Possible quantum kinematics. II. Non-minimal case N.A. Gromov Department of Mathematics, 0 1 Komi Science Center UrD RAS 0 Chernova st., 3a, Syktyvkar, 167982, Russia 2 E-mail: gromov@dm.komisc.ru n a J Abstract 2 2 The quantum analogs of the N-dimensional Cayley-Klein spaces with different combinations of quantum and Cayley-Klein structures are described ] h for non-minimal multipliers, which include the first and the second powers of p - contraction parameters in thetransformation of deformationparameter. The h noncommutative analogs of (N − 1)-dimensional constant curvature spaces t a are introduced. Part of these spaces for N = 5 are interpreted as the non- m commutative analogs of (1 + 3) space-time models. As a result the wide [ variety of the quantum deformations of realistic kinematics are suggested. 1 PACS: 02.40.Gh, 02.40.Yy, 04.20.Gz v 8 7 9 1 Introduction 3 . 1 0 Space-time is a fundamental conception that underlines the most significant 0 physical theories. Therefore the analysis of a possible space-time models 1 has the fundamental meaning for physics. In the previous papers [1],[2] : v the noncommutative analogs of the possible commutative kinematics [3] was i X constructed starting from the mathematical theory of quantum groups and r a quantum vector spaces [4]. Cayley-Klein scheme of contractions and ana- lytical continuations was applied to the five dimensional q-Euclidean space. Different combinations of quantum structure and Cayley-Klein scheme was described with the help of permutations σ. The transformation of the deformation parameter z = Jv under contraction is added in the quantum case as compared with the commutative one. The analysis in [1],[2] was confined to the minimal multiplier J, which has the first power multiplication of contraction parameters. This restriction imply that for certain combinations of quantum structure and Cayley-Klein scheme kinematics contractions do not exist. In order to all Cayley-Klein 1 contractions was possible for all permutations σ it is necessary to regard non-minimal multiplier J, which include the first and the second powers of contractionparameters. Inthispaperwefindnon-minimal multipliers forthe general quantum Cayley-Klein spaces (Sec. 2) and describe noncommutative four-dimensional space-time models (Sec. 3). 2 Quantum Cayley-Klein spaces It was shown in the previous paper [1] that the quantum N-dimensional Cayley-Klein space ON(j;σ;C) is characterized by the following commuta- v tion relations of the Cartesian generators (1,σk′) ′ ′ ξ ξ = ξ ξ coshJv −iξ ξ sinhJv, k < m < k , k 6= m, σk σm σm σk σm σk′ (1,σ ) k (1,σm′) ′ ′ ξ ξ = ξ ξ coshJv −iξ ξ sinhJv, m < k < m, k 6= m, σk σm σm σk σm′ σk (1,σ ) m (1) Jv (1,σ )2 [ξ ,ξ ] = 2iǫsinh( )(coshJv)n−kξ2 n+1 + σk σk′ 2 σn+1(1,σk)(1,σk′) sinh(Jv) n +i(coshJv)k+1(1,σk)(1,σk′) m=k+1(coshJv)m((1,σm)2ξσ2m +(1,σm′)2ξσ2m′), X (2) ′ where k,m = 1,2,...,n, N = 2n+1 or N = 2n, k = N +1−k, permutation σ = (σ ,...,σ ) describes definite combination of the quantum group 1 N structure and Cayley-Klein scheme of group contraction. The invariant form under the coaction of the corresponding quantum orthogonal group on the Cayley-Klein space ON(j;σ;C) is written as v (coshJv)n inv(j) = cosh(Jvρ ) ǫ(1,σ )2ξ2 + 1 n+1 σn+1cosh(Jv/2) n + ((1,σk)2ξσ2k +(1,σk′)2ξσ2k′)(coshJv)k−1!. (3) k=1 X The multiplier J in the transformation z = Jv of the deformation parameter need be chosen in such a way that all indefinite relations in commutators, which appear under nilpotent values of the contraction parameters are canceled . As far as multipliers (1,σk) and (1,σk′) enter sym- metrically into the commutators (1),(2), we can put σk < σk′ without loss 2 of generality. Then indefinite relations in commutators (1) take the form (1,σk)(1,σk′)−1 = (σk,σk′), where k = 1,2,...,n for N = 2n + 1 and k = 1,2,...,n−1 for N = 2n. They are eliminated by the multiplier n J0 = (σk,σk′). (4) k=1 [ Ithasthefirst power multiplicationofcontractionparametersandisthemin- imal multiplier, which guarantees the existence of the Hopf algebra structure for the associated quantum group SO (N;j;σ). The analysis in the previous v paper [1] was confined to this minimal case. The “union” of two multipliers is understood as the multiplication of all parameters j , which occur at least in one multiplier and the power of j in k k the “union” is equal to its maximal power in both multipliers, for example, (j j2) (j j ) = j j2j . 1 2 2 3 1 2 3 If we take into account indefinite relations in commutators (2), then we S come to the non-minimal multiplier J, which consist of contraction parameters in the first and the second powers. The indefinite relations in commutators (2) have the form nm=k+1[(1,σm)2 +(1,σm′)2] = nm=k+1(1,σm)2, (5) P (1,σk)(1,σk′) P(1,σk)2(σk,σk′) k = 1,...,n−1 for even N = 2n and (1,σn+1)+ nm=k+1[(1,σm)2 +(1,σm′)2] = (1,σn+1)+ nm=k+1(1,σm)2, P(1,σk)(1,σk′) (1,σkP)2(σk,σk′) (6) k = 1,...,n for odd N = 2n+1. Let us introduce numbers i = min{σ ,...,σ }, k k+1 n then kth expression in (5) or (6) is equal to (1,i )2 (ik,σk)−2(σk,σk′)−1, ik < σk k =  (σk,ik)(ik,σk′)−1, σk < ik < σk′ (1,σk)2(σk,σk′)  (σk,σk′)(σk′,ik)2, ik > σk′ and compensative multiplierfor this expression is as follows (ik,σk)2(σk,σk′), ik < σk (k) J1 =  (ik,σk′), σk < ik < σk′ (7)  1, ik > σk′.  3 For all expressions in (5) or (6) compensative multiplier J is obtained by 1 (k) the union J = J . Therefore the non-minimal multiplier J in the trans- 1 k 1 formation z = Jv of the deformation parameter is given by S (k) J = J J = J J (8) 0 1 0 1 k [ [ and include the first and the second powers of contraction parameters. The quantum orthogonal Cayley-Klein sphere S(N−1)(j;σ) is obtained v as the quotient of ON(j;σ) by inv(j) = 1. The quantum analogs of the v intrinsic Beltrami coordinates on this quantum sphere are given by the sets of independent right or left generators rσi−1 = ξσiξ1−1, rˆσi−1 = ξ1−1ξσi, i = 1,...,N, i 6= k, σk = 1. (9) Quantum kinematics are realized as quantum orthogonal sphere for N = 5 and certain values of contraction parameters. 3 Quantum kinematics Quantum vector spaces O5(j;σ) are generated by ξ , l = 1,...,5 with com- v σl mutation relations (k = 2,3,4) (1,σ ) 5 ξ ξ = ξ ξ cosh(Jv)−iξ ξ sinh(Jv), σ1 σk σk σ1 σk σ5(1,σ ) 1 (1,σ ) 1 ξ ξ = ξ ξ cosh(Jv)−iξ ξ sinh(Jv), σk σ5 σ5 σk σ1 σk(1,σ ) 5 (1,σ ) 4 ξ ξ = ξ ξ cosh(Jv)−iξ ξ sinh(Jv), σ2 σ3 σ3 σ2 σ3 σ4(1,σ ) 2 (1,σ ) 2 ξ ξ = ξ ξ cosh(Jv)−iξ ξ sinh(Jv), σ3 σ4 σ4 σ3 σ2 σ3(1,σ ) 4 (1,σ )2 [ξ ,ξ ] = 2iξ2 3 sinh(Jv/2), σ2 σ4 σ3(1,σ )(1,σ ) 2 4 [ξ ,ξ ] = 2i ξ2 (1,σ )2cosh(Jv)+(ξ2 (1,σ )2+ σ1 σ5 σ3 3 σ2 2 4 sinh(Jv/2) +ξ2 (1,σ )2)cosh(Jv/2) . (10) σ4 4 !(1,σ1)(1,σ5) By the analysis of the multiplier (8) for N = 5 and commutation relations (10) of the quantum space generators we have find three different non-minimal J, namely J = j2j , J = j j2 and J = j2j2. In the first 1 2 1 2 1 2 two cases all different permutations give in result equivalent kinematics, so we can regard only one permutation, say σˆ = (2,1,3,4,5) for J = j2j and 1 2 σˇ = (1,3,2,4,5)forJ = j j2. InthecaseofJ = j2j2 therearefournonequiv- 1 2 1 2 alent quantum kinematics corresponding to the following permutations: σ˜ = (2,3,1,4,5), σ = (3,1,5,2,4), σ = (3,1,2,4,5), σ = (3,2,1,4,5). I II III In order to clarify the relation with the standard Ino¨nu¨–Wigner contraction procedure [5], the mathematical parameter j is replaced by the physical 1 one ˜j T−1, and the parameter j is replaced by ic−1, where the standard de 1 2 Sitter kinematics S(−) with constant negative curvature is realized for ˜j = i 4 1 (+) and anti de Sitter kinematics S with positive curvature is obtained for 4 ˜j = 1. 1 The limit T → ∞ corresponds to the contraction j = ι , and the limit 1 1 c → ∞ corresponds to j = ι . The parameter T is interpreted as the cur- 2 2 vature radius and has the physical dimension of time [T] = [time], the parameter c is the light velocity [c] = [length][time]−1. As far as argument Jv of hyperbolic function must be dimensionless, then the physical dimension of deformation parameter is equal to those of J, i.e. [v] = [J]. For J = j2j 1 2 we obtain [v] = [cT2] = [length][time], for J = j j2 we have [v] = [c2T] = 1 2 [length]2[time]−1 = [length][velocity] and for J = j2j2 the physical dimension 1 2 of the deformation parameter is given by [v] = [c2T2] = [length]2. The generator ξ does not commute with others, therefore it is con- 1 venient to introduce right and left time t = ξ ξ−1,tˆ = ξ−1ξ and space 2 1 1 2 r = ξ ξ−1,rˆ = ξ−1ξ ,k = 1,2,3, generators. The reason for this defini- k k+2 1 k 1 k+2 tionis thesimplification ofexpressions forcommutation relationsofquantum kinematics. It is possible to use only, say, right generators, but its commutators are cumbersome in the case of the (anti) de Sitter kinematics. 3.1 Quantum (anti) de Sitter kinematics For σˆ = (2,1,3,4,5) the mathematical multiplier J = j2j is rewritten in the 1 2 form J = ij˜12 = iJˆ. Commutation relations of space and time generators of cT2 5 the (anti) de Sitter kinematics are 1 S4(±)(σˆ) = t,r| tˆr = rˆ tcosJˆv +ir sinJˆv , v ( p p(cid:18) 3c (cid:19) ˜j2 ˜j2 Jˆv cT2 Jˆv tˆr −rˆ t = 2i − 1 rˆ r cosJˆv + 1− 1 rˆ r cos sin , 3 3 c2T2 1 1 c2T2 2 2! 2 ! ˜j2 2 1 cT rˆ r = rˆ cosJˆv −itˆcsinJˆv r , rˆ r = rˆ cosJˆv −itˆ sinJˆv r . p 3 3 p 1 2 2 j˜1 ! 1) (cid:16) (cid:17) (11) The right and left generators are connected as follows 1 ˜j tˆ= tcosJˆv +ir sinJˆv, r = rˆ cosJˆv +ir 1 sinJˆv , 3 1 1 2 c cT ! j˜ Jˆv rˆ −r = 2i 1 rˆ r sin , rˆ = r cosJˆv +itcsinJˆv. (12) 2 2 1 1 3 3 cT 2 For σˇ = (1,3,2,4,5) the mathematical multiplier J = j j2 is rewritten 1 2 in the form J = − j˜1 = −Jˇ. Commutation relations of space and time c2T generators of the (anti) de Sitter kinematics are S4(±)(σˇ) = t,r| rˆt = tˆ r coshJˇv+ir sinhJˇv , v 1 1 2 ( (cid:16) (cid:17) cT tˆr = rˆ coshJˇv +irˆ sinhJˇv t, tˆr = rˆ coshJˇv + sinhJˇv t, 2 2 1 3 3 j˜1 ! (cid:16) (cid:17) Jˇv cT rˆr −r rˆ = 2itˆtc2sinh , rˆ r = rˆ coshJˇv + sinhJˇv r . (13) 1 2 2 1 2 p 3 3 j˜1 ! p) The right and left generators are connected as follows j˜ j˜ t = tˆ coshJˇv − 1 r sinhJˇv , r = rˆ coshJˇv − 1 r sinhJˇv , 3 p p 3 cT ! cT ! 2 j˜2 Jˇv Jˇv rˆ −r = cT tˆtcoshJˇv − 1 (rˆ r +rˆ r )cosh sinh . (14) 3 3 j˜  c2T2 1 1 2 2 2  2 1   6 For σ˜ = (2,3,1,4,5) the mathematical multiplier J = j2j2 is rewritten 1 2 in the form J = − j˜12 = −J˜. Commutation relations of space and time c2T2 generators of the (anti) de Sitter kinematics are 1 S4(±)(σ˜) = t,r| tˆr = rˆ tcoshJ˜v −r sinhJ˜v , v ( p p(cid:18) 3c (cid:19) ˜j2 J˜v cT2 J˜v tˆr −rˆ t = −2 coshJ˜v − 1 (rˆ r +rˆ r )cosh sinh , 3 3 " c2T2 1 1 2 2 2 # ˜j2 2 1 c2T2 J˜v rˆ r = rˆ coshJ˜v+tˆcsinhJ˜v r , rˆ r −rˆ r = −2i sinh , (15) p 3 3 p 1 2 2 1 ˜j2 2 ) (cid:16) (cid:17) 1 where k = 1,2,3, p = 1,2. The left and right generators are connected by the relations 1 tˆ= tcoshJ˜v −r sinhJ˜v, rˆ = r coshJ˜v +ir sinhJ˜v, 3 1 1 2 c rˆ = r coshJ˜v −ir sinhJ˜v, rˆ = r coshJ˜v−tcsinhJ˜v. (16) 2 2 1 3 3 For σ = (3,1,5,2,4)commutation relations of space and time generators I of the (anti) de Sitter kinematics are S4(±)(σ ) = t,r| rˆ t = tˆ r coshJ˜v +ir sinhJ˜v , v I 1 1 2 ( (cid:16) (cid:17) T tˆr = rˆ coshJ˜v +irˆ sinhJ˜v , rˆ t = tˆcoshJ˜v +i sinhJ˜v r , 2 2 1 3 ˜j1 ! 3 (cid:16) (cid:17) rˆ r = rˆ r coshJ˜v +ir sinhJ˜v , rˆ r = rˆ coshJ˜v +irˆ sinhJ˜v r , 1 3 3 1 2 3 2 2 1 3 (cid:16) (cid:17) (cid:16) (cid:17) ˜j2 ˜j2 J˜v c2T2 J˜v rˆ r −rˆ r = 2i − 1 rˆ r coshJ˜v + 1+ 1 tˆt cosh sinh . 1 2 2 1 " c2T2 3 3 T2 ! 2 # ˜j2 2 ) 1 (17) The left and right generators are connected by the relations rˆ = r coshJ˜v +ir sinhJ˜v, r = rˆ coshJ˜v +irˆ sinhJ˜v, 1 1 2 2 2 1 ˜j ˜j J˜v r = rˆ coshJ˜v +i 1tsinhJ˜v , t−tˆ= 2irˆ r 1 sinh . (18) 3 3 T ! 3 3c2T 2 7 Forσ = (3,1,2,4,5)commutationrelationsofspaceandtimegenerators II of the (anti) de Sitter kinematics are S4(±)(σ ) = t,r| rˆ t = tˆ r coshJ˜v+ir sinhJ˜v , v II 1 1 3 ( (cid:16) (cid:17) cT tˆr = rˆ coshJ˜v +i sinhJ˜v t, tˆr = rˆ coshJ˜v+irˆ sinhJ˜v t, 2 2 ˜j1 ! 3 3 1 (cid:16) (cid:17) rˆ r = rˆ r coshJ˜v +ir sinhJ˜v , rˆ r = rˆ coshJ˜v +irˆ sinhJ˜v r , 1 2 2 1 3 2 3 3 1 2 (cid:16) (cid:17) (cid:16) (cid:17) ˜j2 ˜j2 J˜v c2T2 J˜v rˆ r −rˆ r = 2i 1 tˆtcoshJ˜v + 1− 1 rˆ r cosh sinh . 1 3 3 1 "T2 c2T2 2 2! 2 # ˜j2 2 ) 1 (19) The left and right generators are connected by the relations c J˜v rˆ = r coshJ˜v+ir sinhJ˜v, rˆ = r coshJ˜v +2tˆt˜j sinh , 1 1 3 2 2 1 T 2 ˜j r = rˆ coshJ˜v +irˆ sinhJ˜v, t = tˆcoshJ˜v−tˆr 1 sinhJ˜v. (20) 3 3 1 2 cT For σ = (3,2,1,4,5) commutation relations of space and time genera- III tors of the (anti) de Sitter kinematics are S4(±)(σ ) = t,r| rˆ t = tˆ r coshJ˜v +ir sinhJ˜v , v III 1 1 3 ( (cid:16) (cid:17) cT2 tˆr −rˆ t = −2 sinhJ˜v, tˆr = rˆ coshJ˜v +irˆ sinhJ˜v t, 2 2 ˜j2 3 3 1 1 (cid:16) (cid:17) rˆ r = rˆ r coshJ˜v +ir sinhJ˜v , rˆ r = rˆ coshJ˜v +irˆ sinhJ˜v r , 1 2 2 1 3 2 3 3 1 2 (cid:16) (cid:17) (cid:16) (cid:17) ˜j2 1 J˜v c2T2 J˜v rˆ r −rˆ r = 2i coshJ˜v + 1 tˆt− rˆ r cosh sinh . (21) 1 3 3 1 " T2 (cid:18) c2 2 2(cid:19) 2 # ˜j12 2 ) The left and right generators are connected by the relations rˆ = r coshJ˜v +ir sinhJ˜v, r = rˆ coshJ˜v+ctˆsinhJ˜v, 1 1 3 2 2 1 r = rˆ coshJ˜v +irˆ sinhJ˜v, tˆ= tcoshJ˜v −r sinhJ˜v. (22) 3 3 1 2 c 8 3.2 Quantum Minkowski kinematics ZerocurvaturelimitT → ∞transform(anti)deSitterkinematicstoMinkowski one. For permutations σˆ,σ˜,σ ,σ ,σ left and right space-time generators I II III coincide tˆ= t,rˆ = r and commutation relations take the form k k v M4(σ˜) = t,r| [t,r ] = 0, [t,r ] = − , [r ,r ] = −iv, [r ,r ] = 0, p = 1,2 , v ( p 3 c 1 2 p 3 ) M4(σˆ) = t,r| [t,r ] = 0, [t,r ] = iv, [r ,r ] = 0, i,k = 1,2,3 , v p 3 i k ( ) M4(σ ) = t,r| [t,r ] = 0, [r ,r ] = [r ,r ] = 0, [r ,r ] = iv , v I k 1 3 2 3 1 2 ( ) M4(σ ) = t,r| [t,r ] = 0, [r ,r ] = [r ,r ] = 0, [r ,r ] = iv , v II k 1 2 2 3 1 3 ( ) v M4(σ ) = t,r| [t,r ] = [t,r ] = 0, [t,r ] = − , v III ( 1 3 2 c [r ,r ] = [r ,r ] = 0, [r ,r ] = iv , (23) 1 2 2 3 1 3 ) For permutation σˇ we have tˆ= t,rˆ = r , but rˆ = r − vt2 and Minkowski p p 3 3 c kinematics is given by v M4(σˇ) = t,r| [t,r ] = 0, [t,r ] = t(1−t2), [r ,r ] = 0, v ( p 3 c 1 2 v v [r ,r ] = r (1−t2), [r ,r ] = r (1−t2) . (24) 2 3 2 1 3 1 c c ) M4(σ ) and M4(σ ) are connected by replacement of space generators r v I v II 2 with r and vice-versa, therefore they can be regarded as equivalent kinemat- 3 ics. The same is true for M4(σ˜) and M4(σ ). so we obtain four nonequiv- v v III alent Minkowski kinematics. 9 3.3 Quantum Newton kinematics Newton kinematics are obtained from (anti) de Sitter kinematics by the non- relativistic limit c → ∞ and are characterized by the following commutation relations of generators N4(±)(σ˜) = t,r| [t,r ] = 0, [r ,r ] = −iv, [r ,r ] = 0 , v k 1 2 p 3 ( ) v N4(±)(σˇ) = t,r| [t,r ] = 0, [r ,r ] = i˜j t2, [r ,r ] = 0 , v ( k 1 2 1T p 3 ) ˜j2 N4(±)(σ ) = t,r| [t,r ] = 0, [r ,r ] = [r ,r ] = 0, [r ,r ] = iv 1+ 1 t2 , v I ( k 1 3 2 3 1 2 T2 !) N4(±)(σ ) = N4(±)(σ ) = t,r| [t,r ] = 0, [r ,r ] = [r ,r ] = 0, v II v III k 1 2 2 3 ( ˜j2 [r ,r ] = iv 1+ 1 t2 . (25) 1 3 T2 !) Unlike previous cases, where tˆ = t,rˆ = r , for permutation σˆ the third k k left and right space generators are connected as rˆ = r + i˜j2 v , therefore 3 3 1T2 commutation relations in this case are ˜j2 N4(±)(σˆ) = t,r| [t,r ] = 0, [t,r ] = iv 1+ 1 t2 , v ( p 3 T2 ! v [r ,r ] = −i˜j tr , [r ,r ] = 0 . (26) 1 2 1 1 p 3 T ) Nonequivalent (anti) de Sitter kinematics S4(±)(σ ) and S4(±)(σ ) become v II v III identical one N4(±)(σ ) = N4(±)(σ ) in the non-relativistic limit. N4(σ ) v II v III v I and N4(σ ) are connected by replacement of space generators r with r and v II 2 3 vice-versa, therefore they can be regarded as equivalent kinematics. So there are four nonequivalent Newton kinematics. 3.4 Quantum Galilei kinematics Galilei kinematics can be obtained by the non-relativistic limit c → ∞ of the Minkowski kinematics (23),(24) or by the zero curvature limit T → ∞ of 10

Possible quantum kinematics. II. Non-minimal case

Similar Possible quantum kinematics. II. Non-minimal case