Noncommutative Scalar Field Research Articles

Noncommutative scalar field theory in a curved background: Duality between noncommutative and effective commutative description

We study a noncommutative (NC) deformation of a charged scalar field, minimally coupled to a classical (commutative) Reissner–Nordström-like background. The deformation is performed via a particularly chosen Killing twist to ensure that the geometry remains undeformed (commutative). An action describing a NC scalar field minimally coupled to the RN geometry is manifestly invariant under the deformed [Formula: see text] gauge symmetry. We find the equation of motion and conclude that the same equation is obtained from the commutative theory in a modified geometrical background described by an effective metric. This correspondence we call “duality between formal and effective approach”. We also show that a NC deformation via semi-Killing twist operator cannot be rewritten in terms of an effective metric. There is dual description for those particular deformations.

Homotopy Double Copy of Noncommutative Gauge Theories

We discuss the double-copy formulation of Moyal–Weyl-type noncommutative gauge theories from the homotopy algebraic perspective of factorisations of L∞-algebras. We define new noncommutative scalar field theories with rigid colour symmetries taking the role of the zeroth copy, where the deformed colour algebra plays the role of a kinematic algebra; some of these theories have a trivial classical limit but exhibit colour–kinematics duality, from which we construct the double copy theory explicitly. We show that noncommutative gauge theories exhibit a twisted form of colour–kinematics duality, which we use to show that their double copies match with the commutative case. We illustrate this explicitly for Chern–Simons theory, and for Yang–Mills theory where we obtain a modified Kawai–Lewellen–Tye relationship whose momentum kernel is linked to a binoncommutative biadjoint scalar theory. We reinterpret rank-one noncommutative gauge theories as double copy theories and discuss how our findings tie in with recent discussions of Moyal–Weyl deformations of self–dual Yang–Mills theory and gravity.

Intrinsic General Relativity and Clifford Algebra

This paper proposes to go beyond the Einstein General Relativity theory in a noncommutative geometric framework. As a first step, we rewrite the General Relativity theory intrinsically (coordinate-free formulation). As a second step, we rewrite the first and the second Bianchi identities, including torsion, within minimal algebraic hypotheses. Then, in order to extend the General Relativity theory for dealing with noncommutative scalar fields on a manifold \(\mathcal {M}\), we are forced to adapt the definition of vector fields and connections. It leads to the consideration of one associative product p of vector fields and four derivations (\(\nabla \): (vector, vector) to vector, \(\partial \): (vector, scalar) to scalar, \(\delta \): (scalar, vector) to scalar, \(\mathcal {C}\): (scalar, scalar) to scalar). At last, the particular case where scalars are a Clifford numbers motivates future investigations towards a common writing of the Einstein field equations and the Dirac equation.

Quantized noncommutative geometry from multitrace matrix models

In this paper, the geometry of quantum gravity is quantized in the sense of being noncommutative (first quantization) but it is also quantized in the sense of being emergent (second quantization). A new mechanism for quantum geometry is proposed in which noncommutative geometry can emerge from “one-matrix multitrace scalar matrix models” by probing the statistical physics of commutative phases of matter. This is in contrast to the usual mechanism in which noncommutative geometry emerges from “many-matrix singletrace Yang–Mills matrix models” by probing the statistical physics of noncommutative phases of gauge theory. In this novel scenario, quantized geometry emerges in the form of a transition between the two phase diagrams of the real quartic matrix model and the noncommutative scalar phi-four field theory. More precisely, emergence of the geometry is identified here with the emergence of the uniform-ordered phase and the corresponding commutative (Ising) and noncommutative (stripe) coexistence lines. The critical exponents and Wigner’s semicircle law are used to determine the dimension and the metric, respectively. Arguments from the saddle point equation, from Monte Carlo simulation and from the matrix renormalization group equation are provided in support of this scenario.

Noncommutative scalar field in de Sitter space–time and pair creation process

In this work, we address the process of pair creation of scalar particles in [Formula: see text] de Sitter space–time in presence of a constant electromagnetic field by applying the noncommutativity on the scalar field up to first-order in [Formula: see text]. We calculate the density of particles created in the vacuum by the mean of the Bogoliubov transformations. In contrast to a previous result, we show that noncommutativity contributes to the pair creation process. We find that the noncommutativity plays the same role of chemical potential and gives an important interest for studies at high energies.

Noncommutative scalar field in the nonextremal Reissner-Nordström background: Quasinormal mode spectrum

In our previous work [18] we constructed a model of a noncommutative, charged and massive scalar field based on the angular twist. Then we used this model to analyze the motion of the scalar field in the Reissner-Nordstr\"om black hole background. In particular, we determined the QNM spectrum analytically in the near-extremal limit. To broaden our analysis, in this paper we apply a well defined numerical method, the continued fraction method and calculate the QNM spectrum for a non-extremal Reissner-Nordstr\"om black hole. To check the validity of our analytic calculations, we compare results of the continued fraction method in the near extremal limit with the analytic results obtained in the previous paper. We find that the results are in good agreement. For completeness, we also study the QNM spectrum in the WKB approximation.

Damped harmonic oscillator and bosonic string theory through noncommutativity

Some features of noncommutativity are investigated in this work. A noncommutative (NC) versions for the 2D harmonic oscillator and scalar field theory are obtained via the Noncommutative Mapping and fourfold discussion are presented: first, the Lagrangian of the damped harmonic oscillator (DHO), which was an original explanation for the dissipative process presenting in the atomic nuclei collision, is reobtained through noncommutativity. Here, we show that the NC parameter plays the role of the viscous damping coefficient; second, the NC harmonic oscillator is mapped into the bosonic string theory in the light cone frame, which appears as a bosonic string theory attached to a D3-brane; third, we discuss the connection between DHO and bosonic string attached to a D3-brane; fourth, the NC scalar field is indicating into 2D-NC harmonic oscillator.

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.

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.

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.

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.

Entanglement entropy renormalization for the noncommutative scalar field coupled to classical BTZ geometry

In this work, we consider a noncommutative (NC) massless scalar field coupled to the classical nonrotational BTZ geometry. In a manner of the theories where the gravity emerges from the underlying scalar field theory, we study the effective action and the entropy derived from this noncommutative model. In particular, the entropy is calculated by making use of the two different approaches, the brick wall method and the heat kernel method designed for spaces with conical singularity. We show that the UV divergent structures of the entropy, obtained through these two different methods, agree with each other. It is also shown that the same renormalization condition that removes the infinities from the effective action can also be used to renormalize the entanglement entropy for the same system. Besides, the interesting feature of the NC model considered here is that it allows an interpretation in terms of an equivalent system comprising of a commutative massive scalar field, but in a modified geometry; that of the rotational BTZ black hole, the result that hints at a duality between the commutative and noncommutative systems in the background of a BTZ black hole.

Closed star product on noncommutative ℝ 3 and scalar field dynamics

We consider the noncommutative space $\mathbb{R}^3_\theta$, a deformation of $\mathbb{R}^3$ for which the star product is closed for the trace functional. We study one-loop IR and UV properties of the 2-point function for real and complex noncommutative scalar field theories with quartic interactions and Laplacian on $\mathbb{R}^3$ as kinetic operator. We find that the 2-point functions for these noncommutative scalar field theories have no IR singularities in the external momenta, indicating the absence of UV/IR mixing. We also find that the 2-point functions are UV finite with the deformation parameter $\theta$ playing the role of a natural UV cut-off. The possible origin of the absence of UV/IR mixing in noncommutative scalar field theories on $\mathbb{R}^3_\theta$ as well as on $\mathbb{R}^3_\lambda $, another deformation of $\mathbb{R}^3$, is discussed.

Noncommutative scalar quasinormal modes and quantization of entropy of a BTZ black hole

We obtain an exact analytic expression for the quasinormal modes of a noncommutative massless scalar field in the background of a massive spinless BTZ black hole up to the first order in the deformation parameter. We also show that the equations of motion governing these quasinormal modes are identical in form to the equations of motion of a commutative massive scalar field in the background of a fictitious massive spinning BTZ black hole. This results hints at a duality between the commutative and noncommutative systems in the background of a BTZ black hole. Using the obtained results for quasinormal mode frequencies, the area and entropy spectra for the BTZ black hole in the presence of noncommutativity are calculated. In particular, the separations between the neighboring values of these spectra are determined and it is found that they are nonuniform. Therefore, it appears that the noncommutativity leads to a non-equispaced (discrete) area and entropy spectra.

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.

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.

Symmetry breaking in nonuniform noncommutativeλϕ4theory at finite temperature

We consider the two-particle irreducible Cornwall-Jackiw-Tomboulis effective action at finite temperature for a noncommutative real scalar field theory in four dimensions, with noncommutativity among space and time variables. By means of a Rayleig-Ritz variation, we study the solutions of a stripe-type nonuniform background, which depends on space and time, and hence on temperature. The analysis in the first approximation shows that such solutions appear in the planar limit, as already known, but also under normal noncommutativity, in an anisotropic region which has not been considered. Further, we show that the transition from the uniform ordered phase to the nonuniform one is first order.

Discrete nonlocal waves

We study generic waves without rotational symmetry in (2+1) - dimensional noncommutative scalar field theory. In the representation chosen, the radial coordinate is naturally rendered discrete. Nonlocality along this coordinate, induced by noncommutativity, accounts for the angular dependence of the fields. The exact form of standing and propagating waves on such a discrete space is found in terms of finite series. A precise correspondence is established between the degree of nonlocality and the angular momentum of a field configuration. At small distance no classical singularities appear, even at the location of the sources. At large radius one recovers the usual commutative behaviour.