Noncommutative spaces of worldlines

TL;DR

This paper constructs a noncommutative space of worldlines in Minkowski spacetime using quantum group theory, specifically the $ ext{kappa}$-Poincaré deformation, providing a new framework for quantum observers with symmetry.

Contribution

It explicitly constructs a noncommutative space of worldlines from the $ ext{kappa}$-Poincaré deformation, linking quantum group symmetry with the geometry of worldlines.

Findings

The space of worldlines can be quantized to form a noncommutative space with quantum group invariance.

The $ ext{kappa}$-Poincaré deformation leads to a symplectic structure on worldlines.

The quantum space is a product of three Heisenberg-Weyl algebras with $ ext{kappa}^{-1}$ as the quantum parameter.

Abstract

The space of time-like geodesics on Minkowski spacetime is constructed as a coset space of the Poincar\'e group in (3+1) dimensions with respect to the stabilizer of a worldline. When this homogeneous space is endowed with a Poisson homogeneous structure compatible with a given Poisson-Lie Poincar\'e group, the quantization of this Poisson bracket gives rise to a noncommutative space of worldlines with quantum group invariance. As an oustanding example, the Poisson homogeneous space of worldlines coming from the $\kappa$-Poincar\'e deformation is explicitly constructed, and shown to define a symplectic structure on the space of worldlines. Therefore, the quantum space of $\kappa$-Poincar\'e worldlines is just the direct product of three Heisenberg-Weyl algebras in which the parameter $\kappa^{-1}$ plays the very same role as the Planck constant $\hbar$ in quantum mechanics. In this way, noncommutative spaces of worldlines are shown to provide a new suitable and fully explicit arena for the description of quantum observers with quantum group symmetry.

Equations

$$\begin{array}[]{lll}[J_{a},J_{b}]=\epsilon_{abc}J_{c},&\quad[J_{a},P_{b}]=\epsilon_{abc}P_{c},&\quad[J_{a},K_{b}]=\epsilon_{abc}K_{c},\\[2.0pt] \displaystyle{[K_{a},P_{0}]=P_{a}},&\quad\displaystyle{[K_{a},P_{b}]=\delta_{ab}P_{0}},&\quad\displaystyle{[K_{a},K_{b}]=-\epsilon_{abc}J_{c}},\\[2.0pt] [P_{0},P_{a}]=0,&\quad[P_{a},P_{b}]=0,&\quad[P_{0},J_{a}]=0,\end{array}$$

$$\mathfrak{l}=\mathrm{span}\{K_a,J_a\},\mathfrak{h}=\mathrm{span}\{P_0,J_a\},$$

$$\rho(X)=x^{\alpha}\rho(P_{\alpha})+\xi^{a}\rho(K_{a})+\theta^{a}\rho(J_{a})=\left(\begin{array}[]{ccccc}0&0&0&0&0\cr x^{0}&0&\xi^{1}&\xi^{2}&\xi^{3}\cr x^{1}&\xi^{1}&0&-\theta^{3}&\theta^{2}\cr x^{2}&\xi^{2}&\theta^{3}&0&-\theta^{1}\cr x^{3}&\xi^{3}&-\theta^{2}&\theta^{1}&0\end{array}\right).$$

$$\begin{array}{rl}& G_{\mathcal{M}}=\exp x^0\rho (P_0)\exp x^1\rho (P_1)\exp x^2\rho (P_2)\exp x^3\rho (P_3)\\ & \times \exp {\xi }^1\rho (K_1)\exp {\xi }^2\rho (K_2)\exp {\xi }^3\rho (K_3)\exp {\theta }^1\rho (J_1)\exp {\theta }^2\rho (J_2)\exp {\theta }^3\rho (J_3),\end{array}$$

$$\begin{array}{rl}& L=\exp {\xi }^1\rho (K_1)\exp {\xi }^2\rho (K_2)\exp {\xi }^3\rho (K_3)\exp {\theta }^1\rho (J_1)\exp {\theta }^2\rho (J_2)\exp {\theta }^3\rho (J_3).\end{array}$$

$$\mathrm{d}{s}^2=(\mathrm{d}{x}^0{)}^2-(\mathrm{d}{x}^1{)}^2-(\mathrm{d}{x}^2{)}^2-(\mathrm{d}{x}^3{)}^2,$$

$$\displaystyle\begin{split}&G_{\mathcal{M}}=\left(\begin{array}[]{cc}1&\bar{0}\\ \bar{x}^{T}&\mathbf{\Lambda}\\ \end{array}\right),\end{split}$$

$$\begin{array}{rl}& G_{\mathcal{W}}=\exp {\eta }^1\rho (K_1)\exp y^1\rho (P_1)\exp {\eta }^2\rho (K_2)\exp y^2\rho (P_2)\exp {\eta }^3\rho (K_3)\exp y^3\rho (P_3)\\ & \times \exp {\varphi }^1\rho (J_1)\exp {\varphi }^2\rho (J_2)\exp {\varphi }^3\rho (J_3)\exp y^0\rho (P_0),\end{array}$$

$$\begin{array}{rl}& H=\exp {\varphi }^1\rho (J_1)\exp {\varphi }^2\rho (J_2)\exp {\varphi }^3\rho (J_3)\exp y^0\rho (P_0).\end{array}$$

$$\displaystyle\begin{split}&G_{\mathcal{W}}=\left(\begin{array}[]{cc}1&\bar{0}\\ {\bar{f}}^{T}&\mathbf{\Lambda}\\ \end{array}\right),\end{split}$$

$$\begin{array}{rl}& f^0(y^{\alpha },{\eta }^a)=y^1\sinh {\eta }^1+\cosh {\eta }^1\left(y^2\sinh {\eta }^2+\cosh {\eta }^2(y^0\cosh {\eta }^3+y^3\sinh {\eta }^3)\right),\\ & f^1(y^{\alpha },{\eta }^a)=y^1\cosh {\eta }^1+\sinh {\eta }^1\left(y^2\sinh {\eta }^2+\cosh {\eta }^2(y^0\cosh {\eta }^3+y^3\sinh {\eta }^3)\right),\\ & f^2(y^{\alpha },{\eta }^a)=y^2\cosh {\eta }^2+\sinh {\eta }^2(y^0\cosh {\eta }^3+y^3\sinh {\eta }^3),\\ & f^3(y^{\alpha },{\eta }^a)=y^0\sinh {\eta }^3+y^3\cosh {\eta }^3.\end{array}$$

$$\bar{x}=\bar{f}(y^{\alpha },{\eta }^a),{\xi }^a={\eta }^a,{\theta }^a={\varphi }^a.$$

$$\mathrm{d}{s}_{(1)}^2=(\cosh {\eta }^2{)}^2(\cosh {\eta }^3{)}^2(\mathrm{d}{\eta }^1{)}^2+(\cosh {\eta }^3{)}^2(\mathrm{d}{\eta }^2{)}^2+(\mathrm{d}{\eta }^3{)}^2.$$

$$\mathrm{d}{s}_{(2)}^2=(\mathrm{d}{y}^1{)}^2+(\mathrm{d}{y}^2{)}^2+(\mathrm{d}{y}^3{)}^2,\mathbf{\eta }={\mathbf{\eta }}_0,$$

$$\cosh \chi =\cosh {\eta }^1\cosh {\eta }^2\cosh {\eta }^3.$$

$$\mathrm{d}{s}_{(1)}^2=(\mathrm{d}{\eta }^1{)}^2+(\mathrm{d}{\eta }^2{)}^2+(\mathrm{d}{\eta }^3{)}^2,{\chi }^2=({\eta }^1{)}^2+({\eta }^2{)}^2+({\eta }^3{)}^2.$$

$$r=\frac{1}{\kappa }(K_1\wedge P_1+K_2\wedge P_2+K_3\wedge P_3).$$

$$\begin{array}{rl}& \delta (P_0)=\delta (J_a)=0,\\ & \delta (P_a)=\frac{1}{\kappa }{P}_a\wedge P_0,\\ & \delta (K_1)=\frac{1}{\kappa }(K_1\wedge P_0+J_2\wedge P_3-J_3\wedge P_2),\\ & \delta (K_2)=\frac{1}{\kappa }(K_2\wedge P_0+J_3\wedge P_1-J_1\wedge P_3),\\ & \delta (K_3)=\frac{1}{\kappa }(K_3\wedge P_0+J_1\wedge P_2-J_2\wedge P_1).\end{array}$$

$$\begin{array}{rl}& \{f,g\}=r^{ij}\left(X_i^{L}f{X}_j^{L}g-X_i^{R}f{X}_j^{R}g\right),f,g\in \mathcal{C}(G),\end{array}$$

$$X_i^{L}f(h)$$

$$\delta (\mathfrak{l})\subset \mathfrak{l}\wedge \mathfrak{g},$$

$$\begin{array}{rl}& \{x^0,x^a\}=-\frac{1}{\kappa }{x}^a,\{x^a,x^b\}=0.\end{array}$$

$$\begin{array}{rl}& [{\hat{x}}^0,{\hat{x}}^a]=-\frac{1}{\kappa }{\hat{x}}^a,[{\hat{x}}^a,{\hat{x}}^b]=0,\end{array}$$

$$[K_a,P_b]={\delta }_{ab}\left(\frac{\kappa }{2}\left(1-{\mathrm{e}}^{-2P_0/\kappa }\right)+\frac{1}{2\kappa }{\text{P}}^2\right)-\frac{1}{\kappa }{P}_a{P}_b,$$

$$\begin{array}{rl}& \Delta (P_0)=P_0\otimes 1+1\otimes P_0,\\ & \Delta (J_a)=J_a\otimes 1+1\otimes J_a,\\ & \Delta (P_a)=P_a\otimes 1+{\mathrm{e}}^{-P_0/\kappa }\otimes P_a,\\ & \Delta (K_a)=K_a\otimes 1+{\mathrm{e}}^{-P_0/\kappa }\otimes K_a+\frac{1}{\kappa }{ϵ}_{abc}{P}_b\otimes J_c.\end{array}$$

$$\begin{array}{rl}& \{y^1,y^2\}=\frac{1}{\kappa }\left(y^2\sinh {\eta }^1-\frac{y^1\tanh {\eta }^2}{\cosh {\eta }^3}\right),\\ & \{y^1,y^3\}=\frac{1}{\kappa }\left(y^3\sinh {\eta }^1-y^1\tanh {\eta }^3\right),\\ & \{y^2,y^3\}=\frac{1}{\kappa }\left(y^3\cosh {\eta }^1\sinh {\eta }^2-y^2\tanh {\eta }^3\right),\\ & \{y^1,{\eta }^1\}=\frac{1}{\kappa }\frac{\left(\cosh {\eta }^1\cosh {\eta }^2\cosh {\eta }^3-1\right)}{\cosh {\eta }^2\cosh {\eta }^3},\\ & \{y^2,{\eta }^2\}=\frac{1}{\kappa }\frac{\left(\cosh {\eta }^1\cosh {\eta }^2\cosh {\eta }^3-1\right)}{\cosh {\eta }^3},\\ & \{y^3,{\eta }^3\}=\frac{1}{\kappa }\left(\cosh {\eta }^1\cosh {\eta }^2\cosh {\eta }^3-1\right),\\ & \{y^a,{\eta }^b\}=0,a\ne b,\{{\eta }^a,{\eta }^b\}=0.\end{array}$$

$$\displaystyle\begin{split}&\{y^{a},y^{b}\}=\frac{1}{\kappa}\bigl{(}\eta^{a}y^{b}-\eta^{b}y^{a}\bigr{)}+\mathcal{O}(\mathbb{y},\boldsymbol{\eta})^{3},\qquad\{\eta^{a},\eta^{b}\}=0,\\ &\{y^{a},\eta^{b}\}=\delta_{ab}\,\frac{1}{2\kappa}\left((\eta^{1})^{2}+(\eta^{2})^{2}+(\eta^{3})^{2}\right)+\mathcal{O}(\mathbb{y},\boldsymbol{\eta})^{3}.\\ \end{split}$$

$$\begin{array}{rl}& \{y^0,y^1\}=-\frac{1}{\kappa }\left(y^1-y^2\frac{\sinh {\eta }^1\tanh {\eta }^2}{\cosh {\eta }^3}-y^3\sinh {\eta }^1\tanh {\eta }^3\right),\\ & \{y^0,y^2\}=-\frac{1}{\kappa }\left(y^2+y^1\frac{\sinh {\eta }^1\tanh {\eta }^2}{\cosh {\eta }^3}-y^3\cosh {\eta }^1\sinh {\eta }^2\tanh {\eta }^3\right),\\ & \{y^0,y^3\}=-\frac{1}{\kappa }\left(y^3+y^1\sinh {\eta }^1\tanh {\eta }^3+y^2\cosh {\eta }^1\sinh {\eta }^2\tanh {\eta }^3\right),\\ & \{y^0,{\eta }^1\}=-\frac{1}{\kappa }\frac{\sinh {\eta }^1}{\cosh {\eta }^2\cosh {\eta }^3},\\ & \{y^0,{\eta }^2\}=-\frac{1}{\kappa }\frac{\cosh {\eta }^1\sinh {\eta }^2}{\cosh {\eta }^3},\\ & \{y^0,{\eta }^3\}=-\frac{1}{\kappa }\cosh {\eta }^1\cosh {\eta }^2\sinh {\eta }^3,\end{array}$$

$$\begin{array}{rl}& q^1=\frac{\cosh {\eta }^2\cosh {\eta }^3}{\cosh {\eta }^1\cosh {\eta }^2\cosh {\eta }^3-1}{y}^1,p^1={\eta }^1,\\ & q^2=\frac{\cosh {\eta }^3}{\cosh {\eta }^1\cosh {\eta }^2\cosh {\eta }^3-1}{y}^2,p^2={\eta }^2,\\ & q^3=\frac{1}{\cosh {\eta }^1\cosh {\eta }^2\cosh {\eta }^3-1}{y}^3,p^3={\eta }^3,\end{array}$$

$$|\text{J}(y,\mathbf{\eta })|=\frac{\cosh {\eta }^2(\cosh {\eta }^3{)}^2}{(\cosh {\eta }^1\cosh {\eta }^2\cosh {\eta }^3-1{)}^3},$$

Full text

PACS:  02.20.Uw  03.30.+p  04.60.-m

KEYWORDS:

time-like worldlines, quantum groups, Minkowski spacetime, Poisson homogeneous spaces, kappa-deformation, non-commutative spaces, quantum observers

1 Introduction

It is widely assumed that the spacetime underlying a fundamental theory of quantum gravity should include some noncommutative structure (see for instance [1, 2, 3, 4, 5]) that generates the Planck scale uncertainty relations describing the minimum length phenomena arising in the interplay between gravity and quantum theory [6]. In this context, a mathematically self-consistent approach to such ‘quantum geometry’ consists in the construction of noncommutative (‘quantum’) spacetimes which are invariant under quantum Poincaré or (A)dS groups (see [7, 8, 9, 10, 11, 12, 13, 14, 15, 16] and references therein), which are defined as both noncommutative and noncocommutative Hopf algebra deformations [17, 18, 19] of the corresponding classical kinematical groups. In this way, several proposals for the analysis of the role played by quantum Lorentzian symmetries both on momentum space (see [20, 21, 22, 23, 24] and references therein) and on phase space [25, 26] have been presented in connection with deformed or doubly special relativity (DSR) theories [27, 28, 29, 30, 31, 32, 33, 34, 35] where generalizations of special relativity with a new invariant quantity additional to the speed of light and related with the Planck scale were introduced. Both in DSR theories and in the relative locality approach [36, 37, 38] the focus is switched from spacetime to momentum space, under the observation that the only quantities that can truly be measured are momenta and energy of particles.

The aim of this work is to propose noncommutative spaces of time-like worldlines as a new feasible alternative in order to describe the Planck scale effects encoded under quantum group symmetry, and to show that these noncommutative spaces of oriented time-like geodesics can be explicitly constructed in a systematic way. This can be achieved in a rigorous and completely general manner through the approach here presented, where we will introduce the ‘quantum’ structure of the homogeneous space of worldlines corresponding to Lorentzian spacetimes as quantizations of Poisson homogeneous spaces of Lorentzian groups. Since time-like worldlines can be identified with inertial observers, this can be thought of as a first –to the best of our knowledge– explicit model of quantum observers with quantum group symmetry.

Indeed, the construction here presented contains a main simplification, since we are assuming that the space of oriented time-like geodesics is also a homogeneous space, which will be the case for maximally symmetric Lorentzian spacetimes, namely the Minkowski case here studied and the (A)dS ones (see [39]). Recall that for a generic spacetime, i.e. a smooth manifold endowed with a pseudo-Riemannian metric, the space of oriented geodesics is a quite complicated object. In fact, it is a topological space but not necessarily Hausdorff, and even when this is the case the topological manifold could not admit a smoothable atlas and henceforth could not be a smooth manifold. These problems have been previously considered in the literature (see, for instance, [40] and [41], where the space of null geodesics is described). For manifolds whose geodesics are closed many results are known (see [42]).

Our construction will make heavy use of this homogeneous space assumption, since this will allow the construction of the noncommutative version of such spaces as quantum homogeneous spaces invariant under the associated quantum isometry groups. Moreover, it is well-known that most of the structures that can be defined on classical homogeneous spaces of geodesics can be inherited from their associated motion groups. For instance, in [43] all symplectic, complex and metric structures were described, and we recall that in [39] all homogeneous spaces of worldlines corresponding to kinematical groups were studied in detail, including the pseudo-Riemannian metrics defined on them, and in [44] the (2+1) Lorentzian spaces of worldlines were considered. In particular, for the Poincaré case it was found that an invariant foliation exists in the space of worldlines and that the resulting homogeneous space is of negative curvature (see [39] for details). This result provides a neat geometrical description of the hyperbolic nature of the space of velocities in special relativity, and all these classical geometric notions should admit some rigorous generalization to the quantum (noncommutative) setting.

We stress that the construction here presented is fully general and explicit, thus being amenable to be applied to any (coisotropic, as we will explain in the sequel) quantum deformation of the Poincaré and (A)dS kinematical groups. Nevertheless, for the sake of brevity we will restrict here to the case with vanishing cosmological constant, and we will explicitly construct and analyse the noncommutative space of worldlines associated to the $\kappa$-Poincaré quantum group, whose relevance is outstanding (see [7, 8, 9, 22, 31] and references therein).

In the next Section we review the construction of Minkowski spacetime and its space of oriented geodesics as coset spaces of the Poincaré group, by emphasizing the relevance of choosing appropriate (and different) sets of local coordinates on the group for the construction of each of these two spaces. In Section 3 we recall how the $\kappa$-Minkowski noncommutative spacetime arises as the quantization of the Poisson Minkowski spacetime associated with the $\kappa$-Poincaré Poisson-Lie structure, which is provided by the $\kappa$-Poincaré classical $r$-matrix. Section 4 mimics the same approach in order to construct the noncommutative $\kappa$-Poincaré space of worldlines as the quantization of the Poisson homogeneous spacetime of worldlines associated to the same $\kappa$-Poincaré Poisson-Lie group, which however has to be expressed in terms of a different set of local coordinates. As the most relevant result, we prove in Section 5 that the non-trivial Poisson structure on the space of worldlines so obtained is just a canonical symplectic structure. This means that fuzziness can be expressed on the space of worldlines in a precise mathematical way, since $\kappa$-Poincaré quantum observers can be described as the quantization of a canonical phase space of worldline coordinates in which the Planck constant $\hbar$ has been replaced by the quantum deformation parameter $\kappa^{-1}$. A final discussion Section including some open problems closes the paper.

2 The space of time-like geodesics as a coset space

In this section we firstly recall the classical construction of the (3+1)-dimensional Minkowski spacetime as a coset space $\mathcal{M}=G/L$ of the Poincaré group $G$ with respect to the isotropy Lorentz subgroup $L$. In this way we obtain a maximally symmetric homogeneous space which can be identified straightforwardly with Minkowski spacetime. Afterwards, we use the very same approach in order to construct the 6-dimensional space of time-like oriented geodesics of Minkowski spacetime, just considering the coset $\mathcal{W}=G/H$, where now $H$ is the isotropy subgroup of a time-like geodesic that will be described in the sequel. This space inherits its main geometrical structures from the Poincaré group, as described in [39, 43].

Let us consider the Poincaré Lie algebra $\mathfrak{g}=\mathfrak{p}(3+1)\equiv\mathfrak{so}(3,1)\ltimes\mathbb{R}^{3}$, which generates the (3+1) Poincaré group $G=P(3+1)$. In the kinematical basis $\{P_{0},P_{a},K_{a},J_{a}\}$ $(a=1,2,3)$ of generators of time translation, space translations, boosts and rotations, respectively, the commutation rules for $\mathfrak{p}(3+1)$ read

[TABLE]

where sum over repeated indices is assumed, latin indices run from 1 to 3, and the speed of light $c=1$ is assumed. In the rest of the paper the same conventions are followed, together with greek indexes running from 0 to 3 and denoting 3-vectors by $\mathbf{v}=(v^{1},v^{2},v^{3})$ and 4-vectors by $\bar{v}=(v^{0},v^{1},v^{2},v^{3})$. Consider the Lie subalgebras of $\mathfrak{g}$ given by

[TABLE]

corresponding to the Lie subgroups $L$ and $H$ of $G$, respectively. Geometrically, these Lie subgroups are the stabilizers of the origin of Minkowski spacetime (an event) and that of the space of worldlines (a time-like oriented geodesic).

The first non-trivial task is to find appropriate descriptions of both the spacetime $\mathcal{M}$ and the space of worldlines $\mathcal{W}$ in such a way that the projection of Poisson-Lie structures on the Poincaré group $G$ can be easily described in a canonical way. This problem turns out to be equivalent to the exponentiation of a faithful representation of the Lie algebra in two (different) and carefully chosen orders.

Explicitly, let us consider a faithful representation $\rho:\mathfrak{p}(3+1)\rightarrow\text{End}(\mathbb{R}^{5})$ such that a generic element $X$ of the Lie algebra $\mathfrak{p}(3+1)$ is given by:

[TABLE]

In order to construct the (3+1)-dimensional Minkowski spacetime $\mathcal{M}$ as a coset space, we parametrize an element of the Poincaré group $G=P(3+1)$ in the form

[TABLE]

and the Lorentz subgroup $L$ is parametrized by

[TABLE]

In this way the coordinates $x^{\alpha}$ can be understood as Minkowski spacetime coordinates in the standard way, and the usual spacetime metric onto $\mathcal{M}$

[TABLE]

arises. Note that these coordinates are specially adapted to $\mathcal{M}$ because they come from a splitting of the Poincaré group given by $G_{\mathcal{M}}=T\cdot L$, where $T$ is the translations sector and $L$ is the Lorentz one, which locally is always valid. Hence a Poincaré group element takes the standard form

[TABLE]

where $\mathbf{\Lambda}$ is the $4\times 4$ matrix representation of an element of the Lorentz subgroup.

While the previous construction is well-known, the corresponding one for the space of worldlines is more subtle, due to the fact that the coordinates $x^{a}$ and $\xi^{a}$ associated with space translations and boosts do not define a set of coordinates in the coset space $\mathcal{W}=G/H$, because they are not well-defined functions on this space. Recall that for a function to be well-defined on the coset space $\mathcal{W}=G/H$ it must be independent of the representative chosen. For instance, for $w$ being a well-defined set of six coordinates on $\mathcal{W}$ they must verify that $w(gh)=w(gh^{\prime})$ for all $g\in G$ and $h,h^{\prime}\in H$. A direct computation shows that this is not the case for $x^{a}$ and $\xi^{a}$ when taking the group law arising from the exponentiation (LABEL:eq:Gm) and considering the worldlines coset $\mathcal{W}=G/H$ (of course, $x^{\alpha}$ do define a proper set of coordinates for the spacetime coset $\mathcal{M}=G/L$).

Therefore, in order to construct the 6-dimensional space of worldlines as the coset space $\mathcal{W}=G/H$ we need to parametrize $G$ by using a different ordering, which turns out to be

[TABLE]

where the stabilizer $H$ of a worldline passing through the origin of $\mathcal{W}$ is parametrized as

[TABLE]

In this way it is straightforward to check that we have well-defined coordinates $y^{a}$ and $\eta^{a}$ on $\mathcal{W}=G/H$, as they are invariant by right multiplication of an element of $H$. These coordinates correspond to the splitting of the Poincaré group as $G_{\mathcal{W}}=D\cdot H$, and the space of worldlines is just parametrized by the local coordinates dual to the generators of $D$. Now the group element has the form

[TABLE]

where $\mathbf{\Lambda}$ is the same matrix as in (LABEL:eq:Gm_split) and $\bar{f}$ is formed by the functions given by

[TABLE]

We stress that the previous construction allows us to obtain the explicit relationships among the local coordinates of the Poincaré group $G$ in both parametrizations, namely

[TABLE]

Note that the position coordinates $x^{a}\equiv f^{a}$ on the Minkowski spacetime cannot be naively identified with the ‘position’ coordinates $y^{a}$ on the space of worldlines, since they only coincide when all rapidities vanish (i.e. for an observer at rest). Moreover, in the representation (LABEL:eq:Gw_split) the action of the Poincaré group on the space of worldlines is not linear.

The space of worldlines $\mathcal{W}$ has also a metric structure which is rather different from the flat Lorentzian metric on the Minkowskian spacetime $\mathcal{M}$ (6). In particular, the metric on $\mathcal{W}$ is degenerate, and an invariant foliation under the Poincaré group action arises in such a manner that a ‘subsidiary’ metric restricted to each leaf of the foliation has to be considered [39]. It can be shown that in terms of the coordinates $y^{a}$ and $\eta^{a}$ the degenerate ‘main’ metric $g^{(1)}$ on $\mathcal{W}$ has a line element

[TABLE]

This is a Riemannian metric of negative constant curvature (whose value is just $-1/c^{2}$) which only involves rapidities, and thus provides the relative rapidity between two free motions. This, in turn, shows that the three-velocity space is hyperbolic. The invariant foliation is determined by a uniform motion with $\boldsymbol{\eta}={\boldsymbol{\eta}}_{0}=$ constant and the ‘subsidiary’ metric $g^{(2)}$ defined on each leaf reads

[TABLE]

that is, each leaf is isometric to the three-dimensional Euclidean space. We also point out that in the three-velocity space the geodesic distance $\chi$ corresponding to the relative speed from an observer at rest and one with a uniform motion with rapidity $\boldsymbol{\eta}$ is given by

[TABLE]

Finally, notice that in the low rapidity regime (i.e. take $c\to\infty$), the expressions (13) and (15) reduce to the usual ones for velocities in classical mechanics

[TABLE]

3 $\kappa$-Minkowski noncommutative spacetime

For the sake of clarity and self-consistency we now recall the well-known construction of the $\kappa$-Minkowski noncommutative spacetime as the quantization of a coisotropic Poisson homogenous structure on $\mathcal{M}$ obtained from the $\kappa$-Poincaré Poisson-Lie structure on the Poincaré group (which has to be parametrized in the form $G_{\mathcal{M}}$ (LABEL:eq:Gm_split)) through canonical projection.

The $\kappa$-Poincaré Lie bialgebra is a coboundary Lie bialgebra generated by the following $r$-matrix

[TABLE]

This means that the associated cocommutator can be directly computed as $\delta(X)=[X\otimes 1+1\otimes X,r],\;\forall X\in\mathfrak{g}$, and reads

[TABLE]

Coboundary Lie bialgebras are the tangent counterpart of coboundary Poisson-Lie groups [18], where the Poisson structure on $G$ is given by the so-called Sklyanin bracket

[TABLE]

such that $X^{L}_{i}$ and $X^{R}_{j}$ are left- and right-invariant vector fields defined by

[TABLE]

where $f\in\mathcal{C}(G)$, $h\in G$ and $T_{i}\in\mathfrak{g}$.

The Poisson brackets defining the $\kappa$-Minkowski spacetime are just the ones providing the Poisson homogeneous structure on $\mathcal{M}$ associated to the Poisson-Lie structure on the Poincaré group defined by the $r$-matrix (17). In order to obtain them explicitly, it suffices to realize that the $\kappa$-Poincaré Lie bialgebra (LABEL:eq:cocom_kappapoincare) is a coisotropic Lie bialgebra with respect to the Lorentz subgroup, since it fulfills the condition (see [45] and references therein)

[TABLE]

where $\mathfrak{l}$ (2) is the Lie algebra of the Lorentz subgroup $L$. In that case, it turns out that the homogeneous Poisson structure for the coset space $\mathcal{M}=G/L$ is just the canonical projection of the Poisson-Lie bracket (LABEL:eq:sklyanin) to the Minkowski space coordinates $x^{\alpha}$, which is tantamount to say that the (Poisson) $\kappa$-Minkowski spacetime is given by the Sklyanin bracket (LABEL:eq:sklyanin) between the Minkowski coordinates, which can be straightforwardly computed and reads

[TABLE]

This bracket can be quantized just by replacing the Poisson brackets by commutators, since no ordering problems arise due to the linear nature of the Poisson structure [46]

[TABLE]

hence $\hat{x}^{\alpha}$ will be the noncommutative coordinates on this quantum ($\kappa$-Minkowski) spacetime. These relations are directly linked through Hopf algebra duality to the quantum $\kappa$-Poincaré Hopf algebra, which in the bicrossproduct basis [9] is given by the deformed commutation rules

[TABLE]

where $\textbf{P}^{2}=P_{1}^{2}+P_{2}^{2}+P_{3}^{2}$, and the rest of commutators being the undeformed ones (1). The corresponding deformed coproduct is given by

[TABLE]

Recall that the skew-symmetric part of the first-order in $\kappa^{-1}$ of the coproduct (LABEL:eq:coproductkappaPoincare) provides the cocommutator (LABEL:eq:cocom_kappapoincare), where quantum spacetime coordinates are dual generators to the deformed translations sector through the pairing $\langle\hat{x}^{\alpha},P_{\beta}\rangle=\delta_{\beta}^{\alpha}$.

4 $\kappa$-Poincaré homogeneous space of worldlines

The main result presented in this paper consists is showing that the noncommutativity in the space of worldlines induced by the $\kappa$-deformation of Poincaré symmetries can be obtained by mimicking the previous construction of the Poisson homogeneous Minkowski spacetime, but taking into account the appropriate isotropy subgroup of worldlines.

As a first step, we have to check whether the $\kappa$-Poincaré Lie bialgebra structure (LABEL:eq:cocom_kappapoincare) is coisotropic with respect to the Lie subalgebra of the isotropy subgroup of time-like worldlines $\mathfrak{h}=\operatorname{span}\{P_{0},J_{a}\}$ (2). This is indeed the case, since we have that $\delta(P_{0})=\delta(J_{a})=0$ and the coisotropy condition (21) is trivially satisfied. Furthermore, in this case the stronger Poisson-subgroup condition [45] $\delta(\mathfrak{h})\subset\mathfrak{h}\wedge\mathfrak{h}$ is also trivially fulfilled. We stress that this does not occur for the $\kappa$-Minkowski isotropy subalgebra $\mathfrak{l}$, and implies that after quantization the isotropy subgroup $\mathfrak{h}$ is promoted to a Hopf subalgebra (again, this is not the case for the Lorentz sector of the $\kappa$-deformation).

This fact provides a first signature that the $\kappa$-deformation is more naturally realized on the space of worldlines than on Minkowski spacetime. As a consequence, the coisotropy condition guarantees that the homogeneous Poisson structure on the space of worldlines $\mathcal{W}$ can be obtained as a canonical projection from the coboundary Poisson structure on $G$ induced by the $r$-matrix (17). This is directly connected with the precise ordering (LABEL:eq:Gw) chosen for the construction of $G_{\mathcal{W}}$ (LABEL:eq:Gw_split), which is the one that provides the appropriate description of the projected Poisson structure onto $\mathcal{W}$.

Indeed, left- and right-invariant vector fields (20) for $G_{\mathcal{W}}$ have to be computed (we omit their explicit expressions for the sake of brevity), and the Sklyanin bracket (LABEL:eq:sklyanin) for the $\kappa$-Poincaré $r$-matrix (17) has to be written in terms of such vector fields expressed in terms of the local Poincaré coordinates $\{y^{a},\eta^{a},\phi^{a},y^{0}\}$. With all these ingredients at hand, the explicit form of the $\kappa$-Poincaré Poisson homogeneous space of worldlines is just given by (the canonical projection of) the Skyanin bracket for the coordinates $y^{a}$ and $\eta^{a}$ of the space $\mathcal{W}$, and reads

[TABLE]

These expressions for the Poisson version of the noncommutative space of worldlines show that this noncommutative space contains a commutative subalgebra of rapidities $\eta^{a}$, while the ‘position’ worldline coordinates $y^{a}$ are noncommutative. The Poisson bracket between a given rapidity $\eta^{a}$ and its corresponding ‘position’ $y^{a}$ does not vanish and depends on the geodesic distance function (15).

The structure of these Poisson brackets for $\mathcal{W}$ becomes more symmetric and manifestly spatially isotropic if they are expanded as power series in the coordinates of $\mathcal{W}$ up to second-order, namely

[TABLE]

We stress that the quadratic terms in (LABEL:eq:pois_y_eta_ser) are the ones coming from the non-relativistic limit $c\to\infty$, since they are just the angular momenta $(\eta^{a}y^{b}-\eta^{b}y^{a})$ and the relative speed $\chi^{2}$ (16). Note also that the linearization of the brackets (LABEL:eq:pois_y_eta_ser) in terms of the local coordinates $y^{a}$ and $\eta^{b}$ vanish, in contradistinction to the $\kappa$-Minkowski spacetime (LABEL:eq:PoissonkappaMinkowski), which is a purely linear bracket having no higher-order terms in the coordinates.

If we now compute the $\kappa$-Poincaré Sklyanin bracket relations between $y^{0}$ (which does not belong to $\mathcal{W}$) and the coordinates of the space of worldlines we obtain

[TABLE]

which means that the smooth functions on $\mathcal{W}$ enlarged with $y^{0}$ (i.e. the coset $G/R$ where $R$ is the rotations subgroup) still define a Poisson subalgebra where $\mathcal{C}^{\infty}(\mathcal{W})$ is a non-Abelian ideal. All these properties are fully consistent with the transformation (LABEL:eq:falpha) which provides the Minkowski spacetime coordinates $x^{\alpha}$ in terms of the ones for $\mathcal{W}$ and $y^{0}$. Indeed, if we use the transformation (LABEL:eq:falpha) to compute the Poisson structure given by (LABEL:eq:pois_y_eta) and (LABEL:eq:pois_y_eta_y0) for the four Minkowski coordinates $x^{\alpha}\equiv f^{\alpha}$, we just obtain the defining relations of the Poisson version of the $\kappa$-Minkowski spacetime (LABEL:eq:PoissonkappaMinkowski). This means that both the noncommutative $\kappa$-Minkowski spacetime and the noncommutative space of $\kappa$-Poincaré worldlines are just two realisations in two different geometric contexts of the very same noncommutative structure provided by the $\kappa$-deformation.

5 Quantum $\kappa$-Poincaré worldlines

At this point, the space of quantum worldlines would be defined by the quantization of the Poisson algebra of worldline coordinates (LABEL:eq:pois_y_eta), which can be obtained by substituting the Poisson brackets into commutators with exactly the same expressions (LABEL:eq:pois_y_eta), but now in terms of the noncommutative quantum worldline coordinates $\hat{y}^{a}$ and $\hat{\eta}^{a}$. Indeed, no ordering ambiguities appear when the Poisson bracket (LABEL:eq:pois_y_eta) is transformed into a commutator: the quantum rapidities $\hat{\eta}^{a}$ commute, and this implies that the crossed commutators $[\hat{y}^{a},\hat{\eta}^{b}]$ have no ordering problems. Finally, at the r.h.s. of the commutators $[\hat{y}^{a},\hat{y}^{b}]$, the $\hat{y}^{a}$ coordinates are always multiplied by a function depending on the $\hat{\eta}^{b}$ coordinates with $b\neq a$, which means that the latter commute with the former and no quantization ambiguities do exist.

Moreover, the fact that the Poisson brackets $\{y^{a},\eta^{b}\}$ only involve the $\eta^{a}$ coordinates provides a natural ansatz for a new set of classical variables whose quantization is straightforward. Let us consider the following diffeomorphism

[TABLE]

which is well-defined whenever $(\eta^{1},\eta^{2},\eta^{3})\neq(0,0,0)$ since its Jacobian determinant

[TABLE]

is different from zero everywhere on such a domain (note again the presence of the geodesic distance $\chi$ (15) in all these expressions). In fact, in terms of the elements of $g^{(1)}$ (13) the expressions above take the simple form

[TABLE]

where in this case sum over repeated indices should not be assumed. These expressions neatly shows the interconnection among the Poisson homogeneous structure underlying the quantum deformation and the (hyperbolic) geometry of the space of velocities of special relativity.

Surprisingly enough, in terms of these new coordinates on $\mathcal{W}$ the noncommutative (Poisson) algebra of the worldline coordinates turns out to be

[TABLE]

Obviously, the chosen notation $(q^{a},p^{a})$ for these new coordinates is not arbitrary, since what we have found is that the homogeneous Poisson structure induced by the $\kappa$-Poincaré $r$-matrix on the space or worldlines is just a symplectic structure on $\mathcal{W}$ (without the origin). Note that the diffeomorphism (LABEL:eq:canonical_diffeomorphism) is defined everywhere but in a point is a direct consequence of the fact that a Poisson-Lie group is never symplectic [18], because it always vanishes at the identity and this property descends to the quotient through canonical projection. Hence, outside the origin of $\mathcal{W}$ (the projection of the identity element to the coset space) we have obtained a symplectic form onto $\mathcal{W}$ given by

[TABLE]

and in this way our new worldline coordinates $(\mathbb{q},\mathbb{p})$ have a direct interpretation as canonical phase space coordinates. Notice that while the change $\frac{y^{a}}{\{y^{a},\eta^{a}\}}\rightarrow q^{a}$ is clearly suggested by the precise form of the fundamental brackets (LABEL:eq:pois_y_eta), the fact that the new coordinates $q^{a}$ Poisson commute, which is essential for them in order to be Darboux coordinates, is a quite surprising result. Also, it is worth stressing that the symplectic structure (33) is the result of the $\kappa$-deformation of the Poincaré symmetry, and this explains why the deformation parameter $\kappa$ explicitly appears within the symplectic form. Note that taking the limit $\kappa\rightarrow\infty$ implies that the Poisson-Lie structure on the Poincaré group becomes the trivial one and the space of worldlines (LABEL:eq:pois_y_eta) and (32) (as well as the spacetime (LABEL:eq:PoissonkappaMinkowski)) becomes commutative.

Obviously, the quantization of the symplectic algebra (32) is straightforward in terms of canonical worldline position operators $\hat{\mathbb{q}}$ and their conjugate momenta $\hat{\mathbb{p}}$, and taking into account that $\kappa^{-1}$ plays exactly the same role as the Planck constant. Therefore, all quantum gravity effects amenable to be described through the $\kappa$-deformation should be fully understood as a standard deformation-quantization on the space of worldlines with deformation parameter $\kappa^{-1}$, just in the same way as ordinary quantum mechanics arises as a deformation-quantization (with parameter $\hbar$) on the classical mechanical phase space.

6 Discussion and open problems

As a conclusion, we propose that quantum deformations of spaces of worldlines arising as quantisations of Poisson homogeneous spaces of kinematical groups should also be considered as noncommutative spaces amenable to describe quantum gravity effects. We have shown that this is a completely general construction that can be applied to any quantum deformation provided that the coisotropy condition (21) of its associated Lie bialgebra with respect to the isotropy subalgebra of worldlines is fulfilled. In this way, a non-trivial Poisson homogenous structure on the space of worldlines can be introduced, and from the latter the noncommutativity between worldline coordinates arises in a natural way.

Furthermore, when this construction is applied to the $\kappa$-deformation of Poincaré symmetries, the quantum space of worldlines so obtained turns out to be isomorphic to three copies of the Heisenberg-Weyl algebra, where the constant $\kappa^{-1}$ plays the role of $\hbar$. This is a straightforward consequence of the symplectic structure of the space of worldlines induced by the $\kappa$-Poincaré Poisson-Lie structure. This result suggests that the $\kappa$-deformation is a natural one for the space of worldlines, an idea enforced by the fact that the isotropy subgroup of worldlines behaves as a Poisson-Lie subgroup under this deformation. Moreover, the fact that the $\kappa$-Poincaré $r$-matrix (17) is connected in a natural way to a symplectic structure on the space of worldlines seems quite natural if we realize that the relativistic Newton-Wigner position operators [47] are defined in terms of the Poincaré Lie algebra generators as

[TABLE]

since in this way the canonical brackets $[Q_{a},P_{b}]=\delta_{ab}$ are obtained in terms of the generators of the Poincaré Lie algebra (1). Therefore, the bivector

[TABLE]

should define a symplectic form under the appropriate realization, and if we substitute (34) into this expression we get

[TABLE]

which implies that both the symplectic bivector $B$ and the $\kappa$-Poincaré $r$-matrix (17) are closely related (recall that $P_{0}$ is one of the generators of the the stabilizer of the origin of the space of worldlines).

Consequently, the noncommutative spaces of worldlines seem to provide a privileged arena in order to explore the physical role of the $\kappa$-deformation. In particular, once the canonical coordinates have been found, noncommutativity in the space of worldlines could be rephrased in more physical terms as the impossibility of determining simultaneously and with infinite precision the six $(\mathbb{q},\mathbb{p})$ coordinates of a given worldline. In this respect, note that (LABEL:eq:canonical_diffeomorphism) implies that, before introducing the quantum deformation, the $\mathbb{p}$ coordinates are just the usual rapidities $\boldsymbol{\eta}$ and the ‘positions’ $\mathbb{q}$ for a worldline are defined as the product of Poincaré coordinates $\mathbb{y}$ associated to translations with certain functions depending on $\boldsymbol{\eta}$. Indeed, the precise physical meaning of the coordinates (LABEL:eq:canonical_diffeomorphism) has to be studied in detail.

The construction here presented is fully general and can thus be applied to any other quantum deformation (provided it is coisotropic with respect to the isotropy subgroup of worldlines) of any kinematical group. This opens the path to several future investigations, and the first of them consists in the construction of the noncommutative space of worldlines associated with the $\kappa$-deformation of the (A)dS groups. This can be explicitly performed by generalizing the approach here presented to the case of nonvanishing cosmological constant $\Lambda$ through the approach to $\kappa$-(A)dS groups and spaces used in [48, 49, 50], in which $\Lambda$ is included as an explicit parameter whose $\Lambda\to 0$ limit provides the flat Poincaré case presented in this paper.

Finally, the model here introduced provides a purely kinematical framework for a schematic theory of quantum (‘noncommutative’) free observers, and strongly suggests that if the quantum deformation is assumed to encode quantum gravity effects, the latter are simply reflected as a canonical Heisenberg-Weyl noncommutativity on the (phase) space of worldlines, which is the simplest possible algebraic framework to deal with. Work on all these lines is in progress and will be presented elsewhere.

Acknowledgements

This work has been partially supported by Ministerio de Ciencia, Innovación y Universidades (Spain) under grant MTM2016-79639-P (AEI/FEDER, UE), by Junta de Castilla y León (Spain) under grant BU229P18 and by the Action MP1405 QSPACE from the European Cooperation in Science and Technology (COST). I.G-S. acknowledges a predoctoral grant from Junta de Castilla y León and the European Social Fund.