A fully Poincare' covariant model is constructed out of the k-Minkowski spacetime. Covariance is implemented by a unitary representation of the Poincare' group, and thus complies with the original Wigner approach to quantum symmetries. This provides yet another example (besides the DFR model), where Poincare' covariance is realised a` la Wigner in the presence of two characteristic dimensionful parameters: the light speed and the Planck length. In other words, a Doubly Special Relativity (DSR) framework may well be realised without deforming the meaning of "Poincare covariance".
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 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 consider two quantum phase spaces which can be described by two Hopf algebroids linked with the well-known $θ_{μν}$-deformed $D=4$ Poincare-Hopf algebra $\mathbb{H}$. The first algebroid describes $θ_{μν}$-deformed relativistic phase space with canonical NC space-time (constant $θ_{μν}$ parameters) and the second one incorporates dual to $\mathbb{H}$ quantum $θ_{μν}$-deformed Poincare-Hopf group algebra $\mathbb{G}$, which contains noncommutative space-time translations given by $Λ$-dependent $Θ_{μν}$ parameters ($% Λ$ $\equiv Λ_{μν}$ parametrize classical Lorentz group). The canonical $θ_{…
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.
We unify k-Poincare algebra and k-Minkowski spacetime by embeding them into quantum phase space. The quantum phase space has Hopf algebroid structure to which we apply the twist in order to get k- deformed Hopf algebroid structure and k-deformed phase space. We explicitly construct k-Poincare-Hopf algebra and k-Minkowski spacetime from twist. It is outlined how this construction can be extended to k-deformed super algebra including exterior derivative and forms. Our results are relevant for constructing physical theories on noncommutative spacetime by twisting Hopf algebroid phase space struct…
Relativistic treatments of quantum mechanical systems are important for understanding hadronic structure and dynamics at sub-nucleon distance scales. Hadronic states in different inertial reference frames are needed to compute current matrix elements that probe hadronic structure and dynamics. Relativistic invariance is an important consideration as the resolution of the probe is increased. Many different treatments of relativistic dynamics are used in practice, including Poincaré covariant methods, Lorentz covariant methods, Euclidean covariant methods and methods based on quantum fields. Wav…
We elaborate on the Green's functions that appeared in [1,2] when generalizing, from massless to massive particles, various equivalences between soft theorems and Ward identities of large gauge symmetries. We analyze these Green's functions in considerable detail and show that they form a hierarchy of functions which describe `boundary to bulk' propagators for large $U(1)$ gauge parameters, supertranslations and sphere vector fields respectively. As a consistency check we verify that the Green's functions associated to the large diffeomorphisms map the Poincare group at null infinity to the Po…
It is generally believed that it is not possible to have a four dimensional differential calculus in $κ$-Minkowski spacetime, with $κ$-Poincaré relativistic symmetries, covariant under ($κ$-deformed) Lorentz transformations. Thus, one usually introduces a fifth differential form, whose physical interpretation is still challenging, and defines a covariant five dimensional calculus. Nevertheless, the four dimensional calculus is at the basis of several works based on $κ$-Minkowski/$κ$-Poincaré framework that led to meaningful insights on its physical interpretation and phenomenological implicati…
Starting from the coadjoint Poincaré algebra we construct a point particle relativistic model with an interpretation in terms of extra-dimensional variables. The starting coadjoint Poincaré algebra is able to induce a mechanism of dimensional reduction between the usual coordinates of the Minkowski space and the extra-dimensional variables which turn out to form an antisymmetric tensor under the Lorentz group. Analysing the dynamics of this model, we find that, in a particular limit, it is possible to integrate out the extra variables and determine their effect on the dynamics of the material …
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…
There has been substantial calculational progress in the last few years for gauge theory amplitudes which involve massless four dimensional particles. One of the central ingredients in this has been the ability to keep precise track of the Poincare algebra quantum numbers of the particles involved. Technically, this is most easily done using the well-known four dimensional spinor helicity method. In this article a natural generalization to all dimensions higher than four is obtained based on a covariant version of the representation theory of the Poincare algebra. Covariant expressions for all…
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 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 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 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…
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 …
In this article we will use the Drinfeld twist leading to light-like $κ$-deformations of Poincaré algebra. We shall apply the standard Hopf algebra methods in order to define the star-product, which shall be used to formulate a scalar field theory compatible with $κ$-Poincaré-Hopf algebra. Using this approach we show that there is no problem with formulating integration on $κ$-Minkowski space and no need for introducing a new measure. We have shown that the $\star$-product obtained from this twist enables us to define a free scalar field theory on $κ$-Minkowski space that is equivalent to a co…
In 2017, G. P. de Brito and co-workers suggested a covariant generalization of the Kempf-Mangano algebra in a $(D+1)$-dimensional Minkowski space-time [A. Kempf and G. Mangano, Phys. Rev. D \textbf{55}, 7909 (1997); G. P. de Brito, P. I. C. Caneda, Y. M. P. Gomes, J. T. Guaitolini Junior, and V. Nikoofard, Adv. High Energy Phys. \textbf{2017}, 4768341 (2017)]. It is shown that reformulation of a real scalar field theory from the viewpoint of the covariant Kempf-Mangano algebra leads to an infinite derivative Klein-Gordon wave equation which describes two bosonic particles in the free space (a …
Our topic concerns a long standing puzzle: the energy of gravitating systems. More precisely we want to consider, for gravitating systems, how to best describe energy-momentum and angular momentum/center-of-mass momentum (CoMM). It is known that these quantities cannot be given by a local density. The modern understanding is that (i) they are quasi-local (associated with a closed 2-surface), (ii) they have no unique formula, (iii) they have no reference frame independent description. In the first part of this work we review some early history, much of it not so well known, on the subject of gr…
We construct the graviton two-point function for a two-parameter family of linear covariant gauges in n-dimensional de Sitter space. The construction is performed via the mode-sum method in the Bunch-Davies vacuum in the Poincaré patch, and a Fierz-Pauli mass term is introduced to regularize the infrared (IR) divergences. The resulting two-point function is de Sitter-invariant, and free of IR divergences in the massless limit (for a certain range of parameters) though analytic continuation with respect to the mass for the pure-gauge sector of the two-point function is necessary for this result…
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…
