberg algebra are presented. For light-like deformations we get (4D) bicovariant calculus, with κ-Poincaré–Hopf algebra and present the corresponding twist, which is written in a new covariant way, using Poincaré generators only. In the time- and space-like case, this twist leads to κ-Snyder space. Our results might lead to applications in NC quantum field theories (especially electrodynamics and gauge theories), quantum gravity models and Planck scale physics.

NoncommutativeSO(2,3)gauge theory and noncommutative gravity

In this paper noncommutative gravity is constructed as a gauge theory of the noncommutative SO(2,3) group, while the noncommutativity is canonical (constant). The Seiberg-Witten map is used to express noncommutative fields in terms of the corresponding commutative fields. The commutative limit of the model is the Einstein-Hilbert action with the cosmological constant term and the topological Gauss-Bonnet term. We calculate the second order correction to this model and obtain terms that are of zeroth to fourth power in the curvature tensor and torsion. Trying to relate our results with $f(R)$ and $f(T)$ models, we analyze different limits of our model. In the limit of big cosmological constant and vanishing torsion we obtain a $x$-dependent correction to the cosmological constant, i.e. noncommutativity leads to a $x$-dependent cosmological constant. We also discuss the limit of small cosmological constant and vanishing torsion and the teleparallel limit.

α∗-cohomology, and classification of translation-invariant non-commutative quantum field theories

Translation-invariant ⋆ products are studied in the setting of α ∗ -cohomology. It is explicitly shown that all quantum behaviors including Green’s functions and the scattering matrix of translation-invariant non-commutative quantum field theories are thoroughly characterized by α ∗ -cohomology classes of the star products.

Symmetry breaking in noncommutative finite temperature λϕ4 theory with a nonuniform ground state

We consider the CJT effective action at finite temperature for a noncommutative real scalar field theory, with noncommutativity among space and time variables. We study the solutions of a stripe type nonuniform background, which depends on space and time. The analysis in the first approximation shows that such solutions appear in the planar limit, but also under normal anisotropic noncommutativity. Further we show that the transition from the uniform ordered phase to the non uniform one is first order and that the critical temperature depends on the nonuniformity of the ground state.

Gauge fields on non‐commutative spaces and renormalization

AbstractConstructing renormalizable models on non‐commutative spaces constitutes a big challenge. Only few examples of renormalizable theories are known, such as the scalar Grosse‐Wulkenhaar model. Gauge fields are even more difficult, since new renormalization techniques are required which are compatible with the inherently non‐local setting on the one hand, and also allow to properly treat the gauge symmetry on the other hand. In this proceeding (which is based on my talk given at the “Workshop on Noncommutative Field Theory and Gravity” in Corfu/Greece, September 8–15, 2013), I focus on this last point and present new extensions to existing renormalization schemes (which are known to work for gauge field theories in commutative space) adapted to non‐commutative Moyal space.

Quantum Field Theory with a Minimal Length Induced from Noncommutative Space

From the inspection of noncommutative quantum mechanics, we obtain an approximate equivalent relation for the energy dependence of the Planck constant in the noncommutative space, which means a minimal length of the space. We find that this relation is reasonable and it can inherit the main properties of the noncommutative space. Based on this relation, we derive the modified Klein—Gordon equation and Dirac equation. We investigate the scalar field and ϕ4 model and then quantum electrodynamics in our theory, and derive the corresponding Feynman rules. These results may be considered as reasonable approximations to those of noncommutative quantum field theory. Our theory also shows a connection between the space with a minimal length and the noncommutative space.

Dirac equation on coordinate dependent noncommutative space–time

In this paper we discuss classical aspects of spinor field theory on the coordinate dependent noncommutative space–time. The noncommutative Dirac equation describing spinning particle in an external vector field and the corresponding action principle are proposed. The specific choice of a star product allows us to derive a conserved noncommutative probability current and to obtain the energy–momentum tensor for free noncommutative spinor field. Finally, we consider a free noncommutative Dirac fermion and show that if the Poisson structure is Lorentz-covariant, the standard energy–momentum dispersion relation remains valid.

Källén-Lehmann representation of noncommutative quantum electrodynamics

Noncommutative (NC) quantum field theory is the subject of many analyses on formal and general aspects looking for deviations and, therefore, potential noncommutative spacetime effects. Within of this large class, we may now pay some attention to the quantization of NC field theory on lower dimensions and look closely at the issue of dynamical mass generation to the gauge field. This work encompasses the quantization of the two-dimensional massive quantum electrodynamics and three-dimensional topologically massive quantum electrodynamics. We begin by addressing the problem on a general dimensionality making use of the perturbative Seiberg-Witten map to, thus, construct a general action, to only then specify the problem to two and three dimensions. The quantization takes place through the K\"all\'en-Lehmann spectral representation and Yang-Feldman-K\"all\'en formulation, where we calculate the respective spectral density function to the gauge field. Furthermore, regarding the photon two-point function, we discuss how its infrared behavior is related to the term generated by quantum corrections in two dimensions, and, moreover, in three dimensions, we study the issue of nontrivial {\theta}-dependent corrections to the dynamical mass generation.

Towards Noncommutative Topological Quantum Field Theory – Hodge theory for cyclic cohomology

Some years ago we initiated a program to define Noncommutative Topological Quantum Field Theory (see [1]). The motivation came both from physics and mathematics: On the one hand, as far as physics is concerned, following the well-known holography principle of 't Hooft (which in turn appears essentially as a generalisation of the Hawking formula for black hole entropy), quantum gravity should be a topological quantum field theory. On the other hand as far as mathematics is concerned, the motivation came from the idea to replace the moduli space of flat connections with the Gabai moduli space of codim-1 taut foliations for 3 dim manifolds. In most cases the later is finite and much better behaved and one might use it to define some version of Donaldson-Floer homology which, hopefully, would be easier to compute. The use of foliations brings noncommutative geometry techniques immediately into the game. The basic tools are two: Cyclic cohomology of the corresponding foliation C*-algebra and the so called "tangential cohomology" of the foliation. A necessary step towards this goal is to develop some sort of Hodge theory both for cyclic (and Hochschild) cohomology and for tangential cohomology. Here we present a method to develop a Hodge theory for cyclic and Hochschild cohomology for the corresponding C*-algebra of a foliation.

Towards Noncommutative Topological Quantum Field Theory: Tangential Hodge-Witten cohomology

Some years ago we initiated a program to define Noncommutative Topological Quantum Field Theory (see [1]). The motivation came both from physics and mathematics: On the one hand, as far as physics is concerned, following the well-known holography principle of 't Hooft (which in turn appears essentially as a generalisation of the Hawking formula for black hole entropy), quantum gravity should be a topological quantum field theory. On the other hand as far as mathematics is concerned, the motivation came from the idea to replace the moduli space of flat connections with the Gabai moduli space of codim-1 taut foliations for 3 dim manifolds. In most cases the later is finite and much better behaved and one might use it to define some version of Donaldson-Floer homology which, hopefully, would be easier to compute. The use of foliations brings noncommutative geometry techniques immediately into the game. The basic tools are two: Cyclic cohomology of the corresponding foliation C*-algebra and the so called "tangential cohomology" of the foliation. A necessary step towards this goal is to develop some sort of Hodge theory both for cyclic (and Hochschild) cohomology and for tangential cohomology. Here we present a method to develop a Hodge theory for tangential cohomology of foliations by mimicing Witten's approach to ordinary Morse theory by perturbations of the Laplacian.

Forbidden and invisible Z boson decays in a covariant θ-exact noncommutative standard model

The triple neutral gauge boson and direct U(1)Y-neutrino interactions, being absent in ordinary field theory, can arise quite naturally in noncommutative gauge field theories. Using non-perturbative methods and a Seiberg–Witten map-based covariant approach to noncommutative gauge theory, we have found θ-exact expressions for both interactions, thereby eliminating previous restrictions to low-energy phenomena. In particular, we obtain for the first time the covariant, θ-exact, triple neutral gauge boson interactions within the noncommutative standard model gauge sector including an additional gauge-field deformation freedom. Finally, we discuss implications for Z → γγ and decays, and show that our results behave quite reasonably throughout all interaction energy scales.

The Ponzano–Regge Model and Parametric Representation

We give a parametric representation of the effective noncommutative field theory derived from a $\kappa$-deformation of the Ponzano-Regge model and define a generalized Kirchhoff polynomial with $\kappa$-correction terms, obtained in a $\kappa$-linear approximation. We then consider the corresponding graph hypersurfaces and the question of how the presence of the correction term affects their motivic nature. We look in particular at the tetrahedron graph, which is the basic case of relevance to quantum gravity. With the help of computer calculations, we verify that the number of points over finite fields of the corresponding hypersurface does not fit polynomials with integer coefficients, hence the hypersurface of the tetrahedron is not polynomially countable. This shows that the correction term can change significantly the motivic properties of the hypersurfaces, with respect to the classical case.

Effects of Noncommutativity on the Black Hole Entropy

The BTZ black hole geometry is probed with a noncommutative scalar field which obeys theκ-Minkowski algebra. The entropy of the BTZ black hole is calculated using the brick wall method. The contribution of the noncommutativity to the black hole entropy is explicitly evaluated up to the first order in the deformation parameter. We also argue that such a correction to the black hole entropy can be interpreted as arising from the renormalization of the Newton’s constant due to the effects of the noncommutativity.