Seiberg-Witten Map Research Articles

Matrix Regularization for Gauge Theories

Abstract We consider how gauge theories can be described by matrix models. Conventional matrix regularization is defined for scalar functions and is not applicable to gauge fields, which are connections of fiber bundles. We clarify how the degrees of freedom of gauge fields are related to the matrix degrees of freedom, by formulating the Seiberg–Witten map between them.

Study of He–Mckellar–Wilkens Effect in Noncommutative Space

The He–McKellar–Wilkens (HMW) effect in noncommutative space has been explored through two distinct methodologies. One approach treats the neutral particle, which harbors a permanent electric dipole moment, as an unstructured entity, while the other approach considers the neutral particle as a composite system consisting of a pair of oppositely charged particles. To preserve gauge symmetry, we apply the Seiberg–Witten map. Surprisingly, both of these approaches converge on the same result. They independently confirm that, up to the first order of the noncommutative parameter (NCP), no corrections are observed in the phase of the HMW effect. Remarkably, these two approaches, although founded on fundamentally different mechanisms, yield identical conclusions.

Quantum tunneling from Schwarzschild black hole in non-commutative gauge theory of gravity

In this letter, we present the first study of Hawking radiation as a tunneling process within the framework of non-commutative (NC) gauge theory of gravity. First, we reconstruct the non-commutative Schwarzschild black hole (NC SBH) within the gauge theory of gravity, employing the Seiberg-Witten (SW) map and the star product. Then, we compute the emission spectrum of outgoing massless particles using the quantum tunneling mechanism. In the first scenario, we calculate the tunneling rate of massless particles crossing the event horizon of the NC SBH with lower frequencies. Our results reveal pure thermal radiation. Notably, we find that the Hawking temperature remains consistent in both the classical thermodynamics and the quantum tunneling approach, suggesting equivalence between these two approaches in NC spacetime. However, in the case of massless particle emission with higher frequencies, we account for energy conservation resulting in the tunneling rate to deviate from pure thermal radiation. This tunneling rate remains consistent with an underlying unitary quantum theory. We establish a relationship between this deviation and the change in the black hole entropy, revealing a logarithmic correction to the entropy within this geometry. Furthermore, we demonstrate that non-commutativity enhances the correlations between two successively emitted particles. Additionally, we determine the NC density number of particle emission and conclude by discussing the implications of our findings.

What is the Seiberg–Witten map exactly?

We give a conceptual treatment of the Seiberg–Witten map as a quasi-isomorphism of differential graded algebras. The corresponding algebras have a very simple form, leading to explicit recurrence formulas for the quasi-isomorphism. Unlike most previous papers, our recurrence relations are nonperturbative in the parameter of non-commutativity. Using the language of quasi-isomorphisms, we give a homotopy classification of ambiguities in Seiberg–Witten maps. Possible generalizations to the Wess–Zumino complexes and some other algebras are briefly discussed.

Revisiting NCQED and scattering amplitudes

Research progress on the noncommutative gauge theories on the Moyal space is discussed in this minireview. We first present a brief overview on the development of gauge theories on Moyal space, with an emphasis on the role of Seiberg–Witten maps. Two important relations induced by reversible Seiberg–Witten maps, namely the formal equivalence of the on-shell DeWitt background field effective action in general and the explicit identical relation between tree-level scattering amplitudes in noncommutative quantum electrodynamics (NCQED), are described in some detail. We then proceed to the properties of the tree-level two-by-two scattering amplitudes in NCQED, including a forward scattering singularity in NCQED Compton scattering. After covering some phenomenological perspectives of noncommutative Yang–Mills (NCYM)-based models, outlooks for the future are given at the end.

Noncommutative $$SO(2,3)_{\star }$$ gauge theory of gravity

Topological gravity (in the sense that it is metric-independent) in a 2n-dimensional spacetime can be formulated as a gauge field theory for the AdS gauge group $$SO(2,2n-1)$$ by adding a multiplet of scalar fields. These scalars can break the gauge invariance of the topological gravity action, thus making a connection with Einstein’s gravity. This review is about a noncommutative (NC) star-product deformation of the four-dimensional AdS gauge theory of gravity, including Dirac spinors and the Yang–Mills field. In general, NC actions can be expanded in powers of the canonical noncommutativity parameter $$\theta$$ using the Seiberg–Witten map. The leading-order term of the expansion is the classical action, while the higher-order $$\theta$$ -dependent terms are interpreted as new types of coupling between classical fields due to spacetime noncommutativity. We study how these perturbative NC corrections affect the field equations of motion and derive some phenomenological consequences, such as NC-deformed Landau levels of an electron. Finally, we discuss how topological gravity in four dimensions (both classical and noncommutative) appears as a low-energy sector of five-dimensional Chern–Simons gauge theory in the sense of Kaluza–Klein reduction.

An Introduction to Noncommutative Physics

Noncommutativity in physics has a long history, tracing back to classical mechanics. In recent years, many new developments in theoretical physics, and in practical applications rely on different techniques of noncommutative algebras. In this review, we introduce the basic concepts and techniques of noncommutative physics in a range of areas, including classical physics, condensed matter systems, statistical mechanics, and quantum mechanics, and we present some important examples of noncommutative algebras, including the classical Poisson brackets, the Heisenberg algebra, Lie and Clifford algebras, the Dirac algebra, and the Snyder and Nambu algebras. Potential applications of noncommutative structures in high-energy physics and gravitational theory are also discussed. In particular, we review the formalism of noncommutative quantum mechanics based on the Seiberg–Witten map and propose a parameterization scheme to associate the noncommutative parameters with the Planck length and the cosmological constant. We show that noncommutativity gives rise to an effective gauge field, in the Schrödinger and Pauli equations. This term breaks translation and rotational symmetries in the noncommutative phase space, generating intrinsic quantum fluctuations of the velocity and acceleration, even for free particles. This review is intended as an introduction to noncommutative phenomenology for physicists, as well as a basic introduction to the mathematical formalisms underlying these effects.

On a charged spinless point particle minimally coupled to a constant magnetic field in a noncommutative plane

In this paper, we provide a mathematically and physically consistent minimal prescription for a charged spinless point particle coupled to a constant magnetic field in a 2-dimensional noncommutative plane. It turns out to be a gauge invariant prescription in contrast to the widely and carelessly used naive minimal prescription in the context of 2-dimensional quantum mechanics in a noncommutative plane. Besides, we explore the noncommutative U(1) gauge theoretic structure of the underlying noncommutative system by explicitly computing the 1-parameter family of Seiberg–Witten maps.

Klein-Gordon Theory in Noncommutative Phase Space

We extend the three-dimensional noncommutative relations of the position and momentum operators to those in the four dimension. Using the Seiberg-Witten (SW) map, we give the Heisenberg representation of these noncommutative algebras and endow the noncommutative parameters associated with the Planck constant, Planck length and cosmological constant. As an analog with the electromagnetic gauge potential, the noncommutative effect can be interpreted as an effective gauge field, which depends on the Plank constant and cosmological constant. Based on these noncommutative relations, we give the Klein-Gordon (KG) equation and its corresponding current continuity equation in the noncommutative phase space including the canonical and Hamiltonian forms and their novel properties beyond the conventional KG equation. We analyze the symmetries of the KG equations and some observables such as velocity and force of free particles in the noncommutative phase space. We give the perturbation solution of the KG equation.

Time-dependent He–McKellar–Wilkens effect on noncommutative space

In this paper, we study time-dependent He–McKellar–Wilkens (HMW) phase shift, as well as its extension on noncommutative space. We show that for adiabatic varying external magnetic field, the original HMW effect can receive a time-dependent correction. The adiabatic approximation is valid if frequency of the varying external field is smaller enough than the one by which the spinor particle takes a single round. Noncommutative corrections are studied by using Seiberg–Witten map such that the final results are gauge invariant. We find that there are two kinds of contributions, one is purely adiabatic, and the other one is a constant correction and depends on frequency of the external field.

Time-dependent Aharonov–Casher effect on noncommutative space

In this paper, we study the time-dependent Aharonov–Casher effect and its corrections due to spatial noncommutativity. Given that the charge of the infinite line in the Aharonov–Casher effect can adiabatically vary with time, we show that the original Aharonov–Casher phase receives an adiabatic correction, which is characterized by the time-dependent charge density. Based on Seiberg–Witten map, we show that noncommutative corrections to the time-dependent Aharonov–Casher phase contains not only an adiabatic term but also a constant contribution depending on the frequency of the varying electric field.

Implications of Seiberg–Witten map on Type-I superconductors

Guided by Pippard superconductivity, we incorporate the Seiberg–Witten map in the classical London theory of Type-I superconductors when an external magnetic field is applied. After defining the noncommutative Maxwell potentials, we derive the London equation for the supercurrent as a function of the noncommutative parameter, up to second order in gauge fields expansions. Keeping track of the effects of noncommutative gauge fields, we argue that noncommutative magnetic field effects can be cast in the London penetration length similar to nonlocal Pippard superconductivity. Also, we show that the flux quantization remains consistent relative to the commutative case. Our effective London penetration length reduces to the standard one in the commutative limit. These results allow us to argue that the framework of noncommutative electrodynamics can give some insights into the anisotropic and nonlocal structure of superconductivity and Condensed Matter Systems, perhaps out of the ultra-microscopic scale regime.

Vector heavy quarkonia production in gluon fragmentation in the ordinary and noncommutative standard model

Study of the mechanism of heavy quarkonia production is a powerful tool for a better understanding of the dynamics of strong interactions and probably to search for new physics. It is well-known that the fragmentation mechanism is dominant for heavy quarkonia production and forms the nonperturbative aspect of hadron production. Since the main contribution of [Formula: see text] production in high energy processes results from the gluon fragmentation, through this work, we study the fragmentation mechanism of gluon into S-wave vector heavy charmonium [Formula: see text] at lowest order perturbative QCD and extend our work to the noncommutative standard model. To investigate the effects of space–time noncommutativity, we employ the Seiberg–Witten maps. Our results show that the noncommutative effects are considerable, specifically when the gluon transverse momentum is small; [Formula: see text][Formula: see text]GeV.

Poisson gauge models and Seiberg-Witten map

The semiclassical limit of full non-commutative gauge theory is known as Poisson gauge theory. In this work we revise the construction of Poisson gauge theory paying attention to the geometric meaning of the structures involved and advance in the direction of a further development of the proposed formalism, including the derivation of Noether identities and conservation of currents. For any linear non-commutativity, Θab(x) = {f}_c^{ab} xc, with {f}_c^{ab} being structure constants of a Lie algebra, an explicit form of the gauge Lagrangian is proposed. In particular a universal solution for the matrix ρ defining the field strength and the covariant derivative is found. The previously known examples of κ-Minkowski, λ-Minkowski and rotationally invariant non-commutativity are recovered from the general formula. The arbitrariness in the construction of Poisson gauge models is addressed in terms of Seiberg-Witten maps, i.e., invertible field redefinitions mapping gauge orbits onto gauge orbits.

Noncommutative (2+1)-dimensional Anandan effects

By identifying the model of a relativistic spin-half neutral particle which possesses magnetic and electric dipole moments interacting with an electromagnetic field as the [Formula: see text] gauge theory, we study the Anandan phase in [Formula: see text]-dimensional noncommutative space–time by applying the Seiberg–Witten map. We find that up to the first order of the noncommutative parameter, Anandan phase does not receive any corrections due to the spatial noncommutativity. Our result also indicates that phases of either Aharonov–Casher or He–Mckellar–Wilkens effects can receive corrections due to spatial noncommutativity.

Geodesic equation in non-commutative gauge theory of gravity* *Supported by PRFU Research Project (B00L02UN050120190001), Univ. Batna 1, Algeria

In this study, we construct a non-commutative gauge theory of the modified structure of the gravitational field using the Seiberg-Witten map and the general tetrad fields of Schwarzschild space-time to show that the non-commutative geometry removes the singularity at the origin of the black hole, thus obtaining a non-singular Schwarzschild black hole. The geodetic structure of this black hole presents new types of motion next to the event horizon within stable orbits that are not allowed by the ordinary Schwarzschild spacetime. The noncommutative periastron advance of the Mercury orbit is obtained, and with the available experimental data, we find a parameter of non-commutativity on the order of . This result shows that the new fundamental length, , is on the order of .

Noncommutative D=5 Chern–Simons gravity: Kaluza–Klein reduction and chiral gravitational anomaly

Actions for noncommutative (NC) gauge field theories can be expanded perturbatively in powers of the noncommutativity parameter theta using the Seiberg–Witten map between ordinary classical fields and their NC counterparts. The leading order term represents classical (theta =0) action while higher-order terms give us theta -dependent NC corrections that ought to capture some aspects of quantum gravity. Building on previous work of Aschieri and Castellani on NC Chern–Simons (CS) gauge and gravity theories, showing that non-trivial theta -dependence exists only for spacetime dimensions Dge 5, we investigate a correlated effect of these extra spatial dimensions and noncommutativity on four-dimensional physics, up to first-order in theta . Assuming that one spatial dimension is compactified into a circle, we apply the Kaluza–Klein reduction procedure on the NC D=5 CS theory for the conformal gauge group SO(4, 2), to obtain an effective, theta -dependent four-dimensional theory of gravity that has Einstein–Hilbert gravity with negative cosmological constant as its commutative limit. We derive field equations for this modified theory of gravity and study the effect of NC interactions on some classical geometries, such as the AdS-Schwarzschild black hole. We find that this NC background spacetime gives rise to chiral gravitational anomaly due to the nonvanishing theta -dependent Pontryagin density.

Seiberg-Witten map with Lorentz-invariance and gauge-covariant star product

We develop the Seiberg-Witten map using the gauge-covariant star product with the noncommutativity tensor θμν(x). The latter guarantees the Lorentz invariance of the theory. The usual form of this map and its other recent generalizations do not consider such a covariant star product. We construct the Seiberg-Witten map for the gauge parameter, the gauge field and the strength tensor to the first order in the noncommutativity parameter θμν(x). Prescription for the generalization of the map to higher orders is also given. Interestingly, the associativity of the covariant star product both in the first and second orders requires the same constraints, namely, on the θμν(x) and on the space-time connection. This fact suggests that the same constraints could be enough to ensure the associativity in all orders. The resulting Seiberg-Witten map applies both to the internal and space-time gauge theories. Comparisons with the Seiberg-Witten map based on other (non-covariant) star products are given and some characteristic properties are also presented. As an application, we consider the GL(2,C) noncommutative gauge theory of gravitation, in which it is shown that the connection determines a space-time with symplectic structure (as proposed by Zumino et al [33]). This example shows that the constraints required for the associativity of the gauge-covariant star product can be satisfied. The presented GL(2,C) noncommutative gauge theory of gravitation is also compared to the one (given by Chamseddine [44]) with non-covariant star product.

Seiberg-Witten maps and scattering amplitudes of noncommutative QED

The connection between tree-level scattering amplitudes and the Seiberg-Witten (SW) map in the Moyal deformed U(1) noncommutative quantum electrodynamics (NCQED) is studied. We show that in the minimal U(1) NCQED based on a reversible Seiberg-Witten (SW) map, SW map induced interactions cancel each other in all tree-level scattering amplitudes and leave them identical to the Moyal NCQED without SW map. On the other hand, the two-by-two Compton and light-by-light scattering amplitudes deviate from minimal model when irreversible SW map is used. Therefore the reversibility of SW map and equivalence between NCQED before and after SW map manifest as an identity between the tree-level scattering amplitudes.

UV/IR mixing in noncommutative SU(N) Yang\u2013Mills theory

We show that there are one-loop IR singularities arising from UV/IR mixing in noncommutative SU(N) Yang–Mills theory defined by means of the theta -exact Seiberg–Witten map. This is in spite of the fact that there are no ordinary U(1) gauge fields in the theory and this is at variance with the noncommutative U(N) case, where the two-point part of the effective action involving the ordinary SU(N) fields do not suffer from those one-loop IR singularities.