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Twisted acceleration-enlarged Newton-Hooke space-times and breaking classical symmetry phenomena 2 1 0 2 n Marcin Daszkiewicz a J 5 2 ] h Institute of Theoretical Physics p - University of Wroclaw pl. Maxa Borna 9, 50−206 Wroclaw, Poland h e−mail : [email protected] t a m [ 1 v 3 6 2 5 . 1 0 2 1 Abstract : v We find the subgroup of classical acceleration-enlarged Newton-Hooke Hopf al- i X gebra which acts covariantly on the twisted acceleration-enlarged Newton-Hooke r space-times. The case of classical acceleration-enlarged Galilei quantum group is a considered as well. 1 1 Introduction Presently, physicists expect that there exist aberrations from relativistic kinematics in high energy (transplanckian) regime. Such a suggestion follows from many theoretical [1], [2] as well as experimental (see e.g. [3]) investigations performed in the last time. Generally, there exist two approaches to describe the particle kinematics in ultra- high energy regime. First of them assumes that relativistic symmetry becomes broken at Planck’s scale to the proper subgroup of Poincare algebra [4], [5]. The second approach is more sofisticated, i.e. it assumes that relativistic symmetry is still present in high energy regime, but it becomes deformed [6]. The first treatment has been proposed in [4], [5] where authors assumed that the whole Lorentz algebra is broken to the four subgroups: T(2), E(2), HOM(2) and SIM(2) identified with four versions of so-called Very Special Relativity. The second treatment arises from Quantum Group Theory [7], [8] which, in accordance with Hopf-algebraic classification of all relativistic and nonrelativistic deformations [9], [10], provides three types of quantum spaces. First of them corresponds to the well-known canonical type of noncommutativity [ xˆ ,xˆ ] = iθ , (1) µ ν µν with antisymmetric constant tensor θµν. Its relativistic and nonrelativistic Hopf-algebraic counterparts have been proposed in [11] and [12] respectively. The second kind of mentioned deformations introduces the Lie-algebraic type of space- time noncommutativity [ xˆ ,xˆ ] = iθρ xˆ , (2) µ ν µν ρ with particularly chosen coefficients θρ being constants. The corresponding Poincare µν quantum groups have been introduced in [13]-[15], while the suitable Galilei algebras - in [16] and [12]. The last kind of quantum space, so-called quadratic type of noncommutativity [ xˆ ,xˆ ] = iθρτxˆ xˆ ; θρτ = const. , (3) µ ν µν ρ τ µν has been proposed in [17], [18] and [15] at relativistic and in [19] at nonrelativistic level. The links between both (mentioned above) approaches have been investigated recently in articles [20] and [21]. Preciously, it has been demonstrated that the very particular realizations of canonical, Lie-algebraic and quadratic space-time noncommutativity are covariant with respect the action of undeformed T(2), E(2) and HOM(2) subgroups re- spectively. Such a result seems to be quite interesting because it connects two different approaches tothe same problem -to the formofPoincare algebra at Planck’s scale. It also confirms expectation that relativistic symmetry in high energy regime should be modified, while the realizations of such an idea by breaking or deforming of Poincare algebra plays only the secondary role. Inthis articleweextend described aboveinvestigations tothecaseofclassical accelera- tion-enlarged Newton-Hooke Hopf algebras U (NH ) [22], [23]. Particulary, we find their 0 ± d 2 subgroupswhichactcovariantlyonthefollowing(providedin[24]and[25])twist-deformed acceleration-enlarged Newton-Hooke space-times1,2 t [ t,x ] = 0 , [ x ,x ] = if , (4) i i j ± τ (cid:18) (cid:19) with t t t t t t f = f sinh ,cosh , f = f sin ,cos . + − τ τ τ τ τ τ (cid:18) (cid:19) (cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19) (cid:18) (cid:19) (cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19) Further, by contraction limit of obtained results (τ → ∞), we analyze the case of so-called classical acceleration-enlarged Galilei Hopf algebra U (G) proposed in [26]. 0 The paper is organized as follows. In second section we describe the general algorithm used in present article. Sections three and four are devotbed respectively to the subgroups of classical acceleration-enlarged Newton-Hooke as well as classical acceleration-enlarged Galilei Hopf symmetries acting covariantly on the proper (acceleration-enlarged) twist- deformed space-times (4). Final remarks are presented in the last section. 2 General prescription In this section we describe the general algorithm which can be applied to the arbitrary twist deformation of space-time symmetries algebra A. Firstofall,werecallbasicfactsrelatedwiththetwist-deformedquantumgroup U (A) F and with the corresponding quantum space-time. In accordance with general twist pro- cedure [27], the algebraic sector of Hopf structure U (A) remains undeformed, while the F coproducts and antipodes transform as follows ∆ (a) → ∆ (a) = F ◦ ∆ (a) ◦F−1 , S (a) = u S (a)u−1 , (5) 0 F 0 F F 0 F with∆ (a) = a⊗1+1⊗a, S (a) = −aandu = f S (f )(weuseSweedler’s notation 0 0 F (1) 0 (2) F = f ⊗ f ). Present in the above formula twist element F ∈ U (A) ⊗ U (A) (1) (2) F F P satisfies the classical cocycle condition P F ·(∆ ⊗1) F = F ·(1⊗∆ ) F , (6) 12 0 23 0 and the normalization condition (ǫ⊗1) F = (1⊗ǫ) F = 1 , (7) with F = F ⊗1, F = 1⊗F and 12 23 ∆ (a) = a⊗1+1⊗a . (8) 0 1x0 =ct. 2Itshouldbenotedthatsymbolτ playstheroleoftimescaleparameter(cosmologicalconstant),which isresponsibleforoscillationorexpansionofspace-timenoncommutativity(4). Forτ approachinginfinity wereproducethecanonical(1),Lie-algebraic(2)andquadratic(3)typeofspace-timenoncommutativity. 3 The corresponding to the above Hopf structure space-time is defined as quantum representation space (Hopf module) with action of the symmetry generators satisfying suitably deformed Leibnitz rules [28], [11] h⊲ω (f(x)⊗g(x)) = ω (∆ (h)⊲f(x)⊗g(x)) , (9) F F F for h ∈ U (A) or U (A) and F F ω (f(x)⊗g(x)) = ω ◦ F−1 ⊲f(x)⊗g(x) ; ω ◦(a⊗b) = a·b . (10) F The action of U (A) algebra on(cid:0)its Hopf module of(cid:1)functions depending on space-time F coordinates x is given by µ h⊲f(x) = h(x ,∂ )f(x) , (11) µ µ while the ⋆ -multiplication of arbitrary two functions is defined as follows F f(x)⋆ g(x) := ω ◦ F−1 ⊲f(x)⊗g(x) . (12) F It should be also noted that the commutati(cid:0)on relations (cid:1) [ x ,x ] = x ⋆ x −x ⋆ x , (13) µ ν ⋆F µ F ν ν F µ are covariant (by definition) with respect the action of Hopf algebra generators (see de- formed Leibnitz rules (9)). In this article we consider the action of undeformed acceleration-enlarged Newton- Hooke as well as classical acceleration-enlarged Galilei Hopf algebras on the commutation relations (13) (A = NH or G). It is given by the particular realizations of differential ± representation (16) and new classical Leibnitz rules d b h⊲ω (f(x)⊗g(x)) = ω (∆ (h)⊲f(x)⊗g(x)) , (14) F F 0 associated with coproduct (8). Further, we demonstrate that in such a case the relations (13) are not invariant with respect the action of the whole algebras U (NH ) and U (G), 0 ± 0 but only with respect their proper subgroups. Such an effect can be identified with the breaking classical symmetry phenomena associated with twist-deformeddspace-times (1b3). 3 Breaking of classical acceleration-enlarged Newton- Hooke symmetry In this section we turn to the case of undeformed acceleration-enlarged Newton-Hooke Hopf algebra U (NH ) defined by the following algebraic sector3 0 ± i [M ,Md] = i(δ M −δ M +δ M −δ M ) , [H,P ] = ± K , ij kl il jk jl ik jk il ik jl i τ2 i 3The both Hopf structures U0(NH±) contain, apart from rotation(Mij), boost (Ki) and space-time translation (P ,H) generators,the additional ones denoted by F , responsible for constant acceleration. i i d 4 [M ,K ] = i(δ K −δ K ) , [M ,P ] = i(δ P −δ P ) , ij k jk i ik j ij k jk i ik j [M ,H] = [K ,K ] = [K ,P ] = 0 , [K ,H] = −iP , [P ,P ] = 0 , (15) ij i j i j i i i j [F ,F ] = [F ,P ] = [F ,K ] = 0 , [M ,F ] = i(δ F −δ F ) , i j i j i j ij k jk i ik j [H,F ] = 2iK , i i and classical coproduct (8). One can check that the above structure is represented on Hopf module of functions as follows (see formula (11)) t H ⊲f(t,x) = i∂ f(t,x) , P ⊲f(t,x) = iC ∂ f(t,x) , (16) t i ± i τ (cid:18) (cid:19) t M ⊲f(t,x) = i(x ∂ −x ∂ )f(t,x) , K ⊲f(t,x) = iτ S ∂ f(t,x) , (17) ij i j j i i ± i τ (cid:18) (cid:19) and t F ⊲f(t,x) = ±2iτ2 C −1 ∂ f(t,x) , (18) i ± i τ (cid:18) (cid:18) (cid:19) (cid:19) with t t t t C = cosh/cos and S = sinh/sin . +/− +/− τ τ τ τ (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) As it was already mentioned in Introduction the twist deformations of quantum group U (NH ) have been provided in [24]. Here, we take under consideration the twisted 0 ± acceleration-enlarged Newton-Hooke space-times defined by the following twist factors d 2 i F = F = exp αklP ∧P [ αkl = −αlk = α ] , (19) α1 4 1 k l 1 1 1 " # k,l=1 X 2 i F = F = exp αklK ∧P [ αkl = −αlk = α ] , (20) α2 4 2 k l 2 2 2 " # k,l=1 X 2 i F = F = exp αklK ∧K [ αkl = −αlk = α ] , (21) α3 4 3 k l 3 3 3 " # k,l=1 X 2 i F = F = exp αklF ∧F [ αkl = −αlk = α ] , (22) α4 4 4 k l 4 4 4 " # k,l=1 X 2 i F = F = exp αklF ∧P [ αkl = −αlk = α ] , (23) α5 4 5 k l 5 5 5 " # k,l=1 X 2 i F = F = exp αklK ∧F [ αkl = −αlk = α ] . (24) α6 4 6 k l 6 6 6 " # k,l=1 X 5 In other words, we consider spaces of the form [ t,x ] = [ x ,x ] = [ x ,x ] = 0 , [ x ,x ] = if(t) ; i = 1,2,3 , (25) i ⋆F 1 3 ⋆F 2 3 ⋆F 1 2 ⋆F with function f(t) given by t t f(t) = f (t) = f = κ C2 , (26) κ1 ±,κ1 τ 1 ± τ (cid:18) (cid:19) (cid:18) (cid:19) t t t f(t) = f (t) = f = κ τ C S , (27) κ2 ±,κ2 τ 2 ± τ ± τ (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) t t f(t) = f (t) = f = κ τ2S2 , (28) κ3 ±,κ3 τ 3 ± τ (cid:18) (cid:19) (cid:18) (cid:19) 2 t t f(t) = f (t) = f = 4κ τ4 C −1 , (29) κ4 ±,κ4 τ 4 ± τ (cid:18) (cid:19) (cid:18) (cid:18) (cid:19) (cid:19) t t t f(t) = f (t) = f = ±κ τ2 C −1 C , (30) κ5 ±,κ5 τ 5 ± τ ± τ (cid:18) (cid:19) (cid:18) (cid:18) (cid:19) (cid:19) (cid:18) (cid:19) t t t f(t) = f (t) = f = ±κ τ3 C −1 S . (31) κ6 ±,κ6 τ 6 ± τ ± τ (cid:18) (cid:19) (cid:18) (cid:18) (cid:19) (cid:19) (cid:18) (cid:19) Of course, for all parameters κ running to zero the above space-times become com- a mutative. Let us now turn to the covariance of relations (26)-(31) with respect the action of undeformed Hopf algebra U (NH ). Using differential representation (16)-(18), classical 0 ± Leibnitz rules (8) and twist factors (19)-(24), one finds (see prescription (14)) d G ⊲ [ t,x ] = 0 , (32) k i ⋆F G ⊲ [ x ,x ] −if(t)(δ δ −δ δ ) = 0 ; G = P , K , F , (33) k i j ⋆F 1i 2j 1j 2i k k k k h i M ⊲ [ t,x ] = 0 , M ⊲ [ x ,x ] −if(t)(δ δ −δ δ ) = 0 , (34) kl i ⋆F 12 i j ⋆F 1i 2j 1j 2i h i M ⊲ [ x ,x ] −if(t)(δ δ −δ δ ) = f(t)(δ δ −δ δ ) , (35) 13 i j ⋆F 1i 2j 1j 2i 2i 3j 2j 3i h i M ⊲ [ x ,x ] −if(t)(δ δ −δ δ ) = −f(t)(δ δ −δ δ ) , (36) 23 i j ⋆F 1i 2j 1j 2i 1i 3j 1j 3i h i H ⊲ [ x ,x ] −if(t)(δ δ −δ δ ) = h(t)(δ δ −δ δ ) , (37) i j ⋆F 1i 2j 1j 2i 1i 2j 1j 2i h H ⊲i [ t,x ] = 0 , (38) i ⋆F 6 with h(t) = df(t), i.e. dt t κ 2t h(t) = h (t) = h = ± 1 S , (39) κ1 ±,κ1 τ τ ± τ (cid:18) (cid:19) (cid:18) (cid:19) t 2t h(t) = h (t) = h = κ C , (40) κ2 ±,κ2 τ 2 ± τ (cid:18) (cid:19) (cid:18) (cid:19) t 2t h(t) = h (t) = h = κ τ S , (41) κ3 ±,κ3 τ 3 ± τ (cid:18) (cid:19) (cid:18) (cid:19) t t t h(t) = h (t) = h = ±8κ τ3S C −1 , (42) κ4 ±,κ4 τ 4 ± τ ± τ (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:18) (cid:19) (cid:19) t 2t t h(t) = h (t) = h = κ τ S −S , (43) κ5 ±,κ5 τ 5 ± τ ± τ (cid:18) (cid:19) (cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19) t t t h(t) = h (t) = h = 2κ τ2 2C +1 S2 . (44) κ6 ±,κ6 τ 6 ± τ ± 2τ (cid:18) (cid:19) (cid:18) (cid:18) (cid:19) (cid:19) (cid:18) (cid:19) The above result means that the commutation relations (26)-(31) remain invariant with respect the action of P , K , F and M generators. Hence, the ”isometry” condition for i i i 12 considered(twisted)spacesbreaksthewhole U (NH )quantumgroupintoitssubalgebra 0 ± generated by spatial translations, boosts, constant acceleration generators and rotation in (x ,x )-plane. d 1 2 Finally, it should be noted that one can easily extend the above algorithm to the case of usual Newton-Hooke Hopf structure U (NH ) by putting acceleration generators F 0 ± i equal zero. 4 The case of acceleration-enlarged Galilei Hopf al- gebra analyzed in the contraction limit (τ → ∞) of U (NH ) Hopf structure 0 ± Let us now turn to the classical acceleration-enlarged Galilei Hopf algebra U (G) given 0 d by the following algebraic sector b [M ,M ] = i(δ M −δ M +δ M −δ M ) , [H,P ] = 0 , ij kl il jk jl ik jk il ik jl i [M ,K ] = i(δ K −δ K ) , [M ,P ] = i(δ P −δ P ) , (45) ij k jk i ik j ij k jk i ik j [M ,H] = [K ,K ] = [K ,P ] = 0 , [K ,H] = −iP , [P ,P ] = 0 , ij i j i j i i i j [F ,F ] = [F ,P ] = [F ,K ] = 0 , [M ,F ] = i(δ F −δ F ) , i j i j i j ij k jk i ik j [H,F ] = 2iK , i i 7 and trivial coproduct (8). It is well-known that the above Hopf structure can be get by the contraction limit (τ → ∞) of discussed in pervious section quantum group U (NH ). 0 ± The noncommutative space-times associated with twist deformations of Hopf algebra U (G) can be provided by the contraction procedure of spaces (26)-(31); they tadke the 0 form b [ t,x ] = [ x ,x ] = [ x ,x ] = 0 , [ x ,x ] = iw(t) ; i = 1,2,3 , (46) i ⋆F 1 3 ⋆F 2 3 ⋆F 1 2 ⋆F with (w (t) = lim f (t)) κi τ→∞ κi w(t) = w (t) = κ , (47) κ1 1 w(t) = w (t) = κ t , (48) κ2 2 w(t) = w (t) = κ t2 , (49) κ3 3 w(t) = w (t) = κ t4 , (50) κ4 4 1 w(t) = w (t) = κ t2 , (51) κ5 2 5 1 w(t) = w (t) = κ t3 . (52) κ6 2 6 It should be also noted, that the Galileian counterpart of covariance conditions (32)-(38) in τ → ∞ limit looks as follows G ⊲ [ t,x ] = 0 , (53) k i ⋆F G ⊲ [ x ,x ] −iw(t)(δ δ −δ δ ) = 0 ; G = P , K , F , (54) k i j ⋆F 1i 2j 1j 2i k k k k h i M ⊲ [ t,x ] = 0 , M ⊲ [ x ,x ] −iw(t)(δ δ −δ δ ) = 0 , (55) kl i ⋆F 12 i j ⋆F 1i 2j 1j 2i h i M ⊲ [ x ,x ] −iw(t)(δ δ −δ δ ) = w(t)(δ δ −δ δ ) , (56) 13 i j ⋆F 1i 2j 1j 2i 2i 3j 2j 3i h i M ⊲ [ x ,x ] −iw(t)(δ δ −δ δ ) = −w(t)(δ δ −δ δ ) , (57) 23 i j ⋆F 1i 2j 1j 2i 1i 3j 1j 3i h i H ⊲ [ x ,x ] −iw(t)(δ δ −δ δ ) = g(t)(δ δ −δ δ ) , (58) i j ⋆F 1i 2j 1j 2i 1i 2j 1j 2i h H ⊲i [ t,x ] = 0 , (59) i ⋆F where (g (t) = lim h (t)) κi τ→∞ κi g(t) = g (t) = 0 , (60) κ1 g(t) = g (t) = κ , (61) κ2 2 g(t) = g (t) = 2κ t , (62) κ3 3 g(t) = g (t) = 4κ t3 , (63) κ4 4 g(t) = g (t) = κ t , (64) κ5 5 3 g(t) = g (t) = κ t2 . (65) κ6 2 6 8 The above result means that the commutations relations (46) remain invariant with re- spect the action of P , K , F , M and H generators in the case of deformation (47) as i i i 12 well as P , K , F and M for space-times (48)-(52). i i i 12 Finally, let us observe that the above considerations can be applied to the case of classical Galilei quantum group U (G) by neglecting operators F . 0 i 5 Final remarks In this article we provide the subgroups of classical acceleration-enlarged Newton-Hooke U (NH ) as well as classical acceleration-enlarged Galilei U (G) Hopf structures, which 0 ± 0 play the role of ”isometry” groups for twist-deformed space-times (25) and (46). In such a wady, by analogy to the investigations performed in [20], [21]b, we get the link between twisted quantum spaces and the proper undeformed Hopf subalgebras. Consequently, the obtained results admit to analyze the twist-deformed dynamical models [29]-[32] in terms of thecorresponding classical quantum subgroups of thewhole nonrelativistic symmetries. The works in this direction already started and are in progress. Acknowledgments The author would like to thank J. Lukierski for valuable discussions. References [1] A. Kempf and G. Mangano, Phys. Rev. D 55, 7909 (1997); hep-th/9612084 [2] A. Connes, M.R. Douglas, A. Schwarz, JHEP 9802, 003 (1998); hep-th/9711162 [3] R.J. Protheore, H. Meyer, Phys. Lett. B 493, 1 (2000) [4] A.G. Cohen, S.L. Glashow, ”A Lorentz-Violating Origin of Neutrino Mass?”; hep-ph/0605036 [5] A.G. Cohen, S.L. Glashow, Phys. Rev. Lett. 97, 021601 (2006); hep-ph/0601236 [6] J. Kowalski-Glikman, G. Amelino-Camelia (Eds.), ”Planck Scale Effects in Astro- physics and Cosmology”, Lect. Notes Phys. 669, Springer, Berlin Heidelberg, 2005 [7] V. Chari, A. 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