Non-commutative relativistic spacetimes and worldlines from 2+1 quantum (anti)de Sitter groups

0.28 MB·English

by  Angel Ballesteros

Non-commutative relativistic spacetimes and worldlines
from 2+1 quantum (anti)de Sitter groups
Angel Ballesterosa, N. Rossano Brunoa,b and Francisco J. Herranza
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aDepartamento de F´ısica, Universidad de Burgos, Pza. Misael Ban˜uelos s.n.,
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09001 Burgos, Spain
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a e-mail: [email protected], [email protected]
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bDipartimento di Fisica, Universita` di Roma Tre and INFN Sez. Roma Tre,
3
Via Vasca Navale 84, 00146 Roma, Italy

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e-mail: rossano@fis.uniroma3.it
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4 Abstract
0
/ The κ-deformationof the 2+1anti-de Sitter, Poincar´eandde Sitter groupsis studied
h
within a unified approach by considering explicitly the curvature of the spacetime
t
- (cosmological constant) as a parameter. The Drinfel’d-double construction as well
p
e as the Poisson–Lie structure underlying the κ-deformation are explicitly given, and
h the three quantum groups are obtained as Weyl quantizations of such Poisson–Lie
: algebras. As a consequence, the non-commutative 2+1 spacetimes and 4D spaces of
v
i worldlines are derived. The former generalize the κ-Minkowski space to the (anti)de
X
Sitter ones, and the latter can be interpreted as a new possibility to introduce non-
r commutative velocity spaces. Furthermore, provided that the deformation parameter
a
is identified with the Planck length, quantum (anti)de Sitter algebras are presented
both in the known basis arising in 2+1 quantum gravity and in a new one which
generalizes the bicrossproduct κ-Poincar´e basis. Finally, the existence of a kind of
‘duality’betweenthe cosmologicalconstantandthe Plancklengthisalsocommented.
1
1 Introduction
The connection between quantum groups and quantum gravity was early suggested from
the point of view of physics at the Planck scale in [1]. Later quantum groups have been
broadlyappliedintheconstructionofdeformedsymmetriesofspacetimes[2,3,4,5,6,7,8,
9,10]mainlycovering thePoincar´e andGalilei cases, forwhich thedeformation parameter
plays the role of a fundamental scale. Amongst all of these quantum kinematical algebras
the well known κ-Poincar´e [2, 6, 7, 11, 12, 13, 14, 15] arises as the deformation deepest
studied which conveys an associated non-commutative spacetime, the κ-Minkowski space,
defined by the spacetime quantum group coordinates dual to the translation (momenta)
generators.
More recently, a further development of deformed Poincar´e symmetries has led to
the so called doubly special relativity (DSR) theories [16, 17, 18, 19, 20, 21] that make
use of two fundamental scales: the usual observer-independent velocity scale c as well
as an observer-independent length scale l related to the deformation parameter. Since
p
different approaches to quantum gravity [22, 23, 24, 25, 26] also assign to the Planck scale
afundamentalrole,DSRtheoriesseemtoestablishanaturallinkbetweenquantumgroups
and quantum gravity [27, 28].
Nevertheless, the role of spacetime curvature in DSR theories is still an open question.
This problem is equivalent to study in detail the κ-deformation for the (anti)de Sitter
((A)dS) groups that might be further applied in quantum gravity theories with a non-
zero cosmological constant. In this respect, we recall that the Hopf structure for the
κ-deformation of the 2+1 (A)dS and Poincar´e ( ) algebras were collectively obtained
P
in [8], and their connection between their deformed commutation rules and 2+1 quantum
gravity has been recently explored in [27]. The results obtained in [8] correspond to the
lhs commutative diagram:
AdS z U (AdS) Fun(AdS) z Fun (AdS) AdS2+1
z z z
−−−→ −−−→ ⊃
ω=+ 1 ω ω ω
R2
x z Uzx( ) duality Funx( ) z Funz( )x M2z+1 (1.1)
P −−−→ P −−−−→ P −−−→ P⊃
ω=− 1 ω ω ω
R2
dS z Uz(dS) Fun(dS) z Funz(dS) dS2z+1
y −−−→ y y −−−→ y⊃
where vertical arrows indicate a classical deformation that introduces the spacetime cur-
vature ω (or cosmological constant Λ) related to the (A)dS radius R by ω = 1/R2, and
±
the horizontal ones show the quantum deformation with parameter z = 1/κ (related to
the Planck length l ); reversed arrows correspond to the spacetime contraction ω 0 and
p
→
classical limit z 0.
→
However, as far as we know, the construction of non-commutative (A)dS spacetimes
(in terms of intrinsic and ambient spacetime quantum group coordinates) as well as some
proposal for non-commutative spaces of worldlines (even for the Poincar´e case) are still
unsolved questions. The aim of this paper is to work out all these problems for the
three relativistic cases simultaneously, by dealing explicitly with the spacetime curvature
ω that plays the role of a contraction parameter. Hence, we propose to explore the rhs
diagram(1.1)(dualtothelhs)bycomputingthequantumdeformationofthe(A)dSgroups
(e.g. Fun (AdS)) which are obtained by quantizing the Poisson–Lie algebra of smooth
z
2
function on these groups (e.g. Fun(AdS)) coming from the classical r-matrix. Thus non-
commutative spaces (e.g. AdS2+1)can thenbeidentifiedwithincertain subalgebras ofthe
z
quantum groups. We stress that in our approach all the κ-Poincar´e relations (including
its non-commutative spaces) can be directly recovered from the general (A)dS expressions
through the limit ω 0.
→
In the next section we recall the necessary basics on the (A)dS groups and their asso-
ciated homogeneous (2+1)D spacetimes and 4D spaces of worldlines (i.e. time-like lines).
Both typesof spaces aredescribedintermsofintrinsicquantities (related to groupparam-
eters) as well as in ambient coordinates with one and two extra dimensions, respectively,
which will be further used in their non-commutative versions. By starting from the clas-
sical r-matrix, that underlies the κ-deformation, we construct in section 3 the Drinfel’d-
double and obtain some preliminary information on the first-order quantum deformation;
as a byproduct we deduce the first-order non-commutative spaces. On one hand, we
find that the three non-commutative relativistic spacetimes share the same (first-order)
κ-Minkowski space (independently of the curvature), and moreover we show that the de-
formation parameter can be interpreted as a curvature on a classical dS spacetime for the
three cases, thus generalizing the results deduced in [29, 30] for κ-Poincar´e. On the other
hand, we obtain that the first-order non-commutative spaces of worldlines are in fact non-
deformed ones, finding a relationship with the non-relativistic (Newtonian) kinematical
groups as well.
Asanintermediatestageinthesearchofthequantum(A)dSgroups,wefirstlycompute
in section 4 the invariant (A)dS vector fields and secondly the Poisson–Lie structures
coming from the classical r-matrix. These results enable to propose in section 5 the non-
commutative (A)dS spaces; these are written in both intrinsic and ambient coordinates.
Theresultingnon-commutative spacetimes show how thecurvaturemodifytheunderlying
first-order κ-Minkowski space, while for the non-commutative spaces of worldlines we find
that2Dvelocity spaceremainsnon-deformedforκ-Poincar´e butbecomes deformedforthe
(A)dS cases. Hence Lorentz invariance seems to be lost (or somewhat ‘deformed’) when a
non-zero curvature is considered.
Section 6 is devoted to study the (dual) quantum (A)dS algebras and their deformed
Casimirs in two different basis. In particular, starting from the expressions given in [8], a
non-linear transformation involving the generators of the isotopy subgroup of a time-like
line allows us to obtain these quantum algebras in a new basis that generalizes for any ω
the bicrossproduct basis of κ-Poincar´e [12]. These results are analysed in connection with
2+1 quantum gravity [27] and a ‘duality’ between curvature/cosmological constant and
deformation parameter/Planck length is suggested. Finally, some remarks and comments
close the paper.
2 (Anti)de Sitter Lie groups and their homogeneous spaces
The Lie algebras of the three (2+1)D relativistic spacetimes of constant curvature can
collectively be described by means of a real (graded) contraction parameter ω [8], and we
denote them by so (2,2). If J,P ,P = (P ,P ),K = (K ,K ) are, in this order, the
ω 0 1 2 1 2
{ }
generators of rotations, time translations, space translations and boosts, the commutation
3
relations of so (2,2) read
ω
[J,P ]= ǫ P [J,K ] = ǫ K [J,P ] = 0
i ij j i ij j 0
[P ,K ]= δ P [P ,K ] = P [K ,K ]= J (2.1)
i j ij 0 0 i i 1 2
− − −
[P ,P ]= ωK [P ,P ]= ωJ
0 i i 1 2
−
where from now on we assume that Latin indices i,j = 1,2, Greek ones µ,ν = 0,1,2,
~ = c = 1 and ǫ is a skewsymmetric tensor such that ǫ = 1. For a positive, zero and
ij 12
negative value of ω, so (2,2) gives rise to a Lie algebra isomorphic to so(2,2), iso(2,1)
ω
and so(3,1), respectively. The case ω = 0 can also be understood as an In¨onu¨–Wigner
contraction [31]: so(2,2) iso(2,1) so(3,1).
→ ←
Parity Πandtime-reversal Θareinvolutive automorphismsofso (2,2) definedby[32]:
ω
Π: (P ,P,K,J) (P , P, K,J)
0 0
→ − − (2.2)
Θ : (P ,P,K,J) ( P ,P, K,J)
0 0
→ − −
whichtogetherwiththecompositionΠΘandtheidentitydetermineaZ Z Z Abelian
2 2 2
× ×
group of involutions. In fact, the contraction parameter ω is related to the Z -grading
2
associated to ΠΘ.
When the Lie group SO (2,2) is considered, two relevant families of symmetric homo-
ω
geneous spaces [33] can be constructed (these are displayed in table 1):
S = SO (2,2)/SO(2,1) is the (2+1)D spacetime of rank 1 associated to ΠΘ, where
(1) ω
•
SO(2,1) is the Lorentz subgroup spanned by J and K. Thus momenta P characterize
µ
the tangent space at the origin. This space turns out to have constant curvature equal to
the contraction parameter: ω = 1/R2 for (A)dS and ω = 0 (R ) for Minkowski.
± → ∞
S = SO (2,2)/(SO(2) SO (2))isthe4Dspaceoftime-likelinesofrank2associated
(2) ω ω
• ⊗
to Π with constant curvature 1 (i.e., 1/c2), where SO(2) = J and SO (2) = P .
ω 0
− − h i h i
The tangent space is determined by P and K. In fact, S can also be interpreted as a
(2)
(2 2)D relativistic phase space [34] in which position and momentum coordinates are
×
related to the group parameters dual to P and K, respectively.
Table 1: AdS, Minkowskian and dS spacetimes and spaces of time-like lines.
ω Spacetime S with curvatureω Spaceof worldlines S with curvature−1
(1) (2)
+1/R2 AdS2+1 =SO(2,2)/SO(2,1) LAdS2×2 =SO(2,2)/(SO(2)⊗SO(2))
0 M2+1 =ISO(2,1)/SO(2,1) LM2×2 =ISO(2,1)/(SO(2)⊗R)
−1/R2 dS2+1 =SO(3,1)/SO(2,1) LdS2×2 =SO(3,1)/(SO(2)⊗SO(1,1))
On the other hand, the two Casimir invariants of so (2,2) are given by
ω
= P2 P2+ω(J2 K2) = JP +K P K P (2.3)
C 0 − − W − 0 1 2− 2 1
where comes from the Killing–Cartan form, while is the Pauli–Lubanski vector.
C W
2.1 Vector model of the (2+1)D relativistic spacetimes
The action of the (A)dS groups on their homogeneous spaces is not linear. As is well
known, this problem can be circumvented by considering the vector representation of the
4
Lie group SO (2,2) which makes use of an ambient space with an ‘extra’ dimension. In
ω
particular, the 4D real matrix representation of so (2,2) given by
ω
0 ω 0 0 0 0 ω 0 0 0 0 ω
−
1 0 0 0 0 0 0 0 0 0 0 0
P0 =  0 0 0 0  P1 = 1 0 0 0  P2 =  0 0 0 0 
 0 0 0 0   0 0 0 0   1 0 0 0 
      (2.4)
     
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 1
J =  0 0 0 1  K1 =  0 1 0 0  K2 =  0 0 0 0 
−
 0 0 1 0   0 0 0 0   0 1 0 0 
     
     
fulfils
YTI +I Y = 0 Y so (2,2) I = diag(1,ω, ω, ω) (2.5)
(1) (1) ω (1)
∈ − −
(YT is the transpose matrix of Y). The exponential of (2.4) leads to the vector repre-
sentation of SO (2,2) as a Lie group of matrices which acts linearly in a 4D space with
ω
ambient (or Weierstrass) coordinates (s ,s ). The one-parameter subgroups of SO (2,2)
3 µ ω
obtained from (2.4) turn out to be
cosρx ρsinρx 0 0 1 0 0 0
0 0
−
1 sinρx cosρx 0 0 0 1 0 0
ex0P0 =  ρ 0 0  eθJ =  
0 0 1 0 0 0 cosθ sinθ
−
 0 0 0 1   0 0 sinθ cosθ 
   
   
coshρx 0 ρsinhρx 0 1 0 0 0
1 1
0 1 0 0 0 coshξ sinhξ 0
ex1P1 =   eξ1K1 =  1 1 
1 sinhρx 0 coshρx 0 0 sinhξ coshξ 0
ρ 1 1 1 1
 0 0 0 1   0 0 0 1 
   
 coshρx 0 0 ρsinhρx   1 0 0 0 
2 2
0 1 0 0 0 coshξ 0 sinhξ
ex2P2 =   eξ2K2 =  2 2 
0 0 1 0 0 0 1 0
 1 sinhρx 0 0 coshρx   0 sinhξ 0 coshξ 
 ρ 2 2   2 2 
   (2.6)
where hereafter we also express the curvature as ω = ρ2. Hence, ρ= 1/R and ρ = i/R for
AdS2+1 and dS2+1, while the (contraction) limit ρ 0 gives rise to M2+1.
→
Any element G SO (2,2) verifies that GTI G= I . The (2+1)D spacetime S is
ω (1) (1) (1)
∈
identified with the orbit of the origin of the spacetime O = (1,0,0,0) which is contained
in the pseudosphere provided by I :
(1)
Σ : s2+ω(s2 s2) = 1. (2.7)
(1) 3 0−
The metric on S comes from the flat ambient metric divided by the curvature and
(1)
restricted to the above constraint:
1
dσ2 = ds2+ω(ds2 ds2 ds2)
(1) ω 3 0− 1− 2
(cid:12)Σ(1)
(cid:0) (cid:1)(cid:12)
(s ds s(cid:12) ds s ds )2
= ds2 ds2 ds2+ω 0 0− (cid:12)1 1− 2 2 . (2.8)
0− 1− 2 1 ω(s2 s2)
− 0−
5
Ambient coordinates can be parametrized in terms of three intrinsic spacetime coor-
dinates in different ways. We shall introduce the geodesic parallel coordinates x [35]
µ
through the following action of the momenta subgroups (2.6) on the origin:
(s ,s )(x )= exp(x P )exp(x P )exp(x P )O (2.9)
3 µ ν 0 0 1 1 2 2
namely,
s = cosρx coshρx coshρx
3 0 1 2
sinρx
0
s = coshρx coshρx
0 1 2
ρ
sinhρx (2.10)
1
s = coshρx
1 2
ρ
sinhρx
2
s = .
2
ρ
The role of the coordinates x that parametrize a generic point Q under (2.10) in the
µ
(2+1)D spacetime is as follows. Let l a time-like geodesic and l , l two space-like
0 1 2
geodesics such that these three basic geodesics are orthogonal at O. Then x is the
0
geodesic distance from O up to a point Q measured along l ; x is the distance between
1 0 1
′
Q and another point Q along a space-like geodesic l orthogonal to l through Q and
1 2 1 0 1
′
parallel to l ; and x is the distance between Q and Q along a space-like geodesic l
1 2 2 2
′
orthogonal to l and parallel to l . Recall that time-like geodesics (as l ) are compact
1 2 0
in AdS2+1 and non-compact in dS2+1, while space-like ones (as l , l′) are compact in
i i
dS2+1 but non-compact in AdS2+1. Thus the trigonometric functions depending on x
0
are circular in AdS2+1 (ρ = 1/R) and hyperbolic in dS2+1 (ρ = i/R) and, conversely,
those depending on x are circular in dS2+1 but hyperbolic in AdS2+1.
i
Under (2.10), the metric (2.8) now reads
dσ2 = cosh2(ρx )cosh2(ρx )dx2 cosh2(ρx )dx2 dx2. (2.11)
(1) 1 2 0− 2 1− 2
Notice that if ρ 0, the parametrization (2.10) gives the flat Cartesian coordinates
→
s = 1,s = x , and the metric (2.11) reduces to dσ2 = dx2 dx2 dx2 in M2+1.
3 µ µ (1) 0− 1− 2
2.2 Bivector model of the 4D spaces of worldlines
The action of SO (2,2) on the space of time-like lines S is also a non-linear one. As in
ω (2)
the previous case, this problem can be solved by introducing an ambient space, now 6D
with two ‘extra’ dimensions, on which the group acts linearly and where S is embedded.
(2)
Let us consider the so called bivector representation of (2.1) given by [34]:
P = ωe +e ωe +e J = e +e e +e
0 24 42 35 53 23 32 45 54
− − − −
P = ωe e +ωe +e K = e +e +e +e (2.12)
1 14 41 36 63 1 12 21 56 65
− −
P = ωe e ωe e K = e +e e e
2 15 51 26 62 2 13 31 46 64
− − − − − −
where e is the 6 6 matrix with entries (e ) = δ δ . Under this representation any
ab ab ij ai bj
×
generator Y so (2,2) fulfils
ω
∈
YTI +I Y = 0 I = diag(1, 1, 1, ω, ω,ω). (2.13)
(2) (2) (2)
− − − −
By exponentiation of (2.12) we obtain the bivector representation of SO (2,2), that is, a
ω
group of matrices which acts linearly in a 6D space with ambient (or Plu¨cker) coordinates
6
(η ,η ,η ,y ,y ,y ). The origin of S is = (1,0,0,0,0,0) and this space is identified
3 1 2 1 2 3 (2)
O
with the intersection of the pseudosphere Σ determined by I with a quadratic cone
(2) (2)
P known as Plu¨cker or Grassmann relation (invariant under the group action); these
constraints are given by [34]:
Σ : η2 η2+ω(y2 y2)= 1 P : η y η y +η y = 0. (2.14)
(2) 3 − 3 − 3 3− 1 2 2 1
The metric on S follows from the 6D flat ambient metric divided by the negative cur-
(2)
vature of S and subjected to both conditions (2.14):
(2)
1
dσ2 = dη2 dη2 dη2+ω(dy2 dy2 dy2) . (2.15)
(2) 1 3 − 1 − 2 3 − 1 − 2
− (cid:12)Σ(2),P
(cid:0) (cid:1)(cid:12)
(cid:12)
Plu¨cker coordinates can be expressed through four intrinsic qu(cid:12)antities of S . We
(2)
shall consider the space x and rapidities ξ group coordinates. The action of the following
sequenceofone-parametersubgroupson (thosedefiningthetangentspacetoS )under
(2)
O
the representation (2.12),
(η ,η,y,y )(x,ξ)= exp(x P )exp(x P )exp(ξ K )exp(ξ K ) (2.16)
3 3 1 1 2 2 1 1 2 2
O
gives rise to
η = coshρx coshρx coshξ coshξ
3 1 2 1 2
η = coshρx sinhξ coshξ
1 2 1 2
η = coshρx sinhξ sinhρx sinhρx sinhξ coshξ
2 1 2 1 2 1 2
−
sinhρx
1
y = coshρx coshξ coshξ
1 2 1 2 (2.17)
− ρ
sinhρx
2
y = coshξ coshξ
2 1 2
− ρ
sinhρx sinhρx
1 2
y = sinhξ coshρx sinhξ coshξ .
3 2 1 1 2
ρ − ρ
Underthecontraction ρ 0(ω = 0),thisparametrization reducestotheMinkowskian
space of worldlines LM2×2:→
η =coshξ coshξ y = x sinhξ x sinhξ coshξ
3 1 2 3 1 2 2 1 2
−
η =sinhξ coshξ y = x coshξ coshξ (2.18)
1 1 2 1 1 1 2
−
η =sinhξ y = x coshξ coshξ .
2 2 2 2 1 2
−
Such expressions indicate that the Plu¨cker coordinates (y,y ) and (η,η ) can be inter-
3 3
pretedasposition-likeandmomentum-likeones,respectively,withinthephasespace(x,ξ).
In LM2×2 the metric (2.15) is degenerate and reads
(η dη +η dη )2
dσ2 = dη2+dη2 1 1 2 2 = cosh2ξ dξ2+dξ2, (2.19)
(2) 1 2 − 1+η2 2 1 2
whichcorrespondstoa2Dspaceofrank1withnegative curvature, thatis, the2Dvelocity
Minkowskian space is hyperbolic. Nevertheless, we stress that this is no longer true when
ω = 0 (in both LAdS2×2 and LdS2×2) where the complete (2 2)D space structure is
6 ×
required, thus precluding the possibility of using a ‘reduced’ 2D velocity space.
7
3 (Anti)de Sitter Drinfel’d-doubles and first-order
non-commutative spaces
The first-order deformation terms in the coproduct of the κ-Poincar´e algebra [2, 6, 7, 11,
12, 13, 14, 15] are known to be generated by the following classical r-matrix:
r = z(K P +K P ) (3.1)
1 1 2 2
∧ ∧
where denotes the skewsymmetric tensor product; z is related to the usual κ and q
∧
deformation parameters by z = 1/κ = lnq.
Suchaclassicalr-matrixalsoholdsforthe(A)dSalgebras[36],sothatweshallconsider
(3.1) for the whole family so (2,2). Hence this element gives rise to the cocommutator δ
ω
of any generator Y through the relation δ(Y ) = [1 Y +Y 1,r], namely
i i i i
⊗ ⊗
δ(P ) = 0 δ(J) = 0
0
δ(P ) = z(P P ωǫ K J) (3.2)
i i 0 ij j
∧ − ∧
δ(K )= z(K P +ǫ P J).
i i 0 ij j
∧ ∧
Next if we denote by yˆi the quantum group coordinate dual to Y , such that yˆi Y = δi,
i h | ji j
jk
and write the cocommutators as δ(Y ) = f Y Y , then Lie bialgebra duality provides
i i j ∧ k
the so called Drinfel’d-double Lie algebra [37, 38] formed by three sets of generators; these
are the initial Lie algebra, the dual relations and the crossed commutation rules given by
[Y ,Y ]= ckY [yˆi,yˆj] = fijyˆk [yˆi,Y ]= ci yˆk fikY . (3.3)
i j ij k k j jk − j k
In our case, we denote by θˆ,xˆ ,ξˆ the dual non-commutative coordinates of the gener-
µ i
{ }
ators J,P ,K , respectively. Thus the (A)dS and Poincar´e Drinfel’d-doubles are collec-
µ i
{ }
tively given in terms of the curvature ω and deformation parameter z by the initial Lie
algebra so (2,2) (2.1), the dual commutators
ω
[θˆ,xˆ ]= zǫ ξˆ [xˆ ,xˆ ]= zxˆ [xˆ ,xˆ ]= 0, [θˆ,xˆ ] = 0
i ij j 0 i i 1 2 0
− (3.4)
[θˆ,ξˆ] = zωǫ xˆ [xˆ ,ξˆ]= zξˆ [ξˆ ,ξˆ] = 0, [xˆ ,ξˆ]= 0
i ij j 0 i i 1 2 i j
− −
together with the crossed relations
[xˆ ,J] = [xˆ ,P ] = 0 [θˆ,J] = [θˆ,P ] = 0
0 0 0 0
[xˆ ,P ]= (ξˆ zP ) [θˆ,P ] = ωǫ (xˆ +zK )
0 i i i i ij j j
− − −
[xˆ ,K ] = xˆ +zK [θˆ,K ] = ǫ (ξˆ zP )
0 i i i i ij j j
− −
[xˆ ,J] = ǫ xˆ [ξˆ,J] = ǫ ξˆ (3.5)
i ij j i ij j
− −
[xˆ ,P ]= ξˆ [ξˆ,P ] = ωxˆ
i 0 i i 0 i
−
[xˆ ,P ]= ǫ θˆ zδ P [ξˆ,P ]= ω(δ xˆ +zǫ J)
i j ij ij 0 i j ij 0 ij
− −
[xˆ ,K ] = δ xˆ +zǫ J [ξˆ,K ] = ǫ θˆ zδ P .
i j ij 0 ij i j ij ij 0
−
Parity and time-reversal automorphisms (2.2) can be generalized to the full Drinfel’d-
double as follows
Π : (P ,P,K,J; xˆ ,xˆ,ξˆ,θˆ; z) (P , P, K,J; xˆ , xˆ, ξˆ,θˆ; z)
z 0 0 0 0
→ − − − − (3.6)
Θ : (P ,P,K,J; xˆ ,xˆ,ξˆ,θˆ; z) ( P ,P, K,J; xˆ ,xˆ, ξˆ,θˆ; z).
z 0 0 0 0
→ − − − − −
8
Since the first-order structure of the complete quantum deformation of so (2,2) is
ω
describedbythecorrespondingDrinfel’d-double,somepreliminaryinformationconcerning
the physical properties of the associated non-commutative spaces can be extracted from
it. Notice that, in this first-order approach, all the expressions will be linear both on the
generators and on the dual quantum group coordinates.
3.1 Non-commutative spacetimes: linear relations
The usual way to propose a non-commutative spacetime is to consider the commutation
rules involving the quantum coordinates xˆ . Therefore, from (3.4) we find that the three
µ
(A)dS and Minkowskian non-commutative spacetimes are simultaneously defined by the
same first-order relations:
[xˆ ,xˆ ] = zxˆ [xˆ ,xˆ ] = 0 (3.7)
0 i i 1 2
−
which coincide with the κ-Minkowski space, M2+1, [11, 12, 13, 14] for any value of the
z
curvature ω. As we shall see in section 5, further corrections of (3.7) depending on ω will
appear when the full quantum (A)dS groups are considered.
As it was already studied in [29, 30], the action of the generators on the non-commuta-
tive spacetime follows by replacing formally P xˆ , which requires to consider the
µ µ
→
commutators involving J,xˆ ,K within the Drinfel’d-double. Next the change of basis
µ i
{ }
given by [30]:
pˆ = xˆ pˆ = xˆ +zK (3.8)
0 0 i i i
−
provides the following commutation relations
[J,pˆ]= ǫ pˆ [J,K ]= ǫ K [J,pˆ ] =0
i ij j i ij j 0
[pˆ,K ] = δ pˆ [pˆ ,K ]= pˆ [K ,K ] = J (3.9)
i j ij 0 0 i i 1 2
− − −
[pˆ ,pˆ]= z2K [pˆ ,pˆ ]= z2J
0 i i 1 2
−
that can directly be related to the initial Lie algebra (2.1). Consequently, whenever z is
a real deformation parameter, the commutators (3.9) (that do not depend on ω) close the
dS algebra so(3,1) for the three cases, and z2 now plays the role of the curvature of
−
the homogeneous space J,K,pˆ / J,K . We stress that the connection between M2+1
µ z
h i h i
and the dS space was so established in [29] and further developed in [30], so that the
expressions (3.9) generalize such a link for the non-commutative (A)dS cases as well.
3.2 Non-commutative spaces of worldlines: linear relations
Asimilarproceduresuggeststhatthecorrespondingnon-commutative spacesofworldlines
arise within the Drinfel’d-double through the commutators of xˆ and ξˆ (dual to P and K);
these are
[xˆ ,xˆ ] = 0 [ξˆ,ξˆ] =0 [xˆ ,ξˆ]= 0 (3.10)
1 2 1 2 i j
which are trivially independent of z and ω.
The adjoint action on the quantum coordinates xˆ, ξˆ of the isotopy subgroup of a
worldline spanned by J and P gives the following non-deformed commutation rules:
0
[J,xˆ ]= ǫ xˆ [J,ξˆ]= ǫ ξˆ [J,P ] =0
i ij j i ij j 0
[xˆ ,ξˆ] = 0 [P ,ξˆ]= ωxˆ [ξˆ,ξˆ]= 0 (3.11)
i j 0 i i 1 2
−
[xˆ ,xˆ ] = 0 [P ,xˆ ]= ξˆ.
1 2 0 i i
9
To unveil this structure we rename the former generators as: J′ = J, P′ = P , P′ = ξˆ,
′ ′ 0 − 0 i i
K = xˆ . In this way we obtain from (3.11) that the generators Y close the oscillating
i i
Newton–Hooke, Galilei and expanding Newton–Hooke algebras [8, 32, 39] according to
ω >,=,< 0, respectively. This fact is consistent with the known result that establishes
that each of the above non-relativistic Newtonian spaces of constant curvature ω can be
obtained from the corresponding relativistic one through a contraction around a time-like
line (c ). Therefore the classical (non-deformed) picture is preserved for worldlines.
→ ∞
Summing up, the first-order deformation of so (2,2) characterized by the chosen r-
ω
matrix (3.1)conveys non-commutativity onthespacetime butcommutativity onthespace
of worldlines. Furthermore, space isotropy is ensured in both cases as expressions (3.7)
and, obviously, (3.10) do not involve the quantum rotation coordinate θˆ.
4 A Poisson–Lie structure on the (anti)de Sitter groups
Sofarwehavestudiedthefirst-orderquantum(A)dSdeformation. However, theobtention
ofthecomplete(inallordersinz andinthegenerators)deformationofasemisimplegroup
is, in general, a very involved task. A way to study the non-commutative structures is to
compute the Poisson–Lie brackets (derived from (3.1)) for the commutative coordinates
and next to analysing their possible non-commutative version.
In particular, let us consider the 4 4 matrix element of the group SO (2,2) obtained
ω
×
through the following product written under the representation (2.6):
T = exp(x P )exp(x P )exp(x P )exp(ξ K )exp(ξ K )exp(θJ) (4.1)
0 0 1 1 2 2 1 1 2 2
where the group coordinates are commutative ones. Left and right invariant vector fields,
YL and YR, of SO (2,2) deduced from (4.1) are displayed in table 2.
ω
The Poisson–Lie brackets that close the algebra of smooth functions on the (A)dS
groups, Fun(SO (2,2)), associated to an r-matrix r = rijY Y come from the Sklyanin
ω i j
⊗
bracket defined by [40]:
f,g = rij(YLf YLg YRf YRg) f,g Fun(SO (2,2)). (4.2)
i j i j ω
{ } − ∈
Thusby substitutingthevector fieldsof table 2and theclassical r-matrix (3.1)in (4.2)we
obtain the Poisson–Lie brackets between the sixcommutative group coordinates θ,x ,ξ
µ i
{ }
which are splitted in the following three sets.
Those involving spacetime x group coordinates:
µ
•
tanhρx tanhρx
1 2
x ,x = z x ,x = z x ,x = 0. (4.3)
{ 0 1} − ρcosh2ρx { 0 2} − ρ { 1 2}
2
Those that comprise space x and boost ξ coordinates (besides the latter bracket):
•
z coshρx coshξ
2 1
x ,ξ = +tanhρx sinhρx A
1 1 1 2
{ } coshρx coshρx − coshξ
2 (cid:18) 1 2 (cid:19)
coshρx
1
x1,ξ2 = zcoshξ2B x2,ξ2 = z coshξ1 coshξ2 (4.4)
{ } − { } coshρx −
(cid:18) 2 (cid:19)
tanhξ
2
x ,ξ = zA ξ ,ξ = zρsinhρx C .
{ 2 1} − { 1 2} 1 − cosh2ρx
(cid:18) 2(cid:19)
10