Interacting Theory of Chiral Bosons and Gauge Fields on Noncommutative Extended Minkowski Spacetime
Yan-Gang Miao, Ying-Jie Zhao
备注：
19 pages, no figures; v2: clarification added; v3: 26 pages, revised to meet the journal requirement, to appear in Commun. Theor. Phys; v4: minor typos corrected
Several interacting models of chiral bosons and gauge fields are investigated on the noncommutative extended Minkowski spacetime which was recently proposed from a new point of view of disposing noncommutativity. The models include the bosonized chiral Schwinger model, the generalized chiral Schwinger model (GCSM) and its gauge invariant formulation. We establish the Lagrangian theories of the models, and then derive the Hamilton's equations in accordance with the Dirac's method and solve the equations of motion, and further analyze the self-duality of the Lagrangian theories in terms of the parent action approach.
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.
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…
Within the setting of a recently proposed model of quantum fields on noncommutative Minkowski spacetime, the consequences of the consistent application of the proper, untwisted Poincare group as the symmetry group are investigated. The emergent model contains an infinite family of fields which are labelled by different noncommutativity parameters, and related to each other by Lorentz transformations. The relative localization properties of these fields are investigated, and it is shown that to each field one can assign a wedge-shaped localization region of Minkowski space. This assignment is c…
We investigate a link between the energy-momentum dispersion relation and the spectral distance in the context of a Lorentzian almost-commutative spectral geometry, defined by the product of Minkowski spacetime and an internal discrete noncommutative space. Using the causal structure, the almost-commutative manifold can be identified with a pair of four-dimensional Minkowski spacetimes embedded in a five-dimensional Minkowski geometry. Considering fermions travelling within the light cone of the ambient five-dimensional spacetime, we then derive the energy-momentum dispersion relation.
We describe an analytic continuation of the Euclidean Grosse-Wulkenhaar and LSZ models which defines a one-parameter family of duality covariant noncommutative field theories interpolating between Euclidean and Minkowski space versions of these models, and provides an alternative regularization to the usual Feynman prescription. This regularization allows for a matrix model representation of the field theories in terms of a complex generalization of the usual basis of Landau wavefunctions. The corresponding propagators are calculated and identified with the Feynman propagators of the field the…
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.
It is shown that the $n$-point functions of scalar massive free fields on the noncommutative Minkowski space are distributions which are boundary values of analytic functions. Contrary to what one might expect, this construction does not provide a connection to the popular traditional Euclidean approach to noncommutative field theory (unless the time variable is assumed to commute). Instead, one finds Schwinger functions with twistings involving only momenta that are on the mass-shell. This explains why renormalization in the traditional Euclidean noncommutative framework crudely differs from …
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 study the Lie bialgebra structures that can be built on the one-dimensional central extension of the Poincaré and (A)dS algebras in (1+1) dimensions. These central extensions admit more than one interpretation, but the simplest one is that they describe the symmetries of (the noncommutative deformation of) an Abelian gauge theory, $U(1)$ or $SO(2)$ on the (1+1) dimensional Minkowski or (A)dS spacetime. We show that this highlights the possibility that the algebra of functions on the gauge bundle becomes noncommutative. This is a new way in which the Coleman-Mandula theorem could be circumve…
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 …
Within the context of the twisted Poincaré algebra, there exists no noncommutative analogue of the Minkowski space interpreted as the homogeneous space of the Poincaré group quotiented by the Lorentz group. The usual definition of commutative classical fields as sections of associated vector bundles on the homogeneous space does not generalise to the noncommutative setting, and the twisted Poincaré algebra does not act on noncommutative fields in a canonical way. We make a tentative proposal for the definition of noncommutative classical fields of any spin over the Moyal space, which has the d…
The complete classification of classical $r$-matrices generating quantum deformations of the (3+1)-dimensional (A)dS and Poincaré groups such that their Lorentz sector is a quantum subgroup is presented. It is found that there exists three classes of such $r$-matrices, one of them being a novel two-parametric one. The (A)dS and Minkowskian Poisson homogeneous spaces corresponding to these three deformations are explicitly constructed in both local and ambient coordinates. Their quantization is performed, thus giving rise to the associated noncommutative spacetimes, that in the Minkowski case a…
The $κ$-Minkoswki space-time provides a quantum noncommutative-deformation of the usual Minkowski space-time. However, a notion of causality is difficult to be defined in such a space with noncommutative time. In this paper, we define a notion of causality on a (1+1)-dimensional $κ$-Minkoswki space-time using the more general framework of Lorentzian noncommutative geometry. We show that this notion allows specific causal relations, but limited by a general constraint which is a quantum generalization of the traditional speed of light limit.
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…
Within the framework of warped convolutions we deform the massless free scalar field. The deformation is performed by using the generators of the special conformal transformations. The investigation shows that the deformed field turns out to be wedge-local. Furthermore, it is shown that the spacetime induced by the deformation with the special conformal operators is nonconstant noncommutative. The noncommutativity is obtained by calculating the deformed commutator of the coordinates.
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…
General realizations, star products and plane waves for kappa-Minkowski spacetime are considered. Systematic construction of general hermitian realization is presented, with special emphasis on noncommutative plane waves and hermitian star product. Few examples are elaborated and possible physical applications are mentioned.
We revisit the notion of quantum Lie algebra of symmetries of a noncommutative spacetime, its elements are shown to be the generators of infinitesimal transformations and are naturally identified with physical observables. Wave equations on noncommutative spaces are derived from a quantum Hodge star operator. This general noncommutative geometry construction is then exemplified in the case of k-Minkowski spacetime. The corresponding quantum Poincare'-Weyl Lie algebra of infinitesimal translations, rotations and dilatations is obtained. The d'Alembert wave operator coincides with the quadratic …
We interpret, in the realm of relativistic quantum field theory, the tangential operator given by Coleman, Mandula as an appropriate coordinate operator. The investigation shows that the operator generates a Snyder-like noncommutative spacetime with a minimal length that is given by the mass. By using this operator to define a noncommutative spacetime, we obtain a Poincaré invariant noncommutative spacetime and in addition solve the soccer-ball problem. Moreover, from recent progress in deformation theory we extract the idea how to obtain, in a physical and mathematical well-defined manner, an…
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 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, s…
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 study classical scalar field theories on noncommutative curved spacetimes. Following the approach of Wess et al. [Classical Quantum Gravity 22 (2005), 3511 and Classical Quantum Gravity 23 (2006), 1883], we describe noncommutative spacetimes by using (Abelian) Drinfel'd twists and the associated *-products and *-differential geometry. In particular, we allow for position dependent noncommutativity and do not restrict ourselves to the Moyal-Weyl deformation. We construct action functionals for real scalar fields on noncommutative curved spacetimes, and derive the corresponding deformed wave …
Noncommutative spacetimes are widely believed to model some properties of the quantum structure of spacetime at the Planck regime. In this contribution the construction of (anti-)de Sitter noncommutative spacetimes obtained through quantum groups is reviewed. In this approach the quantum deformation parameter $z$ is related to a Planck scale, and the cosmological constant $Λ$ plays the role of a second deformation parameter of geometric nature, whose limit $Λ\to 0$ provides the corresponding noncommutative Minkowski spacetimes.
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.
We consider the $\varrho$-Minkowski spacetime, a model with linear noncommutativity involving the time and the azimuthal angle. We study its quantum symmetries, the $\varrho$-Poincaré quantum group, and analyse the concepts of localizability and quantum observers.
We study the notion of a Dirac operator in the framework of twist-deformed noncommutative geometry. We provide a number of well-motivated candidate constructions and propose a minimal set of axioms that a noncommutative Dirac operator should satisfy. These criteria turn out to be restrictive, but they do not fix a unique construction: two of our operators generally satisfy the axioms, and we provide an explicit example where they are inequivalent. For highly symmetric spacetimes with Drinfeld twists constructed from sufficiently many Killing vector fields, all of our operators coincide. For ge…
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.
Together with collaborators, we introduced a noncommutative Riemannian geometry over Moyal algebras and systematically developed it for noncommutative spaces embedded in higher dimensions in the last few years. The theory was applied to construct a noncommutative version of general relativity, which is expected to capture some essential structural features of spacetime at the Planck scale. Examples of noncommutative spacetimes were investigated in detail. These include quantisations of plane-fronted gravitational waves, quantum Schwarzschild spacetime and Schwarzschild-de Sitter spacetime, and…
In this paper we describe how to implement symmetries on a canonical noncommutative spacetime. We focus on noncommutative Lorentz transformations. We then discuss the structure of the light cone on a canonical noncommutative spacetime and show that field theories formulated on these spaces do not violate mircocausality.
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.
General Relativity in the noncommutative spacetime of ${\cal M}^4 \times Z_2 \times Z_2$ is constructed based on a new type of fermions in addition to the known chiral quark-leptons. While this theory has a finite physical spectrum, all the known interactions of Nature and Higgs fields can be derived from it as components of gravity with only four free parameters.
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 clarify the relation between noncommutative spacetimes and multifractional geometries, two quantum-gravity-related approaches where the fundamental description of spacetime is not given by a classical smooth geometry. Despite their different conceptual premises and mathematical formalisms, both research programs allow for the spacetime dimension to vary with the probed scale. This feature and other similarities led to ask whether there is a duality between these two independent proposals. In the absence of curvature and comparing the symmetries of both position and momentum space, we show t…
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 establish a mapping between fractional and noncommutative spacetimes in configuration space. Depending on the scale at which the relation is considered, there arise two possibilities. For a fractional spacetime with log-oscillatory measure, the effective measure near the fundamental scale determining the log-period coincides with the non-rotation-invariant but cyclicity-preserving measure of κ-Minkowski. At scales larger than the log-period, the fractional measure is averaged and becomes a power-law with real exponent. This can be also regarded as the cyclicity-inducing measure in a noncomm…
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 recently constructed the $R$-Poincaré algebra from an appropriate deformed Poisson brackets which reproduce the Fock coordinate transformation. We showed then that the spacetime of this transformation is the de Sitter one. In this paper, we derive in the $R$-Minkowski spacetime the Dirac equation and show that this is none other than the Dirac equation in the de Sitter spacetime given by its conformally flat metric. Furthermore, we propose a new approach for solving Dirac's equation in the de Sitter spacetime using the Schrödinger picture.
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 …
Noncommutative differential calculus on quantum Minkowski space is not separated with respect to the standard generators, in the sense that partial derivatives of functions of a single generator can depend on all other generators. It is shown that this problem can be overcome by a separation of variables. We study the action of the universal L-matrix, appearing in the coproduct of partial derivatives, on generators. Powers of he resulting quantum Minkowski algebra valued matrices are calculated. This leads to a nonlinear coordinate transformation which essentially separates the calculus. A com…
A general formula is calculated for the connection of a central metric w.r.t.\ a noncommutative spacetime of Lie-algebraic type. This is done by using the framework of linear connections on central bi-modules. The general formula is further on used to calculate the corresponding Riemann tensor and prove the corresponding Bianchi identities and certain symmetries that are essential to obtain a symmetric and divergenceless Einstein Tensor. In particular, the obtained Einstein Tensor is not equivalent to the sum of the noncommutative Riemann tensor and scalar, as in the commutative case, but in a…
Quantum inequalities have been established for various quantum fields in both flat and curved spacetimes. In particular, for spin-3/2 fields, Yu and Wu have explicitly derived quantum inequalities for massive case. Employing the similar method developed by Fewster and colleagues, this paper provides an explicit formula of quantum inequalities for massless spin-3/2 field in four-dimensional Minkowski spacetime.
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…
