Noncommutative Field Research Articles

Noncommutative scalar quasinormal modes of the Reissner–Nordström black hole

Aiming to search for a signal of space-time noncommutativity, we study a quasinormal mode spectrum of the Reissner–Nordström black hole in the presence of a deformed space-time structure. In this context we study a noncommutative (NC) deformation of a scalar field, minimally coupled to a classical (commutative) Reissner–Nordström background. The deformation is performed via a particularly chosen Killing twist to ensure that the geometry remains undeformed (commutative). An action describing a noncommutative scalar field minimally coupled to the RN geometry is manifestly invariant under the deformed gauge symmetry group. We find the quasinormal mode solutions of the equations of motion governing the matter content of the model in some particular range of system parameters which corresponds to a near extremal limit. In addition, we obtain a well defined analytical condition which allows for a detailed numerical analysis. Moreover, there exists a parameter range, rather restrictive though, which allows for obtaining a QNMs spectrum in a closed analytic form. We also argue within a semiclassical approach that NC deformation does not affect the Hawking temperature of thermal radiation.

Noncommutative 3-colour scalar quantum field theory model in 2D

We introduce the 3-colour noncommutative quantum field theory model in two dimensions. For this model we prove a generalised Ward–Takahashi identity, which is special to coloured noncommutative QFT models. It reduces to the usual Ward–Takahashi identity in a particular case. The Ward–Takahashi identity is used to simplify the Schwinger–Dyson equations for the 2-point function and the N-point function. The absence of any renormalisation conditions in the large (mathcal {N},V)-limit in 2D leads to a recursive integral equation for the 2-point function, which we solve perturbatively to sixth order in the coupling constant.

The spectrum of 2+1 dimensional Yang-Mills theory on a twisted spatial torus

We compute and analyse the low-lying spectrum of 2+1 dimensional SU(N) Yang-Mills theory on a spatial torus of size l × l with twisted boundary conditions. This paper extends our previous work [1]. In that paper we studied the sector with non-vanishing electric flux and concluded that the energies only depend on the parameters through two combinations: x = λN l/(4π) (with λ the ’t Hooft coupling) and the twist angle tilde{theta} defined in terms of the magnetic flux piercing the two-dimensional box. Here we made a more complete study and we are able to condense our results, obtained by non-perturbative lattice methods, into a simple expression which has important implications for the absence of tachyonic instabilities, volume independence and non-commutative field theory. Then we extend our study to the sector of vanishing electric flux. We conclude that the onset of the would-be large-volume glueball states occurs at an approximately fixed value of x, much before the stringy torelon states have become very massive.

Hall viscosity and geometric response in the Chern-Simons matrix model of the Laughlin states

We study geometric aspects of the Laughlin fractional quantum Hall (FQH) states using a description of these states in terms of a matrix quantum mechanics model known as the Chern-Simons matrix model (CSMM). This model was proposed by Polychronakos as a regularization of the noncommutative Chern-Simons theory description of the Laughlin states proposed earlier by Susskind. Both models can be understood as describing the electrons in a FQH state as forming a noncommutative fluid, i.e., a fluid occupying a noncommutative space. Here we revisit the CSMM in light of recent work on geometric response in the FQH effect, with the goal of determining whether the CSMM captures this aspect of the physics of the Laughlin states. For this model we compute the Hall viscosity, Hall conductance in a non-uniform electric field, and the Hall viscosity in the presence of anisotropy (or intrinsic geometry). Our calculations show that the CSMM captures the guiding center contribution to the known values of these quantities in the Laughlin states, but lacks the Landau orbit contribution. The interesting correlations in a Laughlin state are contained entirely in the guiding center part of the state/wave function, and so we conclude that the CSMM accurately describes the most important aspects of the physics of the Laughlin FQH states, including the Hall viscosity and other geometric properties of these states which are of current interest.

Quantum noncommutative ABJM theory: first steps

We introduce ABJM quantum field theory in the noncommutative spacetime by using the component formalism and show that it is mathcal{N} = 6 supersymmetric. For the U(1)κ × U(1)−κ case, we compute all one-loop 1PI two and three point functions in the Landau gauge and show that they are UV finite and have well-defined commutative limits θμν → 0, corresponding exactly to the 1PI functions of the ordinary ABJM field theory. This result also holds for all one-loop functions which are UV finite by power counting. It seems that the noncommutative quantum ABJM field theory is free from the noncommutative IR instabilities.

Holographic complexity and noncommutative gauge theory

We study the holographic complexity of noncommutative field theories. The four-dimensional mathcal{N}=4 noncommutative super Yang-Mills theory with Moyal algebra along two of the spatial directions has a well known holographic dual as a type IIB supergravity theory with a stack of D3 branes and non-trivial NS-NS B fields. We start from this example and find that the late time holographic complexity growth rate, based on the “complexity equals action” conjecture, experiences an enhancement when the non-commutativity is turned on. This enhancement saturates a new limit which is exactly 1/4 larger than the commutative value. We then attempt to give a quantum mechanics explanation of the enhancement. Finite time behavior of the complexity growth rate is also studied. Inspired by the non-trivial result, we move on to more general setup in string theory where we have a stack of Dp branes and also turn on the B field. Multiple noncommutative directions are considered in higher p cases.

Skew and linearized Reed–Solomon codes and maximum sum rank distance codes over any division ring

Reed–Solomon codes and Gabidulin codes have maximum Hamming distance and maximum rank distance, respectively. A general construction using skew polynomials, called skew Reed–Solomon codes, has already been introduced in the literature. In this work, we introduce a linearized version of such codes, called linearized Reed–Solomon codes. We prove that they have maximum sum-rank distance. Such distance is of interest in multishot network coding or in singleshot multi-network coding. To prove our result, we introduce new metrics defined by skew polynomials, which we call skew metrics, we prove that skew Reed–Solomon codes have maximum skew distance, and then we translate this scenario to linearized Reed–Solomon codes and the sum-rank metric. The theories of Reed–Solomon codes and Gabidulin codes are particular cases of our theory, and the sum-rank metric extends both the Hamming and rank metrics. We develop our theory over any division ring (commutative or non-commutative field). We also consider non-zero derivations, which give new maximum rank distance codes over infinite fields not considered before.

Quantum spaces, central extensions of Lie groups and related quantum field theories11Based on a talk presented by Poulain T at the XXVth International Conference on Integrable Systems and Quantum symmetries (ISQS-25), Prague, June 6-10 2017.

Quantum spaces with su(2) noncommutativity can be modelled by using a family of SO(3)-equivariant differential *-representations. The quantization maps are determined from the combination of the Wigner theorem for SU(2) with the polar decomposition of the quantized plane waves. A tracial star-product, equivalent to the Kontsevich product for the Poisson manifold dual to su(2) is obtained from a subfamily of differential *-representations. Noncommutative (scalar) field theories free from UV/IR mixing and whose commutative limit coincides with the usual ϕ4 theory on ℛ3 are presented. A generalization of the construction to semi-simple possibly non simply connected Lie groups based on their central extensions by suitable abelian Lie groups is discussed.

Galilei group with multiple central extension, vorticity, and entropy generation: Exotic fluid in 3+1 dimensions

A noncommutative extension of an ideal (Hamiltonian) fluid model in $3+1$-dimensions is proposed. The model enjoys several interesting features: it allows a multi-parameter central extension in Galilean boost algebra (which is significant being contrary to existing belief that similar feature can appear only in $2+1$-dim.); noncommutativity generates vorticity in a canonically irrotational fluid; it induces a non-barotropic pressure leading to a non-isentropic system. (Barotropic fluids are entropy preserving as pressure depends only on matter density.) Our fluid model is termed "Exotic" since it has close resemblance with the extensively studied planar (2+1-dim.) Exotic models and Exotic (noncommutative) field theories.

Dark solitons, D-branes and noncommutative tachyon field theory

In this paper we discuss the boson/vortex duality by mapping the (3[Formula: see text]+[Formula: see text]1)D Gross–Pitaevskii theory into an effective string theory in the presence of boundaries. Via the effective string theory, we find the Seiberg–Witten map between the commutative and the noncommutative tachyon field theories, and consequently identify their soliton solutions with D-branes in the effective string theory. We perform various checks of the duality map and the identification of soliton solutions. This new insight between the Gross–Pitaevskii theory and the effective string theory explains the similarity of these two systems at quantitative level.

Exact solution of matricial [formula omitted] quantum field theory

We apply a recently developed method to exactly solve the Φ3 matrix model with covariance of a two-dimensional theory, also known as regularised Kontsevich model. Its correlation functions collectively describe graphs on a multi-punctured 2-sphere. We show how Ward–Takahashi identities and Schwinger–Dyson equations lead in a special large-N limit to integral equations that we solve exactly for all correlation functions.The solved model arises from noncommutative field theory in a special limit of strong deformation parameter. The limit defines ordinary 2D Schwinger functions which, however, do not satisfy reflection positivity.

Stable solutions of symmetric systems involving hypoelliptic operators

Let X and Y be two noncommuting vector fields in an open set Ω in a manifold M equipped with a sub-Riemannian structure. We examine stable solutions of the following symmetric systemΔXYui=Hi(u1,⋯,um)in Ω for 1≤i≤m, when the operator ΔXY is the Hörmander's operator given by ΔXY(⋅):=X(X⋅)+Y(Y⋅) and Hi∈C1(Rm). We prove the following identity for any w∈C2(Ω)|∇XYXw|2+|∇XYYw|2−|X|∇XYw||2−|Y|∇XYw||2={|∇XYw|2[A2+B2]in {|∇XYw|>0}∩Ω,0a.e.in {|∇XYw|=0}∩Ω, where A is the intrinsic curvature of the level sets of w and B is connected with the intrinsic normal and the intrinsic tangent direction to the level sets and also with the Lie bracket [X,Y]. We then apply this to establish a geometric Poincaré inequality for stable solutions of the above system for general vector fields X and Y. This inequality enables us to analyze the level sets of stable solutions. In addition, we provide certain reduction of dimensions results which can be regarded as counterparts of the classical De Giorgi type results. This is remarkable since the classical one-dimensional symmetry results do not hold for general vector fields. Our approaches can be applied, but not limited, to the Grushin vector fields X=(1,0) and Y=(0,x) in R2 and the Heisenberg vector fields X=(1,0,−y2) and Y=(0,1,x2) in R3 and their multidimensional extensions. These specific vector fields generate nonelliptic operators which are hypoelliptic.

Noncommutative scalar fields in compact spaces: quantization and implications

In this paper we consider a two component scalar field theory, with noncommutativity in its conjugate momentum space. We quantize such a theory in a compact space with the help of dressing transformations and we reveal a significant effect of introducing such noncommutativity as the splitting of the energy levels of each individual mode that constitutes the whole system. We further compute the thermal partition function exactly with predicted deformed dispersion relations from noncommutative theories and compare the results with usual results. It is found that thermodynamic quantities in noncommutative models, irrespective of whether the model is more deformed in infrared/UV region, show deviation from standard results in high temperature region.

Involutive representations of coordinate algebras and quantum spaces

We show that mathfrak{s}mathfrak{u}(2) Lie algebras of coordinate operators related to quantum spaces with mathfrak{s}mathfrak{u}(2) noncommutativity can be conveniently represented by SO(3)-covariant poly-differential involutive representations. We show that the quantized plane waves ob-tained from the quantization map action on the usual exponential functions are determined by polar decomposition of operators combined with constraint stemming from the Wigner theorem for SU(2). Selecting a subfamily of ∗-representations, we show that the resulting star-product is equivalent to the Kontsevich product for the Poisson manifold dual to the finite dimensional Lie algebra mathfrak{s}mathfrak{u}(2) . We discuss the results, indicating a way to extend the construction to any semi-simple non simply connected Lie group and present noncommutative scalar field theories which are free from perturbative UV/IR mixing.

Non-commutative fields and the short-scale structure of spacetime

There is a growing evidence that due to quantum gravity effects the effective spacetime dimensionality might change in the UV. In this letter we investigate this hypothesis by using quantum fields to derive the UV behaviour of the static, two point sources potential. We mimic quantum gravity effects by using non-commutative fields associated to a Lie group momentum space with a Planck mass curvature scale. We find that the static potential becomes finite in the short-distance limit. This indicates that quantum gravity effects lead to a dimensional reduction in the UV or, alternatively, that point-like sources are effectively smoothed out by the Planck scale features of the non-commutative quantum fields.

A note on graviton exchange in the emergent gravity scenario

It is well-known that there exists a close relation between a large $N$ matrix model and noncommutative (NC) field theory: The latter can be naturally obtained from the former by expanding it around a specific background. Because the matrix model can be a constructive formulation of string theory, this relation suggests that NC field theory can also include quantum gravity. In particular, the NC $U(1)$ gauge theory attracts much attention because its low-energy effective action partially contains gravity where the metric is determined by the $U(1)$ gauge field. Thus, the NC $U(1)$ gauge theory could be a quantum theory of gravity. In this paper, we investigate the scenario by calculating the scattering amplitude of massless test particles, and find that the NC $U(1)$ gauge theory correctly reproduces the amplitude of the usual graviton exchange if the noncommutativity that corresponds to the background of the matrix model is appropriately averaged. Although this result partially supports the relation between the NC $U(1)$ gauge theory and gravity, it is desirable to find a mechanism by which such an average is naturally realized.

Discussion on Lorentz invariance violation of noncommutative field theory and neutrino oscillation

Depending on deformed canonical anticommutation relations, massless neutrino oscillation based on Lorentz invariance violation in noncommutative field theory is discussed. It is found that the previous studies about massless neutrino oscillation within deformed canonical anticommutation relations should satisfy the condition of new Moyal product and new nonstandard commutation relations. Furthermore, comparing the Lorentz invariant violation parameters A in the previous studies with new Moyal product and new nonstandard commutation relations, we find that the orders of magnitude of noncommutative parameters (Lorentz invariant violation parameters A) is not self-consistent. This inconsistency means that the previous studies of Lorentz invariance violation in noncommutative field theory may not naturally explain massless neutrino oscillation. In other words, it should be impossible to explain neutrino oscillation by Lorentz invariance violation in noncommutative field theory. This conclusion is supported by the latest atmospheric neutrinos experimental results from the super-Kamiokande Collaboration, which show that no evidence of Lorentz invariance violation on atmospheric neutrinos was observed.

Electric dipole moment induced by CP-violating deformations in the noncommutative Standard Model

The possibility to detect the noncommutative (NC) spacetime in the electric dipole moments (EDM) experiments is studied in the effective field theory of noncommutative Standard Model (NCSM) with many additional deformations. The EDM given by the previous literatures do not have any observable effect since they are spin-independent. In this work, it is found that three of the deformed terms provide extra sources of CP violation contributed to EDM. We show that these EDMs are sensitive to the spin and thus have potential to be measured in the highly precise experiments. In particular, the EDM induced by NC spacetime may not be parallel to the direction of spin, which demonstrates the intrinsic feature of NC field theory.

Three-dimensional noncommutative Yukawa theory: Induced effective action and propagating modes

In this paper, we establish the analysis of noncommutative Yukawa theory, encompassing neutral and charged scalar fields. We approach the analysis by considering carefully the derivation of the respective effective actions. Hence, based on the obtained results, we compute the one-loop contributions to the neutral and charged scalar field self-energy, as well as to the Chern–Simons polarization tensor. In order to properly define the behavior of the quantum fields, the known UV/IR mixing due to radiative corrections is analyzed in the one-loop physical dispersion relation of the scalar and gauge fields.

Matrix models from localization of five-dimensional supersymmetric noncommutative U(1) gauge theory

We study localization of five-dimensional supersymmetric U(1) gauge theory on {mathbb{S}}^3times {mathbb{R}}_{theta}^2 where {mathbb{R}}_{theta}^2 is a noncommutative (NC) plane. The theory can be isomorphically mapped to three-dimensional supersymmetric U(N → ∞) gauge theory on {mathbb{S}}^3 using the matrix representation on a separable Hilbert space on which NC fields linearly act. Therefore the NC space {mathbb{R}}_{theta}^2 allows for a flexible path to derive matrix models via localization from a higher-dimensional supersymmetric NC U(1) gauge theory. The result shows a rich duality between NC U(1) gauge theories and large N matrix models in various dimensions.