The Noncommutative Doplicher-Fredenhagen-Roberts-Amorim Space
Everton M.C. Abreu, Albert C.R. Mendes, Wilson Oliveira, Adriano O. Zangirolami
aa r X i v : . [ m a t h - ph ] O c t Symmetry, Integrability and Geometry: Methods and Applications SIGMA (2010), 083, 37 pages The NoncommutativeDoplicher–Fredenhagen–Roberts–Amorim Space ⋆ Everton M.C. ABREU †‡ , Albert C.R. MENDES § , Wilson OLIVEIRA § and Adriano O. ZANGIROLAMI §† Grupo de F´ısica Te´orica e Matem´atica F´ısica, Departamento de F´ısica,Universidade Federal Rural do Rio de Janeiro, BR 465-07, 23890-971, Serop´edica, RJ, Brazil
E-mail: [email protected] ‡ Centro Brasileiro de Pesquisas F´ısicas (CBPF), Rua Xavier Sigaud 150,Urca, 22290-180, RJ, Brazil § Departamento de F´ısica, ICE, Universidade Federal de Juiz de Fora,36036-330, Juiz de Fora, MG, Brazil
E-mail: albert@fisica.ufjf.br, wilson@fisica.ufjf.br, adrianozangirolami@fisica.ufjf.br
Received March 28, 2010, in final form October 02, 2010; Published online October 10, 2010doi:10.3842/SIGMA.2010.083
Abstract.
This work is an effort in order to compose a pedestrian review of the recentlyelaborated Doplicher, Fredenhagen, Roberts and Amorim (DFRA) noncommutative (NC)space which is a minimal extension of the DFR space. In this DRFA space, the object ofnoncommutativity ( θ µν ) is a variable of the NC system and has a canonical conjugate mo-mentum. Namely, for instance, in NC quantum mechanics we will show that θ ij ( i, j = 1 , , θ µν . We willstudy the symmetry properties of an extended x + θ space-time, given by the group P ′ , whichhas the Poincar´e group P as a subgroup. The Noether formalism adapted to such extended x + θ ( D = 4 + 6) space-time is depicted. A consistent algebra involving the enlarged set ofcanonical operators is described, which permits one to construct theories that are dynami-cally invariant under the action of the rotation group. In this framework it is also possibleto give dynamics to the NC operator sector, resulting in new features. A consistent classicalmechanics formulation is analyzed in such a way that, under quantization, it furnishes a NCquantum theory with interesting results. The Dirac formalism for constrained Hamiltoniansystems is considered and the object of noncommutativity θ ij plays a fundamental role asan independent quantity. Next, we explain the dynamical spacetime symmetries in NCrelativistic theories by using the DFRA algebra. It is also explained about the generalizedDirac equation issue, that the fermionic field depends not only on the ordinary coordinatesbut on θ µν as well. The dynamical symmetry content of such fermionic theory is discussed,and we show that its action is invariant under P ′ . In the last part of this work we analyzethe complex scalar fields using this new framework. As said above, in a first quantized for-malism, θ µν and its canonical momentum π µν are seen as operators living in some Hilbertspace. In a second quantized formalism perspective, we show an explicit form for the ex-tended Poincar´e generators and the same algebra is generated via generalized Heisenbergrelations. We also consider a source term and construct the general solution for the complexscalar fields using the Green function technique. Key words: noncommutativity; quantum mechanics; gauge theories ⋆ E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. Zangirolami
Contents
References 33
Theoretical physics is living nowadays a moment of great excitement and at the same timegreat anxiety before the possibility by the Large Hadron Collider (LHC) to reveal the mysteriesthat make the day-by-day of theoretical physicists. One of these possibilities is that, at thecollider energies, the extra and/or the compactified spatial dimensions become manifest. Thesemanifestations can lead, for example, to the fact that standard four-dimensions spacetimes maybecome NC, namely, that the position four-vector operator x µ obeys the following rule[ x µ , x ν ] = iθ µν , (1.1)where θ µν is a real, antisymmetric and constant matrix. The field theories defined on a space-time with (1.1) have its Lorentz invariance obviously broken. All the underlying issues have beenexplored using the representation of the standard framework of the Poincar´e algebra throughthe Weyl–Moyal correspondence. To every field operator ϕ ( x ) it has been assigned a Weylsymbol ϕ ( x ), defined on the commutative counterpart of the NC spacetime. Through this cor-respondence, the products of operators are replaced by Moyal ⋆ -products of their Weyl symbols ϕ ( x ) ψ ( x ) −→ ϕ ( x ) ⋆ ψ ( x ) , where we can define the Moyal product as ϕ ( x ) ⋆ ψ ( x ) = exp (cid:20) i θ µν ∂∂x µ ∂∂y ν (cid:21) ϕ ( x ) ψ ( y ) (cid:12)(cid:12) x = y (1.2)and where now the commutators of operators are replaced by Moyal brackets as,[ x µ , x ν ] ⋆ ≡ x µ ⋆ x ν − x ν ⋆ x µ = iθ µν . he Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 3From (1.2) we can see clearly that at zeroth-order the NCQFT is Lorentz invariant. Since θ µν is valued at the Planck scale, we use only the first-order of the expansion in (1.2).But it was Heisenberg who suggested, very early, that one could use a NC structure for space-time coordinates at very small length scales to introduce an effective ultraviolet cutoff. Afterthat, Snyder tackled the idea launched by Heisenberg and published what is considered as thefirst paper on spacetime noncommutativity in 1947 [1]. C.N. Yang, immediately after Snyder’spaper, showed that the problems of field theory supposed to be removed by noncommutativitywere actually not solved [2] and he tried to recover the translational invariance broken by Sny-der’s model. The main motivation was to avoid singularities in quantum field theories. However,in recent days, the issue has been motivated by string theory [3] as well as by other issues inphysics [4, 5, 6, 7]. For reviews in NC theory, the reader can find them in [8, 9] (see also [10]).In his work Snyder introduced a five dimensional spacetime with SO(4 ,
1) as a symmetrygroup, with generators M AB , satisfying the Lorentz algebra, where A, B = 0 , , , , ~ = c = 1. Moreover, he introduced the relation between coordinates andgenerators of the SO (4 ,
1) algebra x µ = a M µ (where µ, ν = 0 , , , a has dimension of length), promoting in this way thespacetime coordinates to Hermitian operators. The mentioned relation introduces the commu-tator,[ x µ , x ν ] = ia M µν (1.3)and the identities,[ M µν , x λ ] = i (cid:0) x µ η νλ − x ν η µλ (cid:1) and [ M µν , M αβ ] = i (cid:0) M µβ η να − M µα η νβ + M να η µβ − M νβ η µα (cid:1) , which agree with four dimensional Lorentz invariance.Three decades ago Connes et al. (see [11] for a review of this formalism) brought the conceptsof noncommutativity by generalizing the idea of a differential structure to the NC formalism.Defining a generalized integration [12] this led to an operator algebraic description of NC space-times and hence, the Yang–Mills gauge theories can be defined on a large class of NC spaces.And gravity was introduced in [13]. But radiative corrections problems cause its abandon.When open strings have their end points on D-branes in the presence of a background constantB-field, effective gauge theories on a NC spacetime arise [14, 15]. In these NC field theories(NCFT’s) [9], relation (1.3) is replaced by equation (1.1). A NC gauge theory originates froma low energy limit for open string theory embedded in a constant antisymmetric backgroundfield.The fundamental point about the standard NC space is that the object of noncommutativi-ty θ µν is usually assumed to be a constant antisymmetric matrix in NCFT’s. This violatesLorentz symmetry because it fixes a direction in an inertial reference frame. The violation ofLorentz invariance is problematic, among other facts, because it brings effects such as vacuumbirefringence [16]. However, at the same time it permits to treat NCFT’s as deformations ofordinary quantum field theories, replacing ordinary products with Moyal products, and ordinarygauge interactions by the corresponding NC ones. As it is well known, these theories carriesserious problems as nonunitarity, nonlocalizability, nonrenormalizability, U V × IR mixing etc.On the other hand, the Lorentz invariance can be recovered by constructing the NC spacetime E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. Zangirolamiwith θ µν being a tensor operator with the same hierarchical level as the x ’s. This was done in [17]by using a convenient reduction of Snyder’s algebra. As x µ and θ µν belong in this case to thesame affine algebra, the fields must be functions of the eigenvalues of both x µ and θ µν . In [18]Banerjee et al. obtained conditions for preserving Poincar´e invariance in NC gauge theories anda whole investigation about various spacetime symmetries was performed.The results appearing in [17] are explored by some authors [18, 19, 20, 21, 22, 23]. Some ofthem prefer to start from the beginning by adopting the Doplicher, Fredenhagen and Roberts(DFR) algebra [10], which essentially assumes (1.1) as well as the vanishing of the triple commu-tator among the coordinate operators. The DFR algebra is based on principles imported fromgeneral relativity (GR) and quantum mechanics (QM). In addition to (1.1) it also assumes that[ x µ , θ αβ ] = 0 . (1.4)With this formalism, DFR demonstrated that the combination of QM with the classicalgravitation theory, the ordinary spacetime loses all operational meaning at short distances.An important point in DFR algebra is that the Weyl representation of NC operators obeying(1.1) and (1.4) keeps the usual form of the Moyal product, and consequently the form of theusual NCFT’s, although the fields have to be considered as depending not only on x µ butalso on θ αβ . The argument is that very accurate measurements of spacetime localization couldtransfer to test particles energies sufficient to create a gravitational field that in principle couldtrap photons. This possibility is related with spacetime uncertainty relations that can be derivedfrom (1.1) and (1.4) as well as from the quantum conditions θ µν θ µν = 0 , (cid:18) ∗ θ µν θ µν (cid:19) = λ P , (1.5)where ∗ θ µν = ǫ µνρσ θ ρσ and λ P is the Planck length.These operators are seen as acting on a Hilbert space H and this theory implies in extracompact dimensions [10]. The use of conditions (1.5) in [17, 19, 20, 21, 22, 23] would bringtrivial consequences, since in those works the relevant results strongly depend on the valueof θ , which is taken as a mean with some weigh function W ( θ ). They use in this processthe celebrated Seiberg–Witten [15] transformations. Of course those authors do not use (1.5),since their motivations are not related to quantum gravity but basically with the constructionof a NCFT which keeps Lorentz invariance. This is a fundamental matter, since there is noexperimental evidence to assume Lorentz symmetry violation [16]. Although we will see thatin this review we are not using twisted symmetries [24, 25, 26, 27] there is some considerationsabout the ideas and concepts on this twisted subject that we will make in the near future here.A nice framework to study aspects on noncommutativity is given by the so called NC quantummechanics (NCQM), due to its simpler approach. There are several interesting works in NCQM[5, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]. In most of these papers,the object of noncommutativity θ ij (where i, j = 1 , , θ ij as a constant matrix spoils the Lorentzsymmetry or correspondingly the rotation symmetry for nonrelativistic theories. In NCQM,although time is a commutative parameter, the space coordinates do not commute. However,the objects of noncommutativity are not considered as Hilbert space operators. As a consequencethe corresponding conjugate momenta is not introduced, which, as well known, it is importantto implement rotation as a dynamical symmetry [45]. As a result, the theories are not invariantunder rotations.In his first paper [46], R. Amorim promoted an extension of the DFR algebra to a non-relativistic QM in the trivial way, but keeping consistency. The objects of noncommutativityhe Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 5were considered as true operators and their conjugate momenta were introduced. This permitsto display a complete and consistent algebra among the Hilbert space operators and to constructgeneralized angular momentum operators, obeying the SO ( D ) algebra, and in a dynamical way,acting properly in all the sectors of the Hilbert space. If this is not accomplished, some funda-mental objects usually employed in the literature, as the shifted coordinate operator (see (2.10)),fail to properly transform under rotations. The symmetry is implemented not in a mere alge-braic way, where the transformations are based on the indices structure of the variables, but itcomes dynamically from the consistent action of an operator, as discussed in [45]. This new NCspace has ten dimensions and now is known as the Doplicher–Fredenhagen–Roberts–Amorim(DFRA) space. From now on we will review the details of this new NC space and describe therecent applications published in the literature.We will see in this review that a consistent classical mechanics formulation can be shownin such a way that, under quantization, it gives a NC quantum theory with interesting newfeatures. The Dirac formalism for constrained Hamiltonian systems is strongly used, and theobject of noncommutativity θ ij plays a fundamental role as an independent quantity. Thepresented classical theory, as its quantum counterpart, is naturally invariant under the rotationgroup SO ( D ).The organization of this work is: in Section 2 we describe the mathematical details of this newNC space. After this we describe the NC Hilbert space and construct an harmonic oscillatormodel for this new space. In Section 3 we explore the DFRA classical mechanics and treatthe question of Dirac’s formalism. The symmetries and the details of the extended Poincar´esymmetry group are explained in Section 4. In this very section we also explore the relativisticfeatures of DFRA space and construct the Klein–Gordon equation together with the Noether’sformalism. In Section 5 we analyzed the issue about the fermions in this new structure andwe study the Dirac equation. Finally, in Section 6 we complete the review considering theelaboration of complex scalar fields and the source term analysis of quantum field theories. In this section we will introduce the complete extension of the DFR space formulated byR. Amorim [46, 47, 48, 49, 50] where a minimal extension of the DFR is accomplished throughthe introduction of the canonical conjugate momenta to the variable θ µν of the system.In the last section we talked about the DFR space, its physical motivations and main math-ematical ingredients. Concerning now the DFRA space, we continue to furnish its “missingparts” and naturally its implications in quantum mechanics. The results appearing in [17] motivated some other authors [20, 21, 22, 19, 23]. Some of themprefer to start from the beginning by adopting the Doplicher, Fredenhagen and Roberts (DFR)algebra [10], which essentially assumes (1.1) as well as the vanishing of the triple commutatoramong the coordinate operators,[ x µ , [ x ν , x ρ ]] = 0 , (2.1)and it is easy to realize that this relation constitute a constraint in a NC spacetime. Notice thatthe commutator inside the triple one is not a c -number.The DFR algebra is based on principles imported from general relativity (GR) and quantummechanics (QM). In addition to (1.1) it also assumes that[ x µ , θ αβ ] = 0 , (2.2) E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. Zangirolamiand we consider that space has arbitrary D ≥ x µ and p ν , where i, j =1 , , . . . , D and µ, ν = 0 , , . . . , D , represent the position operator and its conjugate momentum.The NC variable θ µν represent the noncommutativity operator, but now π µν is its conjugatemomentum. In accordance with the discussion above, it follows the algebra[ x µ , p ν ] = iδ µν , (2.3a)[ θ µν , π αβ ] = iδ µναβ , (2.3b)where δ µναβ = δ µα δ νβ − δ µβ δ να . The relation (1.1) here in a space with D dimensions, for example,can be written as[ x i , x j ] = iθ ij and [ p i , p j ] = 0 (2.4)and together with the triple commutator (2.1) condition of the standard spacetime, i.e.,[ x µ , θ να ] = 0 . (2.5)This implies that[ θ µν , θ αβ ] = 0 , (2.6)and this completes the DFR algebra.Recently, in order to obtain consistency R. Amorim introduced [46], as we talked above, thecanonical conjugate momenta π µν such that,[ p µ , θ να ] = 0 , [ p µ , π να ] = 0 . (2.7)The Jacobi identity formed by the operators x i , x j and π kl leads to the nontrivial relation[[ x µ , π αβ ] , x ν ] − [[ x ν , π αβ ] , x µ ] = − δ µναβ . (2.8)The solution, unless trivial terms, is given by[ x µ , π αβ ] = − i δ µναβ p ν . (2.9)It is simple to verify that the whole set of commutation relations listed above is indeed consistentunder all possible Jacobi identities. Expression (2.9) suggests the shifted coordinate operator[29, 31, 38, 39, 42] X µ ≡ x µ + 12 θ µν p ν (2.10)that commutes with π kl . Actually, (2.10) also commutes with θ kl and X j , and satisfies a nontrivial commutation relation with p i depending objects, which could be derived from[ X µ , p ν ] = iδ µν (2.11)and [ X µ , X ν ] = 0 . (2.12)To construct a DFRA algebra in ( x, θ ) space, we can write M µν = X µ p ν − X ν p µ − θ µσ π νσ + θ νσ π µσ , he Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 7where M µν is the antisymmetric generator of the Lorentz-group. To construct π µν we haveto obey equations (2.3b) and (2.9), obviously. From (2.3a) we can write the generators oftranslations as P µ = − i∂ µ . With these ingredients it is easy to construct the commutation relations[ P µ , P ν ] = 0 , [ M µν , P ρ ] = − i (cid:0) η µν P ρ − η µρ P ν (cid:1) , [ M µν , M ρσ ] = − i (cid:0) η µρ M νσ − η µσ M νρ − η νρ M µσ − η νσ M µρ ) , and we can say that P µ and M µν are the generator of the DFRA algebra. These relations areimportant, as we will see in Section 5, because they are essential for the extension of the Diracequation to the extended DFRA configuration space ( x, θ ). It can be shown that the Cliffordalgebra structure generated by the 10 generalized Dirac matrices Γ (Section 5) relies on theserelations.Now we need to remember some basics in quantum mechanics. In order to introduce a con-tinuous basis for a general Hilbert space, with the aid of the above commutation relations, itis necessary firstly to find a maximal set of commuting operators. For instance, let us choosea momentum basis formed by the eigenvectors of p and π . A coordinate basis formed by theeigenvectors of ( X , θ ) can also be introduced, among other possibilities. We observe here that itis in no way possible to form a basis involving more than one component of the original positionoperator x , since their components do not commute.To clarify, let us display the fundamental relations involving those basis, namely eigenvalue,orthogonality and completeness relations X i | X ′ , θ ′ i = X ′ i | X ′ , θ ′ i , θ ij | X ′ , θ ′ i = θ ′ ij | X ′ , θ ′ i , p i | p ′ , π ′ i = p ′ i | p ′ , π ′ i , π ij | p ′ , π ′ i = π ′ ij | p ′ , π ′ i , h X ′ , θ ′ | X ′′ , θ ′′ i = δ D ( X ′ − X ′′ ) δ D ( D − ( θ ′ − θ ′′ ) , h p ′ , π ′ | p ′′ , π ′′ i = δ D ( p ′ − p ′′ ) δ D ( D − ( π ′ − π ′′ ) , Z d D X ′ d D ( D − θ ′ | X ′ , θ ′ ih X ′ , θ ′ | = , (2.13) Z d D p ′ d D ( D − π ′ | p ′ , π ′ ih p ′ , π ′ | = , (2.14)notice that the dimension D means that we live in a framework formed by the spatial coordi-nates and by the θ coordinates, namely, D includes both spaces, D = (spatial coordinates + θ coordinates). It can be seen clearly from the equations involving the delta functions and theintegrals equations (2.13) and (2.14).Representations of the operators in those bases can be obtained in an usual way. For instance,the commutation relations given by equations (2.3) to (2.11) and the eigenvalue relations above,unless trivial terms, give h X ′ , θ ′ | p i | X ′′ , θ ′′ i = − i ∂∂X ′ i δ D ( X ′ − X ′′ ) δ D ( D − ( θ ′ − θ ′′ )and h X ′ , θ ′ | π ij | X ′′ , θ ′′ i = − iδ D ( X ′ − X ′′ ) ∂∂θ ′ ij δ D ( D − ( θ ′ − θ ′′ ) . E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. ZangirolamiThe transformations from one basis to the other one are carried out by extended Fourier trans-forms. Related with these transformations is the “plane wave” h X ′ , θ ′ | p ′′ , π ′′ i = N exp( ip ′′ X ′ + iπ ′′ θ ′ ) , where internal products are represented in a compact manner. For instance, p ′′ X ′ + π ′′ θ ′ = p ′′ i X ′ i + 12 π ′′ ij θ ′ ij . Before discussing any dynamics, it seems interesting to study the generators of the groupof rotations SO ( D ). Without considering the spin sector, we realize that the usual angularmomentum operator l ij = x i p j − x j p i does not close in an algebra due to (2.4). And we have that,[ 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 and so their components can not be SO ( D ) generators in this extended Hilbert space. On thecontrary, the operator L ij = X i p j − X j p i , (2.15)closes in the SO ( D ) algebra. However, to properly act in the ( θ, π ) sector, it has to be generalizedto the total angular momentum operator J ij = L ij − θ il π jl + θ jl π il . (2.16)It is easy to see that not only[ J ij , J kl ] = iδ il J kj − iδ jl J ki − iδ ik J lj + iδ jk J li , (2.17)but J ij generates rotations in all Hilbert space sectors. Actually δ X i = i ǫ kl [ X i , J kl ] = ǫ ik X k , δ p i = i ǫ kl [ p i , J kl ] = ǫ ik p k ,δθ ij = i ǫ kl [ θ ij , J kl ] = ǫ ik θ jk + ǫ jk θ ik , δπ ij = i ǫ kl [ π ij , J kl ] = ǫ ik π jk + ǫ jk π ik (2.18)have the expected form. The same occurs with x i = X i − θ ij p j = ⇒ δ x i = i ǫ kl [ x i , J kl ] = ǫ ik x k . Observe that in the usual NCQM prescription, where the objects of noncommutativity are pa-rameters or where the angular momentum operator has not been generalized, X fails to transformas a vector operator under SO ( D ) [29, 31, 38, 39, 42]. The consistence of transformations (2.18)comes from the fact that they are generated through the action of a symmetry operator and notfrom operations based on the index structure of those variables.We would like to mention that in D = 2 the operator J ij reduces to L ij , in accordancewith the fact that in this case θ or π has only one independent component. In D = 3, itis possible to represent θ or π by three vectors and both parts of the angular momentumoperator have the same kind of structure, and so the same spectrum. An unexpected additionof angular momentum potentially arises, although the ( θ, π ) sector can live in a J = 0 Hilbertsubspace. Unitary rotations are generated by U ( ω ) = exp( − iω · J ), while unitary translations,by T ( λ, Ξ) = exp( − iλ · p − i Ξ · π ).he Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 9 In this section we will consider the isotropic D -dimensional harmonic oscillator where we findseveral possibilities of rotational invariant Hamiltonians which present the proper commutativelimit [31, 32, 39, 40]. The well known expression representing the harmonic oscillator can bewritten as H = 12 m p + mω X , (2.19)since X i commutes with X j , satisfies the canonical relation (2.11) and in the DFRA formalismtransforms according to (2.18). With these results we can construct annihilation and creationoperators in the usual way, A i = r mω (cid:18) X i + i p i mω (cid:19) and A † i = r mω (cid:18) X i − i p i mω (cid:19) , where A i and A † i satisfy the usual harmonic oscillator algebra, and H can be written in termsof the sum of D number operators N i = A † i A i , which have the same spectrum and the samedegeneracies when compared with the ordinary QM case [51].The ( θ, π ) sector, however, is not modified by any new dynamics if H represents the totalHamiltonian. As the harmonic oscillator describes a system near an equilibrium configuration,it seems interesting as well to add to (2.19) a new term like H θ = 12Λ π + ΛΩ θ , (2.20)where Λ is a parameter with dimension of (length) − and Ω is some frequency. Both Hamil-tonians, equations (2.19) and (2.20), can be simultaneously diagonalized, since they commute.Hence, the total Hamiltonian eigenstates will be formed by the direct product of the Hamiltonianeigenstates of each sector.The annihilation and creation operators, considering the ( θ, π ) sector, are respectively definedas A ij = r ΛΩ2 (cid:18) θ ij + iπ ij ΛΩ (cid:19) and A † ij = r ΛΩ2 (cid:18) θ ij − iπ ij ΛΩ (cid:19) , which satisfy the oscillator algebra[ A ij , A † kl ] = δ ij,kl , and now we can construct eigenstates of H θ , equation (2.20), associated with quantum num-bers n ij . As well known, the ground state is annihilated by A ij , and its corresponding wavefunction, in the ( θ, π ) sector, is h θ ′ | n ij = 0 , t i = (cid:18) ΛΩ π (cid:19) D ( D − exp (cid:20) − ΛΩ4 θ ′ ij θ ′ ij (cid:21) exp (cid:20) − iD ( D −
1) Ω4 t (cid:21) . (2.21)However, turning to the basics, the wave functions for excited states can be obtained throughthe application of the creation operator A † kl on the fundamental state. On the other hand, weexpect that Ω might be so big that only the fundamental level of this generalized oscillator couldbe occupied. This will generate only a shift in the oscillator spectrum, which is ∆ E = D ( D − Ωand this new vacuum energy could generate unexpected behaviors.Another point related with (2.21) is that it gives a natural way for introducing the weightfunction W ( θ ) which appears, in the context of NCFT’s, in [17, 19]. W ( θ ) is a normalized0 E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. Zangirolamifunction necessary, for example, to control the θ -integration. Analyzing the ( θ, π ) sector, theexpectation value of any function f ( θ ) over the fundamental state is h f ( θ ) i = h n kl = 0 , t | f ( θ ) | n kl = 0 , t i = (cid:18) ΛΩ π (cid:19) D ( D − Z d D ( D − θ ′ f ( θ ′ ) exp (cid:20) − ΛΩ2 θ ′ rs θ ′ rs (cid:21) ≡ Z d D ( D − θ ′ W ( θ ′ ) f ( θ ′ ) , where W ( θ ′ ) ≡ (cid:18) ΛΩ π (cid:19) D ( D − exp (cid:20) − ΛΩ2 θ ′ rs θ ′ rs (cid:21) , and the expectation values are given by h i = 1 , h θ ij i = 0 , h θ ij θ ij i = h θ i , h θ ij θ kl i = 2 D ( D − δ ij,kl h θ i , (2.22)where h θ i ≡ h x i i = h X i i = 0, but one can find non trivial noncommutativity contributions to theexpectation values for other operators. For instance, it is easy to see from (2.22) and (2.10) that h x i = h X i + 2 D h θ ih p i , where h X i and h p i are the usual QM results for an isotropic oscillator in a given state.This shows that noncommutativity enlarges the root-mean-square deviation for the physicalcoordinate operator, as expected and can be measurable, at first sight. All the operators introduced until now belong to the same algebra and are equal, hierarchicallyspeaking. The necessity of a rotation invariance under the group SO( D ) is a consequence of thisaugmented Hilbert space. Rotation invariance, in a nonrelativistic theory, is the main topic ifone intends to describe any physical system in a consistent way.In NCFT’s it is possible to achieve the corresponding SO ( D,
1) invariance also by promo-ting θ µν from a constant matrix to a tensor operator [17, 18, 19, 20, 21, 22, 23], although in thislast situation the rules are quite different from those found in NCQM, since in a quantum fieldtheory the relevant operators are not coordinates but fields.Now that we got acquainted with the new proposed version of NCQM [46] where the θ ij are tensors in Hilbert space and π ij are their conjugate canonical momenta, we will show inthis section that a possible fundamental classical theory, under quantization, can reproduce thealgebraic structure depicted in the last section.The Dirac formalism [52] for constrained Hamiltonian systems is extensively used for thispurpose. As it is well known, when a theory presents a complete set of second-class constraintsΞ a = 0, a = 1 , , . . . , N , the Poisson brackets { A, B } between any two phase space quanti-ties A , B must be replaced by Dirac brackets { A, B } D = { A, B } − { A, Ξ a } ∆ − ab { Ξ b , B } , (3.1)such that the evolution of the system respects the constraint surface given by Ξ a = 0.he Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 11In (3.1)∆ ab = { Ξ a , Ξ b } (3.2)is the so-called constraint matrix and ∆ − ab is its inverse. The fact that the constraints Ξ a aresecond-class guarantees the existence of ∆ ab . If that matrix were singular, linear combinationsof the Ξ a could be first class. For the first situation, the number of effective degrees of freedomof the theory is given by 2 D − N , where 2 D is the number of phase space variables and 2 N isthe number of second-class constraints.If the phase space is described only by the2 D = 2 D + 2 D ( D − x i , p i , θ ij and π ij , the introduction of second-class constraints generates an over con-strained theory when compared with the algebraic structure given in the last section. Conse-quently, it seems necessary to enlarge the phase space by 2 N variables, and to introduce atthe same time 2 N second-class constraints. An easy way to implement these concepts withoutdestroying the symmetry under rotations is to enlarge the phase space introducing a pair ofcanonical variables Z i , K i , also with (at the same time) a set of second-class constraints Ψ i , Φ i .Considering this set of phase space variables, it follows by construction the fundamental (nonvanishing) Poisson bracket structure { x i , p j } = δ ij , { θ ij , π kl } = δ ijkl , { Z i , K j } = δ ij (3.3)and the Dirac brackets structure is derived in accordance with the form of the second-classconstraints, subject that will be discussed in what follows.Let us assume that Z i has dimension of length L , as x i . This implies that both p i and K i havedimension of L − . As θ ij and π ij have dimensions of L and L − respectively, the expressionfor the constraints Ψ i and Φ i is given byΨ 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 dimensionless parameters α , β , γ , ρ , σ and λ are introduced and any power higher thantwo in phase space variables is discarded. It is possible to display the whole group of parametersduring the computation of the Dirac formalism. After that, at the end of the calculations, theparameters have been chosen in order to generate, under quantization, the commutator structureappearing in equations (2.3) to (2.9). The constraints reduce, in this situation, toΨ i = Z i − θ ij p j , Φ i = K i − p i (3.4)and hence the corresponding constraint matrix (3.2) becomes(∆ ab ) = (cid:18) { Ψ i , Ψ j } { Ψ i , Φ j }{ Φ i , Ψ j } { Φ i , Φ j } (cid:19) = (cid:18) δ ij − δ ji (cid:19) . (3.5)Notice that (3.5) is regular even if θ ij is singular. This fact guarantees that the proper commu-tative limit of the theory can be taken.2 E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. ZangirolamiThe inverse of (3.5) is trivially given by(∆ − ab ) = (cid:18) − δ ji δ ij (cid:19) and it is easy to see that the non-zero Dirac brackets (the others are zero) involving only theoriginal set of phase space variables are { x i , p j } D = δ ij , { x i , x j } D = θ ij , { θ ij , π kl } D = δ ijkl , { x i , π kl } D = − δ ijkl p j , (3.6)which furnish the desired result. If y A represents phase space variables and y A the correspondingHilbert space operators, the Dirac quantization procedure, { y A , y B } D → i [ y A , y B ]results the commutators in (2.3) until (2.9). For completeness, the remaining non-zero Diracbrackets involving Z i and K i are { Z i , x j } D = − θ ij , { K i , x j } D = − δ ji , { Z i , π kl } D = 12 δ ijkl p j . (3.7)In this classical theory the shifted coordinate X i = x i + 12 θ ij p j , which corresponds to the operator (2.10), also plays a fundamental role. As can be verified bythe non-zero Dirac brackets just below, { X i , p j } D = δ ij , { X i , x j } D = 12 θ ij , { X i , Z j } D = − θ ij , { X i , K j } D = δ ij , and the angular momentum tensor J ij = X i p j − X j p i − θ il π jl + θ jl π il (3.8)closes in the classical SO ( D ) algebra, by using Dirac brackets instead of commutators. In fact { J ij , J kl } D = δ il J kj − δ jl J ki − δ ik J lj + δ jk J li , and as in the quantum case, the proper symmetry transformations over all the phase spacevariables are generated by (3.8). Beginning with δA = − ǫ kl { A, J kl } D , one have as a result that δX i = ǫ ij X j , δx i = ǫ ij x j , δp 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 . The last two equations above also furnish the proper result on the constraint surface. Hence,it was possible to generate all the desired structure displayed in the last section by using thehe Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 13Dirac brackets and the constraints given in (3.4). These constraints, as well as the fundamentalPoisson brackets in (3.3), can be easily generated by the first order action S = Z dt L FO , (3.9)where L FO = p · ˙ x + K · ˙ Z + π · ˙ θ − λ a Ξ a − H. (3.10)The 2 D quantities λ a are the Lagrange multipliers introduced conveniently to implement theconstraints Ξ a = 0 given by (3.4), and H is some Hamiltonian. The dots “ · ” between phase spacecoordinates represent internal products. The canonical conjugate momenta for the Lagrangemultipliers are primary constraints that, when conserved, generate the secondary constraintsΞ a = 0. Since these last constraints are second class, they are automatically conserved by thetheory, and the Lagrange multipliers are determined in the process.The general expression for the first-order Lagrangian in (3.10) shows the constraints imple-mentation in this enlarged space, which is a trivial result, analogous to the standard procedurethrough the Lagrange multipliers. To obtain a more illuminating second-order Lagrangian wemust follow the basic pattern and with the help of the Hamiltonian, integrate out the momentumvariables in (3.10).As we explained in the last section, besides the introduction of the referred algebraic struc-ture, a specific Hamiltonian has been furnished, representing a generalized isotropic harmonicoscillator, which contemplates with dynamics not only the usual vectorial coordinates but alsothe noncommutativity sector spanned by the tensor quantities θ and π . The correspondingclassical Hamiltonian can be written as H = 12 m p + mω X + 12Λ π + ΛΩ θ , (3.11)which is invariant under rotations. In (3.11) m is a mass, Λ is a parameter with dimensionof L − , and ω and Ω are frequencies. Other choices for the Hamiltonian can be done withoutspoiling the algebraic structure discussed above.The classical system given by (3.9), (3.10) and (3.11) represents two independent isotropicoscillators in D and D ( D − dimensions, expressed in terms of variables X i , p i , θ ij and π ij . Thesolution is elementary, but when one expresses the oscillators in terms of physical variables x i , p i , θ ij and π ij , an interaction appears between them, with cumbersome equations of motion.In this sense the former set of variables gives, in the phase space, the normal coordinates thatdecouple both oscillators.It was possible to generate a Dirac brackets algebraic structure that, when quantized, re-produce exactly the commutator algebra appearing in the last section. The presented theoryhas been proved to 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 difficultto construct a relativistic 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 , and the constraints (3.4) are generalized toΨ µ = Z µ − θ µν p ν , Φ µ = K µ − p µ , 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) . To finish we can say that the Dirac brackets between the phase space variables can also begeneralized from (3.6), (3.7). The Hamiltonian of course cannot be given by (3.11), but atleast for the free particle, it vanishes identically, as it is usual to appear with covariant classicalsystems [52]. Also it is necessary for a new constraint, which must be first class, to generate thereparametrization transformations. In a minimal extension of the usual commutative case, it isgiven by the mass shell condition χ = p + m = 0 , but other choices are possible, furnishing dynamics to the noncommutativity sector or enlargingthe symmetry content of the relativistic action. In this section we will analyze the dynamical spacetime symmetries in NC relativistic theories byusing the DFRA algebra depicted in Section 2. As explained there, the formalism is constructedin an extended spacetime with independent degrees of freedom associated with the object ofnoncommutativity θ µν . In this framework we can consider theories that are invariant underthe Poincar´e group P or under its extension P ′ , when translations in the extra dimensions areallowed. The Noether formalism adapted to such extended x + θ spacetime will be employed.We will study the algebraic structure of the generalized coordinate operators and their con-jugate momenta, and construct the appropriate representations for the generators of P and P ′ ,as well as for the associated Casimir operators. Next, some possible NCQM actions constructedwith those Casimir operators will be introduced and after that we will investigate the symmetrycontent of one of those theories by using Noether’s procedure. In the usual formulations of NCQM, interpreted here as relativistic theories, the coordinates x µ and their conjugate momenta p µ are operators acting in a Hilbert space H satisfying the fun-damental commutation relations given in Section 2, we can define the operator G = 12 ω µν L µν . Note that, analogously to (2.18), it is possible to dynamically generate infinitesimal transfor-mations on any operator A , following the usual rule δ A = i [ A , G ]. For X µ , p µ and L µν , givenin (2.10) and (2.15), with spacetime coordinates, we have the following results δ X µ = ω µν X ν , δ p µ = ω νµ p ν , δ L µν = ω µρ L ρν + ω ν ρ L µρ . However, the physical coordinates fail to transform in the appropriate way. As can be seen, thesame rule applied on x µ gives the result δ x µ = ω µν (cid:18) x ν + 12 θ ρν p ν (cid:19) − θ µν ω νρ p ρ , (4.1)which is a consequence of θ µν not being transformed. Relation (4.1) probably will break Lorentzsymmetry in any reasonable theory. The cure for these problems can be obtained by conside-ring θ µν as an operator in H , and introducing its canonical momentum π µν as well. The pricehe Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 15to be paid is that θ µν will have to be associated with extra dimensions, as happens with theformulations appearing in [17, 18, 19, 20, 21, 22, 23].Moreover, we have that the commutation relation[ x µ , π ρσ ] = − i δ µνρσ p ν (4.2)is necessary for algebraic consistency under Jacobi identities. The set (4.2) completes the algebradisplayed in Section 2, namely, the DFRA algebra. With this algebra in mind, we can generalizethe expression for the total angular momentum, equations (2.16) and (2.17).The framework constructed above permits consistently to write [25] M µν = X µ p ν − X ν p µ − θ µσ π νσ + θ νσ π µσ (4.3)and consider this object as the generator of the Lorentz group, since it not only closes in theappropriate algebra[ M µν , M ρσ ] = iη µσ M ρν − iη νσ M ρµ − iη µρ M σν + iη νρ M σµ , (4.4)but it generates the expected Lorentz transformations on the Hilbert space operators. Actually,for δ A = i [ A , G ], with G = ω µν M µν , we have that, δ x µ = ω µν x ν , δ X µ = ω µν X ν , δ p µ = ω νµ p ν , δθ µν = ω µρ θ ρν + ω ν ρ θ µρ ,δπ µν = ω ρµ π ρν + ω ρν π µρ , δ M µν = ω µρ M ρν + ω νρ M µρ , (4.5)which in principle should guarantee the Lorentz invariance of a consistent theory. We observethat this construction is possible because of the introduction of the canonical pair θ µν , π µν as independent variables. This pair allows the building of an object like M µν in (4.3), whichgenerates the transformations given just above dynamically [45] and not merely by taking intoaccount the algebraic index content of the variables.From the symmetry structure given above, we realize that actually the Lorentz generator (4.3)can be written as the sum of two commuting objects, M µν = M µν + M µν , where M µν = X µ p ν − X ν p µ and M µν = − θ µσ π νσ + θ νσ π µσ , as in the usual addition of angular momenta. Of course both operators have to satisfy the Lorentzalgebra. It is possible to find convenient representations that reproduce (4.5). In the sector H of H = H ⊗ H associated with ( X , p ), it can be used the usual 4 × D (Λ) = (Λ µα ), such that, for instance X ′ µ = Λ µν X ν . For the sector of H relative to ( θ, π ), it is possible to use the 6 × D (Λ) = (cid:0) Λ [ µα Λ ν ] β (cid:1) , such that, for instance, θ ′ µν = Λ [ µα Λ ν ] β θ αβ . D = D ⊕ D . In the infinitesimal case, Λ µν = δ µν + ω µν ,and (4.5) are reproduced. There are four Casimir invariant operators in this context and theyare given by C j = M j µν M j µν and C j = ǫ µνρσ M j µν M j ρσ , where j = 1 ,
2. We note that although the target space has 10 = 4+ 6 dimensions, the symmetrygroup has only 6 independent parameters and not the 45 independent parameters of the Lorentzgroup in D = 10. As we said before, this D = 10 spacetime comprises the four spacetimecoordinates and the six θ coordinates. In Section 6 the structure of this extended space willbecome clearer.Analyzing the Lorentz symmetry in NCQM following the lines above, once we introducean appropriate theory, for instance, given by a scalar action. We know, however, that theelementary particles are classified according to the eigenvalues of the Casimir operators of theinhomogeneous Lorentz group. Hence, let us extend this approach to the Poincar´e group P . Byconsidering the operators presented here, we can in principle consider G = 12 ω µν M µν − a µ p µ + 12 b µν π µν as the generator of some group P ′ , which has the Poincar´e group as a subgroup. By followingthe same rule as the one used in the obtainment of (4.5), with G replaced by G , we arrive atthe set of transformations δ X µ = ω µν X ν + a µ , δ p µ = ω νµ p ν , δθ µν = ω µρ θ ρν + ω ν ρ θ µρ + b µν ,δπ µν = ω ρµ π ρν + ω ρν π µρ , δ M µν = ω µρ M ρν + ω νρ M µρ + a µ p ν − a ν p µ ,δ M µν = ω µρ M ρν + ω ν ρ M µρ + b µρ π νρ + b νρ π µρ , δ x µ = ω µν x ν + a µ + 12 b µν p ν . (4.6)We observe that there is an unexpected term in the last one of (4.6) system. This is a consequenceof the coordinate operator in (2.10), which is a nonlinear combination of operators that act on H and H .The action of P ′ over the Hilbert space operators is in some sense equal to the action of thePoincar´e group with an additional translation operation on the ( θ µν ) sector. All its generatorsclose in an algebra under commutation, so P ′ is a well defined group of transformations. Asa matter of fact, the commutation of two transformations closes in the algebra[ δ , δ ] y = δ y , (4.7)where y represents any one of the operators appearing in (4.6). The parameters compositionrule is given by ω µ ν = ω µ α ω α ν − ω µ α ω α ν , a µ = ω µ ν a ν − ω µ ν a ν ,b µν = ω µ ρ b ρν − ω µ ρ b ρν − ω ν ρ b ρµ + ω ν ρ b ρµ . (4.8)If we consider the operators acting only on H , we verify that they transform standardlyunder the Poincar´e group P in D = 4, whose generators are p µ and M µν . As it is well known,it is formed by the semidirect product between the Lorentz group L in D = 4 and the trans-lation group T , and have two Casimir invariant operators C = p and C = s , where s µ = ǫ µνρσ M νρ p σ is the Pauli–Lubanski vector. If we include in M terms associated withspin, we will keep the usual classification of the elementary particles based on those invariants.A representation for P can be given by the 5 × D (Λ , A ) = (cid:18) Λ µν A µ (cid:19) acting in the 5-dimensional vector (cid:0) X µ (cid:1) .he Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 17Considering the operators acting on H , we find a similar structure. Let us call the cor-responding symmetry group as G . It has as generators the operators π µν and M µν . As onecan verify, C = π and C = M µν π µν are the corresponding Casimir operators. G can beseen as the semidirect product of the Lorentz group and the translation group T . A possiblerepresentation uses the antisymmetric 6 × D (Λ) already discussed, and is givenby the 7 × D (Λ , B ) = Λ [ µα Λ ν ] β B µν ! acting in the 7-dimensional vector (cid:0) θ µν (cid:1) . Now we see that the complete group P ′ is just theproduct of P and G . It has a 11 ×
11 dimensional representation given by D (Λ , A, B ) = Λ µν A µ [ µα Λ ν ] β B µν (4.9)acting in the 11-dimensional column vector X µ θ µν . A group element needs 6 + 4 + 6 parameters to be determined and P ′ is a subgroup of thefull Poincar´e group P in D = 10. Observe that an element of P needs 55 parameters tobe specified. Here, in the infinitesimal case, when A goes to a , B goes to b and Λ µν goes to δ µν + ω µν , the transformations (4.6) are obtained from the action of (4.9) defined above. It isclear that C , C , C and C are the Casimir operators of P ′ .So far we have been considering a possible algebraic structure among operators in H andpossible sets of transformations for these operators. The choice of an specific theory, however,will give the mandatory criterion for selecting among these sets of transformations, the onethat gives the dynamical symmetries of the action. If the considered theory is not invariantunder the θ translations, but it is by Lorentz transformations and x translations, the set of thesymmetry transformations on the generalized coordinates will be given by (4.6) but effectivelyconsidering b µν as vanishing, which implies that P ′ , with this condition, is dynamically con-tracted to the Poincar´e group. Observe, however, that π µν will be yet a relevant operator,since M µν depends on it in the representation here adopted. An important point related withthe dynamical action of P is that it conserves the quantum conditions (1.5).Now we will consider some points concerning some actions which furnish models for a re-lativistic NCQM in order to derive their equations of motion and to display their symmetrycontent. As discussed in the previous section, in NCQM the physical coordinates do not commute andtheir eigenvectors can not be used in order to form a basis in H = H + H . This does notoccur with the shifted coordinate operator X µ due to (2.5), (2.6) and (2.12). Consequently theireigenvectors can be used in the construction of such a basis. Generalizing what has been donein [46], it is possible to introduce a coordinate basis | X ′ , θ ′ i = | X ′ i ⊗ | θ ′ i in such a way that X µ | X ′ , θ ′ i = X ′ µ | X ′ , θ ′ i , θ µν | X ′ , θ ′ i = θ ′ µν | X ′ , θ ′ i (4.10)8 E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. Zangirolamisatisfying usual orthonormality and completeness relations. In this basis h X ′ , θ ′ | p µ | X ′′ , θ ′′ i = − i ∂∂X ′ µ δ ( X ′ − X ′′ ) δ ( θ ′ − θ ′′ ) (4.11)and h X ′ , θ ′ | π µν | X ′′ , θ ′′ i = − iδ ( X ′ − X ′′ ) ∂∂θ ′ µν δ ( θ ′ − θ ′′ ) (4.12)implying that both momenta acquire a derivative realization.A physical state | φ i , in the coordinate basis defined above, will be represented by the wavefunction φ ( X ′ , θ ′ ) = h X ′ , θ ′ | φ i satisfying some wave equation that we assume that can be de-rived from an action, through a variational principle. As it is well known, a direct route forconstructing an ordinary relativistic free quantum theory is to impose that the physical statesare annihilated by the mass shell condition( p + m ) | φ i = 0 (4.13)constructed with the Casimir operator C = p . In the coordinate representation, this givesthe Klein–Gordon equation. The same result is obtained from the quantization of the classicalrelativistic particle, whose action is invariant under reparametrization [52]. There the generatorof the reparametrization symmetry is the constraint ( p + m ) ≈
0. Condition (4.13) is theninterpreted as the one that selects the physical states, that must be invariant under gauge(reparametrization) transformations. In the NC case, besides (4.13), it is reasonable to assumeas well that the second condition( π + ∆) | φ i = 0 (4.14)must be imposed on the physical states, since it is also an invariant, and it is not affected by theevolution generated by (4.13). It can be shown that in the underlying classical theory [53], thiscondition is also associated with a first-class constraint, which generates gauge transformations,and so (4.14) can also be seen as selecting gauge invariant states. In (4.14), ∆ is some constantwith dimension of M , whose sign and value depend if π is space-like, time-like or null. Bothequations permit to construct a generalized plane wave solution φ ( X ′ , θ ′ ) ≡ h X ′ , θ ′ | φ i ∼ exp (cid:18) ik µ X ′ µ + i K µν θ ′ µν (cid:19) , where k + m = 0 and K + ∆ = 0. In coordinate representation given by equations (4.10)–(4.12), the equation (4.13) gives just the Klein–Gordon equation (cid:0) ✷ X − m (cid:1) φ ( X ′ , θ ′ ) = 0while (4.14) gives the supplementary equation( ✷ θ − ∆) φ ( X ′ , θ ′ ) = 0 , (4.15)where ✷ X = ∂ µ ∂ µ , (4.16)with ∂ µ = ∂∂X ′ µ . (4.17)he Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 19Also ✷ θ = 12 ∂ µν ∂ µν , (4.18)with ∂ µν = ∂∂θ ′ µν . (4.19)Both equations can be derived from the action S = Z d X ′ d θ ′ Ω( θ ′ ) (cid:26)
12 ( ∂ µ φ∂ µ φ + m φ ) − Λ( ✷ θ − ∆) φ (cid:27) . (4.20)In (4.20) Λ is a Lagrange multiplier necessary to impose condition (4.15). Ω( θ ′ ) can be seen asa simple constant θ − to keep the usual dimensions of the fields as S must be dimensionless innatural units, as an even weight function as the one appearing in [17, 18, 19, 20, 21, 22, 23] usedto make the connection between the formalism in D = 4 + 6 and the usual one in D = 4 after theintegration in θ ′ , or a distribution used to impose further conditions as those appearing in (1.5)and adopted in [10].A model not involving Lagrangian multipliers, but two of the Casimir operators of P ′ , C = p and C = π , is given by S = Z d X ′ d θ ′ Ω( θ ′ ) 12 (cid:26) ∂ µ φ∂ µ φ + λ ∂ µν φ∂ µν φ + m φ (cid:27) , (4.21)where λ is a parameter with dimension of length, as the Planck length, which has to be in-troduced by dimensional reasons. If it goes to zero one essentially obtains the Klein–Gordonaction. Although, the Lorentz-invariant weight function Ω( θ ) does not exist we can keep ittemporally in some integrals in order to guarantee explicitly their existence. By borrowing theDirac matrices Γ A , A = 0 , , . . . ,
9, written for spacetime D = 10, and identifying the tensorindices with the six last values of A , it is also possible to construct the “square root” of theequation of motion derived from (4.20), obtaining a generalized Dirac theory involving spin andnoncommutativity [53]. It is an object of current investigation.Next we will construct the equations of motion and analyze the Noether’s theorem derivedfor general theories defined in x + θ space, and specifically for the action (4.20), consideringΩ( θ ) as a well behaved function. Let us consider the action S = Z R d xd θ Ω( θ ) L ( φ i , ∂ µ φ i , ∂ µν φ i , x, θ ) , (4.22)relying on a set of fields φ i , the derivatives with respect to x µ and θ µν and the coordinates x µ and θ µν themselves. From now on we will use x in place of X ′ and θ in place of θ ′ in orderto simplify the notation. Naturally the fields φ i can be functions of x µ and θ µν . The index i permits to treat φ in a general way. In (4.22) we consider, as in (4.20), the integration elementmodified by the introduction of Ω( θ ).By assuming that S is stationary for an arbitrary variation δφ i vanishing on the boundary ∂R of the region of integration R , we can write the Euler–Lagrange equation asΩ (cid:18) ∂ L ∂φ i − ∂ µ ∂ L ∂∂ µ φ i (cid:19) − ∂ µν (cid:18) Ω ∂ L ∂∂ µν φ i (cid:19) = 0 . (4.23)0 E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. ZangirolamiWe will treat the variations δx µ , δθ µν of the generalized coordinates and δφ i of the fields such thatthe integrand transforms as a total divergence in the x + θ space, δ (Ω L ) = ∂ µ (Ω S µ )+ ∂ µν (Ω S µν ).Then the Noether’s theorem assures that, on shell, or when (4.23) is satisfied, there is a conservedcurrent ( j µ , j µν ) defined by j µ = ∂ L ∂∂ µ φ i δφ i + L δx µ , j µν = ∂ L ∂∂ µν φ i δφ i + L δθ µν , (4.24)such thatΞ = ∂ µ (Ω j µ ) + ∂ µν (Ω j µν ) (4.25)vanishes. The corresponding charge Q = Z d xd θ Ω( θ ) j (4.26)is independent of the “time” x . On the contrary, if there exists a conserved current like (4.24),the action (4.22) is invariant under the corresponding symmetry transformations. This is justa trivial extension of the usual version of Noether’s theorem [45] in order to include θ µν asindependent coordinates, as well as a modified integration element due to the presence of Ω( θ ).Notice that Ω has not been included in current definition (4.24) because it is seen as part ofthe element of integration, but it is present in (4.25), which is the relevant divergence. It is alsoinside the charge (4.26) since the charge is an integrated quantity.Let us use the equations from (4.22) until (4.25) to the simple model given by (4.21). TheLagrange equation reads δSδφ = − Ω (cid:0) ✷ − m (cid:1) φ − λ ∂ µν (Ω ∂ µν φ ) = 0 (4.27)and (4.25) can be written asΞ = ∂ µ (cid:26) Ω ∂ µ φδφ + Ω2 (cid:18) ∂ α φ∂ α φ + λ ∂ αβ φ∂ αβ φ + m φ (cid:19) δx µ (cid:27) + ∂ µν (cid:26) Ω λ ∂ µν φδφ + Ω2 (cid:18) ∂ α φ∂ α φ + λ ∂ αβ φ∂ αβ φ + m φ (cid:19) δθ µν (cid:27) . (4.28)Before using (4.28) we observe that the transformation δφ = − ( a µ + ω µν x ν ) ∂ µ φ −
12 ( b µν + 2 ω µρ θ ρν ) ∂ µν φ (4.29)closes in an algebra, as in (4.7), with the same composition rule defined in (4.8). The aboveequation defines how a scalar field transforms in the x + θ space under the action of P ′ .Let us now study a rigid x -translation, given by δ a x µ = a µ , δ a θ µν = 0 , δ a φ = − a µ ∂ µ φ, (4.30)where a µ are constants. We see from (4.28) and (4.30) thatΞ a = a µ ∂ µ φ δSδφ vanishing on shell, when (4.27) is valid.he Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 21For a rigid θ -translation, we have that, δ b x µ = 0 , δ b θ µν = b µν , δ b φ = − b µν ∂ µν φ, where b µν are constants, we can writeΞ b = 12 b µν (cid:18) ∂ µν φ δSδφ + L ∂ µν Ω (cid:19) . The first term on the right vanishes on shell but the second one depends on the form of Ω.Later we will comment this point. To end, let us consider a Lorentz transformation, given by δ ω x µ = ω µν x ν , δ ω θ µν = ω µρ θ ρν + ω νρ θ µρ ,δ ω φ = − (cid:0) ω µν x ν ∂ µ + ω µρ θ ρν ∂ µν (cid:1) φ with constant and antisymmetric ω µν . We obtainΞ ω = ω µν δSδφ ( x ν ∂ µ φ + θ νρ ∂ µρ φ ) + L ∂ µν Ω ω µα θ αν . The first term in the above expression vanishes on shell and the second one also vanishes if Ω isa scalar under Lorentz transformations and depends only on θ .In a complete theory where other contributions for the total action would be present, thesymmetry under θ translations could be broken by different reasons, as in what follows, in thecase of the NC U (1) gauge theory. In this situation P could be the symmetry group of thecomplete theory even considering Ω( θ ) as a constant. The twisted Poincar´e (TP) algebra describes the symmetry of NC spacetime whose coordinatesobey the commutation relation of a canonical type like (1.1). It was suggested [54] as a substitutefor the Poincar´e symmetry in field theories on the NC spacetime. In this formalism point ofview, the Moyal product (1.2) is obtained as a twisted product of a module algebra of the TPalgebra. This result shows the TP invariance of the NC field theories.This Poincar´e algebra P for commutative QFT is given by the so-called Lie algebra composedby the ten generators of the Poincar´e group [54][ P µ , P ν ] = 0 , [ M µν , P ρ ] = − i ( g µρ P ν − g νσ P µ ) , [ M µν , M ρσ ] = − i ( g µρ M νσ − g µσ M νρ − g νρ M µσ − g νσ M µρ ) , (4.31)where the matrix M is antisymmetric M µν = − M νµ . The generators M µν form a closedsubalgebra, which is the Lie algebra of the Lorentz group. The generators of the Lorentz groupcan be divided into the group for boosts K i = M i , the spatial rotations group J i = ǫ ijk M jk and the generators of translations P µ , the last ones form a commutative subalgebra of thePoincar´e algebra (the translation subgroup is Abelian). The presence of the imaginary unit i indicates that the generators of the Poincar´e group are Hermitian.The Poincar´e algebra generators are represented by P µ = Z d d x T µ ( x ) , M µν = Z d d x [ x µ T ν ( x ) − x ν T µ ( x )] , where T µ ( x ) = [ π ( x ) ∂ µ φ ( x ) + ∂ µ φ ( x ) π ( x )] − g µ L ( x ) and π = ∂ φ is the canonical momentumof φ . It can be demonstrated that with this structure we can construct the representation ofthe Poincar´e algebra in terms of the Hopf algebraic structure [55, 24].2 E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. Zangirolami During some time, the problem of the Lorentz symmetry breaking was ignored and the investiga-tions in NCQFT were performed by dealing with the full representation content of the Poincar´ealgebra.Chaichian et al. in [27] analyzed a solution to the problem utilizing the form of a twistedPoincar´e symmetry. The introduction of a twist deformation of the universal enveloping algebraof the Poincar´e algebra provided a new symmetry. The representation content of this twistedalgebra is the same as the representation content of the usual Poincar´e algebra [54].To obtain the TP algebra we use the standard Poincar´e algebra and its representation spaceand twist them. For example, in this twisted algebra the energy-momentum tensor is T µ isgiven by T µ = ∞ X n =0 (cid:18) − (cid:19) n n ! θ i j · · · θi n j n ∂ i · · · ∂ i n × (cid:20)
12 ( π ⋆ ∂ µ φ ( x ) + ∂ µ φ ( x ) ⋆ π ( x )) − g µ L (cid:21) P j · · · P j n , where P j n are the generators of translation in NCQFT.It is important to notice that π , φ , L and P are elements embedded in a NC space. Theresulting operators satisfy commutation relations of Poincar´e algebra (4.31). However, it canbe shown [55, 24] that some identities involving momenta and the Hamiltonian in commutativeand NCQFT are preserved in a deformed NCQFT. Also, we can use the same Hilbert space torepresent the field operator for both commutative and deformed NCQFT [55, 24]. The actionof generators of a Lorentz on a NC field transformation is different from the commutative one.And this action has an exponential form in order to give a finite Lorentz transformation. Thisis, in a nutshell, the structure of the Poincar´e symmetry or of the Hopf algebra represented ona deformed NCQFT. The next step is to twist this algebra and its representation space. ThePoincar´e algebra P ( A ) has a subalgebra (commutative) composed by the translation genera-tors P µ F = exp (cid:18) i θ µν P µ P ν (cid:19) , where θ µν is a real constant antisymmetric matrix. This twist operator obviously satisfies thetwist conditions that preserve the Hopf algebra structure [54].Explicitly let us write the twist operator such as F = exp (cid:20) − i θ µν ∂∂x µ ⊗ ∂∂x ν (cid:21) , F − = exp (cid:20) i θ µν ∂∂x µ ⊗ ∂∂x ν (cid:21) , (4.32)where here ∂∂x µ and ∂∂x ν are generally defined vector fields on space or spacetime. The twistingresult is the twisted Poincar´e algebra. It can be shown [55, 24] that the Poincar´e covarianceof a commutative QFT implies the twisted Poincar´e covariance of a deformed NCQFT. In thisdeformed NCQFT the symmetry is described by a quantum group.An unavoidable comparison with the Moyal ⋆ -product can be made and the conclusion is thatthey are in fact the same NC product of functions. Hence, the NCQFT constructed with Weylquantization and Moyal ⋆ -product possess the twisted Poincar´e symmetry. We can say that, inNC theories, relativistic invariance means invariance under the twisted Poincar´e transformations.he Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 23 As we said through the sections above, the DFRA structure constructs an extension of thePoincar´e group P ′ , which has the Poincar´e group P as a subgroup. Then, obviously, the theoriesconsidered have to be invariant under both P and P ′ .As the considerations described in this section the Poincar´e algebra P has an Abelian subal-gebra which allows us to construct a twist operator F depicted in (4.32) which is an element ofthe quantum group theory.We believe that the subalgebra of the extended Poincar´e algebra P ′ also permits the elabo-ration of a kind of extended twist operator F ′ .This extended twist operator in the DFRA framework has to be able to reproduce the newdeformed generators. Having the structure displayed in (4.32) in mind we have to formulatethis new twist operator adding the ( θ, π ) sector terms in order to have the form of (4.32) withthe twist operator as a special case. The formulation of this DFRA twist operator is beyond thescope of this review paper and is a target for further investigations. By using the DFRA framework where the object of noncommutativity θ µν represents indepen-dent degrees of freedom, we will explain here the symmetry properties of an extended x + θ spacetime, given by the group P ′ , which has the Poincar´e group P as a subgroup. In this sec-tion we use the DFRA algebra to introduce a generalized Dirac equation, where the fermionicfield depends not only on the ordinary coordinates but on θ µν as well. The dynamical symmetrycontent of such fermionic theory will be discussed now and it is shown that its action is invariantunder P ′ .In the last sections above, we saw that the DFRA algebra implemented, in a NCQM frame-work , the Poincar´e invariance as a dynamical symmetry [45]. Of course this represents oneamong several possibilities of incorporating noncommutativity in quantum theories, we sawalso that not only the coordinates x µ and their conjugate momenta p µ are operators acting ina Hilbert space H , but also θ µν and their canonical momenta π µν are considered as Hilbert spaceoperators as well. The proposed DFRA algebra is given by (1.1), (2.2)–(2.6) and (2.8). Whereall these relations above are consistent with all possible Jacobi identities by construction.As said before, an important point is that, due to (1.1) the operator x µ can not be used tolabel possible basis in H . However, as the components of X µ commute, we know from QM thattheir eigenvalues can be used for such purpose. To simplify the notation, let us denote by x and θ the eigenvalues of X and θ in what follows. In [47] R. Amorim considered these points withsome detail and have proposed a way for constructing some actions representing possible fieldtheories in this extended x + θ spacetime. One of such actions has been given by S = − Z d xd θ Ω( θ ) 12 (cid:26) ∂ µ φ∂ µ φ + λ ∂ µν φ∂ µν φ + m φ (cid:27) , (5.1)where λ is a parameter with dimension of length, as the Planck length, which has to be introducedby dimensional reasons and Ω( θ ) is a scalar weight function used in [17, 18, 19, 20, 21, 22, 23]in order to make the connection between the D = 4 + 6 and the D = 4 formalisms, where weused the definitions inequations (4.16)–(4.19) and η µν = diag( − , , , δSδφ = Ω (cid:0) ✷ − m (cid:1) φ + λ ∂ µν (Ω ∂ µν φ ) = 0 (5.2) In [46] and [47] it is possible to find a large amount of references concerning NC quantum mechanics. θ , the above transformation is only a symmetry of (5.1)(when b µν vanishes) which dynamically transforms P ′ to P [47]. We observe that (4.29) closes inan algebra, as in (4.7), with the same composition rule defined in (4.8). That equation defineshow a scalar field transforms in the x + θ space under the action of P ′ .In what follows we are going to show how to introduce fermions in this x + θ extended space.To reach this goal, let us first observe that P ’ is a subgroup of the Poincar´e group P in D = 10. Denoting the indices A, B, . . . as spacetime indices in D = 10, A, B, . . . = 0 , , . . . , Y A would transform under P as δY A = ω AB Y B + ∆ A , where the 45 ω ’s and 10 ∆’s are infinitesimal parameters. If one identifies the last six A, B, . . . indices with the macro-indices µν , µ, ν, . . . = 0 , , ,
3, considered as antisymmetric quantities,the transformation relations given above are rewritten as δY µ = ω µν Y ν + 12 ω µαβ Y αβ + ∆ µ , δY µν = ω µνα Y α + 12 ω µναβ Y αβ + ∆ µν . With this notation, the (diagonal) D = 10 Minkowski metric is rewritten as η AB = ( η µν , η αβ,γδ )and the ordinary Clifford algebra { Γ A , Γ B } = − η AB as { Γ µ , Γ αβ } = 0 , { Γ µ , Γ ν } = − η µν , { Γ µν , Γ αβ } = − η µν,αβ . (5.3)This is just a heavy way of writing usual D = 10 relations [3]. Now, by identifying Y A with( x µ , λ θ αβ ), where λ is some parameter with length dimension, we see from the structure givenabove that the allowed transformations in P ’ are those of P , submitted to the conditions ω µνα = ω αµν = 0 , ω µναβ = 4 ω [ µα δ ν ] β , ∆ µ = a µ , ∆ αβ = 1 λ b αβ obviously keeping the identification between ω AB and ω µν when A = µ and B = ν . Of coursewe have now only 6 independent ω ’s and 10 a ’s and b ’s. With the relations given above it ispossible to extract the “square root” of the generalized Klein–Gordon equation (5.2) (cid:0) ✷ + λ ✷ θ − m (cid:1) φ = 0 (5.4)assuming here that Ω is a constant. We will see in the next section with details that this equationcan be interpreted as a dispersion relation in this D = 4 + 6 spacetime. Hence, this last equationfurnish just the generalized Dirac equation (cid:20) i (cid:18) Γ µ ∂ µ + λ αβ ∂ αβ (cid:19) − m (cid:21) ψ = 0 . (5.5)Let us apply from the left on (5.5) the operator (cid:20) i (cid:18) Γ ν ∂ µ + λ αβ ∂ αβ (cid:19) + m (cid:21) . After using (5.3) we observe that ψ satisfies the generalized Klein–Gordon equation (5.4) aswell. The covariance of the generalized Dirac equation (5.5) can also be proved. First we notethat the operator M µν = i (cid:0) [Γ µ , Γ ν ] + [Γ µα , Γ να ] (cid:1) he Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 25gives the desired representation for the SO (1 ,
3) generators, because it not only closes in theLorentz algebra (4.4), but also satisfies the commutation relations[Γ µ , M αβ ] = 2 iδ µ [ α Γ β ] , [Γ µν , M αβ ] = 2 iδ µ [ α Γ νβ ] − iδ ν [ α Γ µβ ] . With these relations it is possible to prove that (5.5) is indeed covariant under the Lorentztransformations given by ψ ( x ′ , θ ′ ) = exp (cid:18) − i µν M µν (cid:19) ψ ( x, θ ) . By considering the complete P ′ group, we observe that the infinitesimal transformations of ψ are given by δψ = − (cid:20) ( a µ + ω µν x ν ) ∂ µ + 12 ( b µν + 2 ω µρ θ νρ ) ∂ µν + i ω µν M µν (cid:21) ψ, (5.6)which closes in the P ′ algebra with the same composition rule given by (4.8), what can be shownafter a little algebra. At last we can show that also here there are conserved Noether’s currentsassociated with the transformation (5.6), once we observe that the equation (5.5) can be derivedfrom the action S = Z d xd θ Ω( θ ) ¯ ψ (cid:20) i (cid:18) Γ µ ∂ µ + λ αβ ∂ αβ (cid:19) − m (cid:21) ψ, (5.7)where we are considering that Ω = θ − and ¯ ψ = ψ † Γ . First we note that (suppressing trivial θ − trivial factors) δ L Sδ ¯ ψ = (cid:20) i (cid:18) Γ µ ∂ µ + λ αβ ∂ αβ (cid:19) − m (cid:21) ψ, δ R Sδψ = − ¯ ψ (cid:20) i (cid:18) Γ µ ←− ∂ µ + λ αβ ←− ∂ αβ (cid:19) + m (cid:21) , where L and R derivatives act from the left and right respectively. The current ( j µ , j µν ),analogously as in (4.24), is here written as j µ = ∂ R L ∂∂ µ ψ δψ + δ ¯ ψ ∂ L L ∂∂ µ ¯ ψ + L δx µ , j µν = ∂ R L ∂∂ µν ψ δψ + δ ¯ ψ ∂ L L ∂∂ µν ¯ ψ + L δθ µν , (5.8)where δ ¯ ψ = − ¯ ψ (cid:20) ←− ∂ µ ( a µ + ω µν x ν ) + ←− ∂ µν
12 ( b µν + 2 ω µρ θ νρ ) − i ω µν M µν (cid:21) , (5.9) δψ is given in (5.6) and δx µ and δθ µν have the same form found in (4.6). Using these last resultsone can show that, ∂ µ j µ + ∂ µν j µν = − (cid:18) δ ¯ ψ δ L Sδ ¯ ψ + δ R Sδψ δψ (cid:19) , (5.10)which vanishes on shell, and hence the invariance of the action (5.7) under P ′ . And we canconclude that it could be dynamically contracted to P , preserving the usual Casimir invariantstructure characteristic of ordinary quantum field theories.Using (5.10) we can realize that there is a conserved charge Q = Z d xd θ j , t derivativewe have that˙ Q = − Z d xd θ ( ∂ i j i + ∂ µν j µν ) , vanishes as a consequence of the divergence theorem. By considering only x µ translations, wecan write j = j µ a µ , and consequently to define the momentum operator P µ = − Z d xd θj µ . Analyzing θ µν translations and Lorentz transformations, we can derive in a similar way anexplicit form for the other generators of P ′ , here denoted by Π µν and J µν . Under an appropriatebracket structure, following the Noether’s theorem, these conserved charges will generate thetransformations (5.6) and (5.9).Summarizing, we can say that we have been able to introduce fermions satisfying a generalizedDirac equation, which is covariant under the action of the extended Poincar´e group P ′ . ThisDirac equation has been derived through a variational principle whose action is dynamicallyinvariant under P ′ . This can justify possible roles played by theories involving noncommutativityin a way compatible with Relativity. Of course this is just a little step toward a field theoryquantization program in this extended x + θ spacetime, which is our next issue. So far we saw that in a first quantized formalism, θ µν and its canonical momentum π µν areseen as operators living in some Hilbert space. This structure is compatible with DFRA algebraand it is invariant under an extended Poincar´e group of symmetry. In a second quantizationscenario, we will reproduce in this section the results obtained in [50]. An explicit form for theextended Poincar´e generators will be presented and we will see the same algebra is generatedvia generalized Heisenberg relations. We also introduce a source term and construct the generalsolution for the complex scalar fields using the Green’s function technique.As we said before in a different way, at the beginning the original motivation of DFR tostudy the relations (1.1), (2.1) and (2.5) was the belief that an attempt of obtaining exact mea-surements involving spacetime localization could confine photons due to gravitational fields.This phenomenon is directly related to (1.1), (2.1) and (2.5) together with (1.5). In a somehowdifferent scenario, other relevant results are obtained in [17, 20, 21, 22, 19, 23] relying on condi-tions (1.1), (2.1) and (2.5). We saw that the value of θ is used as a mean value with some weightfunction, generating Lorentz invariant theories and providing a connection with usual theoriesconstructed in an ordinary D = 4 spacetime.To clarify a little bit what we saw in the last sections, we can say that, based on [46, 47, 48, 49]a new version of NCQM has been presented, where not only the coordinates x µ and theircanonical momenta p µ are considered as operators in a Hilbert space H , but also the objects ofnoncommutativity θ µν and their canonical conjugate momenta π µν . All these operators belongto the same algebra and have the same hierarchical level, introducing a minimal canonicalextension of the DFR algebra, i.e., the DFRA algebra introduced by R. Amorim in a firstpaper [46] followed by others [47, 48, 49, 50]. This enlargement of the usual set of Hilbert spaceoperators allows the theory to be invariant under the rotation group SO ( D ), as showed in detailin the sections above [46, 49], when the treatment was a nonrelativistic one. Rotation invariancein a nonrelativistic theory, is fundamental if one intends to describe any physical system ina consistent way. In the last sections, the corresponding relativistic treatment was presented,which permits to implement Poincar´e invariance as a dynamical symmetry [45] in NCQM.Inhe Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 27this section we essentially consider the “second quantization” of the model discussed above [47],showing that the extended Poincar´e symmetry here is generated via generalized Heisenbergrelations, giving the same algebra displayed in [47, 48].Now we will study the new NC charged Klein–Gordon theory described above in this D = 10, x + θ space and analyze its symmetry structure, associated with the invariance of the actionunder some extended Poincar´e ( P ′ ) group. This symmetry structure is also displayed insidethe second quantization level, constructed via generalized Heisenberg relations. After that, thefields are shown as expansions in a plane wave basis in order to solve the equations of motionusing the Green’s functions formalism adapted for this new ( x + θ ) D = 4 + 6 space. It isassumed in this section that Ω( θ ) defined in (4.20) is constant. An important point is that, due to (1.1), the operator x µ can not be used to label a possiblebasis in H . However, as the components of X µ commute, as can be verified from the DRFAalgebra and the relations following this one, their eigenvalues can be used for such purpose.From now on let us denote by x and θ the eigenvalues of X and θ .In Section 4 we saw that the relations (1.1), (2.2)–(2.7) and (2.9) allowed us to utilize [25] M µν = X µ p ν − X ν p µ − θ µσ π νσ + θ νσ π µσ as the generator of the Lorentz group, where X µ = x µ + 12 θ µν p ν , and we see that the proper algebra is closed, i.e.,[ M µν , M ρσ ] = iη µσ M ρν − iη νσ M ρµ − iη µρ M σν + iη νρ M σµ . Now M µν generates the expected symmetry transformations when acting on all the operatorsin Hilbert space. Namely, by defining the dynamical transformation of an arbitrary operator A in H in such a way that δ A = i [ A , G ], where G = 12 ω µν M µν − a µ p µ + 12 b µν π µν , and ω µν = − ω νµ , a µ , b µν = − b νµ are infinitesimal parameters, it follows that δ x µ = ω µν x ν + a µ + 12 b µν p ν , (6.1a) δ X µ = ω µν X ν + a µ , (6.1b) δ p µ = ω νµ p ν , (6.1c) δθ µν = ω µρ θ ρν + ω νρ θ µρ + b µν , (6.1d) δπ µν = ω ρµ π ρν + ω ρν π µρ , (6.1e) δ M µν = ω µρ M ρν + ω ν ρ M µρ + a µ p ν − a ν p µ + b µρ π νρ + b νρ π µρ , (6.1f)generalizing the action of the Poincar´e group P in order to include θ and π transformations,i.e., P ′ . The P ′ transformations close in an algebra, such that[ δ , δ ] A = δ A , ω µ ν = ω µ α ω α ν − ω µ α ω α ν , a µ = ω µ ν a ν − ω µ ν a ν ,b µν = ω µ ρ b ρν − ω µ ρ b ρν − ω ν ρ b ρµ + ω ν ρ b ρµ . The symmetry structure displayed in equation (6.1) was discussed before.In the last sections we tried to clarify these points with some detail and we have showed a wayfor constructing actions representing possible field theories in this extended x + θ spacetime.One of such actions, generalized in order to permit the scalar fields to be complex, is given by S = − Z d x d θ (cid:26) ∂ µ φ ∗ ∂ µ φ + λ ∂ µν φ ∗ ∂ µν φ + m φ ∗ φ (cid:27) , (6.2)where λ is a parameter with dimension of length, as the Planck length, which is introduced dueto dimensional reasons. Here we are also suppressing a possible factor Ω( θ ) in the measure,which is a scalar weight function, used in [17, 20, 21, 22, 19, 23], in a NC gauge theory context,to make the connection between the D = 4 + 6 and the D = 4 formalisms.The corresponding Euler–Lagrange equation reads δSδφ = (cid:0) ✷ + λ ✷ θ − m (cid:1) φ ∗ = 0 , (6.3)with a similar equation of motion for φ . The action (6.2) is invariant under the transformation δφ = − ( a µ + ω µν x ν ) ∂ µ φ −
12 ( b µν + 2 ω µρ θ ρν ) ∂ µν φ, (6.4)besides the phase transformation δφ = − iαφ, (6.5)with similar expressions for φ ∗ , obtained from (6.4) and (6.5) by complex conjugation. Weobserve that (6.1c) closes in an algebra, as in (4.7), with the same composition rule definedin (4.8). That equation defines how a complex scalar field transforms in the x + θ spaceunder P ′ . The transformation subalgebra generated by (6.1d) is of course Abelian, although itcould be directly generalized to a more general setting.Associated with those symmetry transformations, we can define the conserved currents [47] j µ = ∂ L ∂∂ µ φ δφ + δφ ∗ ∂ L ∂∂ µ φ ∗ + L δx µ , j µν = ∂ L ∂∂ µν φ δφ + δφ ∗ ∂ L ∂∂ µν φ ∗ + L δθ µν . Actually, by using (6.1c) and (6.1d), as well as (6.1b) and (6.1d), we can show, after somealgebra, that ∂ µ j µ + ∂ µν j µν = − δSδφ δφ − δφ ∗ δSδφ ∗ . The expressions above, as seen before, allow us to derive a specific charge Q = − Z d xd θ j , for each kind of conserved symmetry encoded in (6.1c) and (6.1d), since˙ Q = Z d xd θ (cid:18) ∂ i j i + 12 ∂ µν j µν (cid:19) he Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 29vanishes as a consequence of the divergence theorem in this ( x, θ ) extended space. Let us considereach specific symmetry in (6.1c) and (6.1d). For usual x -translations, we can write j = j µ a µ ,and so we can define the total momentum P µ = − Z d xd θ j µ = Z d xd θ (cid:0) ˙ φ ∗ ∂ µ φ + ˙ φ∂ µ φ ∗ − L δ µ (cid:1) . (6.6)For θ -translations, we can write that j = j µν b µν , and consequently, giving P µν = − Z d xd θ j µν = 12 Z d xd θ (cid:0) ˙ φ ∗ ∂ µν φ + ˙ φ∂ µν φ ∗ (cid:1) . (6.7)In a similar way we define the Lorentz charge. By using the operator∆ µν = x [ µ ∂ ν ] + θ α [ µ ∂ ν ] α , (6.8)and defining j = ¯ j µν ω µν , we can write M µν = − Z d xd θ ¯ j µν = Z d xd θ (cid:0) ˙ φ ∗ ∆ νµ φ + ˙ φ ∆ νµ φ ∗ − L δ µ x ν ] (cid:1) . (6.9)At last, for the symmetry given by (6.1d), we can write the U (1) charge as Q = i Z d xd θ ( ˙ φ ∗ φ − ˙ φφ ∗ ) . (6.10)Now let us show that these charges generate the appropriate field transformations (anddynamics) in a quantum scenario, as generalized Heisenberg relations. To start the quantizationof such theory, we can define as usual the field momenta π = ∂ L ∂ ˙ φ = ˙ φ ∗ , π ∗ = ∂ L ∂ ˙ φ ∗ = ˙ φ, (6.11)satisfying the non vanishing equal time commutators (in what follows the commutators are tobe understood as equal time commutators)[ π ( x, θ ) , φ ( x ′ , θ ′ )] = − iδ ( x − x ′ ) δ ( θ − θ ′ ) , [ π ∗ ( x, θ ) , φ ∗ ( x ′ , θ ′ )] = − iδ ( x − x ′ ) δ ( θ − θ ′ ) . (6.12)The strategy now is just to generalize the usual field theory and rewrite the charges (6.6)–(6.10) by eliminating the time derivatives of the fields in favor of the field momenta. Afterthat we use (6.12) to dynamically generate the symmetry operations. In this spirit, accordinglyto (6.6) and (6.11), the spatial translation is generated by P i = Z d xd θ (cid:0) π∂ i φ + π ∗ ∂ i φ ∗ (cid:1) , and it is trivial to verify, by using (6.12), that[ P i , Y ( x, θ )] = − i∂ i Y ( x, θ ) , where Y represents φ , φ ∗ , π or π ∗ . The dynamics is generated by P = Z d xd θ (cid:18) π ∗ π + ∂ i φ ∗ ∂ i φ + λ ∂ µν φ ∗ ∂ µν φ + m φ ∗ φ (cid:19) accordingly to (6.6) and (6.11).0 E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. ZangirolamiWith the Lagrangian given in (6.2) we can have that π µν = ∂ L ∂ ( ∂ µν φ ) = λ ∂ µν φ ∗ , π ∗ µν = ∂ L ∂ ( ∂ µν φ ∗ ) = λ ∂ µν φ these are the canonical conjugate momenta in θ -space, the θ -momenta. Together with π and π ∗ we have the complete momenta space.At this stage it is convenient to assume that classically ∂ µν φ ∗ ∂ µν φ ≥ H = P is positive definite. This condition can also be writtenas π µν π ∗ µν ≥ , since we always have an even exponential for λ . By using the fundamental commutators (6.12),the equations of motion (6.3) and the definitions (6.11), it is possible to prove the Heisenbergrelation[ P , Y ( x, θ )] = − i∂ Y ( x, θ ) . The θ -translations, accordingly to (6.7) and (6.11), are generated by P µν = 12 Z d xd θ (cid:0) π∂ µν φ + π ∗ ∂ µν φ ∗ (cid:1) , and one obtains trivially by (6.12) that[ P µν , Y ( x, θ )] = − i∂ µν Y ( x, θ ) . Lorentz transformations are generated by (6.9) in a similar way. The spatial rotations gen-erator is given by M ij = Z d xd θ (cid:0) π ∆ ji φ + π ∗ ∆ ji φ ∗ (cid:1) , (6.13)while the boosts are generated by M i = 12 Z d xd θ n π ∗ πx i − x (cid:0) π∂ i φ + π ∗ ∂ i φ ∗ (cid:1) + π (cid:0) θ γ [ i ∂ γ − x ∂ i (cid:1) φ + π ∗ (cid:0) θ γ [ i ∂ γ − x ∂ i (cid:1) φ ∗ + (cid:0) ∂ j φ ∗ ∂ j φ + λ ∂ µν φ ∗ ∂ µν φ + m φ ∗ φ (cid:1) x i o . (6.14)As can be verified in a direct way for (6.13) and in a little more indirect way for (6.14)[ M µν , Y ( x, θ )] = i ∆ µν Y ( x, θ ) , for any dynamical quantity Y , where ∆ µν has been defined in (6.8). At last we can rewrite (6.10)as Q = i Z d xd θ (cid:0) πφ − π ∗ φ ∗ (cid:1) , generating (6.1d) and its conjugate, and similar expressions for π and π ∗ . So, the P ’ and (global)gauge transformations can be generated by the action of the operator G = 12 ω µν M µν − a µ P µ + 12 b µν P µν − α Q he Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 31over the complex fields and their momenta, by using the canonical commutation relations (6.12).In this way the P ′ and gauge transformations are generated as generalized Heisenberg relations.This is a new result that shows the consistence of the DFRA formalism. Furthermore, there arealso four Casimir operators defined with the operators given above, with the same form as thosepreviously defined at a first quantized perspective. So, the structure displayed above is verysimilar to the usual one found in ordinary quantum complex scalar fields. We can go one stepfurther, by expanding the fields and momenta in modes, giving as well some other prescription,to define the relevant Fock space, spectrum, Green’s functions and all the basic structure relatedto free bosonic fields. In what follows we consider some of these issues and postpone others forforthcoming works. In order to evaluate a little more the framework described in the last sections, let us rewrite thegeneralized charged Klein–Gordon action (6.2) with source terms as S = − Z d xd θ (cid:26) ∂ µ φ ∗ ∂ µ φ + λ ∂ µν φ ∗ ∂ µν φ + m φ ∗ φ + J ∗ φ + J φ ∗ (cid:27) . (6.15)The corresponding equations of motion are (cid:0) ✷ + λ ✷ θ − m (cid:1) φ ( x, θ ) = J ( x, θ ) (6.16)as well as its complex conjugate one. We have the following formal solution φ ( x, θ ) = φ J =0 ( x, θ ) + φ J ( x, θ ) , where, clearly, φ J =0 ( x, θ ) is the source free solution and φ J ( x, θ ) is the solution with J = 0.The Green’s function for (6.16) satisfies (cid:0) ✷ + λ ✷ θ − m (cid:1) G ( x − x ′ , θ − θ ′ ) = δ ( x − x ′ ) δ ( θ − θ ′ ) , (6.17)where δ ( x − x ′ ) and δ ( θ − θ ′ ) are the Dirac’s delta functions δ ( x − x ′ ) = 1(2 π ) Z d K (1) e iK (1) · ( x − x ′ ) , (6.18) δ ( θ − θ ′ ) = 1(2 π ) Z d K (2) e iK (2) · ( θ − θ ′ ) . (6.19)Now let us define in D = 10 X = (cid:18) x µ , λ θ µν (cid:19) (6.20)and K = (cid:0) K µ (1) , λK µν (2) (cid:1) , (6.21)where λ is a parameter that carries the dimension of length, as said before. From (6.20)and (6.21) we write that K · X = K (1) µ x µ + 12 K (2) µν θ µν . is introduced in order to eliminate repeated terms. In what follows it will also beconsidered that d K = d K (1) d K (2) and d X = d xd θ. So, from (6.16) and (6.17) we formally have that φ J ( X ) = Z d X ′ G ( X − X ′ ) J ( X ′ ) . (6.22)To derive an explicit form for the Green’s function, let us expand G ( X − X ′ ) in terms ofplane waves. Hence, we can write that G ( X − X ′ ) = 1(2 π ) Z d K ˜ G ( K ) e iK · ( X − X ′ ) . (6.23)Now, from (6.17), (6.18), (6.19) and (6.23) we obtain that, (cid:0) ✷ + λ ✷ θ − m (cid:1) Z d K (2 π ) ˜ G ( K ) e iK · ( x − x ′ ) = Z d K (2 π ) e iK · ( x − x ′ ) giving the solution for ˜ G ( K ) as˜ G ( K ) = − K + m (6.24)where, from (6.21), K = K (1) µ K µ (1) + λ K (2) µν K µν (2) . Substituting (6.24) in (6.23) we obtain G ( x − x ′ , θ − θ ′ ) = 1(2 π ) Z d K Z dK K ) − ω e iK · ( x − x ′ ) , (6.25)where the “frequency” in the ( x + θ ) space is defined to be ω = ω ( ~K (1) , K (2) ) = r ~K (1) · ~K (1) + λ K (2) µν K µν (2) + m , (6.26)which can be understood as the dispersion relation in this D = 4 + 6 space. We can see also,from (6.25), that there are two poles K = ± ω in this framework. Of course we can constructan analogous solution for φ ∗ J ( x, θ ).In general, the poles of the Green’s function can be interpreted as masses for the stableparticles described by the theory. We can see directly from equation (6.26) that the plane wavesin the ( x + θ ) space establish the interaction between the currents in this space and have energygiven by ω ( ~K (1) , K (2) ) since ω = ~K + λ K + m = K , + m , he Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 33where K , = ~K + λ K . So, one can say that the plane waves that mediate the interaction describe the propagation ofparticles in a x + θ spacetime with a mass equal to m . We ask if we can use the Cauchy residuetheorem in this new space to investigate the contributions of the poles in (6.25). Accordinglyto the point described above, we can assume that the Hamiltonian is positive definite andit is directly related to the hypothesis that K , = − m <
0. However if the observablesare constrained to a four dimensional spacetime, due to some kind of compactification, thephysical mass can have contributions from the NC sector. This point is left for a forthcomingwork [53], when we will consider the Fock space structure of the theory and possible schemesfor compactification.For completeness, let us note that substituting (6.22) and (6.25) into (6.15), we arrive at theeffective action S eff = − Z d xd θd x ′ d θ ′ J ∗ ( X ) Z d K (2 π ) Z dK K ) − ω + iε e iK · ( X − X ′ ) J ( X ′ ) , which could be obtained, in a functional formalism, after integrating out the fields.We can conclude this last section saying that the DFRA formulation reviewed here takesinto account noncommutativity without destroying the symmetry content of the correspondingcommutative theories. We expect that the new features associated with the objects of noncom-mutativity will be relevant at high energy scales. Even if excited states in the Hilbert spacesector associated with noncommutativity can not assessed, ground state effects could in principlebe detectable.To end this revision work we have considered in this section, complex scalar fields usinga new framework where the object of noncommutativity θ µν represents independent degrees offreedom.We have started from a first quantized formalism, where θ µν and its canonical mo-mentum π µν are considered as operators living in some Hilbert space. This structure, whichis compatible with the minimal canonical extension of the Doplicher–Fredenhagen–Roberts–Amorim (DFRA) algebra, is also invariant under an extended Poincar´e group of symmetry, butkeeping, among others, the usual Casimir invariant operators. After that, in a second quantizedformalism perspective, we explained an explicit form for the extended Poincar´e generators andthe same algebra of the first quantized description has been generated via generalized Heisenbergrelations. Acknowledgments
ACRM and WO would like to thank CNPq (Conselho Nacional de Desenvolvimento Cient´ıfico eTecnol´ogico) for partial financial support, and AOZ would like to thank CAPES (Coordena¸c˜aode Aperfei¸coamento de Pessoal de N´ıvel Superior) for the financial support. CNPq and CAPESare Brazilian research agencies.
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