aa r X i v : . [ h e p - t h ] A p r Tensor Coordinates in NoncommutativeMechanics
Ricardo Amorim
Instituto de F´ısica, Universidade Federal do Rio de Janeiro,Caixa Postal 68528, 21945-970 Rio de Janeiro, Brazil
Abstract
A consistent classical mechanics formulation is presented in such away that, under quantization, it gives a noncommutative quantum the-ory with interesting new features. The Dirac formalism for constrainedHamiltonian systems is strongly used, and the object of noncommu-tativity θ ij plays a fundamental rule as an independent quantity. Thepresented classical theory, as its quantum counterpart, is naturallyinvariant under the rotation group SO ( D ). [email protected] . Space-time noncommutativity has been a very studied subject. Afterthe first published work[1], a huge amount of papers has appeared in recenttimes, most of them connected with strings[2] and noncommutative fieldtheories (NCFT’s)[3]. Both theories, which are close related [4]-[6], are yetin construction, and any new contribution to the theme is welcome.A nice framework to study aspects on noncommutativity is given by theso called noncommutative quantum mechanics (NCQM), due to its simplerapproach. There are several interesting works in NCQM and I cite someof them[7]-[23]. In most of these papers, the object of noncommutativity θ ij , which essentially is the result of the commutation of two coordinate op-erators, is considered as a constant matrix, although this is not the generalcase[1, 14, 24, 28, 30]. Considering θ ij as a constant matrix spoils the Lorentzsymmetry or correspondingly the rotation symmetry for non relativistic the-ories.In a recent work [31], a version of NCQM has been presented, where notonly the coordinates x i and their canonical momenta p i are considered asoperators in Hilbert space but also the objects of noncommutativity θ ij andtheir canonical conjugate momenta π ij . All of these operators belong to thesame algebra and have the same hierarchical level. This enlargement of theusual set of Hilbert space operators permits the theory to be invariant underthe rotation group SO ( D ), as showed in detail in [31] . Rotation invariance,in a nonrelativistic theory, is fundamental if one intends to describe anyphysical system in a consistent way. In NCFT’s it is possible to achieve thecorresponding SO ( D,
1) invariance also by promoting θ µν from a constantmatrix to a tensor operator[24]-[29], although in this last situation the rulesare quite different from those found in NCQM, since in a quantum fieldtheory the relevant operators are not coordinates but fields.In Ref. [31], accordingly to the discussion given above, it was introducedthe canonical commutator algebra [ x i , p j ] = iδ ij (1)[ θ ij , π kl ] = iδ ijkl (2)where δ ijkl = δ ik δ jl − δ il δ jk . It was also assumed that i, j = 1 , , ..., D ; µ, ν = 0 , , ...., D . Natural units are adopted, where ¯ h = c = 1 . x i , x j ] = iθ ij (3)Expression (3) is fundamental in NCQM, although in its right hand side θ ij is usually considered as a constant antisymmetric matrix, which obviouslydoes not satisfies a relation like (2). For simplicity, in [31][ x i , θ jk ] = 0 (4)and [ θ ij , θ kl ] = 0 (5)Expressions (3-5) are simpler than the corresponding ones, which appear inSnyder’s paper [1]. They have been proposed, in the context of quantumgravity, in [30]. They also appear, in the context of NCFT’s, in Refs. [24]-[29]. Furthermore, it is assumed in [31] that[ p i , θ jk ] = 0 (6)and [ p i , π jk ] = 0 (7)The Jacobi identity formed with the operators x i , x j and π kl leads to[[ x i , π kl ] , x j ] − [[ x j , π kl ] , x i ] = − δ ijkl (8)with solution [ x i , π kl ] = − i δ ijkl p j (9)The algebraic structure given above shows that the shifted coordinate operator[7,9, 17, 18, 21] X i ≡ x i + 12 θ ij p j (10)commutes with π kl , θ ij and X j . The shifted coordinate operator plays afundamental rule in NCQM, since it is possible to form a basis with itseigenvectors. This possibility is forbidden for the usual coordinate operator2ince its components satisfy nontrivial commutation relations among them-selves. In Ref. [31] the algebraic structure (1-10) is discussed, and also it isshown that the generalized angular momentum operator J ij = X i p j − X j p i − θ il π jl + θ jl π il (11)closes in the SO ( D ) algebra[ J ij , J kl ] = iδ il J kj − iδ jl J ki − iδ ik J lj + iδ jk J li (12)and generates the expected symmetry transformations when acting in all theoperators in Hilbert space. This does not happen when one considers theusual angular momentum operator l ij = x i p j − x j p i (13)since [ l ij , l kl ] = iδ il l kj − iδ jl l ki − iδ ik l lj + iδ jk l li + iθ il p k p j − iθ jl p k p i − iθ ik p l p j + iθ jk p l p i (14)even if θ ij is not taken as a Hilbert space operator but just as a constantmatrix. In this letter I present a possible underlying classical theory that, underquantization, reproduce the algebraic structure displayed above. The Diracformalism[32] for constrained Hamiltonian systems is extensively used in thispurpose. As it is well known, this formalism teaches us that when a theorypresents a complete set of second class constraints Ξ a = 0 , a = 1 , ... N , thePoisson brackets { A, B } between any two phase space quantities A , B mustbe replaced by the Dirac brackets { A, B } D = { A, B } − { A, Ξ a } ∆ − ab { Ξ b , B } (15)in order that the evolution of the system respects the constraint surface givenby Ξ a = 0. In (15) ∆ ab = { Ξ a , Ξ b } (16)3s the constraint matrix and ∆ − ab is its inverse. Its existence is related to thefact that the constraints Ξ a are second class. If that matrix were singular,linear combinations of the Ξ a could be first class. For the first situation, thenumber of effective degrees of freedom of the theory is given by 2 D − N ,where 2 D is the number of phase space variables and 2 N is the numberof second class constraints. If phase space is spanned only by the 2 D =2 D + 2 D ( D − variables x i , p i , θ ij and π ij , the introduction of second classconstraints generates an over constrained theory, when compared with thealgebraic structure given in [31]. So it seems necessary to enlarge phasespace by 2 N variables, and to introduce at the same time 2 N second classconstraints. The simpler way to implement these ideas without spoilingsymmetry under rotations is to enlarge phase space by a pair of canonicalvariables Z i , K i , introducing at the same time a set of second class constraintsΨ i , Φ i .Considering this set of phase space variables, it follows by constructionthe fundamental ( non vanishing ) Poisson bracket structure { x i , p j } = δ ij { θ ij , π kl } = δ ijkl { Z i , K j } = δ ij (17)and the Dirac brackets structure is derived in accordance with the form ofthe second class constraints, subject that will be discussed in what follows.I assume that Z i has dimension of length L , as x i . This implies thatboth p i and K i have dimension of L − . As θ ij and π ij have respectivelydimensions of L and L − , the simpler form of the constraints Ψ i and Φ i isgiven by Ψ i = Z i + αx i + βθ ij p j + γθ ij K j and Φ i = K i + ρp i + σπ ij x j + λπ ij Z j , if only adimensional parameters α, β, γ, ρ, σ and λ are introducedand any potence higher than two in phase space variables is discarded. Icould display all of these parameters along the implementation of the Diracformalism. Actually this has been done, and at the end of the calculationsthe parameters have been chosen in order to generate, under quantization,the commutator structure appearing in (1-9). The results are surprisinglysimple. The constraints reduce, in this situation, to4 i = Z i − θ ij p j Φ i = K i − p i (18)and the corresponding constraint matrix (16) becomes(∆ ab ) = (cid:18) { Ψ i , Ψ j } { Ψ i , Φ j }{ Φ i , Ψ j } { Φ i , Φ j } (cid:19) = δ ij − δ ji ! (19)A point to be stressed here is that (19) is regular even if θ ij is singular.This guarantees that the proper commutative limit of the theory can betaken. Now the inverse of (19) is trivially given by(∆ − ab ) = (cid:18) − δ ji δ ij (cid:19) (20)and the Dirac brackets involving only the original set of phase space variablesis { x i , p j } D = δ ij { x i , x j } D = θ ij { p i , p j } D = 0 { θ ij , π kl } D = δ ijkl { θ ij , θkl } D = 0 { π ij , π kl } D = 0 { x i , θ kl } D = 0 { x i , π kl } D = − δ ijkl p j { p i , θ kl } D = 0 { p i , π kl } D = 0 (21)which gives the desired result. Actually, if y A represents phase space vari-ables and y A the corresponding Hilbert space operators, the Dirac quantiza-tion prescription { y A , y B } D → i [ y A , y B ] gives the commutators (1-9). Forcompleteness, the remaining Dirac brackets involving Z i and K i are heredisplayed: { Z i , K j } D = 0 { Z i , Z j } D = 0 { K i , K j } D = 0 { Z i , x j } D = − θ ij { K i , x j } D = − δ ij { Z i , p j } D = 0 { K i , p j } D = 0 { Z i , θ kl } D = 0 { Z i , π kl } D = δ ijkl p j { K i , θ kl } D = 0 { K i , π kl } D = 0 (22)As one can verify, the only non trivial Jacobi identities involving the Diracbrackets appearing in (21-22) are given by5 = {{ x i , π kl } D , x j } D + {{ π kl , x j } D , x i } D + {{ x j , x i } D , π kl } D J = {{ x i , π kl } D , Z j } D + {{ π kl , Z j } D , x i } D + {{ Z j , x i } D , π kl } D (23)and both J and J vanish identically, as expected. Of course, (22) is justan auxiliary set, since due to the constraints (18), Z i and K i can be seen asdependent variables. After using the Dirac brackets, those constraints canbe used in a strong way. In this classical theory the shifted coordinate X i = x i + 12 θ ij P j (24)which corresponds to the operator (10), also plays a fundamental role. Ascan be verified, { X i , X j } D = 0 { X i , p j } D = δ ij { X i , x j } D = θ ij { X i θ kl , π kl } D = 0 { X i , π kl } D = 0 { X i , Z j } D = − θ ij { X i , K j } D = δ ij (25)and so the angular momentum tensor J ij = X i p j − X j p i − θ il π jl + θ jl π il (26)closes in the classical SO ( D ) algebra, by using Dirac brackets in place ofcommutators. Actually { J ij , J kl } D = δ il J kj − δ jl J ki − δ ik J lj + δ jk J li (27)As in the quantum case, the proper symmetry transformations over allthe phase space variables are generated by (26). By writing δA = − ǫ kl { A, J kl } D (28)one arrives at δX i = ǫ ij X j δx i = ǫ ij x j p i = ǫ ji p j δθ ij = ǫ ik θ kj + ǫ jk θ ik δπ ij = ǫ ki π kj + ǫ kj π ik δZ i = 12 ǫ ij θ jk p k δK i = ǫ ji p j (29)The last two equations also give the proper result on the constraint surface.So it was possible to generate all the desired structure displayed in [31] byusing the Dirac brackets and the constraints (18). These constraints, as wellas the fundamental Poisson brackets (17), can be trivially generated by thefirst order action S = Z dt L F O (30)where L F O = p. ˙ x + K. ˙ Z + π. ˙ θ − λ a Ξ a − H (31)The 2 D quantities λ a are Lagrange multipliers to implement the constraintsΞ a = 0 given by (18), and H is some Hamiltonian. The dots ”.” represent in-ternal products. Strictly, the momenta canonically conjugate of the Lagrangemultipliers are primary constraints that, when conserved, generate the sec-ondary constraints Ξ a = 0. Since these last constraints are second class, theyare automatically conserved by the theory, and the Lagrange multipliers aredetermined in the process.In Ref. [31], besides the introduction of the referred algebraic structure,a specific Hamiltonian has been given, representing a generalized isotropicharmonic oscillator, which contemplates with dynamics not only the usualvectorial coordinates but also the noncommutativity sector spanned by thetensor quantities θ and π . The corresponding classical Hamiltonian can bewritten as H = 12 m p + mω X + 12Λ π + ΛΩ θ (32)which is invariant under rotations. In (32) m is a mass, Λ is a parameterwith dimension of L − , and ω and Ω are frequencies. Other choices for the7amiltonian can be done without spoiling the algebraic structure discussedabove.The classical system given by (30-32) represents two independent isotropicoscillators in D and D ( D − dimensions, expressed in terms of variables X i , p i , θ ij and π ij . The solution is elementary, but when one expresses the oscillatorsin terms of physical variables x i , p i , θ ij and π ij , it arises a coupling betweenthem, with cumbersome equations of motion. In this sense the former set ofvariables gives, in phase space, the normal coordinates that decouple bothoscillators. To close this letter, I comment that it was possible to generate a Diracbrackets algebraic structure that, when quantized, exactly reproduce thecommutator algebra appearing in [31]. The presented theory has been provedto be invariant under the action of the rotation group SO ( D ) and could bederived through a variational principle.Once this structure has been given, it is not difficult to construct a rel-ativistic generalization of such a model. The fundamental Poisson bracketsbecome { x µ , p ν } = δ µν { θ µν , π ρσ } = δ µνρσ { Z µ , K ν } = δ µν (33)and the constraints (18) are generalized toΨ µ = Z µ − θ µν p ν Φ µ = K µ − p µ (34)generating the invertible constraint matrix(∆ ab ) = (cid:18) { Ψ µ , Ψ ν } { Ψ µ , Φ ν }{ Φ µ , Ψ ν } { Φ µ , Φ ν } (cid:19) = (cid:18) η µν − η µν (cid:19) (35)The Dirac brackets between the phase space variables can also be generalizedfrom (21-22). The Hamiltonian of course can not be given by (32), but atleast for the free particle, it vanishes identically, as it uses to happen withcovariant classical systems[32]. Also it is necessary a new constraint, which8ust be first class, to generate the reparametrization transformations. In aminimal extension of the usual commutative case, it is given by the massshell condition χ = p + m = 0, but other choices are possible, givingdynamics to the noncommutativity sector or enlarging the symmetry contentof the relativistic action. These ideas are under study and will be publishedelsewhere [33]. References [1] H. S. Snyder, Phys. Rev. (1947) 38.[2] J. Polchinski, String Theory, University Press, Cambridge , 1998; R. Sz-abo, An introduction to String Theory and D-Brane Dynamics, ImperialCollege Press, London, 2004.[3] R. J. Szabo, Phys. Repp (2003) 207.[4] M.R.Douglas and C. Hull, JHEP (1998) 008.[5] M. M. Sheikh-Jabbari, Phys. Lett (1999) 032.[6] N. Seiberg and E. Witten, JHEP (1999) 032.[7] M. Chaichian, M. M. Sheikh-Jabbari and A. Tureanu, Phys. Rev. Lett (2001) 2716.[8] M. Chaichian, A. Demichec, P. Presnajder, M. M. Sheikh-Jabbari andA. Tureanu, Nucl. Phys. B 527 (2002) 149.[9] J. Gamboa, M. Loewe and J. C. Rojas, Phys. Rev.
D 64 (2001) 067901.[10] V. P .Nair and A. P. Polychronakos, Phys. Lett
B 505 (2001) 267.[11] Stefano Bellucci and A. Nersessian, Phys. Lett.
B 542 (2002) 295.[12] P.-M. Ho and H.-C. Kao, Phys. Rev Lett (2002) 151602.[13] O. Espinosa and P. Gaete, ”Symmetry in noncommutative quantummechanics”, het-th/0206066 (2002).914] A. A. Deriglazov, Phys. Lett. B555 (2003) 83; JHEP (2003) 021.[15] A. Smailagic and E. Spallucci, J. Phys.
A36 (2003) L467; J. Phys.
A36 (2003) L517.[16] L. Jonke and S. Meljanac, Eur. Phys. Jour.
C29 (2003) 433.[17] A. Kokado, T. Okamura and T. Saito, Phys.
D 69 (2004) 125007.[18] A. Kijanka and P Kosinski, Phys. Rev.
D 70 (2004) 127702.[19] I. Dadic, L. Jonke and S. Meljanac, Acta Phys. Slov. (2005) 145.[20] S. Bellucci and A. Yeranyan, Phys. Lett. B 609 (2005) 418.[21] X. Calmet, Phys. Rev.
D 71 (2005) 085012; X. Calmet and M. Selvaggi,Phys. Rev
D74 (2006) 037901.[22] F. G. Scholtz, B. Chakraborty, J. Govaerts and S. Vaidya, J. Phys.
A40 (2007) 14581.[23] M. Rosenbaum, J. David Vergara and L R. Juarez, Phys. Lett.
A 267 (2007) 267.[24] C. E. Carlson, C.D. Carone and N. Zobin, Phys. Rev.
D 66 (2002)075001.[25] M. Haghighat and M. M. Ettefaghi, Phys. Rev
D 70 (2004) 034017.[26] C. D. Carone and H. J. Kwee, Phys. Rev.
D 73 (2006) 096005.[27] M. M. Ettefaghi and M. Haghighat, Phys. Rev
D 75 (2007) 125002.[28] H. Kase, K. Morita, Y. Okumura and E. Umezawa, Prog. Theor. Phys. (2003) 663; K. Imai, K. Morita and Y. Okumura, Prog. Theor.Phys. (2203) 989.[29] S. Saxell
On general properties of Lorentz invariant formulation of non-commutative quantum field thery , hep-th 08043341 ( 2008 ).[30] S. Doplicher, K. Fredenhagen and J. E. Roberts, Phys. Lett.
B331 (1994) 29; Commun. Math. Phys.172