Reply to “Comment on ‘Noncommutative gauge theories and Lorentz symmetry’” Rabin Banerjee and Biswajit Chakraborty S.N. Bose National Centre for Basic Sciences, JD Block, Sector 3, Salt Lake, Kolkata 700098, India [email protected],[email protected] Kuldeep Kumar Department of Physics, Panjab University, Chandigarh 160014, India 1 1 [email protected] 0 2 n Abstract a J This is a reply to “Comment on ‘Noncommutative gauge theories and Lorentz sym- 5 metry,’” Phys. Rev. D 77 (2008) 048701 by Alfredo Iorio. ] Journal reference: Phys. Rev. D 77 (2008) 048702 h t - p e This is a reply to the Comment [1] on our paper. The author has criticised our earlier h [ work without precisely pointing out the error. However we remain content with our analysis for the following reasons. 1 v The main confusion of the author concerns a naive textbook application of Noether theo- 7 rem to the problem at hand. There are two distinctions to be noted. First, such applications 7 8 involve only scalar parameters rather than vector or tensor ones as is the case here. Second, 0 and more importantly, textbook applications do not discuss actions that have parameters . 1 that are not included in the configuration space. In that case the Noether procedure gets 0 nontrivially modified as was shown in our paper [2]. We shall here reinforce this point by 1 first discussing a simple example from particle mechanics and finally make the connection 1 : with the field theoretic models considered earlier [2]. v i Consider a nonrelativistic particle of mass m moving in three dimensions, subjected to a X constant external force F~: r a mx¨ = F . (1) i i It follows immediately that the rate of change of its angular momentum J~ = ~x × p~, with p~= m~x˙ being the linear momentum, is precisely the applied torque ~τ =~x×F~: dJ~ =~τ. (2) dt Now observe that Eq.(1) follows from the following action: S = Ldt= 1mx˙2+F~ ·~x dt. (3) 2 Z Z (cid:16) (cid:17) It is natural to expect that the force F transforms covariantly under SO(3) rotation, so i that Eq.(1) has a covariant form. Correspondingly, the action (3) is invariant under SO(3) 1 rotation. But despite its rotational invariance, this action does not yield a conserved charge, which is the angular momentum J~in this case; it rather satisfies Eq.(2), as mentioned above. Let us derive this from the action S using Noether’s approach and make some important and relevant observations on the way, by considering its response to an infinitesimal SO(3) rotation, ′ x → x = x +δx , (4) i i i i where δx = ω x , |ω | ≪ 1, ω = −ω . Under this transformation F also undergoes the i ij j ij ij ji i transformation ′ F → F = F +δF , δF = ω F . (5) i i i i i ij j The invariance of the action S under this transformation implies 0 = δS = dt(mx˙ δx˙ +F δx +x δF ). (6) i i i i i i Z Note that it is important to consider x δF term here. As we shall see later that this will i i play an important role. d Equation (6) can be rewritten, using the fact that δx˙ = (δx ), as i dt i d 0 = δS = dt (mx˙ δx )−(mx¨ −F )δx +x δF . (7) i i i i i i i Z (cid:20) dt (cid:21) We can now get rid of the central term involving δx , using the equation of motion (1), so i that the on-shell version of Eq.(7) becomes d 0 = δS = dt (mx˙ δx )+x δF . (8) i i i i Z (cid:20) dt (cid:21) Before proceeding further, let us note at this stage that we cannot regard the force F as an i auxiliaryvariablebelongingtotheconfigurationspace. VariableslikeF shouldbeinterpreted i as coordinates in an extended space hidden behind the external dynamics. Correspondingly, any of the transformations in F , in particular the rotation, cannot be generated by a naive i Posson bracketting with the angular momentum generator J : i {F ,J } =6 ε F (9) i j ijk k unlike {x ,J } = {x ,ε x p } =ε x . (10) i j i jkm k m ijk k Poisson brackets involving variables like F can be defined, but only in the extended space. i Therefore, for a nondynamical variable like F forcing a requirement of ‘dynamical consis- i tency’ on it and thereby saying that F~ does not transform under rotation does not make sense. For other phase-space variables, of course, the requirement of dynamical consistency δφ(x ,p ) = {φ(x ,p ),G} (11) i i i i must be satisfied for any symmetry transformation generated by G, where the δφ on the left-hand side is the ‘algebraic transformation’ defined in the Comment [1]. 2 Coming back to Eq.(8) we can now introduce a vector ω~ = {ω }, dual to the antisym- k metric tensor ω , as ij 1 ω = ε ω . (12) k 2 ijk ij This enables us to cast Eq.(8) in the form d 0 = dtω − (ε x p )+ε x F . (13) k ijk i j kij i j Z (cid:20) dt (cid:21) Now the arbitrariness of ω readily yields k d (ε x p ) = ε x F (14) ijk i j kij i j dt which is nothing but Eq.(2) in component form. We now make few comments. 1. The covariant forms of the Eqs. (1) and (2) were ensured by the fact that we started with an invariant action S in Eq.(3). 2. This exercise demonstrates that SO(3) invariance of S does notyield aconserved angu- lar momentum J~ anymore. This is clearly in contrast with the translational invariance of S, as the Lagrangian L changes only by a total time-derivative: ~x → x~′ =~x+~a, (15) d L → L′ = L+F~ ·~a = L+ (F~t)·~a . (16) dt (cid:2) (cid:3) (Note that the translational symmetry is not preserved in presence of a nonconstant force F~.) The corresponding conserved charge being (m~x˙−F~t), as follows from Eq.(1) by simple inspection. 3. The nonconservation of angular momentum J~, despite having an SO(3) symmetry in the action (3), is entirely dueto the‘transforming’F~ comingfrom x δF term in Eq.(6) i i which gives rise to the torque ~τ =~x×F~. One can therefore identify Eq.(2) or Eq.(14) as the criterion for SO(3) symmetry. Further, J~ still generates rotation on all the phase-space variables (~x,p~) and functions thereof but not on F , as mentioned earlier. i 4. If F were not to transform, then clearly S is no longer invariant. And even if we insist i onδS = 0underthetransformation(4),thenclearlyweshallmeetwithacontradiction, To see that, set δF = 0 in Eq.(8). Then in place of Eq.(13) we have i d dtω (ε x p ) = 0 (17) k kij i j Z dt d yielding (~x×p~) = 0 as the equation, which appears to be a modified criterion for dt SO(3)invariance of S in presenceof anontransforming F~. Butthis isclearly in conflict with Eq.(2). Besides, note that one cannot actually set δS = 0 in the left hand side of Eq.(8) to begin with as S in Eq.(3) is no longer invariant if δF = 0. This indicates i that the system (3) respects SO(3) symmetry only in presence of a transforming F as i in Eq.(5). 3 Having studied the symmetry aspects of this particle model, we now turn our attention to our field-theory model and try to reassess the comments made by Iorio in the Comment [1]. First of all, the existing similarity between this simple model from particle mechanics, Eq.(3), with the toy model (Eq. (23) in our paper [2]) S = d4xL = − d4x 1F Fµν +jµA (18) 4 µν µ Z Z (cid:0) (cid:1) or the first-order (in θαβ) terminated effective commutative theory (Eq. (76) in our paper [2]) S = − d4x 1F Fµν +θαβ 1F F + 1F F Fµν (19) 4 µν 2 µα νβ 8 βα µν Z h i (cid:0) (cid:1) shouldbeobvious. Theconstantbackgroundfieldslikethevector fieldjµ andthetensor field θµν transformingunder(homogeneous) Lorentz transformations ensureLorentz invariance of the actions. Both jµ and the noncommutative parameter θµν here are the counterparts of the constant force F introduced in Eq.(3). Sheer presence of these ‘transforming’ external i parameters gives rise to nonconserving angular momentum tensors in these two respective theories (Eqs. (38) and (82) in our paper [2]): ∂ Mµλρ −Aλjρ+Aρjλ = 0, (20) µ ∂ Mµλρ −θλ F FµαFνρ+ 1FµνFρα +θρ F FµαFνλ+ 1FµνFλα = 0, (21) µ α µν 4 α µν 4 (cid:16) (cid:17) (cid:0) (cid:1) as happens in the particle model, as we have seen already. These two results were obtained in an ab initio computation carried out in [2]. The main point of derivation is that we had to consider the terms involving δjµ and δθµν in these respective analyses [2] as we did earlier in Eq.(6) for the particle model. In these kinds of situations the currents Mµνλ will fail to satisfy the continuity equation in these field theoretical systems as certain components of jµ or θµν can be readily related to identifiable appropriate external ‘torques’ ~τ. For example, one can get the time-derivative of Ji ≡ 1εijk d3xM0jk from Eq.(20) as, 2 R dJ~ ~τ = = R~ ×~j, (22) dt whereR~ = d3xA~(x). Thishasindeedthesamestructuralformasthatoftheparticlemodel (2), with~j Rindeed playing the role of force F~. The same holds for the space components of the constant background field Pµ introduced in Eq.(7) of the Comment[1]. Likewise one can also relate appropriate components of θ to ~τ as well for the model (19). We will not require the explicit form here. All these analyses clearly show that angular momentum tensor will not satisfy continuity equationinpresenceofsuchtransformingadditionalconstantparametersinthetheorywhich act as constant background fields and give rise to torque on the system. In [3], the variations of the parameters jµ or θµν were not considered. Also, it was demanded that [1, 3] ∆ jµ = {jµ,Q} = 0, ∆ θµν = {θµν,Q} = 0 (23) f f following from their requirement of dynamical consistency. But as we have pointed out earlier the Poisson bracket of any nondynamical variable like F , jµ or θµν can possibly be defined only in an extended space in a more fundamental i 4 theory that goes beyondtheusualphasespacediscussedhere. For example, one can envisage a situation where the additional variables describing the extended space correspond to an appropriate external system which is ‘robust’ enough, i.e. which is very weakly influenced by the dynamics of the original variables. One can then possibly construct the ‘total angular momentum,’ taking into consideration the contribution of these additional variables, which can generate transformations inF , jµ or θµν. Inother words, the‘algebraic’ transformations i of jµ or θµν given by δjµ = ωµ jν, δθµν = ωµ θλν −ων θλµ (24) ν λ λ cannot be generated by the angular momentum Mµν ≡ d3xM0µν, obtained solely from the variables occuring within the theory, through a PoissRon bracket like (23), just as the transformation δF (5) could not be generated by a naive Poisson bracket of F with the i i angular momentum operator. TheSUSYmodel considered in the Comment [1] cannot becompared to any of the above mentioned models, as the equation of motion for the D-field can be used to eliminate it from the Wess–Zumino model (11) in the Comment [1] to yield a meaningful on-shell version (17) in [1]. In contrast, neither our jµ, nor author’s Pµ nor θµν can be eliminated in this manner, as the entire theory collapses—as has been noted in the Comment [1] as well. Consequently, they cannotberegardedas variables intheconfiguration spaceand Euler–Lagrangeequation ofmotionofthesefieldsdoesnotmakeanysense. Onthecontrary,D-fieldhassomesimilarity with Lagrange multipliers which are counted in the configuration space variables and enforce meaningful constraints on the theory, like A0 in the Maxwell theory. Thus we see that this procedurehas to be implemented case by case and only in those models where it can be done consistently. We conclude here by summarising that all we wanted to demonstrate in our paper [2] was that to preserve Lorentz invariance of these models, where jµ or θµν can be regarded as constant background fields, it is necessary to consider a transforming jµ or θµν. This is manifest from the structure of the actions (18) and (19) themselves. We have also seen that despite the Lorentz invariance, the models do not admit angular momentum tensor Mµνλ satisfying ∂ Mµνλ = 0, as these background fields act like forces, thereby generating µ external torque on the system, as can be seen by considering spatial components M0ij. Also one cannot impose the requirement of dynamical consistency on any background fields. Finally, we would like to mention that there are reasons from black hole physics to expect that the length scale determined by |θµν| has a lower bound [4]. It thus becomes necessary to demand θµν to be constant from other physical considerations. But as we have shown this cannotbereconciledwiththeusualPoincar´einvariance. Nevertheless,ithasbeenpointedout recently that a twisted Poincar´e symmetry can be reconciled with constant θµν [5]. However these issues were not discussed by us in [2]. References [1] A. Iorio, Phys. Rev. D 77 (2008) 048701. [2] R. Banerjee, B. Chakraborty and K. Kumar, Phys. Rev. D 70 (2004) 125004. [3] A. Iorio and T. Sy´kora, Int. J. Mod. Phys. A 17 (2002) 2369. [4] S. Doplicher, K. Fredenhagen and J.E. Roberts, Commun. Math. Phys. 172 (1995) 187. [5] M. Chaichian, P.P. Kulish, K. Nishijima and A. Tureanu, Phys. Lett. B 604 (2004) 98. 5