Two $\theta_{\mu \nu }$ -deformed covariant relativistic quantum phase spaces as Poincare-Hopf algebroids

TL;DR

This paper explores two types of $ heta_{ u ho }$-deformed covariant relativistic quantum phase spaces modeled as Poincare-Hopf algebroids, highlighting their algebraic structures and covariance properties under quantum Poincare transformations.

Contribution

It introduces two Hopf algebroid frameworks for $ heta_{ u ho }$-deformed quantum phase spaces, emphasizing their covariance and potential for describing multiparticle quantum structures.

Findings

The first algebroid models $ heta_{ u ho }$-deformed relativistic phase space with canonical noncommutative spacetime.

The second algebroid incorporates dual quantum group algebra with $ heta_{ u ho }$-deformed translations dependent on Lorentz parameters.

Both frameworks are covariant under quantum Poincare transformations.

Abstract

We consider two quantum phase spaces which can be described by two Hopf algebroids linked with the well-known $\theta_{\mu \nu }$-deformed $D=4$ Poincare-Hopf algebra $\mathbb{H}$. The first algebroid describes $\theta_{\mu \nu }$-deformed relativistic phase space with canonical NC space-time (constant $\theta_{\mu \nu }$ parameters) and the second one incorporates dual to $\mathbb{H}$ quantum $\theta_{\mu \nu }$-deformed Poincare-Hopf group algebra $\mathbb{G}$, which contains noncommutative space-time translations given by $\Lambda $-dependent $\Theta_{\mu \nu }$ parameters ($% \Lambda $ $\equiv \Lambda_{\mu \nu }$ parametrize classical Lorentz group). The canonical $\theta_{\mu \nu }$-deformed space-time algebra and its quantum phase space extension is covariant under the quantum Poincare transformations described by $\mathbb{G}$. We will also comment on the use of Hopf algebroids for the description of multiparticle structures in quantum phase spaces.

Equations

$$[{\hat{x}}_{\mu },{\hat{x}}_{\nu }]=i{\lambda }^2{\theta }_{\mu \nu },$$

$${\lambda }_P=\frac{\hslash }{m_{P}c}=\sqrt{\frac{\hslash G}{c^3}},$$

$$\mathcal{F}\equiv {\mathcal{F}}_{(1)}\otimes {\mathcal{F}}_{(2)}=\exp \left(\frac{i}{2}{\lambda }^2{\theta }^{\mu \nu }{p}_{\mu }\otimes p_{\nu }\right).$$

$${\Delta }_{\mathcal{F}}(h)$$

$$S_{\mathcal{F}}(h)$$

$$f(\hat{x})\cdot g(\hat{x})\simeq f(x){\star }_{\mathcal{F}}g(x)=m[{\mathcal{F}}^{-1}\circ (f\otimes g)]=({\mathcal{F}}_{(1)}^{-1}⊳f)({\mathcal{F}}_{(2)}^{-1}⊳g).$$

$${\hat{x}}_{\mu }=m[{\mathcal{F}}^{-1}(⊳\otimes 1)(x_{\mu }\otimes 1)]=({\mathcal{F}}_{(1)}^{-1}⊳x_{\mu }){\mathcal{F}}_{(2)}^{-1}.$$

$$\left[{\hat{x}}_{\mu },{\hat{x}}_{\nu }\right]\simeq {\left[x_{\mu },x_{\nu }\right]}_{{\star }_{\mathcal{F}}}\equiv x_{\mu },{\star }_{\mathcal{F}}{x}_{\nu }-x_{\nu }{\star }_{\mathcal{F}}{x}_{\mu }=i{\lambda }^2{\theta }_{\mu \nu }.$$

$$\left[{\hat{\xi }}_{\mu },{\hat{\xi }}_{\nu }\right]=i{\lambda }^2{\theta }^{\rho \sigma }({\eta }_{\mu \rho }{\eta }_{\nu \sigma }-{\hat{\Lambda }}_{\mu \rho }{\hat{\Lambda }}_{\nu \sigma }):=i{\lambda }^2{\Theta }_{\mu \nu }(\hat{\Lambda }).$$

$${\hat{x}}_{\mu }^{\prime }={\hat{\Lambda }}_{\mu \nu }{\hat{x}}^{\nu }+{\hat{\xi }}_{\mu },$$

$$\hat{\mathbf{X}}⋊\mathbb{H}=\hat{\mathbf{X}}⋊({\mathcal{T}}^4⋊O(3,1)).$$

$$[p_{\mu },p_{\nu }]$$

$$[M_{\mu \nu },p_{\rho }]$$

$$[M_{\mu \nu },M_{\rho \sigma }]$$

$${\Delta }_0(h)=h\otimes 1+1\otimes h,S_0(h)=-h,{ϵ}_0(h)=0.$$

$${\Delta }_{\mathcal{F}}(p_{\mu })$$

$${\Delta }_{\mathcal{F}}(M_{\mu \nu })$$

$$-\frac{1}{2}{\theta }^{\alpha \beta }[({\eta }_{\alpha \mu }{p}_{\nu }-{\eta }_{\alpha \nu }{p}_{\mu })\otimes p_{\beta }+p_{\alpha }\otimes ({\eta }_{\beta \mu }{p}_{\nu }-{\eta }_{\beta \nu }{p}_{\mu })].$$

$${\hat{X}}_A\cdot {\hat{X}}_B\simeq X_A{\star }_{\mathcal{F}}{X}_B=m[{\mathcal{F}}^{-1}\circ (X_A\otimes X_B)]=({\mathcal{F}}_{(1)}^{-1}⊳X_A)({\mathcal{F}}_{(2)}^{-1}⊳X_B)$$

$$h⊳X_A=[h,X_A],h=\{p_{\mu },M_{\mu \nu }\},X_A=\{x_{\mu },{\Lambda }_{\mu \nu }\}$$

$$[p_{\mu },x_{\nu }]$$

$$[p_{\mu },{\Lambda }_{\rho \sigma }]$$

$$f(X){\star }_{\mathcal{F}}g(X^{\prime })=\widehat{f(X)}⊳g(X^{\prime })$$

$$f(\hat{X})\simeq \widehat{f(X)}=m[{\mathcal{F}}^{-1}(⊳\otimes 1)(f(X)\otimes 1)]$$

$${\hat{x}}^{\mu }=x^{\mu }+\frac{1}{2}{\theta }^{\mu \alpha }{p}_{\alpha },{\hat{\Lambda }}_{\rho \sigma }={\Lambda }_{\rho \sigma }$$

$$\left[{\hat{x}}_{\mu },{\hat{x}}_{\nu }\right]=i{\theta }_{\mu \nu }$$

$$[{\hat{x}}_{\mu },{\hat{\Lambda }}_{\rho \sigma }]=[{\hat{\Lambda }}_{\mu \nu },{\hat{\Lambda }}_{\rho \sigma }]=0.$$

$$h\vartriangleright(\hat{X}_{A}\hat{X}_{B})=(h_{(1)}\vartriangleright\hat{X}_{A})(h_{(2)}\vartriangleright\hat{X}_{B})\$$

$$[p_{\mu },{\hat{x}}_{\nu }]$$

$$[p_{\mu },{\hat{\Lambda }}_{\rho \sigma }]$$

$$[M_{\mu \nu },{\hat{\Lambda }}_{\rho \sigma }]$$

$$[M_{\mu \nu },{\hat{x}}_{\rho }]$$

$$\mathcal{R}={\mathcal{F}}^T{\mathcal{F}}^{-1}=\exp [-i{\theta }^{\mu \nu }{p}_{\mu }\otimes p_{\nu }](a\otimes b{)}^T=b\otimes a$$

$$(M_{\mu \nu }{)}_B^A={\delta }_{\mu }^A{\eta }_{\nu B}-{\delta }_{\nu }^A{\eta }_{\mu B}(p_{\mu }{)}_B^A={\delta }_{\mu }^A{\delta }_B^4$$

$$\mathcal{R}=1\otimes 1-i{\theta }^{\mu \nu }{p}_{\mu }\otimes p_{\nu }.$$

Full text

1 Introduction

There have been proposed in recent years various models of noncommutative (NC) space-times which characterizes space-time geometry if the quantum gravity (QG) effects are included (see e.g. [1]-[8]). In this paper we shall study the canonical case of quantum space-times, with NC counterparts $\hat{x}_{\mu}~{}(\mu=0,1,2,3,4)$ of space-time coordinates satisfying the well-known formula

[TABLE]

where $\theta_{\mu\nu}=-\theta_{\nu\mu}$ is a numerical $4\times 4$ matrix111In this paper we shall consider the case $D=3+1$ and $[\lambda]=[l]$ describes an elementary length, which for QG-generated noncommutativity is linked with Planck length

[TABLE]

where $m_{P}$ is the Planck mass, and $G$ describes the Newton constant characterizing the gravitational interactions. If $\lambda$ is proportional to $\lambda_{P}$, from formula (2) due to the presence of Planck constant $\hbar$, one can deduce the quantum-mechanical and QG origin of relation (1) $(\lambda_{P}=0$ if $\hbar\rightarrow 0$ or $G\rightarrow 0)$. Further for simplicity we shall use the choice of units $\hbar=c=1$.

The well-known $\theta_{\mu\nu}$-deformed quantum space-times (see (1)) and associated quantum phase spaces are generated by the following Abelian twist [9],[10]

[TABLE]

It defines $\theta_{\mu\nu}$-deformed quantum Poincare-Hopf algebra $\mathbb{H}$, with nondeformed Poincare algebra sectors however with modified coproducts and antipodes [11]

[TABLE]

The twist (3) can be employed in two ways:

1. i)

We introduce the classical Minkowski space-time coordinates $x_{\mu}\in\mathbf{X}$ as vectorial representation of relativistic symmetries described by classical Poincare algebra $\mathcal{P}$, with the algebraic structure of $\mathcal{P}\otimes\mathbf{X}$ given by the semidirect product $\mathcal{P}\rtimes\mathbf{X}$. In quantum-deformed theory one introduces the NC space-times $(\hat{x}_{\mu}\in$ $\mathbf{\hat{X}}\mathbb{)}$ as the module algebra (NC algebraic representation) of quantum Poincare-Hopf algebra $\mathbb{H}$. In the case of $\theta_{\mu\nu}$-deformation the quantum space-time coordinates $\hat{x}_{\mu}\in\mathbf{\hat{X}}$ are obtained from Drinfeld twisting procedure [11] by star product technique [12],[13].

Let us consider the algebra $\hat{A}$ of functions on $\mathbf{\hat{X}}$ $(f(\hat{x}\mathbb{)\in}\hat{A})$. In the scheme of twist quantization one can represent the algebra $(\hat{A},\cdot)$ by the algebra $(A,\star_{\mathcal{F}})$ of classical functions, with their multiplication defined by the nonlocal star product $\star_{\mathcal{F}}$222Because the twist $\mathcal{F}$ is the function of classical Poincare algebra generators $\hat{g}=(p_{\mu},M_{\mu\nu})$, the action $\hat{g}\vartriangleright f(x)$ in formula (6) is described by the differential realization of classical Poincare algebra on functions of standard Minkowski coordinates $x_{\mu}$.

[TABLE]

Putting in (6) $f(\hat{x})=\hat{x}_{\mu},g(\hat{x})\equiv 1$ one gets

[TABLE]

Formula (7) provides the quantum map expressing the deformed NC space-time coordinates by a nonlocal map in undeformed relativistic phase space $(x_{\mu},p_{\mu}=\frac{1}{i}\partial_{\mu})$. If we put $f(x)=x_{\mu},g(x)\equiv x_{\nu}$ one can calculate from (6) the commutator given by the formula (1).

[TABLE] 2. ii)

By Hopf-algebraic duality one can define the $\theta_{\mu\nu}$-deformed quantum Poincare group $\mathbb{G}$, with generators $\hat{g}=\{\hat{\xi}_{\mu},\hat{\Lambda}_{\mu\nu}\}$, describing generalized NC coordinates on algebraic $\theta_{\mu\nu}$-deformed Poincare group manifold [14] with quantum Lorentz group parameters $\hat{\Lambda}_{\mu\nu}$ and the NC quantum Poincare group translations $\hat{\xi}_{\mu}$ satisfying the following algebra

[TABLE]

Using the Heisenberg double construction [5],[15] given by particular choice of semidirect product $\mathbb{G\rtimes H}$ called smash product, one obtains $(10+10)$-dimensional generalized $\theta_{\mu\nu}$-deformed quantum phase space $\mathcal{H}^{(10+10)}=(\hat{\xi}_{\mu},\hat{\Lambda}_{\mu\nu},p_{\mu},M_{\mu\nu})$. Such phase space can be employed in physical applications for the description of NC dynamics on algebraic $\theta_{\mu\nu}$-deformed quantum Poincare group manifold333The idea of phase space description of the dynamics on the classical group and coset group manifolds is due to Souriau [16] and Kostant [17]..

It appears that both $\theta_{\mu\nu}$-deformed structures presented above (see (1) and (9)) are necessary in order to describe in complete way the NC space-time (8) as describing quantum Poincare-covariant $\mathbb{H-}$module, which transform under quantum Poincare group $\mathbb{G}$ in the following standard way

[TABLE]

where NC translations $\hat{\xi}_{\mu}\in\mathbf{T}$ satisfy the relation (9). We shall show that to describe the quantum Poincare transformations (10) using star-product formula one should extend the star multiplication (6) to the functions of NC variables $\hat{x}_{\mu}$ (NC Minkowski space) and $\hat{\xi}_{\mu}$ (NC Poincare translations) (see [14])444See [18] for the use of star product to represent the quantum group transformations..

The plan of our paper is following. In Sect. 2 we recall the $\theta_{\mu\nu}$-deformed Poincare-Hopf algebra $\mathbb{H}$ extended by NC space-time coordinates $\hat{x}_{\mu}\in\mathbf{\hat{X}}$ which describe the Lorentz group extension of the relativistic phase space $(\hat{x}_{\mu},p_{\mu})\in\mathbf{\hat{X}}\mathbb{\rtimes}\mathcal{T}^{4}$

[TABLE]

In Sect. 3 we describe the $\theta_{\mu\nu}$-deformed Poincare quantum group $\mathbb{G}$ and calculate the $\theta_{\mu\nu}$-deformed Heisenberg double $\mathcal{H}^{(10+10)}=\mathbb{H}\rtimes\mathbb{G}$, with generalized NC Poincare coordinates $\mathbb{\{}\hat{\xi}_{\mu},\hat{\Lambda}_{\mu\nu}\}\in\mathbb{G}$ and generalized momenta $\{p_{\mu},M_{\mu\nu}\}\in\mathbb{H}$. In Sect. 4 we shall derive the covariance under the quantum Poincare group transformations given by formula (10). In Sect. 5 we specify the data which define two Hopf algebroids, first providing the quantum Poincare-covariant $\theta_{\mu\nu}$-deformed phase space $(\hat{x}_{\mu},p_{\mu})\in\mathbf{\hat{X}}\mathbb{\rtimes}\mathcal{T}^{4}$ with positions (coordinates) described by algebra $\mathbf{\hat{X}}$ $\mathbb{(}\hat{x}_{\mu}\mathbb{\in\mathbf{\hat{X}})}$ supplemented with Lorentz transformations (Lorentz parameters $\Lambda_{\mu\nu}$) and the second Hopf algebroid describing quantum $\theta_{\mu\nu}$-deformed Poincare symmetry transformations (see (10)), with coordinate sector described by quantum $\theta_{\mu\nu}$-deformed Poincare group $\mathbb{G}$. In particular in Sect. 5.4. by following earlier applications of Hopf algebroids to the description of $\kappa$-deformed quantum phase spaces [19],[20] we shall consider the applicability of coalgebra sector of $\theta_{\mu\nu}$-deformed Hopf algebroids to the phase space description of the multi-particle states.

2 $\theta_{\mu\nu}$-deformed quantum

Poincare algebra $\mathbb{H}$ and NC space-time as $\mathbb{H}$-module

2.1

Twist deformed quantum Poincare algebra $\mathbb{H}$

The classical $D=4$ Poincare-Hopf algebra looks as follows

[TABLE]

where $\eta_{\mu\nu}=diag(-1,1,...,1)$ and is supplemented by primitive costructure maps

[TABLE]

The twist $\mathcal{F}$ is an element of  $\mathbb{H}\otimes\mathbb{H}$ $(\mathbb{H=}\mathcal{U(P)})$ which has an inverse, satisfies the cocycle condition $\mathcal{F}_{12}\,\left(\Delta_{0}\otimes 1\right)\,\mathcal{F}=\mathcal{F}_{23}\,\left(1\otimes\Delta_{0}\right)\,\mathcal{F}$  and the normalization condition $(\epsilon\otimes 1)\mathcal{F}=(1\otimes\epsilon)\mathcal{F}=1\,$where $\mathcal{F}_{12}=\mathcal{F}\otimes 1$ and $\mathcal{F}_{23}=1\otimes\mathcal{F}$.

The twist $\mathcal{F}$ does not modify the algebraic part and the counit, but changes the coproducts $\Delta:\mathbb{H}\rightarrow\mathbb{H}\otimes\mathbb{H}$ and the antipodes $S:\mathbb{H}\rightarrow\mathbb{H}$ according to formulae (4)-(5). The quantum $\theta$-deformation is generated by the twist (3) . From the formula (4) one gets the coproducts

[TABLE]

From (3) and (5) follows that in considered case of $\theta_{\mu\nu}$-deformations $U=1$ and the antipodes remain unchanged, i.e. $S_{\mathcal{F}}(h)=S_{0}(h)=-{h.}$

2.2 Algebra of generalized coordinates $\mathbb{\hat{X}}$ as twisted

$\mathbb{H}$-module

For twist-deformed case we can introduce the deformed coordinates algebra $\mathbb{\hat{X}\ni}\hat{X}_{A}\mathbb{=}\{\hat{x}_{\mu},\hat{\Lambda}_{\mu\nu}\}$ with the multiplication given by the star product formula

[TABLE]

where

[TABLE]

and in undeformed case we obtain

[TABLE]

If we choose $X_{A}=f(X),X_{B}=g(X)$ the formula (16) can be also written as follows

[TABLE]

where $\widehat{f(X)}$ denotes the noncommutative star representation of $f(\hat{X})$ defined by the formula (see also (7))

[TABLE]

For the twist (3) we get from (21) the following explicit formulas describing generalized coordinates $\hat{X}_{A}=\{\hat{x}_{\mu},\hat{\Lambda}_{\mu\nu}\}$ in terms of undeformed relativistic quantum phase space variables $(x_{\mu},p_{\mu})$ and $\Lambda_{\mu\nu}$

[TABLE]

i.e. Lorentz group parameters remain classical. Due to (18) and (22) one gets the expected algebraic relations

[TABLE]

Using the relation

[TABLE]

following [10] one can check easily that the commutators (23)-(24) are covariant under the action (25) of $\theta_{\mu\nu}$-deformed Poincare-Hopf algebra.

2.3 $\theta_{\mu\nu}$-deformed quantum

phase space $\mathcal{H}_{\theta}^{(10+10)}=(\hat{x}_{\mu},\hat{\Lambda}_{\mu\nu};p_{\mu},M_{\mu\nu})$

Using (22) one can check the following set of cross commutators

[TABLE]

Together with commutators (23)-(24) the set of relations (26)-(2.3) satisfies the Jacobi identities and defines the algebra of $\theta_{\mu\nu}$-deformed quantum phase space $\mathcal{H}_{\theta}^{(10+10)}$.

3 $\theta_{\mu\nu}$-deformed quantum

Poincare matrix group $\mathbb{G}$ and corresponding Heisenberg double $\mathbb{G\rtimes H}$

3.1 $RTT$ quantization method and $\theta_{\mu\nu}$-deformed quantum Poincare group algebra $\mathbb{G}$

The universal $\mathcal{R}$-matrix ($(a\wedge b=a\otimes b-b\otimes a)$)

[TABLE]

can be used for the description of 10-generator deformed $D=4$ Poincare group. Using the $5\times 5$ - matrix realization of the Poincare generators

[TABLE]

we can show that in (30) only the linear term is non-vanishing, i.e.

[TABLE]

To find the matrix quantum group, which provides the Hopf algebra dual to $\mathbb{H}$ in the matrix realization (31), we introduce the following $5\times 5$ - matrices

[TABLE]

where $\hat{\Lambda}_{\mu\nu}$ parametrizes the quantum Lorentz rotation and $\hat{\xi}_{\mu}$ denotes quantum translations. In the framework of the $\mathcal{FRT}$ procedure, the algebraic relations defining such a quantum group $\mathbb{G}$ are described by the following relation

[TABLE]

while the composition law for the coproduct remains classical $\Delta(\hat{\mathcal{T}}_{AB})=\hat{\mathcal{T}}_{AC}\otimes\hat{\mathcal{T}}_{\ B}^{C}$ with $\hat{\mathcal{T}}_{1}=\hat{\mathcal{T}}\otimes 1$, $\hat{\mathcal{T}}_{2}=1\otimes\hat{\mathcal{T}}$ and quantum $\mathcal{R}$-matrix (32) given in the representation (31).

In terms of the basis $(\hat{\xi}_{\mu},\hat{\Lambda}_{\mu\nu})$ of $\mathbb{G}$ the algebraic relations (34), describing the quantum group algebra, can be written as follows

[TABLE]

while the coproduct takes the well known classical form

[TABLE]

One can check that coproducts (37) are homomorphic to the algebra (35)-(36) defining $\theta_{\mu\nu}$-deformed quantum Poincare group.

3.2 Duality between quantum Hopf algebras $\mathbb{H}$ and $\mathbb{G}$ and Heisenberg double $\mathcal{H=}\mathbb{H\rtimes G}$

Two Hopf algebras  $\mathbb{H},\mathbb{G}$ are said to be dual if there exists a nondegenerate bilinear form $\langle\,,\,\rangle\,:\mathbb{H}\mathcal{\times\mathbb{G}\longrightarrow}\mathbb{C},(h,\hat{g})\longrightarrow\langle h,\,\hat{g}\,\rangle\,$such that the duality relations

[TABLE]

are satisfied. In our considerations the following pairing relations

[TABLE]

are appropriate. The basic action of $\mathbb{H}$ on $\mathbb{G}$ promoting $\mathbb{G}$ to the $\mathbb{H}$-module is given by the following relation

[TABLE]

After using (38) one gets the relation

[TABLE]

what establishes that algebra $\mathbb{G}$ is indeed the $\mathbb{H}$-module.

In Heisenberg double framework we can obtain cross commutators between the algebra $\mathbb{H}$ and group $\mathbb{G}$ by the following relation

[TABLE]

In such a way we obtain the quantum phase space algebra555For the case of $\kappa$-deformed quantum phase space see [15].. Using pairing (40), coproducts (14), (15) and formula (37) we get

[TABLE]

The Hopf algebroid $\mathcal{\tilde{H}}^{(10+10)}=(\hat{\xi}_{\mu},\hat{\Lambda}_{\mu\nu};p_{\mu},M_{\mu\nu})$ introduces an alternative model of $\theta_{\mu\nu}$-deformed quantum phase space. It describes the quantum phase space characterizing the dynamical system with the coordinates $(\hat{\xi}_{\mu},\hat{\Lambda}_{\mu\nu})$ which are specified by the NC quantum Poincare group manifold $\mathbb{G}$.

4 The covariance of $\mathbb{\hat{X}}$ under quantum Poincare group $\mathbb{G}$ and the generalized star product

4.1

The covariance of $\widehat{\mathbb{X}}$ under quantum Poincare group $\mathbb{G}$

We recall that $\widehat{\mathbb{X}}$ is the algebra of generalized coordinates $\hat{X}_{A}=\{\hat{x}_{\mu},\hat{\Lambda}_{\mu\nu}\}$ and $\mathbb{G}$ is the algebra of Poincare symmetry parameters $\hat{g}=\{\hat{\xi}_{\mu},\hat{\Lambda}_{\mu\nu}\}$. One performs the quantum Poincare transformations of $\hat{X}_{A}$ in the following way

[TABLE]

The commutators of algebra $\widehat{\mathbb{X}}$ (see (23)-(24)) are invariant under a such transformation provided that the quantum Poincare symmetry parameters $\mathbb{G}\ni\hat{g}=\{\hat{\xi}_{\mu},\hat{\Lambda}_{\mu\nu}\}$ satisfy the relations defining the algebra $\mathbb{G}$ (see (35)-(36)) and $[\hat{X}_{A},\hat{g}]=0$. In particular we have

[TABLE]

We see that the generalized coordinates algebra $\widehat{\mathbb{X}}$ in different $\mathbb{G}$-frames specified by (48)-(49) transform covariantly under the transformations of quantum Poincare group $\mathbb{G}$.

In order to describe effectively the quantum Poincare transformations (48) of NC functions $f(\hat{x}_{\mu})$ it is convenient to introduce the generalized star products (see also [18]) representing the algebra of functions $F(\hat{x}_{\mu},\hat{\xi}_{\mu})$ depending as well on NC translations $\hat{\xi}_{\mu}$.

4.2 Star product on the product $\mathbf{X}\mathbb{\otimes}\mathbf{T}$ with noncommutative translations of coordinates

Let us consider firstly the star product $\star_{\mathcal{F}}^{{}^{\prime}}$ describing the NC product of algebra of functions $F(\hat{\xi}_{\mu})$ which depend on NC translations $\hat{\xi}_{\mu}\in\mathbb{G}$ satisfying the relation (9)

[TABLE]

where in formula (51) one treats the Lorentz group parameters $\Lambda\equiv$ $\Lambda_{\mu\nu}$ as the numerical ones. Discussing the quantum Poincare transformations (10) in field theory we should deal with NC algebra of functions on $\mathbf{X}$ $(x_{\mu}\in\mathbf{X})$ as well as on $\mathbf{T}$ $(\xi_{\mu}\in\mathbf{T})$, and subsequently use the composite star product $\tilde{\star}_{\mathcal{F}}=\star_{\mathcal{F}}\cdot\star_{\mathcal{F}}^{{}^{\prime}}$ (see also [21])

[TABLE]

In formula (4.2) both relations (1), (9) are taken into account and we can represent the NC quantum Poincare transformations given by $\mathbb{G}$ by using such a star product language. In particular one can show that using the relation $\Lambda_{\mu\alpha}\Lambda_{\nu\beta}\theta^{\alpha\beta}+\Theta_{\mu\nu}(\Lambda)=\theta_{\mu\nu}$ (see (9)) we get

[TABLE]

In such a way we expressed in star product language the NC Poincare group transformations by classical Poincare group transformations. We see that the NC structure of quantum Poincare group translations are encoded in the replacement of the star product $\star_{\mathcal{F}}\rightarrow\tilde{\star}_{\mathcal{F}}$. It can be shown by explicit calculation that three star products $\star_{\mathcal{F}},\star_{\mathcal{F}}^{{}^{\prime}}$ and $\tilde{\star}_{\mathcal{F}}$ are associative.

5 Two $\theta_{\mu\nu}$-deformed Hopf

algebroids

5.1 Briefly on Hopf algebroids

The Hopf algebroids, introduced in [22] (see also [23]), are described by bialgebroids with supplemented antipodes. It has been argued (see e.g. [24]-[28]) that the Hopf algebroids are well adjusted to the description of physically important quantum (canonical and noncanonical) phase space.

The bialgebroid $\mathcal{B}$ is specified by the set of the data $(H,A;s,t;m,\tilde{\Delta},\epsilon)$ where $H$ is the total algebra with product $m$ and its subalgebra $A\subset H$ is called the base algebra. The source map $s(a):A\rightarrow H$ is a homomorphism and the target map $t(a):A\rightarrow H$  an antihomomorphism, with their images commuting

[TABLE]

The canonical choice of the source map is $s(a)=a$. One can introduce natural $(A,A)-$bimodule structure on $H$ using source and target maps in the basic $(A,A)$-bimodule formula, namely $ahb=ht(a)s(b)$. The coproducts $\tilde{\Delta}$ are described by the maps $H\rightarrow H\otimes_{A}H$ from $H$ into $(A,A)$ bimodules $H\otimes_{A}H$, satisfying the co-associativity condition

[TABLE]

Because $H\otimes_{A}H$ as the codomain of coproducts $\tilde{\Delta}$ does not inherit the algebra structure from $H\otimes H$, in order to have well defined multiplication one introduces the submodule $H\times_{A}H\subset H\otimes H$ defined by the Takeuchi coproduct [29].

The algebra $H$ with the product $m$ and the coalgebra with Takeuchi coproduct $\tilde{\Delta}$ are compatible, i.e.

[TABLE]

The coproduct $\tilde{\Delta}$ in $H\times_{A}H$ can be expressed in terms of standard tensor product $H\otimes H$ by the equivalence classes satisfying the condition

[TABLE]

where $\mathcal{I}_{L}$ define the left ideal in $H\otimes H.$ The equivalence classes defined by (57) are parametrized by so-called coproduct gauge (see e.g. [20]).

The counit map $\epsilon:H\rightarrow A$ is satisfying $\epsilon(1_{H})=$ $1_{A}$, and

[TABLE]

We get Hopf algebroids if we are able to introduce an antipode (bijective map) $S:H\rightarrow H$ which is an algebra antihomomorphism and satisfies the following properties

[TABLE]

where in general case we need additional linear map $\gamma:H\otimes_{A}H\rightarrow H\otimes H$, so-called anchor map.

5.2 First choice: the coordinates $\widehat{\mathbb{X}}$ as the base

algebra (following [30])

Let us choose the bialgebroid group coproducts for base algebra generators $\hat{X}_{A}=\{\hat{x}_{\mu},\hat{\Lambda}_{\mu\nu}\}$

[TABLE]

satisfies the group and generalized quantum phase space algebra (26)-(2.3).

The source $s(\hat{X}_{A})$ and the target $t(\hat{X}_{A})$ maps are the following

[TABLE]

where $s_{0}(X_{A})=X_{A}$, $t_{0}(X_{A})=X_{A}$. The maps (63)-(65) satisfy the following relations

[TABLE]

The counit has canonical form

[TABLE]

Using (59) one gets explicit formulae for the antipodes

[TABLE]

In our case we have $S^{2}=1.$ One can check that for ideal $\mathcal{I}=t\otimes 1-1\otimes s$ it is true that

[TABLE]

and it follows that we do not need the anchor map (see (60)).

In order to determine the ideal (57) let us start form the nondeformed ideal (for $\theta_{\mu\nu}=0$)

[TABLE]

We can obtain $\widehat{X}_{A}$ from $X_{A}$ by using twisting formula (21). One gets for our twisted algebroid the following twist-deformed ideal (see (63)-(65))

[TABLE]

In particular one can check the following relations

[TABLE]

If we change the bialgebroid coproduct (62) by introducing the following coproduct gauge transformation (see also [19],[24],[26])

[TABLE]

it follows from (74) that the new bialgebroid coproduct (75) describes the homomorphism of the commutation relations (23)-(24) extended by the phase space commutators (26)-(2.3).

The formula (75) after substituting the formulae (63)-(65) for source as well as target map and using the exponential parametrization

[TABLE]

takes for example for $\lambda=-\frac{1}{2}$ the following explicit form

[TABLE]

where (77) describes the $\theta_{\mu\nu}$-deformation of the symmetric formula $\tilde{\Delta}_{-\frac{1}{2}}(x_{\mu})=\frac{1}{2}(x_{\mu}\otimes 1+1\otimes x_{\mu})$666For a pair of free non-relativistic two-particle with the same masses the coproduct $\tilde{\Delta}_{-\frac{1}{2}}(x_{\mu})$ describes the global coordinates describing the center of mass (see [19], Sect. 4)..

5.3 Second choice: quantum group $\mathbb{G}$ as the base algebra

(following [22])

The half-primitive bialgebroid coproducts for $\hat{g}=\{\hat{\xi}_{\mu},\hat{\Lambda}_{\mu\nu}\}$

[TABLE]

together with the coproducts (14)-(15) satisfy the Heisenberg double commutators (see (35)-(36) and (44)-(3.2)).

The source $s(\hat{g})$ and the target $t(\hat{g})$ maps should be consistent with base algebra in the following sense

[TABLE]

where due to (35) the relations (81) describe quadratic algebras. Subsequently, it can be easily shown that analogously to (63)-(65) one gets

[TABLE]

The counit is

[TABLE]

and the antipodes which are given by

[TABLE]

satisfy the required relations (59)-(61). Similarly as in Sect 5.2 $S^{2}=1$ and we do not need the anchor map.

If we consider the ideal

[TABLE]

and use the formulae (83)-(84) one can introduce as well the counterpart of the coproduct gauge transformations (75) and (77)-(5.2).

6 Outlook

In this paper we considered the most popular in the literature Moyal quantum deformation of space-time coordinates and the corresponding quantum-deformed noncanonical phase-spaces, described by $\theta_{\mu\nu}$-deformation of relativistic Heisenberg algebra. Our aim was to present the pair of $\theta_{\mu\nu}$-deformed phase spaces in the language of Hopf algebroid, with the extension of translational sectors $(\hat{x}_{\mu},p_{\mu})$ or $(\hat{\xi}_{\mu},p_{\mu})$ by the rotational Lorentz phase space coordinates $(\hat{\Lambda}_{\mu\nu},M_{\mu\nu})$, describing in relativistic particle models the spin degrees of freedom.

There were introduced two different relativistic quantum phase spaces with Hopf-algebroid structure  described in two ways:

by twist deformation of classical canonical Heisenberg bialgebroid, describing $\theta_{\mu\nu}$-deformed relativistic quantum phase space with NC Minkowski space-time coordinates and dual commuting fourmomenta (see e.g. [23]-[26],[30]),

• -

by considering dual pairs of quantum-deformed Poincare-Hopf algebras (see [20]) which define the Poincare-Heisenberg double as semidirect (smash) product of quantum $\theta_{\mu\nu}$-deformed Poincare group describing generalized coordinate sector and dual quantum Poincare algebra which provides the generalized momenta sector (see e.g. [20],[27]).

These methods were applied in order to obtain two versions of $\theta_{\mu\nu}$-deformed quantum phase spaces. We considered the $(10+10)$-dimensional generalized phase spaces, with generalized coordinates described respectively by $(\hat{x}_{\mu},\hat{\Lambda}_{\mu\nu})$ and $(\hat{\xi}_{\mu},\hat{\Lambda}_{\mu\nu})$ and different noncommutativity for $\hat{x}_{\mu}$ and $\hat{\xi}_{\mu}$: first one characterized by constant $\theta_{\mu\nu}$ (see (1)), and second one with $\Lambda$-dependent $\Theta_{\mu\nu}(\hat{\Lambda})$, with commutation relations quadratic in $\hat{\Lambda}$ (see (35)).

The Hopf algebroids introduce new class of quantum spaces $H$ endowed with bialgebroid structure, suitable for the description of quantum manifolds with symplectic structure. The bialgebroidal coproducts $\tilde{\Delta}$ can be introduced in the framework of standard tensor products $H\otimes H$ as algebras defined modulo so-called coproduct gauges [19],[20]. The coproduct gauge freedom can parametrize various classes of dynamical particle models, with different values of physical parameters, but with the same symplectic algebraic structure in $H$, in $H\otimes H$  and higher multiparticle sectors. A simple example has been provided in [19] where it was shown that the coproduct gauges in the model describing pairs of free NR particles with masses $m_{1},m_{2}$ depend on the mass ratio $\frac{m_{1}}{m_{2}}$, which is a physical parameter. It should be pointed out however that at present the role of coproduct gauges e.g. in phase space description of physically important interacting relativistic particles still is not well understood.

Second question which could be studied is the description and classification of infinitesimal quantum deformations of Hopf algebroids. In particular it would be interesting to introduce for Hopf algebroids the notion analogous to classical $r$-matrices for Hopf algebras, as well as the bialgebroidal counterpart of Yang-Baxter and pentagon equations. Answer to the last problem is linked with previous problem, i.e. the understanding of the physical content of the notion of coproduct gauges.

Acknowledgements

J.L. and M.W. have been supported by Polish National Science Center, project 2017/27/B/ST2/01902.