Fedele Lizzi, Patrizia Vitale
We discuss conformal symmetry on the two dimensional noncommutative plane equipped with Moyal product in the twist deformed context. We show that the consistent use of the twist procedure leads to results which are free from ambiguities. This lends support to the importance of the use of twist symmetries in noncommutative geometry.
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In this talk we recall some concepts of Noncommutative Gauge Theories. In particular, we discuss the q-deformed two-dimensional Euclidean Plane which is covariant with respect to the q-deformed Euclidean group. A Seiberg-Witten map is constructed to express noncommutative fields in terms of their commutative counterparts.
Starting from noncommutative quantum mechanics algebra, we investigate the variances of the deformed two-mode quadrature operators under the evolution of three types of two-mode squeezed states in noncommutative space. A novel conclusion can be found and it may associate the checking of the variances in noncommutative space with homodyne detecting technology. Moreover, we analyze the influence of the scaling parameter on the degree of squeezing for the deformed level and the corresponding consequences.
The isotropy of Newton potential in noncommutative space disappears as it deforms to a momentum dependent one. We generalize the earlier derivation of such a deformed potential to the relativistic regime by calculating the 2 scalar-1 graviton scattering amplitude by taking into account the noncommutativity of space.
In two-dimensional noncommutive space for the case of both position - position and momentum - momentum noncommuting, the consistent deformed bosonic algebra at the non-perturbation level described by the deformed annihilation and creation operators is investigated. A general relation between noncommutative parameters is fixed from the consistency of the deformed Heisenberg - Weyl algebra with the deformed bosonic algebra. A Fock space is found, in which all calculations can be similarly developed as if in commutative space and all effects of spatial noncommutativity are simply represented by p…
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…
We construct the deformed generators of Schroedinger symmetry consistent with noncommutative space. The examples of the free particle and the harmonic oscillator, both of which admit Schroedinger symmetry, are discussed in detail. We construct a generalised Galilean algebra where the second central extension exists in all dimensions. This algebra also follows from the Inonu--Wigner contraction of a generalised Poincare algebra in noncommuting space.
Starting from noncommutative Fermi theory in two-dimensions, we construct a deformed Kac-Moody algebra between its vector and Chiral currents . The higher-order corrections to the deformed Kac-Moody algebra are explicitly calculated. We observe that the ordinary Schwinger terms are not affected by noncommutativity. Finally we conclude that the deformed Kac-Moody algebra can be given in term of ordinary Kac-Moody algebra plus infinitely many Lie algebraic structures at each non-zero power of the antisymmetric coefficient $θ$.
Recent studies on non-perturbation aspects of noncommutative quantum mechanics explored a new type of boson commutation relations at the deformed level, described by deformed annihilation-creation operators in noncommutative space. This correlated boson commutator correlates different degrees of freedom, and shows an essential influence on dynamics. This paper devotes to the development of formalism of deformed two-photon squeezed states in noncommutative space. General representations of deformed annihilation-creation operators and the consistency condition for the electromagnetic wave with a…
A deformed Nahm equation for the BPS equation in the noncommutative N=4 supersymmetric U(2) Yang-Mills theory is obtained. Using this, we constructed explicitly a monopole solution of the noncommutative BPS equation to the linear order of the noncommutativity scale. We found that the leading order correction to the ordinary SU(2) monopole lies solely in the overall U(1) sector and that the overall U(1) magnetic field has an expected long range component of magnetic dipole moment.
In this thesis we study different aspects of noncommutativity in quantum mechanics, field theory and gravity. We give particular emphasis on the underlying symmetries of these theories. Deformations of usual symmetries like the external space time symmetries and the internal gauge symmetry in noncommutative theories are analyzed in detail.
We study the general deformed conformal-Poincare (Galilean) symmetries consistent with relativistic (nonrelativistic) canonical noncommutative spaces. In either case we obtain deformed generators, containing arbitrary free parameters, which close to yield new algebraic structures. We show that a particular choice of these parameters reproduces the undeformed algebra. The modified coproduct rules and the associated Hopf algebra are also obtained. Finally, we show that for the choice of parameters leading to the undeformed algebra, the deformations are represented by twist functions.
The leading order corrections to Reissner--Nordstrom solutions of the Einstein's equations on noncommutative space time have been worked out basing on a noncommutative gauge theory of gravity. From the corrcted metric the horizons have been derived and the curvature scalar is also computed. The introduction of noncommutativity leads to the removal of the coordinate singularities.
We study the spectrum of Hydrogen atom, Lamb shift and Stark effect in the framework of simultaneous space-space and momentum-momentum (s-s, p-p) noncommutative quantum mechanics. The results show that the widths of Lamb shift due to noncommutativity is bigger than the one presented in [1]. We also study the algebras of abservables of systems of identical particles in s-s, p-p noncommutative quantum mechanics. We intoduce $θ$-deformed $su(2)$ algebra.
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 consider a field theoretical model on the noncommutative cylinder which leads to a discrete-time evolution. Its Euclidean version is shown to be equivalent to a model on the complex $q$-plane. We reveal a direct link between the model on a noncommutative cylinder and the deformed Virasoro algebra constructed earlier on an abstract mathematical background. As it was shown, the deformed Virasoro generators necessarily carry a second index (in addition to the usual one), whose meaning, however, remained unknown. The present field theoretical approach allows one to ascribe a clear meaning to th…
We consider, in a superspace, new operator dependent noncommutative (NC) geometries of the nonlinear quantum Hall limit related to classes of f-deformed Landau operators in the spherical harmonic well. Different NC coordinate algebras are determined using unitary representation spaces of Fock-Heisenberg tensored algebras and of the Schwinger-Fock realisation of the su(1,1) Lie algebra. A reduced model allowing an underlying N=2 superalgebra is also discussed.
We introduce new $κ$-star product describing the multiplication of quantized $κ$-deformed free fields. The $κ$-deformation of local free quantum fields originates from two sources: noncommutativity of space-time and the $κ$-deformation of field oscillators algebra - we relate these two deformations. We demonstrate that for suitable choice of $κ$-deformed field oscillators algebra the $κ$-deformed version of microcausality condition is satisfied, and it leads to the deformation of the Pauli-Jordan commutation function defined by the $κ$-deformed mass shell. We show by constructing the $κ$-defor…
Noncommutative Chern - Simons' system is non-perturbatively investigated at a full deformed level. A deformed "commutative" phase space is found by a non-canonical change between two sets of deformed variables of noncommutative space. It is explored that in the "commutative" phase space all calculations are similar to the case in commutative space. Spectra of the energy and angular momentum of the Chern - Simons' system are obtained at the full deformed level. The noncommutative-commutative correspondence is clearly showed. Formalism for the general dynamical system is briefly presented. Some …
Twisting Twistor space
本文研究了扭矢空间上的扭曲非交换规范理论，展示了其与四维非交换规范理论的等价性，并探讨了相关理论与超对称杨-米尔斯理论的关系。
We construct a differential algebra of forms on the kappa-deformed space. For a given realization of the noncommutative coordinates as formal power series in the Weyl algebra we find an infinite family of one-forms and nilpotent exterior derivatives. We derive explicit expressions for the exterior derivative and one-forms in covariant and noncovariant realizations. We also introduce higher-order forms and show that the exterior derivative satisfies the graded Leibniz rule. The differential forms are generally not graded-commutative, but they satisfy the graded Jacobi identity. We also consider…
We consider the deformed Poincare group describing the space-time symmetry of noncommutative field theory. It is shown how the deformed symmetry is related to the explicit symmetry breaking.
For commutative Euclidean time, we study the existence of field configurations that {\it a)} are formal power series expansions in $hθ^{\m\n}$, {\it b)} go to ordinary (anti-)instantons as $hθ^{\m\n}\to 0$, and {\it c)} render stationary the classical action of Euclidean noncommutative SU(3) Yang-Mills theory. We show that the noncommutative (anti-)self-duality equations have no solutions of this type at any order in $hθ^{\m\n}$. However, we obtain all the deformations --called first-order-in-$θ$-deformed instantons-- of the ordinary instanton that, at first order in $hθ^{\m\n}$, satisfy the e…
We use twist deformation techniques to analyse the behaviour under area-preserving diffeomorphisms of quantum averages of Wilson loops in Yang-Mills theory on the noncommutative plane. We find that while the classical gauge theory is manifestly twist covariant, the holonomy operators break the quantum implementation of the twisted symmetry in the usual formal definition of the twisted quantum field theory. These results are deduced by analysing general criteria which guarantee twist invariance of noncommutative quantum field theories. From this a number of general results are also obtained, su…
The simplest possible noncommutative harmonic oscillator in two dimensions is used to quantize the free closed bosonic string in two flat dimensions. The partition function is not deformed by the introduction of noncommutativity, if we rescale the time and change the compactification radius appropriately. The four point function is deformed, preserving, nevertheless, the sl(2,C) invariance. Finally the first Ward identity of the deformed theory is derived.
The space-time symmetry of noncommutative quantum field theories with a deformed quantization is described by the twisted Poincaré algebra, while that of standard commutative quantum field theories is described by the Poincaré algebra. Based on the equivalence of the deformed theory with a commutative field theory, the correspondence between the twisted Poincaré symmetry of the deformed theory and the Poincaré symmetry of a commutative theory is established. As a by-product, we obtain the conserved charge associated with the twisted Poincaré transformation to make the twisted Poincaré symmetry…
In this paper, we analyze the modification of integrable models in the $κ$-deformed space-time. We show that two dimensional isotropic oscillator problem, Kepler problem and MICZ-Kepler problem in $κ$-deformed space-time admit integrals of motion as in the commutative space. We also show that the duality equivalence between $κ$-deformed Kepler problem and $κ$-deformed two-dimensional isotropic oscillator explicitly, by deriving Bohlin-Sundman transformation which maps these two systems. These results are valid to all orders the the deformation parameter.
T-duality of gauge theories on a noncommutative $T^d$ can be extended to include fields with twisted boundary conditions. The resulting T-dual theories contain novel nonlocal fields. These fields represent dipoles of constant magnitude. Several unique properties of field theories on noncommutative spaces have simpler counterparts in the dipole-theories.
The twist-deformation of the Poincaré algebra as symmetry of the field theories on noncommutative space-time with Heisenberg-like commutation relation is discussed in connection to the relation between a sound approach to the twist and the quantization in noncommutative field theory. The recent claims of violation of Pauli's spin-statistics relation and the absence of UV/IR mixing in such theories are shown not to be founded.
This is an introduction to an algebraic construction of a gravity theory on noncommutative spaces which is based on a deformed algebra of (infinitesimal) diffeomorphisms. We start with some fundamental ideas and concepts of noncommutative spaces. Then the $θ$-deformation of diffeomorphisms is studied and a tensor calculus is defined. A deformed Einstein-Hilbert action invariant with respect to deformed diffeomorphisms is given. Finally, all noncommutative fields are expressed in terms of their commutative counterparts up to second order of the deformation parameter using the $\star$-product. T…
The role of quantum universal enveloping algebras of symmetries in constructing a noncommutative geometry of space-time and corresponding field theory is discussed. It is shown that in the framework of the twist theory of quantum groups, the noncommutative (super) space-time defined by coordinates with Heisenberg commutation relations, is (super) Poincaré invariant, as well as the corresponding field theory. Noncommutative parameters of global transformations are introduced.
Studies of noncommutative gauge theory have mainly focused on noncommutative spacetimes with constant noncommutative structure, with little known about actions for noncommutative 4D Yang-Mills theory beyond this case. We construct an action for Yang-Mills theory on a quadratically-noncommutative spacetime, i.e. of quantum-plane type, obtained from a Drinfeld twist, with star-gauge symmetry. Applied to supersymmetric Yang-Mills theory, this gives a candidate AdS/CFT dual of string theory on a related deformation of AdS$_5\times$S$^5$, which is expected to be integrable in the planar limit.
We review recent results on twisted noncommutative quantum field theory by embedding it into a general framework for the quantization of systems with a twisted symmetry. We discuss commutation relations in this setting and show that the twisted structure is so rigid that it is hard to derive any predictions, unless one gives up general principles of quantum theory. It is also shown that the twisted structure is not responsible for the presence or absence of UV/IR-mixing, as claimed in the literature.
We study the twisted bosonization of massive Thirring model to relate to sine-Gordon model in Moyal spacetime using twisted commutation relations. We obtain the relevant twisted bosonization rules. We show that there exists dual rela- tionship between twisted bosonic and fermionic operators. The strong-weak duality is also observed to be preserved as its commutative counterpart.
Spacetime geometry is twisted (deformed) into noncommutative spacetime geometry, where functions and tensors are now star-multiplied. Consistently, spacetime diffeomorhisms are twisted into noncommutative diffeomorphisms. Their deformed Lie algebra structure and that of infinitesimal Poincare' transformations is defined and explicitly constructed. This allows to construct a noncommutative theory of gravity.
We construct a quantum field theory in noncommutative spacetime by twisting the algebra of quantum operators (especially, creation and annihilation operators) of the corresponding quantum field theory in commutative spacetime. The twisted Fock space and S-matrix consistent with this algebra have been constructed. The resultant S-matrix is consistent with that of Filk\cite{Filk}. We find from this formulation that the spin-statistics relation is not violated in the canonical noncommutative field theories.
The concept of twisted Poincaré symmetry, as well as some implications, are reviewed. The spin-statistics relation and the nonlocality of NC QFT are discussed in the light of this quantum symmetry. The possibility of a twisted symmetry principle for quantum field and gauge theories formulated on a noncommutative space-time is also explored.
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 …
Gauge theory on twist-noncommutative spaces
本文构造了具有星-规范对称性的四维非对易 Yang-Mills 理论的作用量，涵盖了基于 Poincaré 代数的 Drinfel'd twist 的所有非对易空间，证明了平面等价定理，并探讨了扭曲对称性及与超弦形变的 AdS/CFT 对偶猜想。
In two-dimensional space a subtle point that for the case of both space-space and momentum-momentum noncommuting, different from the case of only space-space noncommuting, the deformed Heisenberg-Weyl algebra in noncommutative space is not completely equivalent to the undeformed Heisenberg-Weyl algebra in commutative space is clarified. It follows that there is no well defined procedure to construct the deformed position-position coherent state or the deformed momentum-momentum coherent state from the undeformed position-momentum coherent state. Identifications of the deformed position-positio…
We consider the derivatives which appear in the context of noncommutative string theory. First, we identify the correct derivations to use when the underlying structure of the theory is a quasitriangular Hopf algebra. Then we show that this is a specific case of a more general structure utilising the Drinfel'd twist. We go on to present reasons as to why we feel that the low-energy effective action, when written in terms of the original commuting coordinates, should explicitly exhibit this twisting.
A deformed boson algebra is naturally introduced from studying quantum mechanics on noncommutative phase space in which both positions and momenta are noncommuting each other. Based on this algebra, corresponding intrinsic noncommutative coherent and squeezed state representations are constructed, and variances of single- and two-mode quadrature operators on these states are evaluated. The result indicates that in order to maintain Heisenberg's uncertainty relations, a restriction between the noncommutative parameters is required.
In this article, we define the Pauli Hamiltonian function for twist-deformed N-enlarged Newton-Hooke space-time provided in article [12]. Further, we derive its energy spectrum, i.e., we find the corresponding eigenvalues as well as the proper eigenfunctions.
In this article, we investigate the impact of twisted space-time on the photoelectric effect, i.e., we derive the $θ$-deformed threshold frequency. In such a way we indicate that the space-time noncommutativity strongly enhances the photoelectric process.
We twist the Hopf algebra of igl(n,R) to obtain the kappa-deformed spacetime coordinates. Coproducts of the twisted Hopf algebras are explicitly given. The kappa-deformed spacetime obtained this way satisfies the same commutation relation as that of the conventional kappa-Minkowski spacetime, but its Hopf algebra structure is different from the well known kappa-deformed Poincare algebra in that it has larger symmetry algebra than the kappa-Minkowski case. There are some physical models which consider this symmetry. Incidentally, we obtain the canonical (theta-deformed) non-commutative spacetim…
We demonstrate the construction of solitons for a time-space Moyal-deformed integrable U(n) sigma model (the Ward model) in 2+1 dimensions. These solitons cannot travel parallel to the noncommutative spatial direction. For the U(1) case, the rank-one single-soliton configuration is constructed explicitly and is singular in the commutative limit. The projection to 1+1 dimensions reduces it to a noncommutative instanton-like configuration. The latter is governed by a new integrable equation, which describes a Moyal-deformed sigma model with a particular Euclidean metric and a magnetic field.
We briefly discuss the twisting procedure applied to the $κ$-deformed space-time. It appears that one can consider only two kinds of such twistings: in space and time directions. For both types of twisitngs we introduce related phase spaces and consider briefly their properties. We discuss in detail the changes of duality relations under the action of twist. The Jordanian twisted space-time and phase space in D=2 are also commented.
We consider the relativistic phase space coordinates (x_μ,p_μ) as composite, described by functions of the primary pair of twistor coordinates. It appears that if twistor coordinates are canonicaly quantized the composite space-time coordinates are becoming noncommutative. We obtain deformed Heisenberg algebra which in order to be closed should be enlarged by the Pauli-Lubanski four-vector components. We further comment on star-product quantization of derived algebraic structures which permit to introduce spin-extended deformed Heisenberg algebra.
A noncommutative complex scalar field, satisfying the deformed canonical commutation relations proposed by Carmona et al. [27]-[31], is constructed. Using these noncommutative deformed canonical commutation relations, a model describing the dynamics of the noncommutative complex scalar field is proposed. The noncommutative field equations are solved, and the vacuum energy is calculated to the second order in the parameter of noncommutativity. As an application to this model, the Casimir effect, due to the zero point fluctuations of the noncommutative complex scalar field, is considered. It tur…
In this paper we derive both the leading order finite size corrections for twist-2 and twist-3 operators and the next-to-leading order finite-size correction for twist-2 operators in beta-deformed SYM theory. The obtained results respect the principle of maximum transcendentality as well as reciprocity. We also find that both wrapping corrections go to zero in the large spin limit. Moreover, for twist-2 operators we studied the pole structure and compared it against leading BFKL predictions.
We present a comparison of the noncommutative field theories built using two different star products: Moyal and Wick-Voros (or normally ordered). We compare the two theories in the context of the noncommutative geometry determined by a Drinfeld twist, and the comparison is made at the level of Green's functions and S-matrix. We find that while the Green's functions are different for the two theories, the S-matrix is the same in both cases, and is different from the commutative case.
Einstein's Theory of General Relativity was formulated as a gauge theory of Lorentz symmetry by Utiyama in 1956, while the Einstein-Cartan gravitational theory was formulated by Kibble in 1961 as the gauge theory of Poincare transformations. In a noncommutative space-time with canonical commutation relations between the coordintes, Lorentz symmetry is violated and field theories constructed on such space-times have instead the so-called twisted Poincare invariance. In this paper a gauge theory formulation of noncommutative gravity is proposed based on the twisted Poincare symmetry together wit…
We analyze the effects of noncommutativity in conformal quantum mechanics (CQM) using the $κ$-deformed space-time as a prototype. Upto the first order in the deformation parameter, the symmetry structure of the CQM algebra is preserved but the coupling in a canonical model of the CQM gets deformed. We show that the boundary conditions that ensure a unitary time evolution in the noncommutative CQM can break the scale invariance, leading to a quantum mechanical scaling anomaly. We calculate the scaling dimensions of the two and three point functions in the noncommutative CQM which are shown to b…
We consider a quantum group interpretation of the non-anticommutative deformations in Euclidean supersymmetric theories. Twist deformations in the corresponding superspaces and Lie superalgebras are constructed in terms of the left supersymmetry generators. Non-anticommutative $\star$-products of superfields are covariant objects in the twist-deformed supersymmetries, and this covariance guarantees the manifest invariance of superfield actions using $\star$-products.
Field theories on deformed spaces suffer from the IR/UV mixing and renormalization is generically spoiled. In work with R. Wulkenhaar, one of us realized a way to cure this disease by adding one more marginal operator. We review these ideas, show the application to $ϕ^3$ models and use the heat kernel expansion methods for a scalar field theory coupled to an external gauge field on a $θ$-deformed space and derive noncommutative gauge field actions.
In this article we formally investigate the impact of twisted space-time on black-body radiation phenomena, i.e. we derive the $θ$-deformed Planck distribution function as well as we perform its numerical integration to the $θ$-deformed total radiation energy. In such a way we indicate that the space-time noncommutativity very strongly damps the black-body radiation process. Besides we provide for small parameter $θ$ the twisted counterparts of Rayleigh-Jeans and Wien distributions respectively.
The quantum phase space described by Heisenberg algebra possesses undeformed Hopf algebroid structure. The $κ$-deformed phase space with noncommutative coordinates is realized in terms of undeformed quantum phase space. There are infinitely many such realizations related by similarity transformations. For a given realization we construct corresponding coproducts of commutative coordinates and momenta (bialgebroid structure). The $κ$-deformed phase space has twisted Hopf algebroid structure. General method for the construction of twist operator (satisfying cocycle and normalization condition) c…
In this article we find the Zeeman corrections for hydrogen atom in the case of twist-deformed space-time. Particularly, we derive the corresponding orbital and spin $\hat{g}$-factors as well as we notice, that the second one of them remains undeformed.
These lectures fall into two distinct, although tenouously related, parts. The first part is about fuzzy and noncommutative spaces, and particle mechanics on such spaces, in other words, noncommutative mechanics. The second part is a discussion/review of twistors and how they can be used in the calculation of Yang-Mills amplitudes. The point of connection between these two topics, discussed in the last section, is in the realization of holomorphic maps as the lowest Landau level wave functions, or as wave functions of the Hilbert space used for the fuzzy version of the two-sphere. This article…
The Moyal-Weyl quantization procedure is embedded into the twist formalism of vector fields on phase space. Double application of twists provide most general deformations of Minkowskian Heisenberg-algebras and corresponding quantizations of the Lorentz-algebra. Such deformations deliver high-energy extensions of standard relativistic quantum mechanics. These are required to obtain minimal uncertainty properties for high-energy spacetime measurements that standard quantum mechanics lacks. The procedure of double twist application is outlined. We give an instructive and genuine example.
We prove that the Moyal product is covariant under linear affine spacetime transformations. From the covariance law, by introducing an $(x,Θ)$-space where the spacetime coordinates and the noncommutativity matrix components are on the same footing, we obtain a noncommutative representation of the affine algebra, its generators being differential operators in $(x,Θ)$-space. As a particular case, the Weyl Lie algebra is studied and known results for Weyl invariant noncommutative field theories are rederived in a nutshell. We also show that this covariance cannot be extended to spacetime transfor…
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