DFTT-70/93 DIFFERENTIAL CALCULUS ON ISO (N), q QUANTUM POINCARE´ ALGEBRA AND q-GRAVITY 4 9 9 1 Leonardo Castellani n a J Istituto Nazionale di Fisica Nucleare, Sezione di Torino 6 and 2 Dipartimento di Fisica Teorica v 9 Via P. Giuria 1, 10125 Torino, Italy. 7 1 2 1 3 9 / h Abstract t - p We present a general method to deform the inhomogeneous algebras of e the B ,C ,D type, and find the corresponding bicovariant differential cal- h n n n v: culus. The method is based on a projection from Bn+1,Cn+1,Dn+1. For i example we obtain the (bicovariant) inhomogeneous q-algebra ISOq(N) as a X consistent projection of the (bicovariant) q-algebra SO (N +2). This pro- q r a jection works for particular multiparametric deformations of SO(N +2), the so-called “minimal” deformations. Thecase of ISO (4) is studiedindetail: a q realformcorrespondingtoaLorentzsignatureexists onlyforoneofthemini- maldeformations, dependingononeparameter q. ThequantumPoincar´e Lie algebra is given explicitly: it has 10 generators (no dilatations) and contains the classical Lorentz algebra. Only the commutation relations involving the momenta depend on q. Finally, we discuss a q-deformation of gravity based on the “gauging” of this q-Poincar´e algebra: the lagrangian generalizes the usual Einstein-Cartan lagrangian. DFTT-70/93 December 1993 e-mail addresses: decnet=31890::castellani; internet= [email protected] 1 Introduction Perturbative quantum Einstein gravity is known to be mathematically inconsis- tent, since it is plagued by ultraviolet divergences appearing at two-loop order (the absence of one-loop divergencies was found in [1], whereas two-loop divergencies were explicitly computed in [2]). In supergravity the situation is only slightly bet- ter, the divergences starting presumably at three loops 1 . In the last fifteen years or so there have been various proposals to overcome this difficulty, and consistently quantize gravity either alone or as part of a unified theory of the fundamental in- teractions. Such a unified picture is provided by superstrings (see for a review [3]), where Einstein gravity arises as a low-energy effective theory, coupled more or less realistically to gauge fields and leptons, and regulated at the Planck scale by an in- finite number of heavy particles (the superstring massive spectrum). How to make phenomenological predictions from superstrings is still object of current research. Another, more speculative, line of thought deals with the quantization of space- time itself, whose smoothness under distances of the order of the Planck length L 10−33cm is really a mathematical assumption. Indeed if we probe spacetime P ∼ geometry with a test particle, the accuracy of the measure depends on the Comp- ton wavelength of the particle. For higher accuracy we need a higher mass m of the particle, and for m 1/L the mass significantly modifies the curvature it is P ∼ supposed to measure (i.e the curvature radius becomes of the order of the particle wavelength: the particle is no more a test particle). Thus it is not inconceivable that spacetime has an intrinsic cell-like structure: lattice gravity, or Regge calculus may turn out to be something more fundamental than a regularization procedure. Another way to discretization is provided by non- commutative geometry: when spacetime coordinates do not commute the position of a particle cannot be measured exactly. The notion of spacetime point loses its physical meaning, and is to be replaced by the notionof spacetime cell; the question is whether this sort of lattice structure does indeed regularize gravity at short distances. References on non-commutative geometry and its uses for regularization can be found in [4]. Fundamental interactions are described by field theories with an underlying al- gebraic structure given by particular Lie groups, as for ex. unitary Lie groups for the strong and electroweak interactions and the Poincar´e group for gravity. It is natural to consider the so-called quantum groups [5, 6, 7] (continuous deformations of Lie groups whose geometry is non-commutative) as the algebraic basis for gen- eralized gauge and gravity theories. The bonus is that we maintain a rich algebraic structure, more general than Lie groups, in a theory living in a discretized space. This does not happen usually with lattice approaches, where one loses the symme- 1 no explicit calculation like the one of ref. [2] exists, but there is no symmetry principle that excludes them. 1 tries of the continuum. For a review of non-commutative differential geometry on quantum groups see for ex. [8]. This subject, initiated in [9], has been actively developed in recent years: a very short list of references can be found in [10]-[15]. In this paper we address the problem of constructing a non-commutative defor- mation of Einstein gravity. For this we need a q-deformation of the Poincar´e Lie algebra. We obtain it in Section 4 as a special case of the quantum inhomogeneous ISO (N) algebras, whose differential calculus is presented in Section 3. These al- q gebras are obtained as projections from particular multiparametric deformations of SO(N + 2), called “minimal” deformations. Their R matrix is diagonal, and the braiding matrix Rˆ = PR has unit square. On q-groups with diagonal R-matrices see for ex. [16] and references therein. Deformations of Lie algebras whose braiding matrix has unit square were considered some time ago by Gurevich [17]. The projective method to obtain the bicovariant differential calculus on inho- mogeneous quantum groups was introduced in [18] for IGL (N). References on q inhomogeneous q-groups can also be found in [19]. A general discussion on the differential calculus on multiparametric q-groups is giveninSection2. InSection5wediscusstheq-deformationofCartan-Maurerequa- tions, Bianchi identities, diffeomorphisms and propose a lagrangian for q-gravity, based on ISO (3,1). Other deformations of the Poincar´e algebra have been con- q sidered in recent literature [20]. Although interesting in their own right, none of these deformations corresponds to a bicovariant differential calculus on a quantum Poincar´e group. 2 Bicovariant calculus on multiparametric quan- tum groups We recall that (multiparametric) quantum groups are characterized by their R- matrix, which controls the noncommutativity of the quantum group basic elements Ta (fundamental representation): b Rab Te Tf = Tb Ta Ref (2.1) ef c d f e cd and satisfies the quantum Yang-Baxter equation Ra1b1 Ra2c1 Rb2c2 = Rb1c1 Ra1c2 Ra2b2 , (2.2) a2b2 a3c2 b3c3 b2c2 a2c3 a3b3 asufficient conditionfortheconsistency ofthe“RTT”relations(2.1). TheR-matrix components Rab depend continuously on a (in general complex) set of parameters cd q ,r. For q = q,r = q we recover the uniparametric q-groups of ref. [6]. Then ab ab q 1,r 1 is the classical limit for which Rab δaδb : the matrix entries ab → → cd → c d Ta commute and become the usual entries of the fundamental representation. The b 2 multiparametric R matrices for the A,B,C,D series can be found in [21] (other ref.s on multiparametric q-groups are given in [22]). For the B,C,D case they read: Rab = δaδb(1 δan2)[rδab +r−1δab′ +(1 δab′)( r θ(b,a)+ qbaθ(a,b))] cd c d − − qab r (2.3) +(r −r−1)[θ(a,b)δcbδda −θ(a,c)rρa−ρcδba′δdc′]+δna2δnb2δcn2δdn2 where θ(x) = 1 for x > 0 and θ(x) = 0 for x 0; we define n N+1 and primed ′ ≤ 2 ≡ 2 indices as a N + 1 a. The indices run on N values (N=dimension of the ≡ − fundamental representation Ta ), with N = 2n+1 for B [SO(2n+1)], N = 2n for b n C [Sp(2n)], D [SO(2n)]. The terms with the index n are present only for the B n n 2 n series. The ρ vector is given by: (n 1,n 3,..., 1,0, 1,..., n+ 1) for B  − 2 − 2 2 −2 − 2 n (ρ1,...ρN) =  (n,n 1,...1, 1,..., n) for Cn (2.4) − − − (n 1,n 2,...,1,0,0, 1,..., n+1) for D n  − − − − Moreover the following relations reduce the number of independent q parameters ab [21]: r2 q = 1, q = ; (2.5) aa ba q ab r2 r2 qab = = = qa′b′ (a < b) (2.6) qab′ qa′b so that the q with a < b < N give all the q’s. ab 2 Remark 1: if we denote by q,r the set of parameters q ,r, we have ab −1 Rq,r = Rq−1,r−1 (2.7) The inverse R−1 is defined by (R−1)ab Rcd = δaδb = Rab (R−1)cd . Eq. (2.7) cd ef e f cd ef implies that for q = r = 1, R¯ = R−1. | | | | Remark 2: for r = 1, Rˆ2 = 1 where Rˆab Rba . cd ≡ cd Orthogonality conditions can be imposed on the elements Ta , consistently with b the RTT relations (2.1): Ta CbcTd = Cad b c Ta C Tc = C (2.8) b ac d bd where the (antidiagonal) metric is : +1 for B , n  Cab = ǫarρaδab′ with ǫa =  +1 for Cn and a n, (2.9) ≤ 1 for C and a > n. n  − 3 and its inverse Cab satisfies CabC = δa = C Cba. We see that for the orthogonal bc c cb series, the matrix elements of the metric and the inverse metric coincide, while for the symplectic series there is a change of sign. The consistency of (2.8) with the RTT relations is due to the identities: C Rˆbc = (Rˆ−1)cf C (2.10) ab de ad fe Rˆbc Cea = Cbf(Rˆ−1)ca (2.11) de fd These identities hold also for Rˆ Rˆ−1. The co-structures of the B,C,D multi- → parametric quantum groups have the same form as in the uniparametric case: the coproduct ∆, the counit ε and the coinverse κ are given by ∆(Ta ) = Ta Tb (2.12) b b ⊗ c ε(Ta ) = δa (2.13) b b κ(Ta ) = CacTd C (2.14) b c db A conjugation can be defined trivially as T∗ = T or via the metric as T∗ = (κ(T))t. ¯ Inthefirst case, compatibilitywiththeRTT relations(2.1)requires Rq,r = Rq−1,r−1, i.e. q = r = 1, and the corresponding real forms are SO (N;R), SO (n,n;R) q,r q,r and|S|p |(n|;R). In the second case the condition on R is R¯ab = Rdc , which q,r cd ba happens for q q¯ = r2 R. The metric on a “real” basis has compact signature ab ab ∈ (+,+,...+) so that the real form is SO (N;R). q,r There is also a third way to define a conjugation on the orthogonal quantum groups SO (2n,C), which extends to the multiparametric case the one proposed q,r by the authors of ref. [23] for SO (2n,C).The conjugation is defined by: q (Ta )∗ = a Tc d (2.15) b D c dD b being the matrix that exchanges the index n with the index n+1. This conjuga- D ∗ ∗ tion is compatible with the coproduct: ∆(T ) = (∆T) ; for r = 1 it is also com- patible with the orthogonality relations (2.8) (due to C¯ = CT| a|nd also C = C) ∗ ∗ D D and with the antipode: κ(κ(T ) ) = T. Compatibility with the RTT relations is easily seen to require (R¯)n↔n+1 = R−1, (2.16) which implies i) q = r = 1 for a and b both different from n or n+1; ab | | | | ii) q /r R when at least one of the indices a,b is equal to n or n+1. ab ∈ Sincelaterweconsiderthecaser = 1and(R)n↔n+1 = R(andthereforeR¯ = R−1 because of (2.16)), the conditions on the parameters will be: q = 1 for a and b both different from n or n+1 ab | | q = 1 for a or b equal to n or n+1 (2.17) ab 4 This last conjugation leads to the real form SO (n+1,n 1;R), and will in fact q,r − be the one we need in order to obtain ISO (3,1;R), as we discuss in Section 4. q A bicovariant differential calculus [9] on the multiparametric q-groups can be constructed in terms of the corresponding R matrix , in much the same way as for uniparametric q-groups (for which we refer to [11, 13, 8]). Here we concentrate on SO (N +2), but everything holds also for Sp (N +2). For later convenience we q,r q,r adopt upper case indices for the fundamental representation of SO (N + 2) and q,r lower case indices for the fundamental representation of SO (N). q,r The basic object is the braiding matrix Λ A2 D2 C1 B1 dF2d−1RF2B1 (R−1)C1G1 (R−1)A2E1 RG2D2 (2.18) A1 D1 | C2 B2 ≡ C2 C2G1 E1A1 G2D1 B2F2 which is used in the definition of the exterior product of quantum left-invariant one forms ω B: A ω A2 ω D2 ω A2 ω D2 Λ A2 D2 C1 B1 ω C2 ω B2 (2.19) A1 ∧ D1 ≡ A1 ⊗ D1 − A1 D1 | C2 B2 C1 ⊗ B1 and in the q-commutations of the quantum Lie algebra generators χA : B χD1 χC1 Λ E2 F2 D1 C1 χE1 χF1 = CD1 C1 A2χA1 (2.20) D2 C2 − E1 F1 | D2 C2 E2 F2 D2 C2|A1 A2 where the structure constants are explicitly given by: 1 CA1 B1 C2 = [ δB1δA1δC2 +Λ B C2 A1 B1 ]. (2.21) A2 B2|C1 r r−1 − B2 C1 A2 B C1 | A2 B2 − The dA vector in (2.18) is defined via the diagonal matrix DA as dA = DA (no B A sum on A), with D = CCt,or DA = CACC (2.22) B BC A graphical representation of the braiding matrix (2.18) is given in Appendix A. Remark 3: for r = 1 we have Λ2 = 1. This is due to Rˆ2 = 1 and DA = δA. B B The braiding matrix Λ and the structure constants C defined in (2.21) satisfy the conditions C nC s Λkl C nC s = C kC s (q-Jacobi identities) (2.23) ri nj − ij rk nl ij rk Λnm Λik Λjs = Λnk Λms Λij (Yang–Baxter) (2.24) ij rp kq ri kj pq C iΛml Λns +Λil C s = Λpq Λis C l +C mΛis (2.25) mn rj lk rj lk jk lq rp jk rm C mΛns = Λij Λnm C s (2.26) rk ml kl ri mj wheretheindexpairs B andA havebeenreplacedbytheindicesi and respectively. A B i These are the so-called “bicovariance conditions”, see ref.s [9, 10, 8], necessary for 5 the existence of a consistent bicovariant differential calculus, as we discuss further in Appendix B. A metric can be defined in the adjoint representation of the B ,C ,D q-groups n n n as follows: C Cc1 b1 = Cc1f(R−1)b1e C (2.27) ij ≡ c2 b2 fc2 b2e Cij C a2 c2 = C Rea2 Cc2f (2.28) ≡ a1 c1 a1e c1f and satisfies the relations: C Cjk = δk = CkjC (2.29) ij i ji and CikΛsl = (Λ−1)is Cjl (2.30) kr rj Λrj C = C (Λ−1)kr (2.31) is jl ik sl i.e. the analogue of eq.s (2.10)-(2.11). These relations allow to define consistent orthogonalityrelationsfortheq-groupmatrixelements intheadjointrepresentation (see Appendix B). Remark 4: when r = 1 ( Rˆ2 = 1,Λ2 = 1), the following useful identities hold: ⇒ Da CacC = δa, (D−1)a CcaC = δa (2.32) b ≡ bc b b ≡ cb b Di CikC = δi, (D−1)i CkiC = δi (2.33) j ≡ jk j j ≡ kj j Rˆab Rˆce = δbδe = Rˆba Rˆec (2.34) cd af f d dc fa Λri Λsk = δiδk = Λir Λks (2.35) sl rj j l ls jr The first two -conjugations (the “usual ones”) on the T’s we have discussed ∗ earlier in this Section can be extended to the dual space spanned by the q-Lie algebra generators χ as in the uniparametric case. The consistent extension of the third conjugation to the χ space is treated in Appendix C, for the case of minimal deformations (r = 1) of SO(2n). We find that (χa )∗ = a χc d (2.36) b −D c dD b is compatible with the bicovariant differential calculus if the Λ and C tensors are invariant under the exchange of the indices n and n+1, and if the following relation holds: C¯ k = C k (2.37) ij − ji 6 3 Inhomogeneous quantum groups and their dif- ferential calculus In this Section we present a general method of quantizing inhomogeneous groups whose homogeneous subgroup belongs to the BCD series. In particular we concen- trate on the q-deformations of the ISO(N) groups, as these are the groups relevant for the construction of q-gravity theories. The idea is to project SO (N + 2) and its differential calculus on ISO (N), q q much as we did for IGL (N) in ref. [18], where we projected from GL (N +1). q q For this we have to consider the multiparametric deformations of the orthogonal groups SO (N + 2) with r = 1 (minimal deformations). Only for r = 1 we can q,r obtain a consistent projection on ISO (N). q We know that the Rab matrix of SO (N) is contained in the RAB matrix cd q,r CD of SO (N + 2): more precisely it is obtained from the “mother ” R matrix by q,r restricting its indices to the values A,B,..=2,3,...N-1. We therefore split the capital indices as A=( ,a, ). Then the R matrix of SO (N+2) can be rewritten in terms q,r ◦ • of SO (N) quantities: q,r d d c c cd  ◦◦ ◦• •◦ •• ◦ • ◦ •  r 0 0 0 0 0 0 0 0  ◦◦ 0 r−1 0 0 0 0 0 0 0     ◦• 0 f(r) r−1 0 0 0 0 0 C λ   •◦ − cd   0 0 0 r 0 0 0 0 0  RABCD =  ••b 0 0 0 0 r δb 0 0 0 0  (3.1)  ◦ q◦b d   b 0 0 0 0 0 q◦bδb 0 λδb 0   a• 0 0 0 0 λδa r0 d q◦aδa 0c 0   ◦ d r c   a 0 0 0 0 0 0 0 r δa 0   a•b 0 Cabλ 0 0 0 0 0 q◦a0 c Rab  − cd where C is the SO (N) metric, λ r r−1 and f(r) λ(1 r−N) for the ab q,r orthogonal series B,D (f(r) λ(1 r−≡N−2−) for the symplec≡tic seri−es). ≡ − It is not difficult to reexpress the Λ and C tensors in our index convention. Less trivial is to find a subset of these components, containing the Λ and C tensors of SO (N), that satisfies the bicovariance conditions (2.23) - (2.26). This subset in q,r fact exists for r = 1 and is given by: Λ a2 d2 c1 b1 = Rf2b1 (R−1)c1g1 (R−1)a2e1 Rg2d2 (3.2) a1 d1 | c2 b2 c2g1 e1a1 g2d1 b2f2 Λ ◦ d2 c1 b1 = q◦d1Rd2b1 (R−1)c1g1 (3.3) a1 d1| c2 ◦ q◦d2 c2g1 d1a1 Λ a2 ◦ c1 b1 = q◦b2(R−1)c1b1 (R−1)a2e1 (3.4) a1 d1| ◦ b2 q◦b1 e1a1 b2d1 Λ ◦ ◦ c1 b1 = q◦d1(R−1)c1b1 (3.5) a1 d1| ◦ ◦ q◦b1 d1a1 7 Λ a2 d2 c1 • = q◦c1(R−1)a2c1 Rg2d2 (3.6) • d1 | c2 b2 q◦c2 g2d1 b2c2 Λ a2 d2 • b1 = q◦a2Rf2b1 Ra2d2 (3.7) a1 • | c2 b2 q◦a1 c2a1 b2f2 Λ a2 d2 • • = q◦a2Ra2d2 (3.8) • • | c2 b2 q◦c2 b2c2 Λ•ad21◦|c1◦ •b2 = q◦c1q◦b2(R−1)a2cb12d1 (3.9) Λa◦1 d•2|•c2b1◦ = (q◦a1q◦d2)−1(R−1)d2bc12a1 (3.10) Cc1 b1 d2 = structure constants of SO (N) (3.11) c2 b2|d1 q,r=1 Cc1 b1 ◦ = lim 1 [ δb1δc1 + q◦b2(R−1)c1b1 (R−1)ae1 ] (3.12) ◦ b2|d1 r→1 r −r−1 − b2 d1 q◦b1 e1a b2d1 Cc1 b1 ◦ = Rg2b1 (R−1)c1g1 (R−1)ae1 (3.13) c2 ◦|d1 c2g1 e1a g2d1 Cc1 b1 d2 = q−1Cae1Rd2b1 (R−1)c1g1 (3.14) c2 ◦|• ◦d2 c2g1 e1a Cc1 • d2 = q◦c1Rc1d2 (3.15) c2 b2|• q◦c2 b2c2 Cc1 • ◦ = q◦c1 C δc1 (3.16) c2 b2|d1 −q◦c2q◦d2 b2c2 d1 1 C• b1 d2 = lim [ δb1δd2 +Rf2b1 Rad2 ] (3.17) c2 b2|• r→1 r r−1 − b2 c2 c2a b2f2 − This is the key result of this Section, and enables the consistent projection on the ISO (N) algebra by setting: q χ◦ = χa = χ◦ = χ• = χ◦ = χ• = 0 (3.18) b • ◦ • • ◦ The ISO (N) Lie algebra is given explicitly in Table 1. The reason we call this the q ISO (N) Lie algebra will be explained below. q We prove now that the components (3.2)-(3.17) indeed satisfy the bicovariance conditions (2.23)- (2.26). We label by the letter H the subset of indices present in eq.s (3.2) - (3.17), i.e. = a ,a ,• (H = b, ◦, b), and by the letter K all the other H b ◦ b a a • composite indices. We have to prove that, setting equal to H all free indices in (2.23) - (2.26), only H indices enter in the index sums (and therefore the H-tensors of (3.2)-(3.17) satisfy by themselves the bicovariance conditions). This is true i) for the quantum Yang-Baxter eqs. (2.24) since the tensor PΛ is diagonal for r = 1, so that ΛHH = ΛHH = ΛHH = ΛHK = ΛKH = ΛKK = 0; (3.19) HK KH KK HH HH HH ii) for the q-Jacobi eqs. (2.23) because C K can be different from zero only when HH K = ◦, and C H = C H = 0 when = • ; iii) for the last two bicovariant • HK KH K ◦ conditions (2.25) - (2.26) again because of (3.19). Thus far we have shown that there is a subset of χA (the generators of the q- B Lie algebra of SO (N +2) ) closing on the q-algebra of Table 1, namely χa ,χa q,r=1 b ◦ 8 and χ• . This algebra is bicovariant, in the sense that the corresponding Λ and C b tensors satisfy (2.23)-(2.26) . It would seem that the number of momenta is twice what we need, since there are two kinds of “momentum” generators, χa and χ• . ◦ b However by examining in some detail the q-algebra we can conclude that only N combinations of these momenta do survive, and if we rewrite the algebra of Table 1 in terms of these combinations we precisely obtain a deformation of ISO(N). Let us prove this. Consider the structure constants Cc1 b1 ◦. It is not difficult to see from (3.12) ′ ◦ b2|d1 that for c = b ,b = b these constants are vanishing for any value of r (use the 1 2 1 2 explicit expression (2.3)), and thus in particular for r = 1. On the other hand the structure constants Cb1 c1 ◦ and Cb1 c1 d2 are not vanishing for the same values of b2 ◦|d1 b2 ◦|• c ,b ,b , but: 1 1 2 Cb1 c1 ◦ = δc1 (3.20) b2 ◦|d1 d1 Cb1 c1 d2 = Cc1b1q−1δd2 (3.21) b2 ◦|• ◦d2 b2 Thus we have the two commutations: χc1χb1 Λ e2 ◦ c1 b1χe1 χf1 = 0 (3.22) ◦ b2 − e1 f1| ◦ b2 e2 ◦ χb1 χc1 Λ ◦ f2 b1 c1χe1χf1 = χb1 +q−1χ• (3.23) b2 ◦ − e1 f1| b2 ◦ ◦ f2 ◦ ◦b′1 b′1 Next we remark that for Λ2 = I as is the case for r = 1 the two left-hand sides of the above equations are equal up to a minus sign, so that finally we have: χb1 +q−1χ• = 0 (3.24) ◦ ◦b′1 b′1 These N equations reduce the number of independent momenta to N. We can easily rewrite the algebra in Table 1 in terms of the redefined momenta: χa q21 χa q−12χ• (3.25) ≡ ◦a′ ◦ − ◦a′ a′ and we have done so in Table 2. This was possible because the q-commutator of χa with a given generator χ is ◦ −1 • the same as the q-commutator of q χ with χ, because the constraint (3.24) is − ◦a′ a′ consistent with the q-Lie algebra of Table 1. Another way to see it is to remark that the algebra of Table 1 satisfies the q-Jacobi identities (2.23). Then we have an explicit matrix representation of the q-generators χ: the adjoint representation (χ )j C j. Eq. (3.24) means that the generators q12 χa and q−21χ• have i k ≡ ki ◦a′ ◦ − ◦a′ a′ the same matrix representative, and hence the same commutations with the other generators. The q-Lie algebra of Table 2 satisfies the bicovariant conditions (2.23)-(2.26). As discussed in Appendix C, these define a (bicovariant) differential calculus on the 9