Twisted acceleration-enlarged Newton-Hooke space-times and breaking classical symmetry phenomena

0.13 MB·English

by  Marcin Daszkiewicz

Twisted acceleration-enlarged Newton-Hooke
space-times and breaking classical symmetry
phenomena
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Marcin Daszkiewicz
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Institute of Theoretical Physics
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- University of Wroclaw pl. Maxa Borna 9, 50−206 Wroclaw, Poland
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e−mail : [email protected]
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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.
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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.
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