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Classical basis for kappa-Poincare algebra and doubly special relativity theories

Several issues concerning quantum kappa-Poincare algebra are discussed and reconsidered here. We propose two different formulations of kappa-Poincare quantum algebra. Firstly we present a complete Hopf algebra formulae of kappa-Poincare in classical Poincare basis. Further by adding one extra generator, which modifies the classical structure of Poincare algebra, we eliminate non polynomial functions in the kappa-parameter. Hilbert space representations of such algebras make Doubly Special Relativity (DSR) similar to the Stueckelberg's version of (proper-time) relativistic Quantum Mechanics.


I. INTRODUCTION
The concept of a quantum group was introduced more than 20 years ago in Refs. [1]- [4] (see also [5], [6]), and since then the subject has been widely investigated in different approaches and has gained in popularity and all sorts of applications (see, e.g., [7]- [16]). One of the applications is to consider the notion of a quantum group as noncommutative generalization of a symmetry group of the physical system, which means that quantum group takes the place of symmetry group of spacetime, i.e. Poincaré group. Roughly speaking, quantum groups are the deformations of some classical structures as groups or Lie algebras, which are made in the category of Hopf algebras.
Similarly, quantum spaces are noncommutative generalizations (deformations) of ordinary spaces.
The most important in physics and mathematically the simplest one seem to be the canonical and the Lie-algebraic quantum deformations. It has not taken long time to notice that in description of the short-distance structure of spacetime (at the Planck scale) the existing symmetries may be modified including deformation of Poincaré symmetry. Moreover, it has been suggested that the symmetries of the κ -deformed Minkowski space should be described in terms of the Hopf algebra [7]- [11]. The studies on this type of deformations were inspired by [7], where the quantum κ -Poincaré algebra with the masslike deformation parameter was first proposed. The algebraic structure of the κ -Poincaré algebra has gained even more attention and since then has been intensively analyzed from mathematical and physical point of view. For historical review see, e.g., [14]- [16] and the references therein.
A chance for physical application of this theory appeared when an extension of special relativity was proposed in Refs. [17,18], and another one, showing different point of view, in Refs. [19].
This extension includes two observer-independent scales, the velocity of light and the scale of mass, now called "Doubly Special Relativity" (DSR). Also, various phenomenological aspects of DSR theories have been studied in, e.g., [20]. For comparison of these two approaches see, e.g., [21]. The connection between κ -deformation and DSR theory in first formulation (DSR1) has been shown (see, e.g., [18], [22], [23]) including the conclusion that the spacetime of DSR must be noncommutative as the result of Hopf structure of this algebra.
The κ -Poincaré algebra as well as DSR have been studied extensively and have found many applications besides physics at the Planck scale -gravity also in elementary particle physics and quantum field theory (see, e.g., [17]- [25] and references therein). κ -Poincaré Hopf algebra has been discovered in the so-called standard basis [7] inherited from the anty-de Sitter basis by the contraction procedure. For this basis only the rotational sector remains algebraically undeformed.
Introducing bicrossproduct basis allows to leave the lorentzian generators undeformed. This basis is the easier form of the κ -Poincaré algebra basis and was postulated in [10], [11]. In this form, the Lorentz subalgebra of the κ -Poincaré algebra, generated by rotations and boosts is not deformed and the difference is only in the way the boosts act on momenta. There is also a change in co-algebraic sector, the coproducts are no longer trivial, which has the already mentioned consequence: the spacetime of DSR is noncommutative.
It is well known that the Drinfeld-Jimbo quantization algorithm relies on simultaneous deformations of the algebraic and co-algebraic sectors and it is applicable to semisimple Lie algebras [1,2]. In particular, this implies the existence of classical basis for Drinfeld-Jimbo quantized algebras. Strictly speaking, the Drinfeld-Jimbo technique cannot be applied to the Poincaré nonsemisimple algebra which has been obtained by the contraction procedure from the Drinfeld-Jimbo deformation of the anti-deSitter (simple) Lie algebra so (3,2). Nevertheless, κ−Poincaré quantum group shares many properties of the original Drinfeld-Jimbo quantization. These include existence of the classical basis, the square of the antipode and the solution to the specialization problem (see Sect.III).
In the paper we define κ -Poincaré (Hopf) algebra in its classical Poincaré Lie algebra basis. The constructions of such basis were previously investigated in several papers [26], [27]. Particularly, the explicit formulas expressing classical basis in terms of bicrossproduct one have been obtained therein . Explicit formulas for coproducts can be found in different (realization dependent) context in [22], [28], see also [14]. To the best of our knowledge, there are no other examples of Drinfeld-Jimbo type deformation expressed in a classical Lie-algebraic basis.
It is known that different DSR models are defined by different choices of basis in the universal envelop of the Poincaré Lie algebra. They lead to different κ−Poincaré coproducts. Now we are in the position to demonstrate that when these models are compared in the same classical basis, they do differ by different operator realizations in the space of scalar-valued functions on a spacetime manifold. This result in some sense allows us to distinguish between the description of DSR1 and DSR2 theories. For the special choice of realization, we recover a well known bicrossproductform of κ -Poincaré algebra and the standard DSR model. Moreover, according to the formalism developed in our previous paper [16], we have a wide range of models and deformed dispersion relations related with them to our disposal (Sect.IV).
The algebra (M k , N k , P µ ) can be extended in the standard way to a Hopf algebra by defining on the universal enveloping algebra U P 1,3 the coproduct ∆ 0 , the counit ǫ, and the antipode S 0 , where the nondeformed -primitive coproduct, the antipode and the counit are given for X ∈ P 1,3 . In addition ∆ 0 (1) = 1 ⊗ 1, S 0 (1) = 1 and ǫ(1) = 1. For the purpose of deformation one has to extend further this Hopf algebra by considering formal power series in κ −1 , and correspondingly considering the Hopf algebra (U P 1,3 [[κ −1 ]], · , ∆ 0 , S 0 , ǫ) as a topological Hopf algebra with the so-called "h-adic" topology [5,6]. Quantum deformations of this Hopf algebra are controlled by classical r -matrices satisfying the classical Yang-Baxter (YB) equation: homogeneous or inhomogeneous. The relation between classical r -matrix r and a universal (quantum) r -matrix R reads as: where 1 κ denotes the deformation parameter. In the case of r -matrices satisfying homogeneous YB equations the co-algebraic sector is twist-deformed while algebraic one remains classical [1].
Additionally, one can also apply existing twist tensors to related Hopf module-algebras in order to obtain quantized, e.g., spacetimes (see [14]- [16]). In contrast Drinfeld-Jimbo quantization, corresponding to inhomogeneous r− matrices, relies on suitable deformation of the algebraic and co-algebraic sectors simultaneously. Therefore, the classification of quantum deformations is done by means of classification of the corresponding classical r -matrices: homogeneous and inhomogeneous one.
In the case of relativistic symmetries, such classification (complete for the Lorentz and almostcomplete for Poincaré algebras) has been performed in Ref. [29] (see also [30] where this classification scheme has been extended). Particularly, r -matrix which corresponds to κ -deformation of Poincaré algebra is given by and it satisfies the inhomogeneous (modified) Yang-Baxter equation: Therefore, one does not expect to obtain the κ Poincaré coproduct by twist. However, most of the items on that list contain homogeneous r−matrices. Explicit twists for them have been provided in Ref. [31] (for superization see [32,33]); the corresponding quantization has been carried out in [13].
Our purpose in this note is to formulate κ−Poincaré Hopf algebra in classical Poincaré basis.
We would like to mention that complete treatment of this problem was not considered before (see [26], [27]). One defines the deformed (quantized) coproducts ∆ κ and the antipodes S κ on U ≡ ] leaving algebraic sector classical (untouched) like in the case of twisted deformation.
and the antipodes where are just shortcuts; P 2 P µ P µ ≡ P 2 − P 2 0 , and P 2 = P i P i . Let us stress the point that above expressions are formal power series in the parameter 1 κ , e.g., where 0.5 n = 0.5(0.5−1)...(0.5−n+1) n! are binomial coefficients. From the above one calculates as well as To complete the definition one leaves the counit ǫ undeformed. Let us observe that ǫ(Π 0 ) = ǫ(Π −1 0 ) = 1. It is also worth noticing that the square of the antipode (11)-(12) is given by a similarity transformation 1 , i.e.
one gets the deformed coproducts of the form Similarly the commutators of new generators can be obtained as with the remaining one being the same as for Poincaré Lie algebra (1)- (2). This proves that our deformed Hopf algebra (1)-(2), (7)-(10) is Hopf isomorphic to the κ -Poincaré Hopf algebra [7] written in its bicrossproduct basis (M i , N i , P µ ) [10]. From now on we shall denote the Hopf The following immediate comments are now in order: i) Substituting N i = M 0i and ǫ i jk M k = M i j the above result easily generalizes to the case of κ -Poincaré Hopf algebra in an arbitrary spacetime dimension n (with the Lorentzian signature).
ii) Although P 1,n−1 is Lie subalgebra of P 1,n the corresponding Hopf algebra U(1, n − 1)[[κ −1 ]] 1 In the case of twisted deformation the antipode itself is given by the similarity transformation.
iii) Changing the generators by a similarity transformation X → S XS −1 for X ∈ (M i , N i , P µ ) leaves the algebraic sector (1)-(2) unchanged but in general it changes coproducts (7) Magueijo-Smolin type DSR theories [19], [34] (DSR2) even if we do not intend to take into account coproducts. But the coproducts are there and can be used, e.g., in order to introduce an additional law for four-momenta. In this situation the κ-deformed coproducts are not necessary the privileged one and the additional law can be determined by, e.g., the twisted coproducts [13].

III. NEW ALGEBRAIC FORM OF κ-POINCARÉ HOPF ALGEBRA
The mathematical formalism of quantum groups requires us to deal with formal power series.
Therefore the parameter κ has to stay formal, i.e., undetermined. Particularly, we can not assign any particular numerical value to it and consequently any fundamental constant of nature, like, e.g., the Planck mass cannot be related with it. There are in principle two methods to remedy this situation and allow κ to admit constant value.
The first one is to reformulate algebra in such a way that all infinite series will be eliminated on the abstract level. In the traditional Drinfeld-Jimbo approach this is always possible by using the so-called specialization method or q-deformation (see e.g. [5], [6]) 2 . The idea is to replace P 0 by two group-like elements Π 0 , Π −1 0 and "forget" relation (13). This provides, for any specific (complex) numerical value κ 0, a new quantum algebra U κ (1, 3). It is defined as a universal, unital associative algebra generated by eleven generators (M i , N i , P i , Π 0 , Π −1 0 ) being a subject of the standard Poincaré Lie-algebra commutation relations (1)-(2) except those containing P 0 . These last should be replaced by the following new ones (Π 0 and Π −1 0 are considered mutually inverse) The Hopf algebra structure is determined on U κ (1, 3) by the same formulae (7)-(15) except those containing P 0 . It should be noticed that these formulas contain only finite powers of the numerical parameter κ. The generator P 0 can be now introduced as Thus subalgebra generated by elements (M i , N i , P i , P 0 ) is, of course, isomorphic to the universal envelope of the Poincaré Lie algebra, i.e. U P 1,3 ⊂ U κ (1, 3). But this is not a Hopf subalgebra.

IV. HILBERT SPACE REALIZATIONS
The second method allowing to specify value of κ relies on representation theory. Let us con-  [36] (see [11] for different representation). There are canonical commutation relations between (local) momentum and position operators represented by standard multiplication and differentiation operators: x µ and p µ = −ı∂ µ . Representation of the Poincaré Lie algebra in this Hilbert space can be chosen, for example, as: where △ = − p 2 denotes the Laplace operator. Now all operators in the above formulas are well defined for constant value of κ as Hilbert space operators. Moreover, it turns out that operators (M i , N i , p µ ) constitute the bicrossproduct basis. Therefore, dispersion relations expressed in canonical momenta p µ are the standard DSR.
One can notice that boost generators (27) in this representation are not Hermitian, because of the last term. However Hermitian representation of the κ -Poincaré algebra can be determined as: and the dispersion relation is In both cases above the representation of the element Π 0 in the Hilbert space realization is given by the same formula Π 0 = e P 0 κ (cf. also (17)) (see [11], [12]).
In the minimal case, connected with Weyl-Poincaré algebra, in physical n = 4 dimensions [16] the representation of the Poincaré algebra (M i , N i , P µ ) reads as The generator Π 0 has now the form Π 0 = 1 + p 0 κ and the deformed Casimir operator leads to following dispersion relation which is not deformed for (free) massless particles.
We close this note with an open question concerning choice of "physical" Casimir operator leading to correct dispersion relation and its operator realization.

Conclusions
In this letter we have introduced two different Hopf algebras of κ− Poincaré as quantum deformation of the Drinfeld-Jimbo type. The first one is related to "h-adic" topology which forces the parameter κ to stay abstract and undetermined. All formulae for coproducts have been written intrinsically in a classical Lie algebra basis which is very typical for twisted Drinfeld deformation technique. As it has been already explained, in the Introduction, the existence of such classical basis also for the Drinfeld -Jimbo deformations is a direct consequence of their formalism. However the explicit construction is highly nontrivial mathematical problem and to the best our knowledge it was investigated mainly for the case of κ−Poincaré [26], [27]. Particularly, the formulas expressing classical basis in terms of bicrossproduct one have been obtained therein .
The second definition relies on reformulating the Hopf algebra structure in such a way that infinite series disappear: it provides the one-parameter family of mutually isomorphic Hopf algebras labeled by a numerical (complex in general) parameter κ. So, the particular value of κ becomes irrelevant. From the physical point of view one is allowed to work in the system of natural (Planck) units with = G = c = 1 without changing mathematical properties of the underlaying quantum model. In this way the so-called specialization problem for the deformation parameter κ has been solved 3 . Finally, it has been shown that different (proper-time) Hilbert space representation of this algebra can be understood as the corresponding to different DSR type models providing different dispersion relations. Therefore, we believe that our research might be also helpful to distinguish between two approaches to doubly special relativity theories.