The Quantum Spheres and their Embedding into Quantum Minkowski Space-Time

We recast the Podle\`s spheres in the noncommutative physics context by showing that they can be regarded as slices along the time coordinate of the different regions of the quantum Minkowski space-time. The investigation of the transformations of the quantum sphere states under the left coaction of the ${\cal SO}_{q}(3)$ group leads to a decomposition of the transformed Hilbert space states in terms of orthogonal subspaces exhibiting the periodicity of the quantum sphere states.


Introduction
A great variety of works based on the quantum spheres have been developped since the appearence of Podles spheres [1] and their symmetries [2].Most part of these studies have been done either in the quantum bundle formalism where the quantum spheres provide concrete examples to test the different structures of this formalism [3,4,5] or, more recently, in quantum field theories on quantum spheres which should respect the SU q (2) quantum symmetries (see for example [6,7,8,9] and references therein).In the other hand, the evolution of a free particle in the quantum Minkowski space-time has been analysed in [10] and the transformations of its quantum velocity under the Lorentz subgroup of boost transformations in [11].
In section 1 of this paper we persue these studies by recasting the quantum spheres in the noncommutative special relativity where we show that we can regard them as quantum manifolds embedded into the quantum Minkowski space-time.This embedding preserves the reality structure and the commutations rules of the quantum Minkowski space-time coordinates.In particular, we show that in the time-like region of the quantum Minkowski space-time the Hilbert space H (L) of states describing the noncommutative relativistic evolution of a free particle having a quantum velocity of length |v| 2 q = (1 + Q 2 c(L)), c(L) = − 1 (q (L+1) +q −(L+1) ) 2 with L ≥ 1 is an integer, q is the deformation parameter and Q = q + q −1 are precisely, for fixed time, the space of irreducible representations of the Podles quantum spheres S 2 qc with c = c(L).We also show in this section that the Hilbert space of representations of the space-like region of the quantum Minkowski space-time corresponds, for particular fixed time, to the Hilbert space of representations of the quantum spheres S 2 qc where c ∈]0, [∞ or S 2 q∞ .In section 3, we shox that the state transformations under the coaction of the SO q (3) group exhibites the periodicity of the quantum sphere states through a decomposition of the transformed Hilbert space in terms of orthogonal subspaces each describes the same quantum sphere.

The quantum spheres
Before embedding the different quantum spheres into quantum Minkowski space-time, let us recall briefly some properties of the noncommutative special relativity presented in [10].First it was shown in [12] that the generators Λ M N (N, M = 0, 1, 2, 3) of quantum Lorentz group may be written in terms of those of quantum SL(2, C) group as where M β α (α, β = 1, 2) and M β α = (M β α ) ⋆ are the generators of the quantum SL(2, C) group subject to the unimodularity conditions and the spinor metrics are taken to be The R-matrices are given by The Lorentz group generators are real, (Λ M N ) ⋆ = Λ M N , and generate a Hopf algebra L endowed with a coaction ∆, a counit ε and an antipode S acting as ∆(Λ is an invertible and hermitian quantum metric given by of the generators of the quantum Lorentz group.The quantum metric G N M can be considered as a metric of a quantum Minkowski space-time M 4 equipped with real coordinates X N , (X N ) ⋆ = X N .X 0 represents the time operator and X i (i = 1, 2, 3) represent the space right invariant coordinates, ∆ R (X I ) = X I ⊗ I, which transform under the left coaction as From the hermiticity of the Minkowskian metric and the orthogonality conditions we can see that the four-vector length G N M X N X M = −τ 2 is real and invariant.It was also shown in [12] that τ 2 is central, it commutes with the Minkowski space-time coordinates and the quantum Lorentz group generators.Λ M N and X N are subject to the commutation rules controlled by the R N M P Q matrix as: and where the R-matrix of the Lorentz group is constructed out of those of SL(2, C) group and satisfyes the relations show the quantum symmetrization of the Minkowskian metric G N M and its inverse.To make an explicit calculation of the different commutation rules of the generators of the quantum Lorentz group, we take the following choice of Pauli hermitian matrices This choice leads us to a quantum metric form G LK exhibiting two independent blocks, one for the time index and the others for space components indices (k = 1, 2, 3) whose nonvanishing elements are . In the classical limit q = 1, this metric reduces to the classical Minkowski metric with signature (−, +, +, +).Explicitly, the length of the four-vector X N reads where The Pauli matrices satisfy which make explicit the restriction of the quantum Lorentz group to the quantum subgroup of the three dimensional space rotations by restricting the quantum SL(2, C) group generators to those of the SU(2) group.In fact when we impose the unitarity conditions, which lead us to the restriction of the Minkowski space-time transformations under the quantum Lorentz group to the orthogonal transformations group SO q (3).This subgroup leaves invariant the three dimensional quantum subspace R 3 ⊂ M 4 equipped with the real coordinate system X i (i = 1, 2, 3) and the Euclidian metric G ij .More precisely as a consequence of (7), (2) reduces to where ∆ (L) is the restriction of (2) to the three dimensional quantum subspace which are the quantum symmetrization of the Euclidian metric G ij and its inverse.The form of the antipode S(Λ j i ) implies the orthogonality properties of the generators of the quantum subgroup SO q (3) as The commutation rules of the coordinate X i of R 3 satisfy the same commutation rules (6) where X 0 is taken to be a constant parameter, recall that it commutes with the spacial coordinates X i .Therefore, Λ with the commutation relation and SO q (3) group.In the three dimensional space R 3 spanned by the basis X z , X z and X 3 , where the SO q (3) coacts, the generators Λ where the indices i, j run over z = 1 + i2, z = 1 − i2 and 3.
In the case τ 2 > 0, time-like region, it was shown in [10] that the evolution of a free particle in the Minkowski space-time is described by states belonging to the Hilbert space H (L) whose basis is spanned by common eingenstate of X 0 and X 3 where Q , L = 0, 1, 2, ....∞ and n runs by integer steps over the range 0 ≤ n ≤ L. X z and X z act on the basis elements of H (L) as respectively where λλ = 1.In the following we take λ = −1.The length of velocity of the particle is given by | v| 2 q = q 2 ( (qVz q = 1 which is the velocity of the light.In this region the evolution of the particle is described by states |t, n (n = 0, 1, ..., ∞) satisfying In the following we take the length of the quantum three-vector as 3 .The quantum group SO q (3) acts on the spatial coordinates X i as (8) and lives invariant both X 0 and τ 2 , then q and the relations (11,12) for finite L ≥ 1 and and the relations (13) for L = ∞ where the orthonormal states |L, n and |n , L, n ′ |L, n = δ n ′ ,n , n ′ , n = 0, 1, ..., L, and n ′ |n = δ n ′ ,n , n ′ , n = 0, 1, ..., ∞, denote the states satisfying (11,12) and ( 13) respectively.The unique state |t, 0, 0 corresponding to L = 0 describes a particle at rest at the origin of the spacial coordinate system, for t = 0 then τ 2 = 0 it representes the origin of the four coordinate system of the quantum Minkowski space-time, X N |0, 0, 0 = 0|0, 0, 0 .Therefore, The L + 1 dimensional Hilbert subspace H (L) of states describing the evolution of a free particle of a given length of the velocity in the noncommutative Minkowski space-time can be identified, for fixed time, with the Hilbert space q of irreducible representations of the quantum spheres of radius R . This observation leads us to state Theorem: the Podles spheres S 2 qλρ are slices along the time coordinate of the different regions of the quantum Minkowski space-time M 4 The quantum spheres S 2 qc where c = c(n) = − 1 (q n +q −n ) 2 n = 2, 3, ..., ∞ correspond to slices at t 0 = −q and τ 2 0 = q 1 2 γ 2(L) where n = L + 1, L ≥ 1.The quantum spheres S 2 qc where c ∈]0, ∞[ correspond to slices at t 0 = −q and τ 2 0 = −q 1 2 Q 2 c and S 2 q∞ corresponds to a slice at t 0 = 0 and τ 2 0 = −q proof: Due to the fact that X 0 and τ 2 commute with the spacial coordinates X z , X z and X 3 , X 0 and τ 2 can be taken to be constants without contradict the commutation rules (6).If we set X z = Qe 1 , X 3 = e 0 , X z = Qe −1 , λ = (q − q −1 )X 0 = (q − q −1 )t and ρ = q −2 t 2 − q − 1 2 τ 2 into (5-6), we see that we recover the algebra generators of the quantum sphere A(S 2 q ) given by (2a-e) in [1].
For t 0 = −q and τ 2 0 = q where n = L + 1.These constraints correspond to slices of the past time-like region for finite L or a slice of the past light-cone region for L = ∞.They fit with the quantum spheres S 2 qc with c = c(n) ≤ 0 which are described by states satisfying (11,12,14) for finite L ≥ 1 and (13) and (15) for L = ∞.
We follow the same procedure presented in [10] to invstigate the Hilbert space H s of space of representations of the space-like region of the Minkowski space-time.The elements |t, n n = 0, 1, ..., ∞ of the basis of H s satisfy )) − α) and where If we put in (18) t = t 0 = −q the Hilbert space states H s can be identified to the space of irreducible representations of the Podles quantum spheres S 2 qc where c ∈]0, ∞[ and if we put in (17) t 0 = 0 and α = −q 2 Q 2 we obtain the space of representations of the Podlés quantum sphere S 2 q∞ .Q.E.D.
We may also consider, as for the time-like region, that the Hilbert space H s spanned by |t, n satisfying (18) as space of states describing the evolution in the space-like region of the quantum Minkowski space-time of a free particle moving with an operator velocity of components V z = X x /t, V z = X z /t and V 3 = X 3 /t.In this case we obtain from (18) a length of the velocity [10].Note that from (18), (11,12) and (13) we have To investigate the transformations of the quantum sphere states under the SO q (3) quantum group we have to construct the Hilbert space states H SOq(3) where the generators Λ fulfil the same commutation rules (6) the transformed states of the quantum sphere satisfy the same relations (11,12) in the time-like region, (13) in the light-cone and (17) in the space-like region.Since the coordinates X i transform under the tensorial product of SO q (3) and S 2 q , the transformed states also belong to the tensorial product H SOq(3) ⊗ H S 2 q [10] which needs the construction of the Hilbert space H SOq(3) .
To construct H SOq(3) we are not obliged to compute explicitly the complicated commutation rules (3) where we impose (7) but we simply consider the action of the SU q (2) generators on the orthonormal Hilbert space states; γ|n = q n |n , (10), give the action of the generators of SO q (3) on the basis |n of the Hilbert space H SOq(3) as Now we are ready to investigate the transformations of the quantum sphere states under the SO q (3) group.

1 2 m
+ 2, L, n + 1|L, p (30) from (27).The relations (28-30) are the recursion formulas which permit to compute the coefficients m, L, n|L, p giving the transformed states |L, p in terms of basis elements of the Hilbert space states H SOq(3) ⊗ H