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A kappa deformed Clifford Algebra, Hopf Algebras and Quantum Gravity

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Explicit deformations of the Lorentz (Conformal) algebra are performed by recurring to Clifford algebras. In particular, deformations of the boosts generators are possible which still retain the form of the Lorentz algebra. In this case there is an
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  A kappa deformed Clifford Algebra, Hopf Algebras and Quantum Gravity Carlos Castro PerelmanJune 2015 Universidad Tecnica Particular de Loja, San Cayetano Alto, Loja, 1101608 EcuadorCenter for Theoretical Studies of Physical Systems, Clark Atlanta University, Atlanta,GA. 30314perelmanc@hotmail.com Abstract Explicit deformations of the Lorentz (Conformal) algebra are performed by re-curring to Clifford algebras. In particular, deformations of the boosts generatorsare possible which still retain the form of the Lorentz algebra. In this case there isan invariant value of the energy that is set to be equal to the Planck energy. A dis-cussion of Clifford-Hopf   κ -deformed quantum Poincare algebra follows. To finalizewe provide further deformations of the Clifford geometric product based on Moyalstar products associated with noncommutative spacetime coordinates. Keywords : Clifford Algebras, Relativity,  κ -deformed Poincare algebra.Sometime ago Magueijo and Smolin [5] proposed a modification of special relativityin which a physical energy, which may be the Planck energy, joins the speed of lightas an invariant. This was accomplished by a  nonlinear  modification of the action of theLorentz group on momentum space, generated by adding a dilatation to each boost in sucha way that the Planck energy remains invariant. The associated algebra has unmodifiedstructure constants, and they highlighted the similarities between the group action foundand a transformation previously proposed by Fock [6].In this work we shall take a different approach and construct deformations of theLorentz (Conformal) algebra by recurring solely to Clifford algebras and leading to aninvariant value of the energy. A discussion of Clifford-Hopf   κ -deformed quantum Poincarealgebra follows based on the work by [2].We begin by reviewing [7] how the conformal algebra in four dimensions admits aClifford algebra realization; i.e. the generators of the conformal algebra can be expressedin terms of the Clifford algebra basis generators. The conformal algebra in four dimensions so (4 , 2) is isomorphic to the  su (2 , 2) algebra.1  Let  η ab  = ( − , + , + , +) be the Minkowski spacetime (flat) metric in  D  = 3 + 1-dimenisons. The epsilon tensors are defined as  ǫ 0123  =  − ǫ 0123 = 1, The real Clifford Cl (3 , 1 ,R ) algebra associated with the tangent space of a 4 D  spacetime  M  is defined bythe anticommutators {  Γ a ,  Γ b  } ≡  Γ a  Γ b  + Γ b  Γ a  = = 2  η ab  (1 . 1 a )such that[Γ a , Γ b ] = 2Γ ab ,  Γ 5  =  − i  Γ 0  Γ 1  Γ 2  Γ 3 ,  (Γ 5 ) 2 = 1;  { Γ 5 , Γ a }  = 0; (1 . 1 b )Γ abcd  =  ǫ abcd  Γ 5 ; Γ ab  = 12(Γ a Γ b − Γ b Γ a ) .  (1 . 2 a )Γ abc  =  ǫ abcd  Γ 5  Γ d ; Γ abcd  =  ǫ abcd  Γ 5 .  (1 . 2 b )Γ a  Γ b  = Γ ab  +  η ab ,  Γ ab  Γ 5  = 12 ǫ abcd  Γ cd ,  (1 . 2 c )Γ ab  Γ c  =  η bc  Γ a − η ac  Γ b  +  ǫ abcd  Γ 5  Γ d (1 . 2 d )Γ c  Γ ab  =  η ac  Γ b − η bc  Γ a  +  ǫ abcd  Γ 5  Γ d (1 . 2 e )Γ a  Γ b  Γ c  =  η ab  Γ c  +  η bc  Γ a − η ac Γ b  +  ǫ abcd  Γ 5  Γ d (1 . 2 f  )Γ ab Γ cd  =  ǫ abcd  Γ 5  − 4 δ  [ a [ c  Γ b ] d ]  − 2 δ  abcd .  (1 . 2 g ) δ  abcd  = 12 ( δ  ac  δ  bd  −  δ  ad  δ  bc  ) .  (1 . 2 .h )the generators Γ ab , Γ abc , Γ abcd  are defined as usual by a signed-permutation sum of theanti-symmetrizated products of the gammas.At this stage we may provide the relation among the  Cl (3 , 1) algebra generators andthe the conformal algebra  so (4 , 2)  ∼  su (2 , 2) in 4 D  . It is well known to the experts thatthe operators of the Conformal algebra can be written in terms of the Clifford algebragenerators as P  a  = 12Γ a  (1  −  Γ 5 );  K  a  = 12Γ a  (1 + Γ 5 );  D  =  −  12 Γ 5 , L ab  = 12 Γ ab .  (1 . 3) P  a  (  a  = 1 , 2 , 3 , 4) are the translation generators;  K  a  are the conformal boosts;  D  isthe dilation generator and  L ab  are the Lorentz generators. In order to match the physicaldimensions of momentum in (1.3) a mass parameter should be introduced. For convenienceit is set to unity as well as  c  = 1. The total number of generators is respectively 4 + 4 +1 + 6 = 15. From the above realization of the conformal algebra generators (1.3), theexplicit evaluation of the commutators yields[ P  a , D ] =  P  a ; [ K  a , D ] =  − K  a ; [ P  a , K  b ] =  − 2 g ab  D  + 2  L ab 2  [ P  a , P  b ] = 0; [ K  a ,K  b ] = 0[ L ab , L cd ] =  g bc  L ad  −  g ac  L bd  +  g ad  L bc  −  g bd  L ac , ...  (1 . 4)which is consistent with the  su (2 , 2)  ∼  so (4 , 2) commutation relations. We should noticethat the  K  a ,P  a  generators in (1.3) are both comprised of Hermitian Γ a  and  anti-Hermitian ± Γ a Γ 5  generators, respectively. The dilation  D  operator is Hermitian, while the Lorentzgenerator  L ab  is anti-Hermitian. The fact that Hermitian and anti-Hermitian generatorsare required is consistent with the fact that  U  (2 , 2) is a pseudo-unitary groupIf one wishes to deform the Clifford generators Γ A  one may choose for deformationgenerator the following F   =  P  0 2 κ  Γ 5  (1 . 5)such that upon exponentiation it yields˜Γ A  =  e F  Γ A  e − F  =  e [ F,  ] Γ A  =Γ A  + [  F,  Γ A  ] + 12! [  F,  [  F,  Γ A  ] ] + 13! [  F,  [  F,  [  F,  Γ A  ] ] ] +  ...  (1 . 6)The first order deformations of the Lorentz generators Γ µν   are˜Γ µν   = Γ µν   + [ P  0 2 κ  Γ 5 , Γ µν  ] = Γ µν   + 12 κ  (  η µ 0 P  ν   −  η ν  0 P  µ  )Γ 5  (1 . 7)From eq-(1.7) one can infer that ˜Γ ij  = Γ ij  so that the rotation generators remain unde-formed to first oder. Due to the vanishing commutator [ P  0  Γ 5 ,  Γ ij ] = 0 the higher ordercontributions remain zero and the rotation generators remain undeformed to  all  ordersΓ ij  = ˜Γ ij  . The second order contributions to the deformed boosts are14 κ 2  [  P  0  Γ 5  ,  [  P  0  Γ 5 ,  Γ 0 i  ] ] = 14 κ 2 [  P  0  Γ 5 , P  i  Γ 5  ] = 0 (1 . 8)and similar findings occur with the higher order nested commutators. Therefore, thehigher order contributions to the deformed boost generators are  zero  and one has thenthat the deformed boosts are given by˜Γ 0 i  = Γ 0 i  −  12 κ P  i  Γ 5  (1 . 9)and the following commutator becomes[˜Γ 0 i , P  0 ] =  P  i  (1 −  P  0 κ  ) (1 . 10)so that  P  0  =  κ  is an  invariant  energy under deformed boosts because the commutator(1.10)  vanishes  when  P  0  =  κ . The deformed boosts (1.9) can also be rewritten in termsof the dilatation generator  D  =  − 12 Γ 5  of eq-(1.3) as ˜Γ 0 i  = Γ 0 i  +  1 κ  P  i  D  in agreementwith the results of [5].3  From eqs-(1.6, 1.7,1.9) one learns that[˜Γ 0 i ,  ˜Γ 0  j ] = Γ ij  = ˜Γ ij  (1 . 11)and[˜Γ ij ,  ˜Γ 0 k ] = [Γ ij ,  ˜Γ 0 k ] =  −  η ik  Γ 0  j  +  η  jk  Γ 0 i  + 12 κ η ik P   j Γ 5  −  12 κ η  jk P  i Γ 5  = η  jk  ˜Γ 0 i  −  η ik  ˜Γ 0  j  (1 . 12)such that the Lorentz algebra (1.11, 1.12) remains  unmodified  despite having deformedthe boost generators in eq-(1.9).The procedure in this work is based entirely on Clifford algebras and differs from theapproach made by [5]. Many other deformations of the boosts/rotation generators arepossible. Some of these deformations will not affect the Lorentz (Conformal) algebra whileothers will also deform the Lorentz (Conformal) algebra. Let us provide examples wherethe Lorentz (Conformal) algebra is also modified. Choosing for instance the followingexponential operator e F  =  e P  0 Γ 03 /κ (1 . 13)it leads to[ ˜Γ 03 , P  3  ] = [ Γ 03  +  P  3 κ  Γ 30 , P  3  ] =  P  0  (1 −  P  3 κ  ) (1 . 14)Let us study the  higher order  contributions to the  deformed  boost generators ˜Γ 0 i ,i  =1 , 2 , 3. Given(Γ 0 ) 2 =  − 1 ,  (Γ 1 ) 2 = (Γ 2 ) 2 = (Γ 3 ) 2 = 1 ,  (Γ 5 ) 2 = 1 ,  { Γ 5 ,  Γ a }  = 0 (1 . 15)one can show that( P  0 ) 2 = 14 (Γ 0 − Γ 0 Γ 5 ) (Γ 0 − Γ 0 Γ 5 ) = 0 (1 . 16)and similarly( P  1 ) 2 = ( P  2 ) 2 = ( P  3 ) 2 = 0 (1 . 17)consequently the Clifford algebraic realization of the conformal algebra generators (1.3)yields  nilpotent  momentum generators. As a result of the nilpotent conditions of eqs-(1.16,1.17) the higher order contributions to the deformed boost generators are  zero 12 κ 2  [  P  0  Γ 03  ,  [  P  0  Γ 03 ,  Γ 03  ] ] = 0 (1 . 18)and similar findings occur with the higher order nested commutators. Hence the deformedboost generator is given by ˜Γ 03  = Γ 03 − P  3 κ  Γ 03 , and the deformed commutator to all ordersbecomes4  [ ˜Γ 03 , P  3  ] = [ Γ 03  −  P  3 κ  Γ 03 , P  3  ] =  P  0  (1 −  P  3 κ  ) (1 . 19)such that  P  3  =  κ  is now an  invariant  momentum value under the  deformed  boostgenerators ˜Γ 03  to all orders since the commutator (1.19) vanishes when  P  3  =  κ . Despitethat there is an invariant momentum, there is now no condition on the energy because[ ˜Γ 03 , P  0  ] =  P  3  (1 . 20)as it occurs under ordinary Lorentz boost transformations.Conversely, one can find a deformed boost generator such that there is an invariantenergy given by  κ , with  no  no condition on the momentum. For example, let us choosein this case  F   =  e P  3 Γ 03 /κ instead of   F   =  e P  0 Γ 03 /κ . Repeating the same process with thenew value of   F   one gets[ ˜Γ 03 , P  0  ] =  P  3  (1 −  P  0 κ  ) (1 . 21)such that the commutator (1.21) vanishes when  P  0  =  κ  leading to an  invariant  energy P  0  =  κ , with  no  condition on the momentum since[ ˜Γ 03 , P  3  ] =  P  0  (1 . 22)as it occurs under ordinary Lorentz boost transformations. The  key  difference amongthese three examples is that only in the case of deformed boost generators given by eq-(1.5, 1.9) the Lorentz algebra (1.11, 1.12) still remains  unmodified .Next we will discuss why the procedure in this work (based entirely on Clifford al-gebras) differs from the work [5]. Besides the nilpotent momentum generator conditionsgiven by eqs-(1.16, 1.17) one can also show that P  a  P  b  = 14 (Γ a − Γ a Γ 5 ) (Γ b − Γ b Γ 5 ) = 0 (1 . 23)after using eq-(1.15). Consequently the commutator in eq-(1.10) will then  reduce  to  P  i ,as in ordinary boost transformations, since the term  P  i P  0  = 0 in eq-(1.23) vanishes.The latter result is also compatible with the condition  P  i Γ 5  =  − P  i  resulting from themomentum operators realization in eq-(1.3) and leading to a deformed boost ˜Γ 0 i  = Γ 0 i − 12 κ P  i Γ 5  = Γ 0 i  +  12 κ P  i  given by a  linear  combination of boosts and momentum generators.Hence one must distinguish the momentum  operator  realization of   P  a  in terms of Clifford algebra generators and the approach taken by [5] ; i.e. the operators  P  0 ,P  1 ,P  2 ,P  3 are realized in terms 4 × 4 matrices associated with the Clifford algebra generators insteadof being the four momentum components corresponding to the momentum four-vector  p a  = (  p o ,p 1 ,p 2 ,p 3 ) in momentum space. Upper case letters  P  a  belong to the momentumoperators realized in terms of Clifford generators (matrices) while lower case letters  p a belong to the momentum four-vector.The latter approach by [5] recurs to a  nonlinear  modification of the action of theLorentz group on momentum space, generated by adding a dilatation to each boost in5
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