We present Lie-algebraic deformations of Minkowski space with undeformed Poincaré algebra. These deformations interpolate between Snyder and $κ$-Minkowski space. We find realizations of noncommutative coordinates in terms of commutative coordinates and derivatives. Invariants and tensors with respect to Lorentz algebra are discussed. A general mapping from $κ$-deformed Snyder to Snyder space is constructed. Deformed Leibniz rule, the coproduct structure and star product are found. Special cases, particularly Snyder and $κ$-Minkowski in Maggiore-type realizations are discussed.
We construct realizations of the generators of the $κ$-Minkowski space and $κ$-Poincaré algebra as formal power series in the $h$-adic extension of the Weyl algebra. The Hopf algebra structure of the $κ$-Poincaré algebra related to different realizations is given. We construct realizations of the exterior derivative and one-forms, and define a differential calculus on $κ$-Minkowski space which is compatible with the action of the Lorentz algebra. In contrast to the conventional bicovariant calculus, the space of one-forms has the same dimension as the $κ$-Minkowski space.
Unified graded differential algebra, generated by $κ$-Minkowski noncommutative (NC) coordinates, Lorentz generators and anticommuting one-forms, is constructed. It is compatible with $κ$-Poincaré-Hopf algebra. For time- and space-like deformations, the super-Jacobi identities are not satisfied. By introducing additional generator, interpreted as exterior derivative, we find a new unique algebra that satisfies all super-Jacobi identities. It is universal and valid for all type of deformations (time-, space-, and light-like). For time-like deformations this algebra coincides with the one in \cit…
We consider new Abelian twists of Poincare algebra describing non-symmetric generalization of the ones given in [1], which lead to the class of Lie-deformed quantum Minkowski spaces. We apply corresponding twist quantization in two ways: as generating quantum Poincare-Hopf algebra providing quantum Poincare symmetries, and by considering the quantization which provides Hopf algebroid describing the class of quantum relativistic phase spaces with built-in quantum Poincare covariance. If we assume that Lorentz generators are orbital i.e.do not describe spin degrees of freedom, one can embed the …
We unify kappa-Minkowki spacetime and Lorentz algebra in unique Lie algebra. Introducing commutative momenta, a family of kappa-deformed Heisenberg algebras and kappa-deformed Poincare algebras are defined. They are specified by the matrix depending on momenta. We construct all such matrices. Realizations and star product are defined and analyzed in general and specially, their relation to coproduct of momenta is pointed out. Hopf algebra of the Poincare algebra, related to the covariant realization, is presented in unified covariant form. Left-right dual realizations and dual algebra are intr…
The three quantum groups dual to the generalized twist deformed Poincare Hopf algebras are provided with use of FRT procedure. Their Galilean counterparts are obtained by nonrelativistic contraction scheme.
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…
We deform Heisenberg algebra and corresponding coalgebra by twist. We present undeformed and deformed tensor identities. Coalgebras for the generalized Poincaré algebras have been constructed. The exact universal $R$-matrix for the deformed Heisenberg (co)algebra is found. We show, up to the third order in the deformation parameter, that in the case of $κ$-Poincaré Hopf algebra this $R$-matrix can be expressed in terms of Poincaré generators only. This implies that the states of any number of identical particles can be defined in a $κ$-covariant way.
We propose canonical and Lie-algebraic twist deformations of $κ$-deformed Poincare Hopf algebra which leads to the generalized $κ$-Minkowski space-time relations. The corresponding deformed $κ$-Poincare quantum groups are also calculated. Finally, we perform the nonrelativistic contraction limit to the corresponding twisted Galilean algebras and dual Galilean quantum groups.
A classical deformation procedure, based on universal enveloping algebras, Casimirs and curvatures of symmetrical homogeneous spaces, is applied to several cases of physical relevance. Starting from the (3+1)D Galilei algebra, we describe at the level of representations the process leading to its two physically meaningful deformed neighbours. The Poincare algebra is obtained by introducing a negative curvature in the flat Galilean phase space (or space of worldlines), while keeping a flat spacetime. To be precise, starting from a representation of the Galilei algebra with both Casimirs differe…
The three new deformed Poincare Hopf algebras are constructed with use of twist procedure. The corresponding relativistic space-times providing the sum of canonical and Lie-algebraic type of noncommutativity are proposed. Finally, the nonrelativistic contraction limits to the corresponding Galilei Hopf algebras are performed.
We discuss the construction of $κ$-Poincaré invariant actions for gauge theories on $κ$-Minkowski spaces. We consider various classes of untwisted and (bi)twisted differential calculi. Starting from a natural class of noncommutative differential calculi based on a particular type of twisted derivations belonging to the algebra of deformed translations, combined with a twisted extension of the notion of connection, we prove an algebraic relation between the various twists and the classical dimension d of the $κ$-Minkowski space(-time) ensuring the gauge invariance of the candidate actions for g…
We present Lie-algebraic deformations of Minkowski space with undeformed Poincaré algebra. These deformations interpolate between Snyder and $κ$-Minkowski space. We find realizations of noncommutative coordinates in terms of commutative coordinates and derivatives. By introducing modules, it is shown that although deformed and undeformed structures are not isomorphic at the level of vector spaces, they are however isomorphic at the level of Hopf algebraic action on corresponding modules. Invariants and tensors with respect to Lorentz algebra are discussed. A general mapping from $κ$-deformed S…
We investigate the observational consequences of the light-like deformations of the Poincaré algebra induced by the jordanian and the extended jordanian classes of Drinfel'd twists. Twist-deformed generators belonging to a Universal Enveloping Algebra close nonlinear algebras. In some cases the nonlinear algebra is responsible for the existence of bounded domains of the deformed generators. The Hopf algebra coproduct implies associative nonlinear additivity of the multi-particle states. A subalgebra of twist-deformed observables is recovered whenever the twist-deformed generators are either he…
Two one-parameter families of twists providing kappa-Minkowski * -product deformed spacetime are considered: Abelian and Jordanian. We compare the derivation of quantum Minkowski space from two perspectives. The first one is the Hopf module algebra point of view, which is strictly related with Drinfeld's twisting tensor technique. The other one relies on an appropriate extension of "deformed realizations" of nondeformed Lorentz algebra by the quantum Minkowski algebra. This extension turns out to be de Sitter Lie algebra. We show the way both approaches are related. The second path allows us t…
We construct a non-commutative kappa-Minkowski deformation of U(1) gauge theory, following a general approach, recently proposed in JHEP 2008 (2020) 041. We obtain an exact (all orders in the non-commutativity parameter) expression for both the deformed gauge transformations and the deformed field strength, which is covariant under these transformations. The corresponding Yang-Mills Lagrangian is gauge covariant and reproduces the Maxwell Lagrangian in the commutative limit. Gauge invariance of the action functional requires a non-trivial integration measure which, in the commutative limit, do…
$κ$-Poincaré invariant gauge theories on $κ$-Minkowski space-time, which are noncommutative analogs of the usual $U(1)$ gauge theory, exist only in five dimensions. These are built from noncommutative twisted connections on a hermitian right module over the algebra coding the $κ$-Minkowski space-time. We show that twisting the action of this algebra on the hermitian module, assumed to be a copy of it, affects neither the value of the above dimension nor the noncommutative gauge group defined as the unitary automorphisms of the module leaving the hermitian structure unchanged. Only the hermitic…
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 present a large class of supersymmetric classical r-matrices, describing the supertwist deformations of Poincare and Euclidean superalgebras. We consider in detail new family of four supertwists of N=1 Poincare superalgebra and provide as well their Euclidean counterpart. The proposed supertwists are better adjusted to the description of deformed D=4 Euclidean supersymmetries with independent left-chiral and right-chiral supercharges. They lead to new quantum superspaces, obtained by the superextension of twist deformations of spacetime providing Lie-algebraic noncommutativity of space-time…
We present the star-product algebra of the kappa-deformation of Minkowski space and the formulation of Poincare covariant differential calculus. We use these tools to construct a twisted K-cycle over the algebra and a twisted cyclic cocycle.
Some classes of Deformed Special Relativity (DSR) theories are reconsidered within the Hopf algebraic formulation. For this purpose we shall explore a minimal framework of deformed Weyl-Heisenberg algebras provided by a smash product construction of DSR algebra. It is proved that this DSR algebra, which uniquely unifies $κ$-Minkowski spacetime coordinates with Poincaré generators, can be obtained by nonlinear change of generators from undeformed one. Its various realizations in terms of the standard (undeformed) Weyl-Heisenberg algebra opens the way for quantum mechanical interpretation of DSR…
We show how some classical r-matrices for the D=4 Poincare algebra can be supersymmetrized by an addition of part depending on odd supercharges. These r-matrices for D=4 super-Poincare algebra can be presented as a sum of the so-called subordinated r-matrices of super-Abelian and super-Jordanian type. Corresponding twists describing quantum deformations are obtained in an explicit form. These twists are the super-extensions of twists obtained in the paper arXiv:0712.3962.
Twisted deformations of the conformal symmetry in the Hopf algebraic framework are constructed. The first one is obtained by a Jordanian twist built up from dilatation and momenta generators. The second is the light-like $κ$-deformation of the Poincare algebra extended to the conformal algebra, obtained by a twist corresponding to the extended Jordanian r-matrix. The $κ$-Minkowski spacetime is covariant quantum space under both of these deformations. The extension of the conformal algebra by the noncommutative coordinates is presented in two cases. The differential realizations for $κ$-Minkows…
We discussed twisted quantum deformations of D=4 Lorentz and Poincare algebras. In the case of Poincare algebra it is shown that almost all classical r-matrices of S.Zakrzewski classification can be presented as a sum of subordinated r-matrices of Abelian and Jordanian types. Corresponding twists describing quantum deformations are obtained in explicit form. This work is an extended version of the paper \url{arXiv:0704.0081v1 [math.QA]}.
A natural star product for 4-d $κ$-Minkowski space is used to investigate various classes of $κ$-Poincaré invariant scalar field theories with quartic interactions whose commutative limit coincides with the usual $ϕ^4$ theory. $κ$-Poincaré invariance forces the integral involved in the actions to be a twisted trace, thus defining a KMS weight for the noncommutative (C*-)algebra modeling the $κ$-Minkowski space. The associated modular group and Tomita modular operator are characterized. In all the field theories, the twist generates different planar one-loop contributions to the 2-point functio…
We give a quantum deformation of the chiral super Minkowski space in four dimensions as the big cell inside a quantum super Grassmannian. The quantization is performed in such way that the actions of the Poincaré and conformal quantum supergroups on the quantum Minkowski and quantum conformal superspaces are presented.
In this paper, we consider the Poincare group (space time). In mathematics, the Poincaré group of spacetime, named after Henri Poincaré, is the group of isometries of Minkowski spacetime, introduced by Hermann Minkowski. It is a non-abelian Lie group with ten generators. Spacetime, in physical science, single concept that recognizes the union of space and time, posited by Albert Einstein in the theories of relativity. One of the interesting problems for Mathematicians and Physicists is. Can we do the Fourier analysis on space time. The purpose of this paper is to define the Fourier transform t…
We propose a new deformed Rieffel product for functions in de Sitter spacetime, and study the resulting deformation of quantum field theory in de Sitter using warped convolutions. This deformation is obtained by embedding de Sitter in a higher-dimensional Minkowski spacetime, deforming there using the action of translations and subsequently projecting back to de Sitter. We determine the two-point function of a deformed free scalar quantum field, which differs from the undeformed one, in contrast to the results in deformed Minkowski spacetime where they coincide. Nevertheless, we show that in t…
We use the decomposition of o(3,1)=sl(2;C)_1\oplus sl(2;C)_2 in order to describe nonstandard quantum deformation of o(3,1) linked with Jordanian deformation of sl(2;C}. Using twist quantization technique we obtain the deformed coproducts and antipodes which can be expressed in terms of real physical Lorentz generators. We describe the extension of the considered deformation of D=4 Lorentz algebra to the twist deformation of D=4 Poincare algebra with dimensionless deformation parameter.
We extend our previous study of Hopf-algebraic $κ$-deformations of all inhomogeneous orthogonal Lie algebras ${\rm iso}(g)$ as written in a tensorial and unified form. Such deformations are determined by a vector $τ$ which for Lorentzian signature can be taken time-, light- or space-like. We focus on some mathematical aspects related to this subject. Firstly, we describe real forms with connection to the metric's signatures and their compatibility with the reality condition for the corresponding $κ$-Minkowski (Hopf) module algebras. Secondly, $h$-adic vs $q$-analog (polynomial) versions of def…
Minkowski space, conformal group, compactification, conformal infinity, conformal inversion, light cone at infinity, SU(2,2), SO(4,2), Hodge star operator, Clifford algebra, spinors, twistors, antilinear operators, exterior algebra, bivectors, isotropic subspaces, null geodesics, Lie spheres, Dupin cyclides, gravitation} \abstract{Maxwell's equations are invariant not only under the Lorentz group but also under the conformal group. Irving E. Segal has shown that while the Galilei group is a deformation of the Poincaré group, and the Poincaré group is a deformation of the conformal group, the c…
Algebraic properties of the BRST symmetry associated to the twisted gauge symmetry occurring in the $κ$-Poincaré invariant gauge theories on the $κ$-Minkowski space are investigated. We find that the BRST operation associated to the gauge invariance of the action functional can be continuously deformed together with its corresponding Leibniz rule, into a nilpotent twisted BRST operation, leading to a twisted BRST symmetry algebra which may be viewed as a noncommutative analog of the usual Yang-Mills BRST algebra.
We consider a superextension of the extended Jordanian twist, describing nonstandard quantization of anti-de-Sitter ($AdS$) superalgebra $osp(1|4)$ in the form of Hopf superalgebra. The super-Jordanian twisting function and corresponding basic coproduct formulae for the generators of $osp(1|4)$ are given in explicit form. The nonlinear transformation of the classical superalgebra basis not modifying the defining algebraic relations but simplifying coproducts and antipodes is proposed. Our physical application is to interpret the new super-Jordanian deformation of $osp(1|4)$ superalgebra as def…
Employing the algebraic framework of local quantum physics, vacuum states in Minkowski space are distinguished by a property of geometric modular action. This property allows one to construct from any locally generated net of observables and corresponding state a continuous unitary representation of the proper Poincare group which acts covariantly on the net and leaves the state invariant. The present results and methods substantially improve upon previous work. In particular, the continuity properties of the representation are shown to be a consequence of the net structure, and surmised cohom…
We construct a phase space for a three dimensional cellular complex with decorations on edges and faces using crossed modules (strict 2-groups) equipped with a (non-trivial) Poisson structure. We do not use the most general crossed module, but only the ones where the target map (t-map) is trivial. As a particular case, we recover that deformations of the Poincare group can be exported to deformations of the Poincare 2-group. The kappa-deformation case provides a natural candidate for describing discrete geometries with curved edge decorations (but still flat face decorations). Our construction…
Several related operator-algebraic constructions for quantum field theory models on Minkowski spacetime are reviewed. The common theme of these constructions is that of a Borchers triple, capturing the structure of observables localized in a Rindler wedge. After reviewing the abstract setting, we discuss in this framework i) the construction of free field theories from standard pairs, ii) the inverse scattering construction of integrable QFT models on two-dimensional Minkowski space, and iii) the warped convolution deformation of QFT models in arbitrary dimension, inspired from non-commutative…
We recall the construction of non-formal deformation quantization of the Poincare Group ISO(1,1) on its coadjoint orbit and exhibit the associated non-formal star-exponentials.
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…
We study Yang-Baxter deformations of 4D Minkowski spacetime. The Yang-Baxter sigma model description was originally developed for principal chiral models based on a modified classical Yang-Baxter equation. It has been extended to coset curved spaces and models based on the usual classical Yang-Baxter equation. On the other hand, for flat space, there is the obvious problem that the standard bilinear form degenerates if we employ the familiar coset Poincaré group/Lorentz group. Instead we consider a slice of AdS$_5$ by embedding the 4D Poincaré group into the 4D conformal group $SO(2,4)$. With …
We establish duality between real forms of the quantum deformation of the 4-dimensional orthogonal group studied by Fioresi et al. and the classification work made by Borowiec et al.. Classically these real forms are the isometry groups of $\mathbb{R}^4$ equipped with Euclidean, Kleinian or Lorentzian metric. A general deformation, named $q$-linked, of each of these spaces is then constructed, together with the coaction of the corresponding isometry group.
Starting from formal deformation quantization we use an explicit formula for a star product on the Poincaré disk D_n to introduce a Fréchet topology making the star product continuous. To this end a general construction of locally convex topologies on algebras with countable vector space basis is introduced and applied. Several examples of independent interest are investigated as e.g. group algebras over finitely generated groups and infinite matrices. In the case of the star product on D_n the resulting Fréchet algebra is shown to have many nice features: it is a strongly nuclear Köthe space,…
We discussed quantum deformations of D=4 Lorentz and Poincare algebras. In the case of Poincare algebra it is shown that almost all classical r-matrices of S. Zakrzewski classification correspond to twisted deformations of Abelian and Jordanian types. A part of twists corresponding to the r-matrices of Zakrzewski classification are given in explicit form.
The Minkowski spacetime quantum Clifford algebra structure associated with the conformal group and the Clifford-Hopf alternative k-deformed quantum Poincare algebra is investigated in the Atiyah-Bott-Shapiro mod 8 theorem context. The resulting algebra is equivalent to the deformed anti-de Sitter algebra U_q(so(3,2)), when the associated Clifford-Hopf algebra is taken into account, together with the associated quantum Clifford algebra and a (not braided) deformation of the periodicity Atiyah-Bott-Shapiro theorem.
The concept of deformation of Riemannian geometry is reviewed, with applications to gravitation and cosmology. Starting with an analysis of the cosmological constant problem, it is shown that space-times are deformable in the sense of local change of shape. These deformations leave an observable signature in the space-time, characterized by a conserved tensor, associated with a tangent acceleration, defined by the extrinsic curvature of the space-time. In the applications to cosmology, we find that the accelerated expansion of the universe is the observable effect of the deformation, dispensin…
We define a family of star products and involutions associated with $κ$-Minkowski space. Applying corresponding quantization maps we show that these star products restricted to a certain space of Schwartz functions have isomorphic Banach algebra completions. For two particular star products it is demonstrated that they can be extended to a class of polynomially bounded smooth functions allowing a realization of the full Hopf algebra structure on $κ$-Minkowski space. Furthermore, we give an explicit realization of the action of the $κ$-Poincaré algebra as an involutive Hopf algebra on this repr…
