We show that the realizations of noncommutative coordinates that are linear in the Lorentz generators form a closed Lie algebra under certain conditions. The star product and the coproduct for the momentum generators are obtained for these Lie algebras and the corresponding twist satisfies the cocycle and normalization conditions. We also obtain the twisted flip operator and the $\mathcal R$-matrix that define the statistics of particles or quantum fields propagating in these noncommutative spacetimes. The Lie algebra obtained in this work contains a special case which has been used in the literature to put bounds on noncommutative parameters from the experimental limits on Pauli forbidden transitions. The general covariant framework presented here is suitable for analyzing the properties of particles or quantum fields at the Planck scale.
We describe the local D=4 field theory on $κ$--deformed Minkowski space as nonlocal relativistic field theory on standard Minkowski space--time. For simplicity the case of $κ$-deformed scalar field $ϕ$ with the interaction $λϕ^{4}$ is considered, and the $κ$--deformed interaction vertex is described. It appears that fundamental mass parameter $κ$ plays a role of regularizing imaginary Pauli--Villars mass in $κ$--deformed propagator.
Some physical aspects of $q$-deformed spacetimes are discussed. It is pointed out that, under certain standard assumptions relating deformation and quantization, the classical limit (Poisson bracket description) of the dynamics is bound to contain unusual features. At the same time, it is argued that the formulation of an associated $q$-deformed field theory is fraught with serious difficulties.
In this paper, we investigate the twisted algebra of the fermionic oscillators associated with Dirac field defined in $κ$-Minkowski space-time. Starting from $κ$-deformed Dirac theory, which is invariant under the undeformed $κ$-Poincare algebra, using the twisted flip operator, we derive the deformed algebra of the creation and annihilation operators corresponding to the Dirac field quanta in $κ$-Minkowski space-time. In the limit $a\rightarrow 0$, the deformed algebra reduces to the commutative result.
We describe the extension of the Wigner`s infinite-dimensional unitary representations of Poincaré group to the case of $κ$-deformed Poincaré group. We show that the corresponding coordinate wave functions on noncommutative space-time are described by free field equations on $κ$-deformed Minkowski space. The cases of Klein--Gordon, Dirac, Proca and Maxwell fields are considered. Finally some aspects of second quantization are also discussed.
We review the application of twist deformation formalism and the construction of noncommutative gauge theory on $κ$-Minkowski space-time. We compare two different types of twists: the Abelian and the Jordanian one. In each case we provide the twisted differential calculus and consider ${U}(1)$ gauge theory. Different methods of obtaining a gauge invariant action and related problems are thoroughly discussed.
The differential structure on the kappa-Minkowski spacetime from Jordanian twist of Weyl algebra is constructed, and it is shown to be closed in 4-dimensions in contrast to the conventional formulation. Based on this differential structure, we have formulated a scalar field theory in this kappa-Minkowski spacetime.
In this paper, we derive the Dirac equation in the $κ$-deformed Minkowski space-time. We start with $κ$-deformed Minkowski space-time and investigate the undeformed $κ$-Lorentz transformation valid to all order in the deformation parameter $a$. Using the undeformed $κ$-Lorentz algebra, we obtain the $κ$-deformed Dirac equation, valid to all order in the deformation parameter $a$. In limit $a\rightarrow$0, we get back the correct commutative result.
We consider two new classes of twisted D=4 quantum Poincaré symmetries described as the dual pairs of noncocommutative Hopf algebras. Firstly we investigate a two-parameter class of twisted Poincaré algebras which provide the examples of Lie-algebraic noncommutativity of the translations. The corresponding associative star-products and new deformed Lie-algebraic Minkowski spaces are introduced. We discuss further the twist deformations of Poincaré symmetries generated by the twist with its carrier in Lorentz algebra. We describe corresponding deformed Poincaré group which provides the quadrati…
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 consider the issue of statistics for identical particles or fields in kappa-deformed spaces, where the system admits a symmetry group G. We obtain the twisted flip operator compatible with the action of the symmetry group, which is relevant for describing particle statistics in presence of the noncommutativity. It is shown that for a special class of realizations, the twisted flip operator is independent of the ordering prescription.
In this paper we study the deformed statistics and oscillator algebras of quantum fields defined in $κ$-Minkowski spacetime. The twisted flip operator obtained from the twist associated with the star product requires an enlargement of the Poincaré algebra to include the dilatation generators. Here we propose a novel notion of a fully covariant flip operator and show that to the first order in the deformation parameter it can be expressed completely in terms of the Poincaré generators alone. The $R$-matrices corresponding to the twisted and the covariant flip operators are compared up to first …
We present Lie-algebraic deformations of Minkowski space with undeformed Poincare algebra. These deformations interpolate between Snyder and kappa-Minkowski space. We find realizations of noncommutative coordinates in terms of commutative coordinates and derivatives. Deformed Leibniz rule, the coproduct structure and star product are found. Special cases, particularly Snyder and kappa-Minkowski in Maggiore-type realizations are discussed. Our construction leads to a new class of deformed special relativity theories.
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 …
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…
Deformed gauge transformations on deformed coordinate spaces are considered for any Lie algebra. The representation theory of this gauge group forces us to work in a deformed Lie algebra as well. This deformation rests on a twisted Hopf algebra, thus we can represent a twisted Hopf algebra on deformed spaces. That leads to the construction of Lagrangian invariant under a twisted Lie algebra.
The Hilbert space representations of a non-commutative q-deformed Minkowski space, its momenta and its Lorentz boosts are constructed. The spectrum of the diagonalizable space elements shows a lattice-like structure with accumulation points on the light-cone.
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…
We present the Casimir energy of spherical shell, for the symmetrically deformed scalar field in $κ$-Minkowski space-time, satisfying Dirichlet boundary condition. The Casimir energy shows the particle anti-particle symmetry contrary to the asymmetrically deformed case. In addition, the deformation effect starts from $O(1/κ)$ term unlike in the parallel plates.
This study of gauge field theories on kappa-deformed Minkowski spacetime extends previous work on field theories on this example of a noncommutative spacetime. We construct deformed gauge theories for arbitrary compact Lie groups using the concept of enveloping algebra-valued gauge transformations and the Seiberg-Witten formalism. Derivative-valued gauge fields lead to field strength tensors as the sum of curvature- and torsion-like terms. We construct the Lagrangians explicitly to first order in the deformation parameter. This is the first example of a gauge theory that possesses a deformed L…
In recent years, different views on the interpretation of Lorentz covariance of non commuting coordinates were discussed. Here, by a general procedure, we construct the minimal canonical central covariantisation of the k-Minkowski spacetime. We then show that, though the usual k-Minkowski spacetime is covariant under deformed (or twisted) Lorentz action, the resulting framework is equivalent to taking a non covariant restriction of the covariantised model. We conclude with some general comments on the approach of deformed covariance.
We construct the spectral curve of gamma-deformed AdS/CFT from the strong coupling scaling limit of the T-system. As we interpret the twisted T-functions in the classical limit as characters of the highest weight representations of the psu(2,2|4) symmetry group, we compute the twisted quasimomenta which characterize classical integrability and analyze their analytic and asymptotic properties. These twisted quasimomenta are compared to Beisert-Roiban Bethe ansatz equations and classical string solutions.
Derivation of $κ$-Poincare bicovariant commutation relations between coordinates and 1-forms on $κ$-Minkowski space is given using Dirac operator and Allain Connes formula. The deformed U(1) gauge theory and appearance of an additional spin 0 gauge field is discussed.
We study the spectral dimension associated with diffusion processes on Euclidean $κ$-Minkowski space. We start by describing a geometric construction of the "Euclidean" momentum group manifold related to $κ$-Minkowski space. On such space we identify various candidate Laplacian functions, i.e. deformed Casimir invariants, and calculate the corresponding spectral dimension for each case. The results obtained show a variety of running behaviours for the spectral dimension according to the choice of deformed Laplacian, from dimensional reduction to super-diffusion.
We discuss boosts in a deformed Minkowski space, i.e. a four-dimensional space-time with metric coefficients depending on non-metric coordinates (in particular on the energy). The general form of a boost in an arbitrary direction is derived in the case of space anisotropy. Two maximal 3-vector velocities are mathematically possible, an isotropic and an anisotropic one. However, only the anisotropic velocity has physical meaning, being invariant indeed under deformed boosts.
We derive the simplest commutation relations of operator algebras associated to M2 branes and an M5 brane in the $Ω$-deformed M-theory, which is a natural set-up for Twisted holography. Feynman diagram 1-loop computations in the twisted-holographic dual side reproduce the same algebraic relations.
We investigate a Lie algebra-type $ κ$-deformed Minkowski space-time with undeformed Lorentz algebra and mutually commutative vector-like Dirac derivatives. There are infinitely many realizations of $ κ$-Minkowski space. The coproduct and the star product corresponding to each of them are found. Utilizing the properties of the {\em{natural}} realization, we construct a scalar field theory on $ κ$-deformed Minkowski space and show that it is equivalent to the scalar, nonlocal, relativistically invariant field theory on the ordinary Minkowski space.
We consider non-supersymmetric two-dimensional CP(N-1) model deformed by a term presenting the bosonic part of the twisted mass deformation of N=2 supersymmetric version of the model. Our deformation has a special form preserving a Z_N symmetry at the Lagrangian level. In the large mass limit the model is weakly coupled. Its dynamics is described by the Higgs phase, with Z_N spontaneously broken. At small masses it is in the strong coupling Coulomb/confining phase. The Z_N symmetry is restored. Two phases are separated by a phase transition. We find the phase transition point in the large-N li…
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…
The differential calculus on `non-standard' $h$-Minkowski spaces is given. In particular it is shown that, for them, it is possible to introduce coordinates and derivatives which are simultaneously hermitian.
A Hamiltonian formulation of gauge symmetries on noncommutative ($θ$ deformed) spaces is discussed. Both cases- star deformed gauge transformation with normal coproduct and undeformed gauge transformation with twisted coproduct- are considered. While the structure of the gauge generator is identical in either case, there is a difference in the computation of the graded Poisson brackets that yield the gauge transformations. Our analysis provides a novel interpretation of the twisted coproduct for gauge transformations.
Using general but simple covariance arguments, we classify the `quantum' Minkowski spaces for dimensionless deformation parameters. This requires a previous analysis of the associated Lorentz groups, which reproduces a previous classification by Woronowicz and Zakrzewski. As a consequence of the unified analysis presented, we give the commutation properties, the deformed (and central) length element and the metric tensor for the different spacetime algebras.
We discuss a deformation of superspace based on a hermitian twist. The twist implies a $\star$-product that is noncommutative, hermitian and finite when expanded in power series of the deformation parameter. The Leibniz rule for the twisted SUSY transformations is deformed. A minimal deformation of the Wess-Zumino action is proposed and its renormalizability properties are discussed. There is no tadpole contribution, but the two-point function diverges. We speculate that the deformed Leibniz rule, or more generally the twisted symmetry, interferes with renormalizability properties of the model…
We investigate the integrable aspects of the planar $γ$-deformed ABJM theory and propose the twisted asymptotic Bethe ansatz equations. A more general method through a twisted generating functional is discussed, based on which, the asymptotic large L solution of Y-system is modified in order to match the asymptotic Bethe ansatz equations. Several applications of our method were made in the $sl(2)$-like sector and some important examples in $β$-deformed ABJM are presented as well.
We propose a generalized description for the kappa-Poincare-Hopf algebra as a symmetry quantum group of underlying kappa-Minkowski spacetime. We investigate all the possible implementations of (deformed) Lorentz algebras which are compatible with the given choice of kappa-Minkowski algebra realization. For the given realization of kappa-Minkowski spacetime there is a unique kappa-Poincare-Hopf algebra with undeformed Lorentz algebra. We have constructed a three-parameter family of deformed Lorentz generators with kappa-Poincare algebras which are related to kappa-Poincare-Hopf algebra with und…
In this paper we construct Seiberg-Witten maps for superfields on canonically deformed N=1, d=4 Minkowski and Euclidean superspace. On Minkowski superspace we show that the Seiberg-Witten map is not compatible with locality, (anti)chirality and supersymmetry at the same time. On Euclidean superspace we show that there exists a local, chiral and supersymmetric Seiberg-Witten map for chiral superfields if we take the noncommutativity parameter to be selfdual, and a local, antichiral and supersymmetric Seiberg-Witten map for antichiral superfields if we take the noncommutativity parameter to be a…
We study the worldsheet scattering theory of the $η$ deformation of the AdS$_\mathsf{5} \times $S$^\mathsf{5}$ superstring corresponding to the purely fermionic Dynkin diagram. This theory is a Weyl-invariant integrable deformation of the AdS$_\mathsf{5} \times $S$^\mathsf{5}$ superstring, with trigonometric quantum-deformed symmetry. We compute the two-body worldsheet S matrix of this string in the light-cone gauge at tree level to quadratic order in fermions. The result factorizes into two elementary blocks, and solves the classical Yang-Baxter equation. We also determine the corresponding e…
Non-anticommutative deformations have been studied in the context of supersymmetry (SUSY) in three and four space-time dimensions, and the general picture is that highly nontrivial to deform supersymmetry in a way that still preserves some of its important properties, both at the formal algebraic level (e.g., preserving the associativity of the deformed theory) as well as at the physical level (e.g., maintaining renormalizability). The Hopf algebra formalism allows the definition of algebraically consistent deformations of SUSY, but this algebraic consistency does not guarantee that physical m…
We investigate the properties of kappa-Minkowski spacetime by using representations of the corresponding deformed algebra in terms of undeformed Heisenberg-Weyl algebra. The deformed algebra consists of kappa-Poincare algebra extended with the generators of the deformed Weyl algebra. The part of deformed algebra, generated by rotation, boost and momentum generators, is described by the Hopf algebra structure. The approach used in our considerations is completely Lorentz covariant. We further use an adventages of this approach to consistently construct a star product which has a property that u…
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.
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.
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.
$κ$-deformed commutation relation between quantum operators is constructed via abelian twist deformation in $κ$-Minkowski spacetime. The commutation relation is written in terms of universal $R$-matrix satisfying braided statistics. The equal-time commutator function turns out to vanish in this framework.
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 study the twisted N=4 super Yang-Mills theories in the Omega-background with the constant R-symmetry Wilson line gauge field. Based on the classification of topological twists of N=4 supersymmetry (the half, the Vafa-Witten and the Marcus twists), we construct the deformed off-shell supersymmetry associated with the scalar supercharges for these twists. We find that the Omega-deformed action is written in the exact form with respect to the scalar supercharges as in the undeformed case.
We study a deformed $su(m|n)$ algebra on a quantum superspace. Some interesting aspects of the deformed algebra are shown. As an application of the deformed algebra we construct a deformed superconformal algebra. {}From the deformed $su(1|4)$ algebra, we derive deformed Lorentz, translation of Minkowski space, $iso(2,2)$ and its supersymmetric algebras as closed subalgebras with consistent automorphisms.
We consider pulsating strings in Lunin-Maldacena backgrounds, specifically in deformed Minkowski spacetime and deformed AdS_5xS^5. We find the relation between the energy and the oscillation number of the pulsating string when the deformation is small. Since the oscillation number is an adiabatic invariant it can be used to explore the regime of highly excited string states. We then quantize the string and look for such a sector. For the deformed Minkowski background we find a precise match with the classical results if the oscillation number is quantized as an even number. For the deformed Ad…
The dissertation presents possibilities of applying noncommutative spacetimes description, particularly kappa-deformed Minkowski spacetime and Drinfeld's deformation theory, as a mathematical formalism for Doubly Special Relativity theories (DSR), which are thought as phenomenological limit of quantum gravity theory. Deformed relativistic symmetries are described within Hopf algebra language. In the case of (quantum) kappa-Minkowski spacetime the symmetry group is described by the (quantum) kappa-Poincare Hopf algebra. Deformed relativistic symmetries were used to construct the DSR algebra, wh…
We discuss the description of generic excited states in the quantum deformed AdS_5 x S^5 mirror thermodynamic Bethe ansatz and derive the associated Y-system. This Y-system shows an interesting new feature; it depends explicitly on the excited state under consideration. Similarly, it also depends on twisted boundary conditions. We construct the asymptotic solution of these TBA and Y-system equations by deriving the twisted transfer matrix for the quantum deformed Hubbard model and finding the deformed mirror bound state dressing phase. This asymptotic construction is insensitive to the precise…
We proceed to study Yang-Baxter deformations of 4D Minkowski spacetime based on a conformal embedding. We first revisit a Melvin background and argue a Lax pair by adopting a simple replacement law invented in 1509.00173. This argument enables us to deduce a general expression of Lax pair. Then the anticipated Lax pair is shown to work for arbitrary classical $r$-matrices with Poincaé generators. As other examples, we present Lax pairs for pp-wave backgrounds, the Hashimoto-Sethi background, the Spradlin-Takayanagi-Volovich background.
We construct a new class of vacuum black hole solutions whose geometry is deformed and twisted by the presence of NUT charges. The solutions are obtained by `unspinning' the general Kerr-NUT-(A)dS spacetimes, effectively switching off some of their rotation parameters. The resulting geometry has a structure of warped space with the Kerr-like Lorentzian part warped to a Euclidean metric of deformed and/or twisted sphere, with the deformation and twist characterized by the `Euclidean NUT' parameters. In the absence of NUTs, the solution reduces to a well known Kerr-(A)dS black hole with several …
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 consider a class of non-unitary Toda theories based on the Lie superalgebras $A^{(1)}(n,n)$ in two-dimensional Minkowski spacetime, which can be twisted into a topological sector. In particular we study the simplest example with $n=1$ and real fields, and show how this theory can be treated as an integrable perturbation of the $A(1,0)$ superconformal model. We construct the conserved currents and the vertex operators which are chiral primary fields in the conformal theory. We then define the chiral ring of the $A^{(1)}(1,1)$ Toda theory and compute topological correlation functions in the t…
We study the propagation of quantum fields on $κ$-Minkowsi spacetime. Starting from the non-commutative partition function for a free field written in momentum space we derive the Feynman propagator and analyze the non-trivial singularity structure determined by the group manifold geometry of momentum space. The additional contributions due to such singularity structure result in a deformed field propagation which can be alternatively described in terms of an ordinary field propagation determined by a source with a blurred spacetime profile. We show that the $κ$-deformed Feynman propagator can…
A bicovariant calculus on the twisted inhomogeneous multiparametric $q$-groups of the $B_n,C_n,D_n$ type, and on the corresponding quantum planes, is found by means of a projection from the bicovariant calculus on $B_{n+1}$, $C_{n+1}$, $D_{n+1}$. In particular we obtain the bicovariant calculus on a dilatation-free $q$-Poincaré group $ISO_q (3,1)$, and on the corresponding quantum Minkowski space. The classical limit of the $B_n,C_n,D_n$ bicovariant calculus is discussed in detail.
Realizations of $κ$-Minkowski space linear in momenta are studied for time-, space- and light-like deformations. We construct and classify all such linear realizations and express them in terms of $\mathfrak{gl}(n)$ generators. There are three one-parameter families of linear realizations for time-like and space-like deformations, while for light-like deformations, there are only four linear realizations. The relation between deformed Heisenberg algebra, star product, coproduct of momenta and twist operator is presented. It is proved that for each linear realization there exists Drinfeld twist…
Using a representation of the q-deformed Lorentz algebra as differential operators on quantum Minkowski space, we define an algebra of observables for a q-deformed relativistic quantum mechanics with spin zero. We construct a Hilbert space representation of this algebra in which the square of the mass $ p^2 $ is diagonal.
