aa r X i v : . [ phy s i c s . g e n - ph ] M a r Geometry of the Non-Compact G(2)
Merab Gogberashvili , and Alexandre Gurchumelia Javakhishvili Tbilisi State University, 3 Chavchavadze Avenue, Tbilisi 0179, Georgia Andronikashvili Institute of Physics, 6 Tamarashvili Street, Tbilisi 0177, Georgia
March 13, 2019
Abstract
Geometrical applications of the non-compact form of Cartan’s exceptional Lie group G(2) is considered.This group generates specific rotations of 7-dimensional Minkowski-like space with three extra time-likecoordinates and in some limiting cases imitates standard Poincare transformations. In this model space-time translations are non-commutative and are represented by the rotations towards the extra time-likecoordinates. The second order Casimir element of non-compact G(2) and its expression by the Casimiroperators of the Lorentz and Poincare groups are found.PACS numbers: 02.20.Sv; 03.65.Fd; 11.30.LyKeywords: Non-compact Lie group G2; Casimir operators; Extra time-like dimension
In our previous papers geometrical applications of split octonions over the field of real numbers wasconsidered [1–5]. Space-time symmetries in this model are represented by the 14-parameter automorphismgroup of split octonions, G NC . Generators of G NC were first provided by ´Elie Cartan in the followingform [6]: Y kk = − z k ∂∂z k + y k ∂∂y k + 13 X i (cid:18) z i ∂∂z i − y i ∂∂y i (cid:19) ,Y k = − t ∂∂z k + y k ∂∂t + 12 X i,j ǫ ijk (cid:18) z i ∂∂y j − z j ∂∂y i (cid:19) ,Y k = − t ∂∂y k + z k ∂∂t + 12 X i,j ǫ ijk (cid:18) y i ∂∂z j − y j ∂∂z i (cid:19) ,Y ij = − z j ∂∂z i + y i ∂∂y j . ( i, j, k = 1 , ,
3) (1)Due to the constraint, Y + Y + Y = 0 , (2)only 14 of 15 operators (1) are linearly independent.One can write the generators (1) as the 7 × A ( α ) 2 d B ( b ) − b T − d T B ( d ) 2 b − A T ( α ) = α k Y k + α k Y k + α ij Y ij , (3)which act on a 7 dimensional vector p = y k tz k . ( k = 1 , ,
3) (4)1n (3) α µν ( µ, ν = 0 , , ,
3) are group parameters, the quantities b and d denote column vectors, b = (cid:0) α α α (cid:1) T , d = (cid:0) α α α (cid:1) T , (5)the 3 × A is a SU (3) group generator, A ( α ) = 13 − α + α + α − α − α − α α − α + α − α − α − α α + α − α , (6)and the two 3 × B are SO (3) generators, B ( b ) = α − α − α α α − α , B ( d ) = α − α − α α α − α . (7)In the representation (1) the invariant quadratic form, which is conserved under the G NC transformationshas the form: p T g p = t + z k y k , (8)where g = 12 , (9)is the 7 × denotes the 3 × G NC group and itis convenient to transform (8) in the Minkowski-like diagonal form. For this purpose let us perform thesimilarity transformation of the generators (3), X = Ω Y Ω − , (10)where Ω = 12 − . (11)This transforms the 7-vector (4) as Ω p = q , i.e. y k tz k Ω −→ ( y k + z k ) t ( y k − z k ) def = λ k tx k = q . ( k = 1 , ,
3) (12)Cartan’s coordinates y k and z k and their differential operators have following expressions in terms of thenew coordinates x k and λ k , y k = λ k + x k , ∂∂y k = 12 (cid:18) ∂∂λ k + ∂∂x k (cid:19) ,z k = λ k − x k , ∂∂z k = 12 (cid:18) ∂∂λ k − ∂∂x k (cid:19) , (13)and the invariant quadratic form (8) obtains the form: p T g p = q T g q = λ + t − x , (14)2here g = Ω − g Ω − = diag(1 , , , , − , − , − | {z } Minkowski metric ) (15)is the metric tensor in our representation and λ = X k λ k λ k , x = X k x k x k . (16)We want to associate t and x k with the coordinates of ordinary Minkowski space-time, while λ k may corre-spond to some extra time-like dimensions.Let us also write out Cartan’s operators (1) in new coordinates, X kk = (cid:18) x k ∂∂λ k + λ k ∂∂x k (cid:19) − X i (cid:18) x i ∂∂λ i + λ i ∂∂x i (cid:19) , ( i, j, k = 1 , , X k = (cid:18) λ k ∂∂t − t ∂∂λ k (cid:19) + (cid:18) x k ∂∂t + t ∂∂x k (cid:19) + 12 X i,j ǫ ijk (cid:0) λ i − x i (cid:1) (cid:18) ∂∂λ j + ∂∂x j (cid:19) ,X k = (cid:18) λ k ∂∂t − t ∂∂λ k (cid:19) − (cid:18) x k ∂∂t + t ∂∂x k (cid:19) + 12 X i,j ǫ ijk (cid:0) λ i + x i (cid:1) (cid:18) ∂∂λ j − ∂∂x j (cid:19) ,X ij = 12 ( λ i + x i ) (cid:18) ∂∂λ j + ∂∂x j (cid:19) −
12 ( λ j − x j ) (cid:18) ∂∂λ i − ∂∂x i (cid:19) , (17)and for the convenience introduce the five classes of G NC generators,Θ k = X k − X k = − (cid:18) x k ∂∂t + t ∂∂x k (cid:19) − X i,j ǫ ijk (cid:18) λ i ∂∂x j − x j ∂∂λ i (cid:19) ,B k = − X k − X k = − (cid:18) λ k ∂∂t + t ∂∂λ k (cid:19) − X i,j ǫ ijk (cid:18) λ i ∂∂λ j − x j ∂∂x i (cid:19) , Γ k = X i,j | ǫ ijk | X ij = X i,j | ǫ ijk | (cid:18) x i ∂∂λ j + λ j ∂∂x i (cid:19) , ( i, j, k = 1 , , R k = X i,j ǫ ijk X ij = X i,j ǫ ijk (cid:18) λ i ∂∂λ j + x i ∂∂x j (cid:19) , Φ k = X kk = (cid:18) x k ∂∂λ k + λ k ∂∂x k (cid:19) − X i (cid:18) x i ∂∂λ i + λ i ∂∂x i (cid:19) . (18)If we denote corresponding group parameters by θ k , β k , γ k , ρ k and ϕ k , then in the new basis thetransformations matrix (3) can be written as: B ( ρ ) − B ( β ) − β M ( γ, ϕ ) − B ( θ ) − β T θ T M ( γ, ϕ ) − B ( θ ) 2 θ B ( ρ ) + B ( β ) = θ k Θ k + β k B k + γ k Γ k + ρ k R k + ϕ k Φ k . (19)In this expression β = (cid:0) β β β (cid:1) T , θ = (cid:0) θ θ θ (cid:1) T , (20)are the column vectors, the 3 × B correspond to the SO (3) group generators, as in (7), and M has the form: M ( γ, ϕ ) = 13 − ϕ + ϕ + ϕ − γ − γ − γ ϕ − ϕ + ϕ − γ − γ − γ ϕ + ϕ − ϕ . (21)3hen infinitesimal transformations of coordinates of the 7-space (14) can be written in the form: λ ′ k = λ k + X i,j ǫ ijk (cid:0) β i − ρ i (cid:1) λ j − β k t − X i,j (cid:0) ǫ ijk θ i + | ǫ ijk | γ i (cid:1) x j − ϕ k − X i ϕ i ! x k ,t ′ = t + 2 X i (cid:0) β i λ i + θ i x i (cid:1) ,x ′ k = x k − X i,j ǫ ijk (cid:0) β i + ρ i (cid:1) x j + 2 θ k t + X i,j (cid:0) ǫ ijk θ i − | ǫ ijk | γ i (cid:1) λ j − ϕ k − X i ϕ i ! λ k . (22)If we exponentiate the generator matrices (19) separately for each five group parameters, ρ k , β k , θ k , ϕ k and γ k , we will get finite group transformations with either trigonometric or hyperbolic functions. We havetwo classes of compact rotations by the angles ρ k and β k , 6 rotations in total. Also there are three typesof boost-like G NC -transformations generated by the angles θ k , ϕ k and γ k , 8 boosts in total because of theconstraint (2), which in our basis is X k Φ k = 0 , (23)implying only two boost-type φ k -transformation are linearly independent. • Rotations.
The 3 parameters ρ k (corresponding to the generators R k ), out of 14 of G NC transforma-tions, are compact 3-angles that simultaneously rotate the spatial and time-like coordinates x i and λ i around k -th axis ( k = i ) within each of these 3-spaces. For example, the finite transformations for R are: λ ′ = λ , λ ′ = λ cos ρ + λ sin ρ , λ ′ = λ cos ρ − λ sin ρ ,t ′ = t ,x ′ = x , x ′ = x cos ρ + x sin ρ , x ′ = x cos ρ − x sin ρ . (24)The parameters β k (generators B k ) also are compact 3-angles that rotate x i and λ i β k in time-like 4-space of t and λ k .For example, the finite transformations for B are: λ ′ = λ cos (2 β ) − t sin (2 β ) , λ ′ = λ cos β − λ sin β , λ ′ = λ cos β + λ sin β ,t ′ = t cos (2 β ) + λ sin (2 β ) ,x ′ = x , x ′ = x cos β + x sin β , x ′ = x cos β − x sin β . (25) • Boosts.
The parameters θ k (generators Θ k ) of G NC are hyperbolic 3-angles of rotation between t and x k , which in fact are Lorentz boost with the double angle 2 θ k . However, these transformations are notpure Lorentz boosts because they also generate hyperbolic rotations by the angles θ k between λ i and x j ( i = j = k ). For demonstrating this let us write out finite transformations for θ : λ ′ = λ , λ ′ = λ cosh θ + x sinh θ , λ ′ = λ cosh θ − x sinh θ ,t ′ = t cosh (2 θ ) + x sinh (2 θ ) ,x ′ = x cosh (2 θ ) + t sinh (2 θ ) , x ′ = x cosh θ − λ sinh θ , x ′ = x cosh θ + λ sinh θ . (26)We get ordinary Lorentz boosts when x = x = λ = λ = 0.Another boost-like G NC -transformations are done by the generators Γ k with the hyperbolic 3-angles γ k . For example, the finite transformations for Γ are: λ ′ = λ , λ ′ = λ cosh γ − x sinh γ , λ ′ = λ cosh γ − x sinh γ ,t ′ = t ,x ′ = x , x ′ = x cosh γ − λ sinh γ , x ′ = x cosh γ − λ sinh γ . (27)4he last class of hyperbolic rotations between regular space coordinates x k and extra time-like coordi-nates λ k are done by the angles ϕ k . They generate specific diagonal boosts of the spatial coordinates x k towards the corresponding λ k . For example, the finite boost generated by Φ are: λ ′ = λ cosh (cid:0) ϕ (cid:1) − x sinh (cid:0) ϕ (cid:1) ,λ ′ = λ cosh (cid:0) ϕ (cid:1) + x sinh (cid:0) ϕ (cid:1) ,λ ′ = λ cosh (cid:0) ϕ (cid:1) + x sinh (cid:0) ϕ (cid:1) ,t ′ = t ,x ′ = x cosh (cid:0) ϕ (cid:1) − λ sinh (cid:0) ϕ (cid:1) ,x ′ = x cosh (cid:0) ϕ (cid:1) + λ sinh (cid:0) ϕ (cid:1) ,x ′ = x cosh (cid:0) ϕ (cid:1) + λ sinh (cid:0) ϕ (cid:1) . (28)We notice that if we assume that ϕ k = 0 and consider only the transformation rules of the Minkowskispace-time coordinates, x k and t , then (22) will imitate the ordinary infinitesimal Poincar´e transformations, x ′ k = x k − X i,j ε kij α i x j + φ k t + a k , t ′ = t + X i φ i x i + a , (29)where for the convenience we have introduced the new angles, α i = β i + ρ i , φ k = 2 θ k . (30)In (29) the parameters for space-time ’translations’, a k = X i,j (cid:0) ǫ ijk θ i − | ǫ ijk | γ i (cid:1) λ j , a = 2 X i β i λ i , (31)are represented by the G NC -rotations into the extra time-like directions λ k . So in the language of octonionicgeometry [1–5], any translation in ordinary space-time is generated by the boosts with λ k . Time translations a are smooth, since β k are compact angles. However, the angles θ k and γ k are hyperbolic and for anyspatial translation a k there exists a horizon (analogue to the Rindler horizon). Also, unlike the translationsof Poincar´e group, the ’ G NC -translations’ are rotations and thus are non-commutative. For example, com-mutator between different spatial ’translations’ (31) generated by the rotations with the angles γ i producerotation generators, [Γ i , Γ j ] = − ǫ ijk R k . (32)Limits on the parameter of the non-commutativity of coordinates, [ x i , x j ] < − Gev − [7], in the theorieswith non-commutative coordinates [8, 9], put restrictions on the values of [ a i , a j ], i.e. on the intervals ofchange of the extra time-like coordinates ∆ λ k < − Gev − .Now let us calculate the Casimir operator of G NC in our basis and compare it with the Casimirs thatare relevant for the Minkowski space-time. Casimir invariants are of primordial importance for a physicalmodel, since they allow us to label the irreducible representations. Eigenvalues of Casimir operators oftenrepresent significant dynamical physical quantities, such as angular momentum, elementary particle massand spin, Hamiltonians of various physical systems etc [10, 11].The rank-2 exceptional Cartan’s group G has second and sixths order Casimir operators [12]. In Cartan’sbasis (1) the second order Casimir of G NC has the form: C = 2 X ij X ij − (cid:16) X k X k + X k X k (cid:17) , (33)5hich in our basis (18) translates as C = X k (cid:18)
13 Θ k − B k + Γ k − R k + 2Φ k (cid:19) == 6 " t ∂∂t + X k (cid:18) λ k ∂∂λ k + x k ∂∂x k (cid:19) + x ∂ ∂t + X k t ∂ ∂x k + 2 t X k x k ∂ ∂t∂x k ++ X i,j,k | ǫ ijk | x i x j ∂ ∂x i ∂x j − x i ∂ ∂x j ! − λ ∂ ∂t − X k ∂ ∂x k ! − X k (cid:0) t − x (cid:1) ∂ ∂λ k ++ 2 t X k λ k ∂ ∂t∂λ k + X i,j λ i x j ∂ ∂λ i ∂x j + X i,j,k | ǫ ijk | λ i λ j ∂∂λ i ∂λ j − λ i ∂∂λ j ! . (34)We note that the first order operator in the first term, c = t ∂∂t + X k (cid:18) λ k ∂∂λ k + x k ∂∂x k (cid:19) , (35)itself commutes with all G NC generators (18), therefore putting any other constant coefficient before it in(34) will not change the Casimir operators property of commuting with all generators. For the conveniencewe will assume the coefficient in front of the operator (35) in (34) to be 3 instead of 6. Then for the case ofthe constant extra time-like coordinates, λ k = const , i.e. when derivatives with the respect of λ k vanish, wefind that (34) can be expressed as a sum of the second order Casimir operators of the Lorentz group [13], C L = 3 t ∂∂t + X k x k ∂∂x k ! + x ∂ ∂t + X k t ∂ ∂x k ++ 2 t X k x k ∂ ∂t∂x k + X i,j,k | ǫ ijk | x i x j ∂ ∂x i ∂x j − x i ∂ ∂x j ! , (36)and of the Poincar´e group [13], C P = ∂ ∂t − X k ∂ ∂x k , (37)in the form: C = C L − λ C P . (38)This shows that, if one does not consider transformations with extra time-like coordinates λ k , the geometricalinterpretation of the group G NC is consistent with the main properties of the Minkowski space-time.To conclude, we have considered geometrical applications of Cartan’s smallest non-compact exceptionalLie group, G NC . This group generates specific rotations of the (3+4)-space with the ordinary Minkowski andthree extra time-like coordinates. It is shown that in some limiting cases G NC -rotations imitate standardPoincar´e transformations. In this picture space-time translations are non-commutative and are representedby the rotations towards the extra time-like coordinates. The second order Casimir operator of G NC inMinkowski-like basis is found and it is shown that in the case of constant extra dimensions it can be expressedby the Casimir operators of the Lorentz and Poincar´e groups. Acknowledgements:
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