We study field theories on the noncommutative Minkowski space with noncommuting time. The focus lies on dispersion relations in quantized interacting models in the Yang-Feldman formalism. In particular, we compute the two-point correlation function of the field strength in noncommutative quantum electrodynamics to second order. At this, we take into account the covariant coordinates that allow the construction of local gauge invariant quantities (observables). It turns out that this does not remove the well-known severe infrared problem, as one might have hoped. Instead, things become worse, since nonlocal divergences appear. We also show that these cancel in a supersymmetric version of the theory if the covariant coordinates are adjusted accordingly.
Furthermore, we study the phi^3 and the Wess-Zumino model and show that the distortion of the dispersion relations is moderate for parameters typical for the Higgs field. We also disuss the formulation of gauge theories on noncommutative spaces and study classical electrodynamics on the noncommutative Minkowski space using covariant coordinates. In particular, we compute the change of the speed of light due to nonlinear effects in the presence of a background field. Finally, we examine the so-called twist approach to quantum field theory on the noncommutative Minkowski space and point out some conceptual problems of this approach.
It is shown that the Hamiltonian approach for a $ϕ^3$-interaction on the 4-dimensional noncommutative Minkowski space leads to an ultraviolet finite $S$-matrix if the noncommutativity is averaged at each vertex.
We study the lowest-order modifications of the static potential for Born-Infeld electrodynamics and for the $θ$-expanded version of the noncommutative U(1) gauge theory, within the framework of the gauge-invariant but path-dependent variables formalism. The calculation shows a long-range correction ($1/r^5$-type) to the Coulomb potential in Born-Infeld electrodynamics. However, the Coulomb nature of the potential (to order $e^2$) is preserved in noncommutative electrodynamics.
We study Noncommutative Electrodynamics using the concept of covariant coordinates. We propose a scheme for interpreting the formalism and construct two basic examples, a constant field and a plane wave. Superposing these two, we find a modification of the dispersion relation. Our results differ from those obtained via the Seiberg-Witten map.
Using the gauge-invariant but path-dependent variables formalism, we compute the static quantum potential for noncommutative axionic electrodynamics, and find a radically different result than the corresponding commutative case. We explicitly show that the static potential profile is analogous to that encountered in both non-Abelian axionic electrodynamics and in Yang-Mills theory with spontaneous symmetry breaking of scale symmetry.
In the preceding paper [arXiv:hep-th/0604217], we construct the Dirac operator and the integral on the canonical noncommutative space. As a matter of fact, they are ones on the noncommutative torus. In the present article, we introduce the method to extend to the Minkowskian and Euclidean cases. As a concluding remark, we present a geometrical notion of our gauge theory.
We consider local field theory on $κ$-deformed Minkowski space which is an example of solvable Lie-algebraic noncommutative structure. Using integration formula over $κ$-Minkowski space and $κ$-deformed Fourier transform we consider for deformed local fields the reality conditions as well as deformation of action functionals in standard Minkowski space. We present explicite formulas for two equivalent star products describing CBH quantization of field theory on $κ$-Minkowski space. We express also via star product technique the noncommutative translations in $κ$-Minkowski space by commutative …
We exploit the Seiberg-Witten maps for fields and currents in a U(1) gauge theory relating the noncommutative and commutative (usual) descriptions to obtain the O(θ) structure of the commutator anomalies in noncommutative electrodynamics. These commutators involve the (covariant) current-current algebra and the (covariant) current-field algebra. We also establish the compatibility of the anomalous commutators with the noncommutative covariant anomaly through the use of certain consistency conditions derived here.
In this paper we calculate the divergent part of the one loop effective action for QED on noncommutative space using the background field method. The effective action is obtained up to the second order in the noncommutativity parameter $θ$ and in the classical fields.
It has been conjectured in the literature that renormalizability of the $θ$-expanded noncommutative gauge theories improves when one takes into account full nonuniqueness of the Seiberg-Witten expansion, which relates noncommutative (`high-energy') with commutative (`low-energy') fields. In order to check this conjecture we analyze renormalizability of the noncommutative chiral electrodynamics: we quantize the action which contains all possible terms implied by the SW map. After renormalization we arrive at a different theory in which the relation between the coupling constants is changed. Thi…
We perform a Noether analysis for a description of translation transformations in 4D kappa-Minkowski noncommutative spacetime which is based on the structure of a 5D differential calculus. Taking properly into account the properties of the differential calculus we arrive at an explicit formula for the conserved charges. We also propose a choice of basis for the 5D calculus which leads to an intuitive description of time derivatives.
Following the construction of the $κ$-Minkowski space from the bicrossproduct structure of the $κ$-Poincare group, we investigate possible differential calculi on this noncommutative space. We discuss then the action of the Lorentz quantum algebra and prove that there are no 4D bicovariant differential calculi, which are Lorentz covariant. We show, however, that there exist a five-dimensional differential calculus, which satisfies both requirements. We study also a toy example of 2D $κ$-Minkowski space and and we briefly discuss the main properties of its differential calculi.
We describe the generalized kappa-deformations of D=4 relativistic symmetries with finite masslike deformation parameter kappa and an arbitrary direction in kappa-deformed Minkowski space being noncommutative. The corresponding bicovariant differential calculi on kappa-deformed Minkowski spaces are considered. Two distinguished cases are discussed: 5D noncommutative differential calculus (kappa-deformation in time-like or space-like direction), and 4D noncommutative differential calculus having the classical dimension (noncommutative kappa-deformation in light-like direction). We introduce als…
Quantum field theory on the noncommutative two-dimensional Minkowski space with Grosse-Wulkenhaar potential is discussed in two ways: In terms of a continuous set of generalised eigenfunctions of the wave operator, and directly in position space. In both settings, we find a new type of divergence in planar graphs. It is present at and above the self-dual point. This new kind of divergence might make the construction of a Minkowski space version of the Grosse-Wulkenhaar model impossible.
We show that free $κ$-Minkowski space field theory is equivalent to a relativistically invariant, non local, free field theory on Minkowski space-time. The field theory we obtain has in spectrum a relativistic mode of arbitrary mass $m$ and a Planck mass tachyon. We show that while the energy momentum for the relativistic mode is essentially the standard one, it diverges for the tachyon, so that there are no asymptotic tachyonic states in the theory. It also follows that the dispersion relation is not modified, so that, in particular, in this theory the speed of light is energy-independent.
Chiral gauge anomalies on noncommutative Minkowski Space-time are computed and have their origin elucidated. The consistent form and the covariant form of the anomaly are obtained. Both Fujikawa's method and Feynman diagram techniques are used to carry out the calculations
We prove that a scalar quantum field theory defined on noncommutative Minkowski spacetime with noncommuting momentum coordinates is covariant with respect to the UV/IR duality which exchanges coordinates and momenta. The proof is based on suitable resonance expansions of charged noncommutative scalar fields in a background electric field, which yields an effective description of the field theory in terms of a coupled complex two-matrix model. The two independent matrix degrees of freedom ensure unitarity and manifest CT-invariance of the field theory. The formalism describes an analytic contin…
We have investigated some issues relevant for the possibility to construct physical theories on the $κ$-Minkowski noncommutative spacetime. The notion of field in $κ$-Minkowski has been introduced by generalizing the Weyl system/map formalism and a comparative study of the star products arising from this generalization has been done. A line of analysis of the symmetries of $κ$-Minkowski has been proposed that relies on the possibility to find a "maximally"-symmetric action which is invariant under a 10-generator Poincaré-like symmetry algebra. The equation of motion for scalar particles has be…
The matching method is employed to analytically investigate the properties of holographic superconductors in higher dimensions in the framework of power Maxwell electrodynamics taking into account the effects of spacetime noncommutativity. The relationship between the critical temperature and the charge density and the value of the condensation operator is obtained first. The Meissner like effect is then studied. The analysis indicate that larger values of the noncommutative parameter and the parameter $q$ appearing in the power Maxwell theory makes the condensate difficult to form. The critic…
We analyze quantization of noncommutative chiral electrodynamics in the enveloping algebra formalism in linear order in noncommutativity parameter $θ$. Calculations show that divergences exist and cannot be removed by ordinary renormalization, however they can be removed by the Seiberg-Witten redefinition of fields. Performing the redefinitions explicitly, we obtain renormalizable lagrangian and discuss the influence of noncommutativity on field propagation. Noncommutativity affects the propagation of chiral fermions only: half of the fermionic modes become massive and birefringent.
We consider the fermionic contributions to the free energy of noncommutative QED at finite temperature $T$. This analysis extends the main results of our previous investigation where we have considered the pure bosonic sector of the theory. For large values of $θT^2$ ($θ$ is the magnitude of the noncommutative parameters) the fermionic contributions decrease the value of the critical temperature, above which there occurs a thermodynamic instability.
This is an exposition of joint work with S. Doplicher, K. Fredenhagen, and G. Piacitelli on field theory on the noncommutative Minkowski space. The limit of coinciding points is modified compared to ordinary field theory in a suitable way which allows for the definition of so-called regularized field monomials as interaction terms. Employing these in the Hamiltonian formalism results in an ultraviolet finite S-matrix.
We study a complex free scalar field theory on a noncommutative background spacetime called $κ$-Minkowski. In particular we address the problem of second quantization. We obtain the algebra of creation and annihilation operators in an explicitly covariant way. Our procedure does not use canonical/Hamiltonian formulations, which turn out to be ill-defined in our context. Instead we work in a spacetime covariant way by introducing a noncommutative Pauli-Jordan function. This function is obtained as a generalization of the ordinary, commutative, one by taking into account the constraints imposed …
We perform a Noether analysis for a description of translation transformations in 4D $κ$-Minkowski noncommutative spacetime which is based on the structure of a 5D differential calculus. The techniques that some of us had previously developed (hep-th/0607221) for a description of translation transformations based on a 4D differential calculus turn out to be applicable without any modification, and they allow us to show that the basis usually adopted for the 5D calculus does not take into account certain aspects of the structure of time translations in $κ$-Minkowski. We propose a change of basi…
We discuss both the UV and IR origin of the one-loop triangle gauge anomalies for noncommutative nonabelian chiral gauge theories with fundamental, adjoint and bi-fundamental fermions for U(N) groups. We find that gauge anomalies only come from planar triangle diagrams, the non-planar triangle contributions giving rise to no breaking of the Ward identies. Generally speaking, theories with fundamental and bi-fundamental chiral matter are anomalous. Theories with only adjoint chiral fermions are anomaly free.
In this paper we define a causal Lorentz covariant noncommutative (NC) classical Electrodynamics. We obtain an explicit realization of the NC theory by solving perturbatively the Seiberg-Witten map. The action is polynomial in the field strenght $F$, allowing to preserve both causality and Lorentz covariance. The general structure of the Lagrangian is studied, to all orders in the perturbative expansion in the NC parameter $θ$. We show that monochromatic plane waves are solutions of the equations of motion to all orders. An iterative method has been developed to solve the equations of motion a…
In the context of the Feynman's derivation of electrodynamics, we show that noncommutativity allows other particle dynamics than the standard formalism of electrodynamics.
We study quantum field theory on the two-dimensional Noncommutative Minkoswki space with a Grosse-Wulkenhaar potential. We explicitly construct the retarded propagator and show that it is not a tempered distribution. This leads to problems when trying to define planar products of such distributions, as they appear in the Yang-Feldman series. At and above the self-dual point, these can no longer be defined, not even at different points. This shows that we do not deal with an ordinary ultraviolet divergence.
There is much discussion of scenarios where the space-time coordinates x^μare noncommutative. The discussion has been extended to include nontrivial anticommutation relations among spinor coordinates in superspace. A number of authors have studied field theoretical consequences of the deformation of N=1 superspace arising from nonanticommutativity of coordinates θ, while leaving \bar{theta}'s anticommuting. This is possible in Euclidean superspace only. In this note we present a way to extend the discussion by making both θand \bar{theta} coordinates non-anticommuting in Minkowski superspace. …
We construct field theory on noncommutative $κ$-Minkowski space-time. Having the Lorentz action on the noncommutative space-time coordinates we show that the field lagrangian is invariant. We show that noncommutativity requires replacing the Leibnitz rule with the coproduct one.
We study quantum electrodynamics on the noncommutative Minkowski space in the Yang-Feldman formalism. Local observables are defined by using covariant coordinates. We compute the two-point function of the interacting field strength to second order and find the infrared divergent terms already known from computations using the so-called modified Feynman rules. It is shown that these lead to nonlocal renormalization ambiguities. Also new nonlocal divergences stemming from the covariant coordinates are found. Furthermore, we study the supersymmetric extension of the model. For this, the supersymm…
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 dimensi…
We study infrared divergences due to ultraviolet-infrared mixing in quantum field theory on Moyal space with Lorentzian signature in the Yang-Feldman formalism. Concretely, we are considering the phi^4 and the phi^3 model in arbitrary even dimension. It turns out that the situation is worse than in the Euclidean setting, in the sense that we find infrared divergences in graphs that are finite there. We briefly discuss the problems one faces when trying to adapt the nonlocal counterterms that render the Euclidean model renormalizable.
We examine some alternative possibilities for an action functional for $κ$-Minkowski noncommutative spacetime, with an approach which should be applicable to other spacetimes with coordinate-dependent commutators of the spacetime coordinates ($[x_μ,x_ν]=f_{μ,ν}(x)$). Early works on $κ$-Minkowski focused on $κ$-Poincaré covariance and the dependence of the action functional on the choice of Weyl map, renouncing to invariance under cyclic permutations of the factors composing the argument of the action functional. A recent paper (hep-th/0307149), by Dimitrijevic, Jonke, Moller, Tsouchnika, Wess …
We develop a gauged Wess-Zumino model in noncommutative Minkowski superspace. This is the natural extension of the work of Carlson and Nazaryan, which extended N=1/2 supersymmetry written over deformed Euclidean superspace to Minkowski superspace. We investigate the interaction of the vector and chiral superfields. Noncommutativity is implemented by replacing products with star products. Although, in general, our star product is nonassociative, we prove that it is associative to the first order in the deformation parameter. We show that our model reproduces the N=1/2 theory in the appropriate …
In this paper, we derive Lorentz force and Maxwell's equations on kappa-Minkowski space-time up to the first order in the deformation parameter. This is done by elevating the principle of minimal coupling to non-commutative space-time. We also show the equivalence of minimal coupling prescription and Feynman's approach. It is shown that the motion in kappa space-time can be interpreted as motion in a background gravitational field, which is induced by this non-commutativity. In the static limit, the effect of kappa deformation is to scale the electric charge. We also show that the laws of elec…
In this talk we present a field theoretical model constructed in Minkowski N=1 superspace with a deformed supercoordinate algebra. Our study is motivated in part by recent results from super-string theory, which show that in a particular scenario in Euclidean superspace the spinor coordinates θdo not anticommute. Field theoretical consequences of this deformation were studied in a number of articles. We present a way to extend the discussion to Minkowski space, by assuming non-vanishing anticommutators for both θ, and \barθ. We give a consistent supercoordinate algebra, and a star product that…
Kappa-Minkowski space-time is an example of noncommutative space-time with potentially interesting phenomenology. However, the construction of field theories on this space is plagued with ambiguities. We propose to resolve certain ambiguities by clarifying the geometrical picture of gauge transformations on the kappa-Minkowski space-time in the twist approach. We construct the action for the noncommutative U(1) gauge fields in a geometric way, as an integral of a maximal form. The effective action with the first order corrections in the deformation parameter is obtained using the Seiberg-Witte…
We consider noncommutative analogs of scalar electrodynamics and N=2 D=4 SUSY Yang-Mills theory. We show that one-loop renormalizability of noncommutative scalar electrodynamics requires the scalar potential to be an anticommutator squared. This form of the scalar potential differs from the one expected from the point of view of noncommutative gauge theories with extended SUSY containing a square of commutator. We show that fermion contributions restore the commutator in the scalar potential. This provides one-loop renormalizability of noncommutative N=2 SUSY gauge theory. We demonstrate a pre…
In this paper we calculate the divergent part of the one loop effective action for QED on noncommutative space using the background field method. The effective action is obtained up to the second order in the noncommutative parameter theta and in the classical fields.
The realization that forthcoming experimental studies, such as the ones planned for the GLAST space telescope, will be sensitive to Planck-scale deviations from Lorentz symmetry has increased interest in noncommutative spacetimes in which this type of effects is expected. We focus here on $κ$-Minkowski spacetime, a much-studied example of Lie-algebra noncommutative spacetime, but our analysis appears to be applicable to a more general class of noncommutative spacetimes. A technical controversy which has significant implications for experimental testability is the one concerning the $κ$-Minkows…
An analogy with real Clifford algebras on even-dimensional vector spaces suggests to assign a couple of space and time dimensions modulo 8 to any algebra (represented over a complex Hilbert space) containing two self-adjoint involutions and an anti-unitary operator with specific commutation relations. It is shown that this assignment is compatible with the tensor product: the space and time dimensions of the tensor product are the sums of the space and time dimensions of its factors. This could provide an interpretation of the presence of such algebras in PT-symmetric Hamiltonians or the descr…
This is the written version of a talk I gave at the 35th Symposium Ahrenshoop in Berlin, Germany, August 2002. It is an exposition of joint work with S. Doplicher, K. Fredenhagen, and Gh. Piacitelli [1]. The violation of unitarity found in quantum field theory on noncommutative spacetimes in the context of the so-called modified Feynman rules is linked to the notion of time ordering implicitely used in the assumption that perturbation theory may be done in terms of Feynman propagators. Two alternative approaches which do not entail a violation of unitarity are sketched. An outlook upon our mor…
Using the gauge-invariant but path-dependent variables formalism, we compute the static quantum potential for noncommutative axionic electrodynamics (or axionic electrodynamics in the presence of a minimal length). Accordingly, we obtain an ultraviolet finite static potential which is the sum of a Yukawa-type and a linear potential, leading to the confinement of static charges. Interestingly, it should be noted that this calculation involves no theta expansion at all. The present result makes manifest the key role played by the new quantum of length in our analysis.
The problem of finding a formulation of Noether's theorem in noncommutative geometry is very important in order to obtain conserved currents and charges for particles in noncommutative spacetimes. In this paper, we formulate Noether's theorem for translations of kappa-Minkowski noncommutative spacetime on the basis of the 5-dimensional kappa-Poincare' covariant differential calculus. We focus our analysis on the simple case of free scalar theory. We obtain five conserved Noether currents, which give rise to five energy-momentum charges. By applying our result to plane waves it follows that the…
In our previous work we have constructed a model of noncommutative (NC) gravity based on $SO(2,3)_\star$ gauge symmetry. In this paper we extend the model by adding matter fields: fermions and a $U(1)$ gauge field. Using the enveloping algebra approach and the Seiberg-Witten map we construct actions for these matter fields and expand the actions up to first order in the noncommutativity (deformation) parameter. Unlike in the case of pure NC gravity, first non-vanishing NC corrections are linear in the noncommutativity parameter. In the flat space-time limit we obtain a non-standard NC Electrod…
In this article an energy correction is calculated in the time independent perturbation setup using a regularised ultraviolet finite Hamiltonian on the noncommutative Minkowski space. The correction to the energy is invariant under rotation and translation but is not Lorentz covariant and this leads to a distortion of the dispersion relation. In the limit where the noncommutativity vanishes the common quantum field theory on the commutative Minkowski space is reobtained.
We have constructed noncommutative phi^4 field theory on kappa-Minkowski spacetime. Quantum properties via 2-point functions were analized, and effect of birefringence for the massive scalar field has been found.
Kappa-Minkowski space-time is an example of noncommutative space-time with potentially interesting phenomenological consequences. However, the construction of field theories on this space, although operationally well-defined, is plagued with ambiguities. A part of ambiguities can be resolved by clarifying the geometrical picture of gauge transformations on the kappa-Minkowski space-time. To this end we use the twist approach to construct the noncommutative U(1) gauge theory coupled to fermions. However, in this approach we cannot maintain the kappa-Poincaré symmetry; the corresponding symmetry…
After recalling Snyder's idea of using vector fields over a smooth manifold as `coordinates on a noncommutative space', we discuss a two dimensional toy-model whose `dual' noncommutative coordinates form a Lie algebra: this is the well known $κ$-Minkowski space. We show how to improve Snyder's idea using the tools of quantum groups and noncommutative geometry. We find a natural representation of the coordinate algebra of $κ$-Minkowski as linear operators on an Hilbert space study its `spectral properties' and discuss how to obtain a Dirac operator for this space. We describe two Dirac operator…
A useful concept in the development of physical models on the $κ$-Minkowski noncommutative spacetime is that of a curved momentum space. This structure is not unique: several inequivalent momentum space geometries have been identified. Some are associated to a different assumption regarding the signature of spacetime (i.e. Lorentzian vs. Euclidean), but there are inequivalent momentum spaces that can be associated to the same signature and even the same group of symmetries. Moreover, in the literature there are two approaches to the definition of these momentum spaces, one based on the right- …
We propose a new relativistic Lorentz-invariant spin-noncommutative algebra. Using the Weyl ordering of noncommutative position operators, we build an analogue of the Moyal-Groenewald product for the proposed algebra. The Lagrange function of an electromagnetic field in the space with spin noncommutativity is constructed. In such a space electromagnetic field becomes non-abelian. A gauge transformation law of this field is also obtained. Exact nonlinear field equations of noncommutative electromagnetic field are derived from the least action principle. Within the perturbative approach we consi…
The star product technique translates the framework of local fields on noncommutative space-time into nonlocal fields on standard space-time. We consider the example of fields on $κ$- deformed Minkowski space, transforming under $κ$-deformed Poincaré group with noncommutative parameters. By extending the star product to the tensor product of functions on $κ$-deformed Minkowski space and $κ$-deformed Poincaré group we represent the algebra of noncommutative parameters of deformed relativistic symmetries by functions on classical Poincaré group.
We propose a noncommutative extension of the Minkowski spacetime by introducing a well-defined proper time from the kappa-deformed Minkowski spacetime related to the standard basis. The extended Minkowski spacetime is commutative, i.e. it is based on the standard Heisenberg commutation relations, but some information of noncommutativity is encoded through the proper time to it. Within this framework, by simply considering the Lorentz invariance we can construct field theory models that comprise noncommutative effects naturally. In particular, we find a kind of temporal fuzziness related to non…
We study the issue of the electrodynamics theory in noncommutative curved space time (NCCST) with a new star-product. In this paper, the motion equation of electrodynamics and canonical energy-momentum tensor in noncommutative curved space time will be found. The most important point is the assumption of the noncommutative parameter ($θ$) be $x^{\m}$-independent.
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.
We study the radiation reaction acting on an accelerating charge moving in noncommutative spacetime and obtain an expression for it. Radiation reaction, due to a nonrelativistic point charge, is found to receive a small noncommutative correction term. The Abraham-Lorentz equation for a point charge in noncommutative spacetime suffers from the pre-acceleration and the runaway problems. We explore as an application the radiation reaction experienced by a charge which undergoes harmonic oscillations in a noncommutative plane.
We discuss some exact Seiberg--Witten-type maps for noncommutative electrodynamics. Their implications for anomalies in different (noncommutative and commutative) descriptions are also analysed.
