The Noncommutative Doplicher-Fredenhagen-Roberts-Amorim Space

This work is an effort in order to compose a pedestrian review of the recently elaborated Doplicher, Fredenhagen, Roberts and Amorim (DFRA) noncommutative (NC) space which is a minimal extension of the DFR space. In this DRFA space, the object of noncommutativity (θμν) is a variable of the NC system...

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Дата:	2010
Автори:	Everton M.C. Abreu, Albert C.R. Mendes, Oliveira, W., Zangirolami, A.O.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2010
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	The Noncommutative Doplicher-Fredenhagen-Roberts-Amorim Space / Everton M.C. Abreu, Albert C.R. Mendes, W. Oliveira, A.O. Zangirolami // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 55 назв. — англ.

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spelling	irk-123456789-1465172019-02-10T01:24:52Z The Noncommutative Doplicher-Fredenhagen-Roberts-Amorim Space Everton M.C. Abreu Albert C.R. Mendes Oliveira, W. Zangirolami, A.O. This work is an effort in order to compose a pedestrian review of the recently elaborated Doplicher, Fredenhagen, Roberts and Amorim (DFRA) noncommutative (NC) space which is a minimal extension of the DFR space. In this DRFA space, the object of noncommutativity (θμν) is a variable of the NC system and has a canonical conjugate momentum. Namely, for instance, in NC quantum mechanics we will show that θij (i,j=1,2,3) is an operator in Hilbert space and we will explore the consequences of this so-called ''operationalization''. The DFRA formalism is constructed in an extended space-time with independent degrees of freedom associated with the object of noncommutativity θμν. We will study the symmetry properties of an extended x+θ space-time, given by the group P', which has the Poincaré 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 of canonical operators is described, which permits one to construct theories that are dynamically invariant under the action of the rotation group. In this framework it is also possible to give dynamics to the NC operator sector, resulting in new features. A consistent classical mechanics formulation is analyzed in such a way that, under quantization, it furnishes a NC quantum theory with interesting results. The Dirac formalism for constrained Hamiltonian systems is considered and the object of noncommutativity θij plays a fundamental role as an independent quantity. Next, we explain the dynamical spacetime symmetries in NC relativistic theories by using the DFRA algebra. It is also explained about the generalized Dirac equation issue, that the fermionic field depends not only on the ordinary coordinates but on θμν 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 analyze the complex scalar fields using this new framework. As said above, in a first quantized formalism, θμν and its canonical momentum πμν are seen as operators living in some Hilbert space. In a second quantized formalism perspective, we show an explicit form for the extended Poincaré generators and the same algebra is generated via generalized Heisenberg relations. We also consider a source term and construct the general solution for the complex scalar fields using the Green function technique. 2010 Article The Noncommutative Doplicher-Fredenhagen-Roberts-Amorim Space / Everton M.C. Abreu, Albert C.R. Mendes, W. Oliveira, A.O. Zangirolami // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 55 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 70S05; 70S10; 81Q65; 81T75 DOI:10.3842/SIGMA.2010.083 http://dspace.nbuv.gov.ua/handle/123456789/146517 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	This work is an effort in order to compose a pedestrian review of the recently elaborated Doplicher, Fredenhagen, Roberts and Amorim (DFRA) noncommutative (NC) space which is a minimal extension of the DFR space. In this DRFA space, the object of noncommutativity (θμν) is a variable of the NC system and has a canonical conjugate momentum. Namely, for instance, in NC quantum mechanics we will show that θij (i,j=1,2,3) is an operator in Hilbert space and we will explore the consequences of this so-called ''operationalization''. The DFRA formalism is constructed in an extended space-time with independent degrees of freedom associated with the object of noncommutativity θμν. We will study the symmetry properties of an extended x+θ space-time, given by the group P', which has the Poincaré 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 of canonical operators is described, which permits one to construct theories that are dynamically invariant under the action of the rotation group. In this framework it is also possible to give dynamics to the NC operator sector, resulting in new features. A consistent classical mechanics formulation is analyzed in such a way that, under quantization, it furnishes a NC quantum theory with interesting results. The Dirac formalism for constrained Hamiltonian systems is considered and the object of noncommutativity θij plays a fundamental role as an independent quantity. Next, we explain the dynamical spacetime symmetries in NC relativistic theories by using the DFRA algebra. It is also explained about the generalized Dirac equation issue, that the fermionic field depends not only on the ordinary coordinates but on θμν 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 analyze the complex scalar fields using this new framework. As said above, in a first quantized formalism, θμν and its canonical momentum πμν are seen as operators living in some Hilbert space. In a second quantized formalism perspective, we show an explicit form for the extended Poincaré generators and the same algebra is generated via generalized Heisenberg relations. We also consider a source term and construct the general solution for the complex scalar fields using the Green function technique.
format	Article
author	Everton M.C. Abreu Albert C.R. Mendes Oliveira, W. Zangirolami, A.O.
spellingShingle	Everton M.C. Abreu Albert C.R. Mendes Oliveira, W. Zangirolami, A.O. The Noncommutative Doplicher-Fredenhagen-Roberts-Amorim Space Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Everton M.C. Abreu Albert C.R. Mendes Oliveira, W. Zangirolami, A.O.
author_sort	Everton M.C. Abreu
title	The Noncommutative Doplicher-Fredenhagen-Roberts-Amorim Space
title_short	The Noncommutative Doplicher-Fredenhagen-Roberts-Amorim Space
title_full	The Noncommutative Doplicher-Fredenhagen-Roberts-Amorim Space
title_fullStr	The Noncommutative Doplicher-Fredenhagen-Roberts-Amorim Space
title_full_unstemmed	The Noncommutative Doplicher-Fredenhagen-Roberts-Amorim Space
title_sort	noncommutative doplicher-fredenhagen-roberts-amorim space
publisher	Інститут математики НАН України
publishDate	2010
url	http://dspace.nbuv.gov.ua/handle/123456789/146517
citation_txt	The Noncommutative Doplicher-Fredenhagen-Roberts-Amorim Space / Everton M.C. Abreu, Albert C.R. Mendes, W. Oliveira, A.O. Zangirolami // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 55 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 6 (2010), 083, 37 pages The Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space? Everton M.C. ABREU †‡, Albert C.R. Mendes §, Wilson OLIVEIRA § and Adriano O. ZANGIROLAMI § † Grupo de F́ısica Teórica e Matemática F́ısica, Departamento de F́ısica, Universidade Federal Rural do Rio de Janeiro, BR 465-07, 23890-971, Seropédica, RJ, Brazil E-mail: evertonabreu@ufrrj.br ‡ Centro Brasileiro de Pesquisas F́ısicas (CBPF), Rua Xavier Sigaud 150, Urca, 22290-180, RJ, Brazil § Departamento de F́ısica, 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, 2010 doi:10.3842/SIGMA.2010.083 Abstract. This work is an effort in order to compose a pedestrian review of the recently elaborated Doplicher, Fredenhagen, Roberts and Amorim (DFRA) noncommutative (NC) space which is a minimal extension of the DFR space. In this DRFA space, the object of noncommutativity (θµν) 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 θij (i, j = 1, 2, 3) is an operator in Hilbert space and we will explore the consequences of this so-called “ope- rationalization”. The DFRA formalism is constructed in an extended space-time with in- dependent degrees of freedom associated with the object of noncommutativity θµν . We will study the symmetry properties of an extended x+θ space-time, given by the group P ′, which has the Poincaré 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 of canonical 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 possible to give dynamics to the NC operator sector, resulting in new features. A consistent classical mechanics formulation is analyzed in such a way that, under quantization, it furnishes a NC quantum theory with interesting results. The Dirac formalism for constrained Hamiltonian systems is considered and the object of noncommutativity θij plays a fundamental role as an independent quantity. Next, we explain the dynamical spacetime symmetries in NC rel- ativistic theories by using the DFRA algebra. It is also explained about the generalized Dirac equation issue, that the fermionic field depends not only on the ordinary coordinates but on θµν 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 analyze the complex scalar fields using this new framework. As said above, in a first quantized for- malism, θµν and its canonical momentum πµν are seen as operators living in some Hilbert space. In a second quantized formalism perspective, we show an explicit form for the ex- tended Poincaré generators and the same algebra is generated via generalized Heisenberg relations. We also consider a source term and construct the general solution for the complex scalar fields using the Green function technique. Key words: noncommutativity; quantum mechanics; gauge theories 2010 Mathematics Subject Classification: 70S05; 70S10; 81Q65; 81T75 ?This paper is a contribution to the Special Issue “Noncommutative Spaces and Fields”. The full collection is available at http://www.emis.de/journals/SIGMA/noncommutative.html mailto:evertonabreu@ufrrj.br mailto:albert@fisica.ufjf.br mailto:wilson@fisica.ufjf.br mailto:adrianozangirolami@fisica.ufjf.br http://dx.doi.org/10.3842/SIGMA.2010.083 http://www.emis.de/journals/SIGMA/noncommutative.html 2 E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. Zangirolami Contents 1 Introduction 2 2 The noncommutative quantum mechanics 5 2.1 The Doplicher–Fredenhagen–Roberts–Amorim space . . . . . . . . . . . . . . . . 5 2.2 The noncommutative harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . 9 3 Tensor coordinates in noncommutative mechanics 10 4 Dynamical symmetries in NC theories 14 4.1 Coordinate operators and their transformations in relativistic NCQM . . . . . . 14 4.2 Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.3 Equations of motion and Noether’s theorem . . . . . . . . . . . . . . . . . . . . . 19 4.4 Considerations about the twisted Poincaré symmetry and DFRA space . . . . . . 21 4.4.1 The twisted Poincaré algebra . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.4.2 The DFRA analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5 Fermions and noncommutative theories 23 6 Quantum complex scalar fields and noncommutativity 26 6.1 The action and symmetry relations . . . . . . . . . . . . . . . . . . . . . . . . . . 27 6.2 Plane waves and Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . 31 References 33 1 Introduction Theoretical physics is living nowadays a moment of great excitement and at the same time great anxiety before the possibility by the Large Hadron Collider (LHC) to reveal the mysteries that make the day-by-day of theoretical physicists. One of these possibilities is that, at the collider energies, the extra and/or the compactified spatial dimensions become manifest. These manifestations can lead, for example, to the fact that standard four-dimensions spacetimes may become NC, namely, that the position four-vector operator xµ obeys the following rule [xµ,xν ] = iθµν , (1.1) where θµν 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 been explored using the representation of the standard framework of the Poincaré algebra through the Weyl–Moyal correspondence. To every field operator ϕ(x) it has been assigned a Weyl symbol 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 [ i 2 θµν ∂ ∂xµ ∂ ∂yν ] ϕ(x)ψ(y) ∣∣ x=y (1.2) and where now the commutators of operators are replaced by Moyal brackets as, [xµ, xν ]? ≡ xµ ? xν − xν ? xµ = iθµν . The Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 3 From (1.2) we can see clearly that at zeroth-order the NCQFT is Lorentz invariant. Since θµν 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. After that, Snyder tackled the idea launched by Heisenberg and published what is considered as the first paper on spacetime noncommutativity in 1947 [1]. C.N. Yang, immediately after Snyder’s paper, showed that the problems of field theory supposed to be removed by noncommutativity were 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 in physics [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 symmetry group, with generators MAB, satisfying the Lorentz algebra, where A,B = 0, 1, 2, 3, 4 and using natural units, i.e., ~ = c = 1. Moreover, he introduced the relation between coordinates and generators of the SO(4, 1) algebra xµ = aM4µ (where µ, ν = 0, 1, 2, 3 and the parameter a has dimension of length), promoting in this way the spacetime coordinates to Hermitian operators. The mentioned relation introduces the commu- tator, [xµ,xν ] = ia2Mµν (1.3) and the identities, [Mµν ,xλ] = i ( xµηνλ − xνηµλ ) and [Mµν ,Mαβ ] = i ( Mµβηνα −Mµαηνβ + Mναηµβ −Mνβηµα ) , which agree with four dimensional Lorentz invariance. Three decades ago Connes et al. (see [11] for a review of this formalism) brought the concepts of 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 constant B-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 from a low energy limit for open string theory embedded in a constant antisymmetric background field. The fundamental point about the standard NC space is that the object of noncommutativi- ty θµν is usually assumed to be a constant antisymmetric matrix in NCFT’s. This violates Lorentz symmetry because it fixes a direction in an inertial reference frame. The violation of Lorentz invariance is problematic, among other facts, because it brings effects such as vacuum birefringence [16]. However, at the same time it permits to treat NCFT’s as deformations of ordinary quantum field theories, replacing ordinary products with Moyal products, and ordinary gauge interactions by the corresponding NC ones. As it is well known, these theories carries serious problems as nonunitarity, nonlocalizability, nonrenormalizability, UV × IR mixing etc. On the other hand, the Lorentz invariance can be recovered by constructing the NC spacetime 4 E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. Zangirolami with θµν 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 xµ and θµν belong in this case to the same affine algebra, the fields must be functions of the eigenvalues of both xµ and θµν . In [18] Banerjee et al. obtained conditions for preserving Poincaré invariance in NC gauge theories and a 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 of them 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 from general relativity (GR) and quantum mechanics (QM). In addition to (1.1) it also assumes that [xµ, θαβ ] = 0. (1.4) With this formalism, DFR demonstrated that the combination of QM with the classical gravitation 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 the usual NCFT’s, although the fields have to be considered as depending not only on xµ but also on θαβ . The argument is that very accurate measurements of spacetime localization could transfer to test particles energies sufficient to create a gravitational field that in principle could trap photons. This possibility is related with spacetime uncertainty relations that can be derived from (1.1) and (1.4) as well as from the quantum conditions θµνθ µν = 0, ( 1 4 ∗θµνθµν )2 = λ8 P , (1.5) where ∗θµν = 1 2εµνρσθ ρσand λP is the Planck length. These operators are seen as acting on a Hilbert space H and this theory implies in extra compact dimensions [10]. The use of conditions (1.5) in [17, 19, 20, 21, 22, 23] would bring trivial consequences, since in those works the relevant results strongly depend on the value of θ2, which is taken as a mean with some weigh function W (θ). They use in this process the 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 construction of a NCFT which keeps Lorentz invariance. This is a fundamental matter, since there is no experimental evidence to assume Lorentz symmetry violation [16]. Although we will see that in this review we are not using twisted symmetries [24, 25, 26, 27] there is some considerations about 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 quantum mechanics (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 θij (where i, j = 1, 2, 3), which essentially is the result of the commutation of two coordinate operators, is considered as a constant matrix, although this is not the general case [1, 5, 17, 19, 10]. Considering θij as a constant matrix spoils the Lorentz symmetry 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 consequence the corresponding conjugate momenta is not introduced, which, as well known, it is important to implement rotation as a dynamical symmetry [45]. As a result, the theories are not invariant under 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 noncommutativity The Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 5 were considered as true operators and their conjugate momenta were introduced. This permits to display a complete and consistent algebra among the Hilbert space operators and to construct generalized 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 it comes dynamically from the consistent action of an operator, as discussed in [45]. This new NC space 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 the recent applications published in the literature. We will see in this review that a consistent classical mechanics formulation can be shown in such a way that, under quantization, it gives a NC quantum theory with interesting new features. The Dirac formalism for constrained Hamiltonian systems is strongly used, and the object of noncommutativity θij plays a fundamental role as an independent quantity. The presented classical theory, as its quantum counterpart, is naturally invariant under the rotation group SO(D). The organization of this work is: in Section 2 we describe the mathematical details of this new NC space. After this we describe the NC Hilbert space and construct an harmonic oscillator model for this new space. In Section 3 we explore the DFRA classical mechanics and treat the question of Dirac’s formalism. The symmetries and the details of the extended Poincaré symmetry group are explained in Section 4. In this very section we also explore the relativistic features of DFRA space and construct the Klein–Gordon equation together with the Noether’s formalism. In Section 5 we analyzed the issue about the fermions in this new structure and we study the Dirac equation. Finally, in Section 6 we complete the review considering the elaboration of complex scalar fields and the source term analysis of quantum field theories. 2 The noncommutative quantum mechanics In this section we will introduce the complete extension of the DFR space formulated by R. Amorim [46, 47, 48, 49, 50] where a minimal extension of the DFR is accomplished through the introduction of the canonical conjugate momenta to the variable θµν 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 “missing parts” and naturally its implications in quantum mechanics. 2.1 The Doplicher–Fredenhagen–Roberts–Amorim space The results appearing in [17] motivated some other authors [20, 21, 22, 19, 23]. Some of them 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 commutator among the coordinate operators, [xµ, [xν ,xρ]] = 0, (2.1) and it is easy to realize that this relation constitute a constraint in a NC spacetime. Notice that the commutator inside the triple one is not a c-number. The DFR algebra is based on principles imported from general relativity (GR) and quantum mechanics (QM). In addition to (1.1) it also assumes that [xµ, θαβ ] = 0, (2.2) 6 E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. Zangirolami and we consider that space has arbitrary D ≥ 2 dimensions. As usual xµ and pν , where i, j = 1, 2, . . . , D and µ, ν = 0, 1, . . . , D, represent the position operator and its conjugate momentum. The NC variable θµν represent the noncommutativity operator, but now πµν is its conjugate momentum. In accordance with the discussion above, it follows the algebra [xµ,pν ] = iδµ ν , (2.3a) [θµν , παβ ] = iδµν αβ , (2.3b) where δµν αβ = δµ αδν β − δ µ βδ ν α. The relation (1.1) here in a space with D dimensions, for example, can be written as [xi,xj ] = iθij and [pi,pj ] = 0 (2.4) and together with the triple commutator (2.1) condition of the standard spacetime, i.e., [xµ, θνα] = 0. (2.5) This implies that [θµν , θαβ ] = 0, (2.6) and this completes the DFR algebra. Recently, in order to obtain consistency R. Amorim introduced [46], as we talked above, the canonical conjugate momenta πµν such that, [pµ, θ να] = 0, [pµ, πνα] = 0. (2.7) The Jacobi identity formed by the operators xi, xj and πkl leads to the nontrivial relation [[xµ, παβ ],xν ]− [[xν , παβ ],xµ] = −δµν αβ . (2.8) The solution, unless trivial terms, is given by [xµ, παβ ] = − i 2 δµν αβpν . (2.9) It is simple to verify that the whole set of commutation relations listed above is indeed consistent under all possible Jacobi identities. Expression (2.9) suggests the shifted coordinate operator [29, 31, 38, 39, 42] Xµ ≡ xµ + 1 2 θµνpν (2.10) that commutes with πkl. Actually, (2.10) also commutes with θkl and Xj , and satisfies a non trivial commutation relation with pi depending objects, which could be derived from [Xµ,pν ] = iδµ ν (2.11) and [Xµ,Xν ] = 0. (2.12) To construct a DFRA algebra in (x, θ) space, we can write Mµν = Xµpν −Xνpµ − θµσπ ν σ + θνσπ µ σ , The Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 7 where Mµν is the antisymmetric generator of the Lorentz-group. To construct πµν we have to obey equations (2.3b) and (2.9), obviously. From (2.3a) we can write the generators of translations as Pµ = −i∂µ. With these ingredients it is easy to construct the commutation relations [Pµ,Pν ] = 0, [Mµν ,Pρ] = −i ( ηµνPρ − ηµρPν ) , [Mµν ,Mρσ] = −i ( ηµρMνσ − ηµσMνρ − ηνρMµσ − ηνσMµρ), and we can say that Pµ and Mµν are the generator of the DFRA algebra. These relations are important, as we will see in Section 5, because they are essential for the extension of the Dirac equation to the extended DFRA configuration space (x, θ). It can be shown that the Clifford algebra structure generated by the 10 generalized Dirac matrices Γ (Section 5) relies on these relations. 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, it is necessary firstly to find a maximal set of commuting operators. For instance, let us choose a momentum basis formed by the eigenvectors of p and π. A coordinate basis formed by the eigenvectors of (X, θ) can also be introduced, among other possibilities. We observe here that it is in no way possible to form a basis involving more than one component of the original position operator x, since their components do not commute. To clarify, let us display the fundamental relations involving those basis, namely eigenvalue, orthogonality and completeness relations Xi\|X ′, θ′〉 = X ′i\|X ′, θ′〉, θij \|X ′, θ′〉 = θ′ ij \|X ′, θ′〉, pi\|p′, π′〉 = p′i\|p′, π′〉, πij \|p′, π′〉 = π′ij \|p′, π′〉, 〈X ′, θ′\|X ′′, θ′′〉 = δD(X ′ −X ′′)δ D(D−1) 2 (θ′ − θ′′), 〈p′, π′\|p′′, π′′〉 = δD(p′ − p′′)δ D(D−1) 2 (π′ − π′′),∫ dDX ′ d D(D−1) 2 θ′\|X ′, θ′〉〈X ′, θ′\| = 1, (2.13)∫ dDp′ d D(D−1) 2 π′\|p′, π′〉〈p′, π′\| = 1, (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 the integrals 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 〈X ′, θ′\|pi\|X ′′, θ′′〉 = −i ∂ ∂X ′i δ D(X ′ −X ′′)δ D(D−1) 2 (θ′ − θ′′) and 〈X ′, θ′\|πij \|X ′′, θ′′〉 = −iδD(X ′ −X ′′) ∂ ∂θ′ij δ D(D−1) 2 (θ′ − θ′′). 8 E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. Zangirolami The transformations from one basis to the other one are carried out by extended Fourier trans- forms. Related with these transformations is the “plane wave” 〈X ′, θ′\|p′′, π′′〉 = N exp(ip′′X ′ + iπ′′θ′), where internal products are represented in a compact manner. For instance, p′′X ′ + π′′θ′ = p′′iX ′i + 1 2 π′′ijθ ′ij . Before discussing any dynamics, it seems interesting to study the generators of the group of rotations SO(D). Without considering the spin sector, we realize that the usual angular momentum operator lij = xipj − xjpi does not close in an algebra due to (2.4). And we have that, [lij , lkl] = iδillkj − iδjllki − iδikllj + iδjklli − iθilpkpj + iθjlpkpi + iθikplpj − iθjkplpi and so their components can not be SO(D) generators in this extended Hilbert space. On the contrary, the operator Lij = Xipj −Xjpi, (2.15) closes in the SO(D) algebra. However, to properly act in the (θ, π) sector, it has to be generalized to the total angular momentum operator Jij = Lij − θilπ j l + θjlπ i l . (2.16) It is easy to see that not only [Jij ,Jkl] = iδilJkj − iδjlJki − iδikJlj + iδjkJli, (2.17) but Jij generates rotations in all Hilbert space sectors. Actually δXi = i 2 εkl[Xi,Jkl] = εikXk, δpi = i 2 εkl[pi,Jkl] = εikpk, δθij = i 2 εkl[θij ,Jkl] = εikθ j k + εjkθi k, δπij = i 2 εkl[πij ,Jkl] = εikπ j k + εjkπi k (2.18) have the expected form. The same occurs with xi = Xi − 1 2 θijpj =⇒ δxi = i 2 εkl[xi,Jkl] = εikxk. 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 transform as 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 not from operations based on the index structure of those variables. We would like to mention that in D = 2 the operator Jij reduces to Lij , in accordance with the fact that in this case θ or π has only one independent component. In D = 3, it is possible to represent θ or π by three vectors and both parts of the angular momentum operator have the same kind of structure, and so the same spectrum. An unexpected addition of angular momentum potentially arises, although the (θ, π) sector can live in a J = 0 Hilbert subspace. Unitary rotations are generated by U(ω) = exp(−iω · J), while unitary translations, by T (λ,Ξ) = exp(−iλ · p− iΞ · π). The Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 9 2.2 The noncommutative harmonic oscillator In this section we will consider the isotropic D-dimensional harmonic oscillator where we find several possibilities of rotational invariant Hamiltonians which present the proper commutative limit [31, 32, 39, 40]. The well known expression representing the harmonic oscillator can be written as H0 = 1 2m p2 + mω2 2 X2, (2.19) since Xi commutes with Xj , satisfies the canonical relation (2.11) and in the DFRA formalism transforms according to (2.18). With these results we can construct annihilation and creation operators in the usual way, Ai = √ mω 2 ( Xi + ipi mω ) and A†i = √ mω 2 ( Xi − ipi mω ) , where Ai and A†i satisfy the usual harmonic oscillator algebra, and H0 can be written in terms of the sum of D number operators Ni = A†iAi, which have the same spectrum and the same degeneracies when compared with the ordinary QM case [51]. The (θ, π) sector, however, is not modified by any new dynamics if H0 represents the total Hamiltonian. 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 Hθ = 1 2Λ π2 + ΛΩ2 2 θ2, (2.20) where Λ is a parameter with dimension of (length)−3 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 Hamiltonian eigenstates of each sector. The annihilation and creation operators, considering the (θ, π) sector, are respectively defined as Aij = √ ΛΩ 2 ( θij + iπij ΛΩ ) and A†ij = √ ΛΩ 2 ( θij − iπij ΛΩ ) , which satisfy the oscillator algebra [Aij ,A†kl] = δij,kl, and now we can construct eigenstates of Hθ, equation (2.20), associated with quantum num- bers nij . As well known, the ground state is annihilated by Aij , and its corresponding wave function, in the (θ, π) sector, is 〈θ′\|nij = 0, t〉 = ( ΛΩ π )D(D−1) 8 exp [ −ΛΩ 4 θ′ijθ ′ij ] exp [ −iD(D − 1) Ω 4 t ] . (2.21) However, turning to the basics, the wave functions for excited states can be obtained through the application of the creation operator A†kl on the fundamental state. On the other hand, we expect that Ω might be so big that only the fundamental level of this generalized oscillator could be occupied. This will generate only a shift in the oscillator spectrum, which is ∆E = D(D−1) 4 Ω and this new vacuum energy could generate unexpected behaviors. Another point related with (2.21) is that it gives a natural way for introducing the weight function W (θ) which appears, in the context of NCFT’s, in [17, 19]. W (θ) is a normalized 10 E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. Zangirolami function necessary, for example, to control the θ-integration. Analyzing the (θ, π) sector, the expectation value of any function f(θ) over the fundamental state is 〈f(θ)〉 = 〈nkl = 0, t\|f(θ)\|nkl = 0, t〉 = ( ΛΩ π )D(D−1) 4 ∫ d D(D−1) 2 θ′f(θ′) exp [ −ΛΩ 2 θ′rsθ ′rs ] ≡ ∫ d D(D−1) 2 θ′W (θ′)f(θ′), where W (θ′) ≡ ( ΛΩ π )D(D−1) 4 exp [ −ΛΩ 2 θ′rsθ ′rs ] , and the expectation values are given by 〈1〉 = 1, 〈θij〉 = 0, 1 2 〈θijθij〉 = 〈θ2〉, 〈θijθkl〉 = 2 D(D − 1) δij,kl〈θ2〉, (2.22) where 〈θ2〉 ≡ 1 2ΛΩ and now we can calculate the expectation values for the physical coordinate operators. As one can verify, 〈xi〉 = 〈Xi〉 = 0, but one can find non trivial noncommutativity contributions to the expectation values for other operators. For instance, it is easy to see from (2.22) and (2.10) that 〈x2〉 = 〈X2〉+ 2 D 〈θ2〉〈p2〉, where 〈X2〉 and 〈p2〉 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 physical coordinate operator, as expected and can be measurable, at first sight. 3 Tensor coordinates in noncommutative mechanics All the operators introduced until now belong to the same algebra and are equal, hierarchically speaking. The necessity of a rotation invariance under the group SO(D) is a consequence of this augmented Hilbert space. Rotation invariance, in a nonrelativistic theory, is the main topic if one 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 θµν from a constant matrix to a tensor operator [17, 18, 19, 20, 21, 22, 23], although in this last situation the rules are quite different from those found in NCQM, since in a quantum field theory the relevant operators are not coordinates but fields. Now that we got acquainted with the new proposed version of NCQM [46] where the θij are tensors in Hilbert space and πij are their conjugate canonical momenta, we will show in this section that a possible fundamental classical theory, under quantization, can reproduce the algebraic structure depicted in the last section. The Dirac formalism [52] for constrained Hamiltonian systems is extensively used for this purpose. As it is well known, when a theory presents a complete set of second-class constraints Ξa = 0, a = 1, 2, . . . , 2N , 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,Ξa}∆−1 ab {Ξ b, B}, (3.1) such that the evolution of the system respects the constraint surface given by Ξa = 0. The Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 11 In (3.1) ∆ab = {Ξa,Ξb} (3.2) is the so-called constraint matrix and ∆−1 ab is its inverse. The fact that the constraints Ξa are second-class guarantees the existence of ∆ab. If that matrix were singular, linear combinations of the Ξa could be first class. For the first situation, the number of effective degrees of freedom of the theory is given by 2D −2N , where 2D is the number of phase space variables and 2N is the number of second-class constraints. If the phase space is described only by the 2D = 2D + 2 D(D − 1) 2 variables xi, pi, θij and πij , 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 2N variables, and to introduce at the same time 2N second-class constraints. An easy way to implement these concepts without destroying the symmetry under rotations is to enlarge the phase space introducing a pair of canonical variables Zi, Ki, also with (at the same time) a set of second-class constraints Ψi, Φi. Considering this set of phase space variables, it follows by construction the fundamental (non vanishing) Poisson bracket structure {xi, pj} = δi j , {θij , πkl} = δij kl, {Zi,Kj} = δi j (3.3) and the Dirac brackets structure is derived in accordance with the form of the second-class constraints, subject that will be discussed in what follows. Let us assume that Zi has dimension of length L, as xi. This implies that both pi and Ki have dimension of L−1. As θij and πij have dimensions of L2 and L−2 respectively, the expression for the constraints Ψi and Φi is given by Ψi = Zi + αxi + βθijpj + γθijKj and Φi = Ki + ρpi + σπijx j + λπijZ j , if only dimensionless parameters α, β, γ, ρ, σ and λ are introduced and any power higher than two in phase space variables is discarded. It is possible to display the whole group of parameters during the computation of the Dirac formalism. After that, at the end of the calculations, the parameters have been chosen in order to generate, under quantization, the commutator structure appearing in equations (2.3) to (2.9). The constraints reduce, in this situation, to Ψi = Zi − 1 2 θijpj , Φi = Ki − pi (3.4) and hence the corresponding constraint matrix (3.2) becomes (∆ab) = ( {Ψi,Ψj} {Ψi,Φj} {Φi,Ψj} {Φi,Φj} ) = ( 0 δi j −δj i 0 ) . (3.5) Notice that (3.5) is regular even if θij is singular. This fact guarantees that the proper commu- tative limit of the theory can be taken. 12 E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. Zangirolami The inverse of (3.5) is trivially given by (∆−1 ab ) = ( 0 −δ j i δi j 0 ) and it is easy to see that the non-zero Dirac brackets (the others are zero) involving only the original set of phase space variables are {xi, pj}D = δi j , {xi, xj}D = θij , {θij , πkl}D = δij kl, {xi, πkl}D = −1 2 δij kl pj , (3.6) which furnish the desired result. If yA represents phase space variables and yA the corresponding Hilbert space operators, the Dirac quantization procedure, {yA, yB}D → 1 i [yA,yB] results the commutators in (2.3) until (2.9). For completeness, the remaining non-zero Dirac brackets involving Zi and Ki are {Zi, xj}D = −1 2 θij , {Ki, x j}D = −δj i , {Zi, πkl}D = 1 2 δij klpj . (3.7) In this classical theory the shifted coordinate Xi = xi + 1 2 θijpj , which corresponds to the operator (2.10), also plays a fundamental role. As can be verified by the non-zero Dirac brackets just below, {Xi, pj}D = δi j , {Xi, xj}D = 1 2 θij , {Xi, Zj}D = −1 2 θij , {Xi,Kj}D = δi j , and the angular momentum tensor J ij = Xipj −Xjpi − θilπ j l + θjlπ i l (3.8) closes in the classical SO(D) algebra, by using Dirac brackets instead of commutators. In fact {J ij , Jkl}D = δilJkj − δjlJki − δikJ lj + δjkJ li, and as in the quantum case, the proper symmetry transformations over all the phase space variables are generated by (3.8). Beginning with δA = −1 2 εkl{A, Jkl}D, one have as a result that δXi = εijX j , δxi = εijx j , δpi = ε j i pj , δθij = εikθ kj + εjkθ ik, δπij = ε k i πkj + ε k j πik, δZi = 1 2 εijθ jkpk, δKi = ε j i pj . 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 the The Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 13 Dirac brackets and the constraints given in (3.4). These constraints, as well as the fundamental Poisson brackets in (3.3), can be easily generated by the first order action S = ∫ dtLFO, (3.9) where LFO = p · ẋ+K · Ż + π · θ̇ − λaΞa −H. (3.10) The 2D quantities λa are the Lagrange multipliers introduced conveniently to implement the constraints Ξa = 0 given by (3.4), and H is some Hamiltonian. The dots “·” between phase space coordinates represent internal products. The canonical conjugate momenta for the Lagrange multipliers are primary constraints that, when conserved, generate the secondary constraints Ξa = 0. Since these last constraints are second class, they are automatically conserved by the theory, 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 procedure through the Lagrange multipliers. To obtain a more illuminating second-order Lagrangian we must follow the basic pattern and with the help of the Hamiltonian, integrate out the momentum variables 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 harmonic oscillator, which contemplates with dynamics not only the usual vectorial coordinates but also the noncommutativity sector spanned by the tensor quantities θ and π. The corresponding classical Hamiltonian can be written as H = 1 2m p2 + mω2 2 X2 + 1 2Λ π2 + ΛΩ2 2 θ2, (3.11) which is invariant under rotations. In (3.11) m is a mass, Λ is a parameter with dimension of L−3, and ω and Ω are frequencies. Other choices for the Hamiltonian can be done without spoiling the algebraic structure discussed above. The classical system given by (3.9), (3.10) and (3.11) represents two independent isotropic oscillators in D and D(D−1) 2 dimensions, expressed in terms of variables Xi, pi, θij and πij . The solution is elementary, but when one expresses the oscillators in terms of physical variables xi, pi, θij and πij , 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 that decouple 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 theory has been proved to be invariant under the action of the rotation group SO(D) and could be derived through a variational principle. Once this structure has been given, it is not difficult to construct a relativistic generalization of such a model. The fundamental Poisson brackets become {xµ, pν} = δµ ν , {θµν , πρσ} = δµν ρσ, {Zµ,Kν} = δµ ν , and the constraints (3.4) are generalized to Ψµ = Zµ − 1 2 θµνpν , Φµ = Kµ − pµ, 14 E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. Zangirolami generating the invertible constraint matrix (∆ab) = ( {Ψµ,Ψν} {Ψµ,Φν} {Φµ,Ψν} {Φµ,Φν} ) = ( 0 ηµν −ηµν 0 ) . To finish we can say that the Dirac brackets between the phase space variables can also be generalized from (3.6), (3.7). The Hamiltonian of course cannot be given by (3.11), but at least for the free particle, it vanishes identically, as it is usual to appear with covariant classical systems [52]. Also it is necessary for a new constraint, which must be first class, to generate the reparametrization transformations. In a minimal extension of the usual commutative case, it is given by the mass shell condition χ = p2 +m2 = 0, but other choices are possible, furnishing dynamics to the noncommutativity sector or enlarging the symmetry content of the relativistic action. 4 Dynamical symmetries in NC theories In this section we will analyze the dynamical spacetime symmetries in NC relativistic theories by using the DFRA algebra depicted in Section 2. As explained there, the formalism is constructed in an extended spacetime with independent degrees of freedom associated with the object of noncommutativity θµν . In this framework we can consider theories that are invariant under the Poincaré group P or under its extension P ′, when translations in the extra dimensions are allowed. 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 constructed with those Casimir operators will be introduced and after that we will investigate the symmetry content of one of those theories by using Noether’s procedure. 4.1 Coordinate operators and their transformations in relativistic NCQM In the usual formulations of NCQM, interpreted here as relativistic theories, the coordinates xµ and their conjugate momenta pµ are operators acting in a Hilbert space H satisfying the fun- damental commutation relations given in Section 2, we can define the operator G1 = 1 2 ωµνLµν . Note that, analogously to (2.18), it is possible to dynamically generate infinitesimal transfor- mations on any operator A, following the usual rule δA = i[A,G1]. For Xµ, pµ and Lµν , given in (2.10) and (2.15), with spacetime coordinates, we have the following results δXµ = ωµ νX ν , δpµ = ω ν µ pν , δLµν = ωµ ρL ρν + ων ρL µρ. However, the physical coordinates fail to transform in the appropriate way. As can be seen, the same rule applied on xµ gives the result δxµ = ωµ ν ( xν + 1 2 θρνpν ) − 1 2 θµνωνρpρ, (4.1) which is a consequence of θµν not being transformed. Relation (4.1) probably will break Lorentz symmetry in any reasonable theory. The cure for these problems can be obtained by conside- ring θµν as an operator in H, and introducing its canonical momentum πµν as well. The price The Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 15 to be paid is that θµν will have to be associated with extra dimensions, as happens with the formulations appearing in [17, 18, 19, 20, 21, 22, 23]. Moreover, we have that the commutation relation [xµ, πρσ] = − i 2 δµν ρσpν (4.2) is necessary for algebraic consistency under Jacobi identities. The set (4.2) completes the algebra displayed in Section 2, namely, the DFRA algebra. With this algebra in mind, we can generalize the expression for the total angular momentum, equations (2.16) and (2.17). The framework constructed above permits consistently to write [25] Mµν = Xµpν −Xνpµ − θµσπ ν σ + θνσπ µ σ (4.3) and consider this object as the generator of the Lorentz group, since it not only closes in the appropriate algebra [Mµν ,Mρσ] = iηµσMρν − iηνσMρµ − iηµρMσν + iηνρMσµ, (4.4) but it generates the expected Lorentz transformations on the Hilbert space operators. Actually, for δA = i[A,G2], with G2 = 1 2ωµνMµν , we have that, δxµ = ωµ νx ν , δXµ = ωµ νX ν , δpµ = ω ν µ pν , δθµν = ωµ ρθ ρν + ων ρθ µρ, δπµν = ω ρ µ πρν + ω ρ ν πµρ, δMµν = ωµ ρM ρν + ων ρM µρ, (4.5) which in principle should guarantee the Lorentz invariance of a consistent theory. We observe that this construction is possible because of the introduction of the canonical pair θµν , πµν as independent variables. This pair allows the building of an object like Mµν in (4.3), which generates the transformations given just above dynamically [45] and not merely by taking into account 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, Mµν = Mµν 1 + Mµν 2 , where Mµν 1 = Xµpν −Xνpµ and Mµν 2 = −θµσπ ν σ + θνσπ µ σ , as in the usual addition of angular momenta. Of course both operators have to satisfy the Lorentz algebra. It is possible to find convenient representations that reproduce (4.5). In the sector H1 of H = H1 ⊗ H2 associated with (X,p), it can be used the usual 4 × 4 matrix representation D1(Λ) = (Λµ α), such that, for instance X′µ = Λµ νX ν . For the sector of H2 relative to (θ, π), it is possible to use the 6 × 6 antisymmetric product representation D2(Λ) = ( Λ[µ αΛν] β ) , such that, for instance, θ′ µν = Λ[µ αΛν] βθ αβ . 16 E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. Zangirolami The complete representation is given by D = D1⊕D2. In the infinitesimal case, Λµ ν = δµ ν +ωµ ν , and (4.5) are reproduced. There are four Casimir invariant operators in this context and they are given by Cj1 = Mj µνMjµν and Cj2 = εµνρσMj µνMj ρσ, where j = 1, 2. We note that although the target space has 10 = 4+6 dimensions, the symmetry group has only 6 independent parameters and not the 45 independent parameters of the Lorentz group in D = 10. As we said before, this D = 10 spacetime comprises the four spacetime coordinates and the six θ coordinates. In Section 6 the structure of this extended space will become clearer. Analyzing the Lorentz symmetry in NCQM following the lines above, once we introduce an appropriate theory, for instance, given by a scalar action. We know, however, that the elementary particles are classified according to the eigenvalues of the Casimir operators of the inhomogeneous Lorentz group. Hence, let us extend this approach to the Poincaré group P. By considering the operators presented here, we can in principle consider G3 = 1 2 ωµνMµν − aµpµ + 1 2 bµνπ µν as the generator of some group P ′, which has the Poincaré group as a subgroup. By following the same rule as the one used in the obtainment of (4.5), with G2 replaced by G3, we arrive at the set of transformations δXµ = ωµ νX ν + aµ, δpµ = ω ν µ pν , δθµν = ωµ ρθ ρν + ων ρθ µρ + bµν , δπµν = ω ρ µ πρν + ω ρ ν πµρ, δMµν 1 = ωµ ρM ρν 1 + ων ρM µρ 1 + aµpν − aνpµ, δMµν 2 = ωµ ρM ρν 2 + ων ρM2 µρ + bµρπ ν ρ + bνρπµ ρ, δxµ = ωµ νx ν + aµ + 1 2 bµνpν . (4.6) We observe that there is an unexpected term in the last one of (4.6) system. This is a consequence of the coordinate operator in (2.10), which is a nonlinear combination of operators that act onH1 and H2. The action of P ′ over the Hilbert space operators is in some sense equal to the action of the Poincaré group with an additional translation operation on the (θµν) sector. All its generators close in an algebra under commutation, so P ′ is a well defined group of transformations. As a matter of fact, the commutation of two transformations closes in the algebra [δ2, δ1]y = δ3y, (4.7) where y represents any one of the operators appearing in (4.6). The parameters composition rule is given by ωµ 3ν = ωµ 1αω α 2ν − ω µ 2αω α 1ν , aµ 3 = ωµ 1νa ν 2 − ω µ 2νa ν 1 , bµν 3 = ωµ 1ρb ρν 2 − ω µ 2ρb ρν 1 − ω ν 1ρb ρµ 2 + ων 2ρb ρµ 1 . (4.8) If we consider the operators acting only on H1, we verify that they transform standardly under the Poincaré group P in D = 4, whose generators are pµ and Mµν 1 . 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 T4, and have two Casimir invariant operators C1 = p2 and C2 = s2, where sµ = 1 2εµνρσM νρ 1 pσ is the Pauli–Lubanski vector. If we include in M1 terms associated with spin, we will keep the usual classification of the elementary particles based on those invariants. A representation for P can be given by the 5× 5 matrix D3(Λ, A) = ( Λµ ν Aµ 0 1 ) acting in the 5-dimensional vector ( Xµ 1 ) . The Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 17 Considering the operators acting on H2, we find a similar structure. Let us call the cor- responding symmetry group as G. It has as generators the operators πµν and Mµν 2 . As one can verify, C3 = π2 and C4 = Mµν 2 πµν are the corresponding Casimir operators. G can be seen as the semidirect product of the Lorentz group and the translation group T6. A possible representation uses the antisymmetric 6×6 representation D2(Λ) already discussed, and is given by the 7× 7 matrix D4(Λ, B) = ( Λ[µ αΛν] β Bµν 0 1 ) acting in the 7-dimensional vector ( θµν 1 ) . Now we see that the complete group P ′ is just the product of P and G. It has a 11× 11 dimensional representation given by D5(Λ, A,B) =  Λµ ν 0 Aµ 0 Λ[µ αΛν] β Bµν 0 0 1  (4.9) acting in the 11-dimensional column vector Xµ θµν 1  . A group element needs 6 + 4 + 6 parameters to be determined and P ′ is a subgroup of the full Poincaré group P10 in D = 10. Observe that an element of P10 needs 55 parameters to be specified. Here, in the infinitesimal case, when A goes to a, B goes to b and Λµ ν goes to δµ ν + ωµ ν , the transformations (4.6) are obtained from the action of (4.9) defined above. It is clear that C1, C2, C3 and C4 are the Casimir operators of P ′. So far we have been considering a possible algebraic structure among operators in H and possible 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 one that gives the dynamical symmetries of the action. If the considered theory is not invariant under the θ translations, but it is by Lorentz transformations and x translations, the set of the symmetry transformations on the generalized coordinates will be given by (4.6) but effectively considering bµν as vanishing, which implies that P ′, with this condition, is dynamically con- tracted to the Poincaré group. Observe, however, that πµν will be yet a relevant operator, since Mµν depends on it in the representation here adopted. An important point related with the 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 symmetry content. 4.2 Actions As discussed in the previous section, in NCQM the physical coordinates do not commute and their eigenvectors can not be used in order to form a basis in H = H1 + H2. This does not occur with the shifted coordinate operator Xµ due to (2.5), (2.6) and (2.12). Consequently their eigenvectors can be used in the construction of such a basis. Generalizing what has been done in [46], it is possible to introduce a coordinate basis \|X ′, θ′〉 = \|X ′〉 ⊗ \|θ′〉 in such a way that Xµ\|X ′, θ′〉 = X ′µ\|X ′, θ′〉, θµν \|X ′, θ′〉 = θ′ µν \|X ′, θ′〉 (4.10) 18 E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. Zangirolami satisfying usual orthonormality and completeness relations. In this basis 〈X ′, θ′\|pµ\|X ′′, θ′′〉 = −i ∂ ∂X ′µ δ 4(X ′ −X ′′)δ6(θ′ − θ′′) (4.11) and 〈X ′, θ′\|πµν \|X ′′, θ′′〉 = −iδ4(X ′ −X ′′) ∂ ∂θ′µν δ6(θ′ − θ′′) (4.12) implying that both momenta acquire a derivative realization. A physical state \|φ〉, in the coordinate basis defined above, will be represented by the wave function φ(X ′, θ′) = 〈X ′, θ′\|φ〉 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 for constructing an ordinary relativistic free quantum theory is to impose that the physical states are annihilated by the mass shell condition (p2 +m2)\|φ〉 = 0 (4.13) constructed with the Casimir operator C1 = p2. In the coordinate representation, this gives the Klein–Gordon equation. The same result is obtained from the quantization of the classical relativistic particle, whose action is invariant under reparametrization [52]. There the generator of the reparametrization symmetry is the constraint (p2 + m2) ≈ 0. Condition (4.13) is then interpreted 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 assume as well that the second condition (π2 + ∆)\|φ〉 = 0 (4.14) must be imposed on the physical states, since it is also an invariant, and it is not affected by the evolution generated by (4.13). It can be shown that in the underlying classical theory [53], this condition 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 constant with dimension of M4, whose sign and value depend if π is space-like, time-like or null. Both equations permit to construct a generalized plane wave solution φ(X ′, θ′) ≡ 〈X ′, θ′\|φ〉 ∼ exp ( ikµX ′µ + i 2 Kµνθ ′µν ) , where k2 + m2 = 0 and K2 + ∆ = 0. In coordinate representation given by equations (4.10)– (4.12), the equation (4.13) gives just the Klein–Gordon equation( 2X −m2 ) φ(X ′, θ′) = 0 while (4.14) gives the supplementary equation (2θ −∆)φ(X ′, θ′) = 0, (4.15) where 2X = ∂µ∂µ, (4.16) with ∂µ = ∂ ∂X ′µ . (4.17) The Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 19 Also 2θ = 1 2 ∂µν∂µν , (4.18) with ∂µν = ∂ ∂θ′µν . (4.19) Both equations can be derived from the action S = ∫ d4X ′d6θ′ Ω(θ′) { 1 2 (∂µφ∂µφ+m2φ2)− Λ(2θ −∆)φ } . (4.20) In (4.20) Λ is a Lagrange multiplier necessary to impose condition (4.15). Ω(θ′) can be seen as a simple constant θ−6 0 to keep the usual dimensions of the fields as S must be dimensionless in natural units, as an even weight function as the one appearing in [17, 18, 19, 20, 21, 22, 23] used to make the connection between the formalism in D = 4+6 and the usual one in D = 4 after the integration 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 ′, C1 = p2 and C3 = π2, is given by S = ∫ d4X ′d6θ′Ω(θ′) 1 2 { ∂µφ∂µφ+ λ2 4 ∂µνφ∂µνφ+m2φ2 } , (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–Gordon action. Although, the Lorentz-invariant weight function Ω(θ) does not exist we can keep it temporally in some integrals in order to guarantee explicitly their existence. By borrowing the Dirac matrices ΓA, A = 0, 1, . . . , 9, written for spacetime D = 10, and identifying the tensor indices with the six last values of A, it is also possible to construct the “square root” of the equation of motion derived from (4.20), obtaining a generalized Dirac theory involving spin and noncommutativity [53]. It is an object of current investigation. Next we will construct the equations of motion and analyze the Noether’s theorem derived for general theories defined in x + θ space, and specifically for the action (4.20), considering Ω(θ) as a well behaved function. 4.3 Equations of motion and Noether’s theorem Let us consider the action S = ∫ R d4xd6θΩ(θ)L(φi, ∂µφ i, ∂µνφ i, x, θ), (4.22) relying on a set of fields φi, the derivatives with respect to xµ and θµν and the coordinates xµ and θµν themselves. From now on we will use x in place of X ′ and θ in place of θ′ in order to simplify the notation. Naturally the fields φi can be functions of xµ and θµν . The index i permits to treat φ in a general way. In (4.22) we consider, as in (4.20), the integration element modified by the introduction of Ω(θ). By assuming that S is stationary for an arbitrary variation δφi vanishing on the boundary ∂R of the region of integration R, we can write the Euler–Lagrange equation as Ω ( ∂L ∂φi − ∂µ ∂L ∂∂µφi ) − ∂µν ( Ω ∂L ∂∂µνφi ) = 0. (4.23) 20 E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. Zangirolami We will treat the variations δxµ, δθµν of the generalized coordinates and δφi of the fields such that the integrand transforms as a total divergence in the x+θ space, δ(ΩL) = ∂µ(ΩSµ)+∂µν(ΩSµν). Then the Noether’s theorem assures that, on shell, or when (4.23) is satisfied, there is a conserved current (jµ, jµν) defined by jµ = ∂L ∂∂µφi δφi + Lδxµ, jµν = ∂L ∂∂µνφi δφi + Lδθµν , (4.24) such that Ξ = ∂µ(Ωjµ) + ∂µν(Ωjµν) (4.25) vanishes. The corresponding charge Q = ∫ d3xd6θΩ(θ)j0 (4.26) is independent of the “time” x0. 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 just a trivial extension of the usual version of Noether’s theorem [45] in order to include θµν as independent 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 of the element of integration, but it is present in (4.25), which is the relevant divergence. It is also inside 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). The Lagrange equation reads δS δφ = −Ω ( 2−m2 ) φ− λ2 2 ∂µν(Ω∂µνφ) = 0 (4.27) and (4.25) can be written as Ξ = ∂µ { Ω∂µφδφ+ Ω 2 ( ∂αφ∂ αφ+ λ2 4 ∂αβφ∂ αβφ+m2φ2 ) δxµ } + ∂µν { Ωλ2∂µνφδφ+ Ω 2 ( ∂αφ∂ αφ+ λ2 4 ∂αβφ∂ αβφ+m2φ2 ) δθµν } . (4.28) Before using (4.28) we observe that the transformation δφ = −(aµ + ωµ νx ν) ∂µφ− 1 2 (bµν + 2ωµ ρθ ρν) ∂µνφ (4.29) closes in an algebra, as in (4.7), with the same composition rule defined in (4.8). The above equation 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 δax µ = aµ, δaθ µν = 0, δaφ = −aµ∂µφ, (4.30) where aµ are constants. We see from (4.28) and (4.30) that Ξa = aµ∂µφ δS δφ vanishing on shell, when (4.27) is valid. The Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 21 For a rigid θ-translation, we have that, δbx µ = 0, δbθ µν = bµν , δbφ = −1 2 bµν∂µνφ, where bµν are constants, we can write Ξb = 1 2 bµν ( ∂µνφ δS δφ + L∂µνΩ ) . 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 δωx µ = ωµ νx ν , δωθ µν = ωµ ρθ ρν + ων ρθ µρ, δωφ = − ( ωµ νx ν∂µ + ωµ ρθ ρν∂µν ) φ with constant and antisymmetric ωµ ν . We obtain Ξω = ωµ ν δS δφ (xν∂µφ+ θνρ∂µρφ) + L∂µνΩωµ αθ αν . The first term in the above expression vanishes on shell and the second one also vanishes if Ω is a scalar under Lorentz transformations and depends only on θ. In a complete theory where other contributions for the total action would be present, the symmetry under θ translations could be broken by different reasons, as in what follows, in the case of the NC U(1) gauge theory. In this situation P could be the symmetry group of the complete theory even considering Ω(θ) as a constant. 4.4 Considerations about the twisted Poincaré symmetry and DFRA space The twisted Poincaré (TP) algebra describes the symmetry of NC spacetime whose coordinates obey the commutation relation of a canonical type like (1.1). It was suggested [54] as a substitute for the Poincaré symmetry in field theories on the NC spacetime. In this formalism point of view, the Moyal product (1.2) is obtained as a twisted product of a module algebra of the TP algebra. This result shows the TP invariance of the NC field theories. This Poincaré algebra P for commutative QFT is given by the so-called Lie algebra composed by the ten generators of the Poincaré group [54] [Pµ,Pν ] = 0, [Mµν ,Pρ] = −i(gµρPν − gνσPµ), [Mµν ,Mρσ] = −i(gµρMνσ − gµσMνρ − gνρMµσ − gνσMµρ), (4.31) where the matrix M is antisymmetric Mµν = −Mνµ. The generators Mµν form a closed subalgebra, which is the Lie algebra of the Lorentz group. The generators of the Lorentz group can be divided into the group for boosts Ki = M0i, the spatial rotations group Ji = 1 2εijkMjk and the generators of translations Pµ, the last ones form a commutative subalgebra of the Poincaré algebra (the translation subgroup is Abelian). The presence of the imaginary unit i indicates that the generators of the Poincaré group are Hermitian. The Poincaré algebra generators are represented by Pµ = ∫ ddxT0µ(x), Mµν = ∫ ddx [xµT0ν(x)− xνT0µ(x)], where T0µ(x) = 1 2 [π(x)∂µφ(x)+∂µφ(x)π(x)]−g0µL(x) and π = ∂0 φ is the canonical momentum of φ0. It can be demonstrated that with this structure we can construct the representation of the Poincaré algebra in terms of the Hopf algebraic structure [55, 24]. 22 E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. Zangirolami 4.4.1 The twisted Poincaré algebra 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 Poincaré algebra. Chaichian et al. in [27] analyzed a solution to the problem utilizing the form of a twisted Poincaré symmetry. The introduction of a twist deformation of the universal enveloping algebra of the Poincaré algebra provided a new symmetry. The representation content of this twisted algebra is the same as the representation content of the usual Poincaré algebra [54]. To obtain the TP algebra we use the standard Poincaré algebra and its representation space and twist them. For example, in this twisted algebra the energy-momentum tensor is T0µ is given by T0µ = ∞∑ n=0 ( −1 2 )n 1 n! θi1j1 · · · θinjn∂i1 · · · ∂in × [ 1 2 (π ? ∂µφ(x) + ∂µφ(x) ? π(x))− g0µL ] Pj1 · · ·Pjn , where Pjn 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. The resulting operators satisfy commutation relations of Poincaré algebra (4.31). However, it can be shown [55, 24] that some identities involving momenta and the Hamiltonian in commutative and NCQFT are preserved in a deformed NCQFT. Also, we can use the same Hilbert space to represent the field operator for both commutative and deformed NCQFT [55, 24]. The action of 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. This is, in a nutshell, the structure of the Poincaré symmetry or of the Hopf algebra represented on a deformed NCQFT. The next step is to twist this algebra and its representation space. The Poincaré algebra P(A) has a subalgebra (commutative) composed by the translation genera- tors Pµ F = exp ( i 2 θµνPµPν ) , where θµν is a real constant antisymmetric matrix. This twist operator obviously satisfies the twist conditions that preserve the Hopf algebra structure [54]. Explicitly let us write the twist operator such as F = exp [ − i 2 θµν ∂ ∂xµ ⊗ ∂ ∂xν ] , F−1 = exp [ i 2 θµν ∂ ∂xµ ⊗ ∂ ∂xν ] , (4.32) where here ∂ ∂xµ and ∂ ∂xν are generally defined vector fields on space or spacetime. The twisting result is the twisted Poincaré algebra. It can be shown [55, 24] that the Poincaré covariance of a commutative QFT implies the twisted Poincaré covariance of a deformed NCQFT. In this deformed NCQFT the symmetry is described by a quantum group. An unavoidable comparison with the Moyal ?-product can be made and the conclusion is that they are in fact the same NC product of functions. Hence, the NCQFT constructed with Weyl quantization and Moyal ?-product possess the twisted Poincaré symmetry. We can say that, in NC theories, relativistic invariance means invariance under the twisted Poincaré transformations. The Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 23 4.4.2 The DFRA analysis As we said through the sections above, the DFRA structure constructs an extension of the Poincaré group P ′, which has the Poincaré group P as a subgroup. Then, obviously, the theories considered have to be invariant under both P and P ′. As the considerations described in this section the Poincaré algebra P has an Abelian subal- gebra which allows us to construct a twist operator F depicted in (4.32) which is an element of the quantum group theory. We believe that the subalgebra of the extended Poincaré 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 new deformed generators. Having the structure displayed in (4.32) in mind we have to formulate this new twist operator adding the (θ, π) sector terms in order to have the form of (4.32) with the twist operator as a special case. The formulation of this DFRA twist operator is beyond the scope of this review paper and is a target for further investigations. 5 Fermions and noncommutative theories By using the DFRA framework where the object of noncommutativity θµν 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 Poincaré group P as a subgroup. In this sec- tion we use the DFRA algebra to introduce a generalized Dirac equation, where the fermionic field depends not only on the ordinary coordinates but on θµν as well. The dynamical symmetry content of such fermionic theory will be discussed now and it is shown that its action is invariant under P ′. In the last sections above, we saw that the DFRA algebra implemented, in a NCQM frame- work1, the Poincaré invariance as a dynamical symmetry [45]. Of course this represents one among several possibilities of incorporating noncommutativity in quantum theories, we saw also that not only the coordinates xµ and their conjugate momenta pµ are operators acting in a Hilbert space H, but also θµν and their canonical momenta πµν are considered as Hilbert space operators as well. The proposed DFRA algebra is given by (1.1), (2.2)–(2.6) and (2.8). Where all 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 xµ can not be used to label possible basis in H. However, as the components of Xµ commute, we know from QM that their 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 with some detail and have proposed a way for constructing some actions representing possible field theories in this extended x+ θ spacetime. One of such actions has been given by S = − ∫ d4xd6θΩ(θ) 1 2 { ∂µφ∂µφ+ λ2 4 ∂µνφ∂µνφ+m2φ2 } , (5.1) where λ is a parameter with dimension of length, as the Planck length, which has to be introduced by 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 we used the definitions inequations (4.16)–(4.19) and ηµν = diag(−1, 1, 1, 1). The corresponding Lagrange equation reads, analogously as in (4.27), δS δφ = Ω ( 2−m2 ) φ+ λ2 2 ∂µν(Ω∂µνφ) = 0 (5.2) 1In [46] and [47] it is possible to find a large amount of references concerning NC quantum mechanics. 24 E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. Zangirolami and the action (5.1) is invariant under the transformation (4.29), where Ω is considered as a constant. If Ω is a scalar function of θ, the above transformation is only a symmetry of (5.1) (when bµν vanishes) which dynamically transforms P ′ to P [47]. We observe that (4.29) closes in an algebra, as in (4.7), with the same composition rule defined in (4.8). That equation defines how 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 Poincaré group P10 in D = 10. Denoting the indices A,B, . . . as spacetime indices in D = 10, A,B, . . . = 0, 1, . . . , 9, a vector Y A would transform under P10 as δY A = ωA BY 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, 1, 2, 3, considered as antisymmetric quantities, the transformation relations given above are rewritten as δY µ = ωµ νY ν + 1 2 ωµ αβY αβ + ∆µ, δY µν = ωµν αY α + 1 2 ωµν αβY αβ + ∆µν . With this notation, the (diagonal) D = 10 Minkowski metric is rewritten as ηAB = (ηµν , ηαβ,γδ) and the ordinary Clifford algebra {ΓA,ΓB} = −2ηAB as {Γµ,Γαβ} = 0, {Γµ,Γν} = −2ηµν , {Γµν ,Γαβ} = −2ηµν,αβ . (5.3) This is just a heavy way of writing usual D = 10 relations [3]. Now, by identifying Y A with (xµ, 1 λθ αβ), where λ is some parameter with length dimension, we see from the structure given above that the allowed transformations in P’ are those of P10, submitted to the conditions ωµν α = ω α µν = 0, ωµν αβ = 4ω[µ αδ ν] β, ∆µ = aµ, ∆αβ = 1 λ bαβ obviously keeping the identification between ωAB and ωµν when A = µ and B = ν. Of course we have now only 6 independent ω’s and 10 a’s and b’s. With the relations given above it is possible to extract the “square root” of the generalized Klein–Gordon equation (5.2)( 2 + λ22θ −m2 ) φ = 0 (5.4) assuming here that Ω is a constant. We will see in the next section with details that this equation can be interpreted as a dispersion relation in this D = 4+6 spacetime. Hence, this last equation furnish just the generalized Dirac equation[ i ( Γµ∂µ + λ 2 Γαβ∂αβ ) −m ] ψ = 0. (5.5) Let us apply from the left on (5.5) the operator[ i ( Γν∂µ + λ 2 Γαβ∂αβ ) +m ] . After using (5.3) we observe that ψ satisfies the generalized Klein–Gordon equation (5.4) as well. The covariance of the generalized Dirac equation (5.5) can also be proved. First we note that the operator Mµν = i 4 ( [Γµ,Γν ] + [Γµα,Γν α] ) The Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 25 gives the desired representation for the SO(1, 3) generators, because it not only closes in the Lorentz algebra (4.4), but also satisfies the commutation relations [Γµ,Mαβ ] = 2iδµ [αΓβ], [Γµν ,Mαβ ] = 2iδµ [αΓ ν β] − 2iδν [αΓ µ β] . With these relations it is possible to prove that (5.5) is indeed covariant under the Lorentz transformations given by ψ(x′, θ′) = exp ( − i 2 ΛµνMµν ) ψ(x, θ). By considering the complete P ′ group, we observe that the infinitesimal transformations of ψ are given by δψ = − [ (aµ + ωµ νx ν)∂µ + 1 2 (bµν + 2ωµ ρθ νρ)∂µν + i 2 ωµνMµν ] ψ, (5.6) which closes in the P ′ algebra with the same composition rule given by (4.8), what can be shown after a little algebra. At last we can show that also here there are conserved Noether’s currents associated with the transformation (5.6), once we observe that the equation (5.5) can be derived from the action S = ∫ d4xd6θΩ(θ)ψ̄ [ i ( Γµ∂µ + λ 2 Γαβ∂αβ ) −m ] ψ, (5.7) where we are considering that Ω = θ−6 0 and ψ̄ = ψ†Γ0. First we note that (suppressing trivial θ−6 0 trivial factors) δLS δψ̄ = [ i ( Γµ∂µ + λ 2 Γαβ∂αβ ) −m ] ψ, δRS δψ = −ψ̄ [ i ( Γµ←−∂ µ + λ 2 Γαβ←−∂ αβ ) +m ] , where L and R derivatives act from the left and right respectively. The current (jµ, jµν), analogously as in (4.24), is here written as jµ = ∂RL ∂∂µψ δψ + δψ̄ ∂LL ∂∂µψ̄ + Lδxµ, jµν = ∂RL ∂∂µνψ δψ + δψ̄ ∂LL ∂∂µνψ̄ + Lδθµν , (5.8) where δψ̄ = −ψ̄ [ ←− ∂ µ(aµ + ωµ νx ν) + ←− ∂ µν 1 2 (bµν + 2ωµ ρθ νρ)− i 2 ωµνMµν ] , (5.9) δψ is given in (5.6) and δxµ and δθµν have the same form found in (4.6). Using these last results one can show that, ∂µj µ + ∂µνj µν = − ( δψ̄ δLS δψ̄ + δRS δψ δψ ) , (5.10) which vanishes on shell, and hence the invariance of the action (5.7) under P ′. And we can conclude that it could be dynamically contracted to P, preserving the usual Casimir invariant structure characteristic of ordinary quantum field theories. Using (5.10) we can realize that there is a conserved charge Q = ∫ d3xd6θ j0, 26 E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. Zangirolami for each one of the specific transformations encoded in (5.8). Performing a simple t derivative we have that Q̇ = − ∫ d3xd6θ (∂i j i + ∂µνj µν), vanishes as a consequence of the divergence theorem. By considering only xµ translations, we can write j0 = j0µa µ, and consequently to define the momentum operator Pµ = − ∫ d3xd6θj0µ. Analyzing θµν translations and Lorentz transformations, we can derive in a similar way an explicit form for the other generators of P ′, here denoted by Πµν and Jµν . Under an appropriate bracket structure, following the Noether’s theorem, these conserved charges will generate the transformations (5.6) and (5.9). Summarizing, we can say that we have been able to introduce fermions satisfying a generalized Dirac equation, which is covariant under the action of the extended Poincaré group P ′. This Dirac equation has been derived through a variational principle whose action is dynamically invariant under P ′. This can justify possible roles played by theories involving noncommutativity in a way compatible with Relativity. Of course this is just a little step toward a field theory quantization program in this extended x+ θ spacetime, which is our next issue. 6 Quantum complex scalar fields and noncommutativity So far we saw that in a first quantized formalism, θµν and its canonical momentum πµν are seen as operators living in some Hilbert space. This structure is compatible with DFRA algebra and it is invariant under an extended Poincaré group of symmetry. In a second quantization scenario, we will reproduce in this section the results obtained in [50]. An explicit form for the extended Poincaré generators will be presented and we will see the same algebra is generated via generalized Heisenberg relations. We also introduce a source term and construct the general solution 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 to study 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 somehow different 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 weight function, generating Lorentz invariant theories and providing a connection with usual theories constructed 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 xµ and their canonical momenta pµ are considered as operators in a Hilbert space H, but also the objects of noncommutativity θµν and their canonical conjugate momenta πµν . All these operators belong to the same algebra and have the same hierarchical level, introducing a minimal canonical extension of the DFR algebra, i.e., the DFRA algebra introduced by R. Amorim in a first paper [46] followed by others [47, 48, 49, 50]. This enlargement of the usual set of Hilbert space operators allows the theory to be invariant under the rotation group SO(D), as showed in detail in the sections above [46, 49], when the treatment was a nonrelativistic one. Rotation invariance in a nonrelativistic theory, is fundamental if one intends to describe any physical system in a consistent way. In the last sections, the corresponding relativistic treatment was presented, which permits to implement Poincaré invariance as a dynamical symmetry [45] in NCQM.In The Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 27 this section we essentially consider the “second quantization” of the model discussed above [47], showing that the extended Poincaré symmetry here is generated via generalized Heisenberg relations, 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 action under some extended Poincaré (P ′) group. This symmetry structure is also displayed inside the second quantization level, constructed via generalized Heisenberg relations. After that, the fields are shown as expansions in a plane wave basis in order to solve the equations of motion using the Green’s functions formalism adapted for this new (x + θ) D = 4 + 6 space. It is assumed in this section that Ω(θ) defined in (4.20) is constant. 6.1 The action and symmetry relations An important point is that, due to (1.1), the operator xµ can not be used to label a possible basis in H. However, as the components of Xµ commute, as can be verified from the DRFA algebra 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] Mµν = Xµpν −Xνpµ − θµσπ ν σ + θνσπ µ σ as the generator of the Lorentz group, where Xµ = xµ + 1 2 θµνpν , and we see that the proper algebra is closed, i.e., [Mµν ,Mρσ] = iηµσMρν − iηνσMρµ − iηµρMσν + iηνρMσµ. Now Mµν generates the expected symmetry transformations when acting on all the operators in Hilbert space. Namely, by defining the dynamical transformation of an arbitrary operator A in H in such a way that δA = i[A,G], where G = 1 2 ωµνMµν − aµpµ + 1 2 bµνπµν , and ωµν = −ωνµ, aµ, bµν = −bνµ are infinitesimal parameters, it follows that δxµ = ωµ νx ν + aµ + 1 2 bµνpν , (6.1a) δXµ = ωµ νX ν + aµ, (6.1b) δpµ = ω ν µ pν , (6.1c) δθµν = ωµ ρθ ρν + ων ρθ µρ + bµν , (6.1d) δπµν = ω ρ µ πρν + ω ρ ν πµρ, (6.1e) δMµν = ωµ ρM ρν + ων ρM µρ + aµpν − aνpµ + bµρπ ν ρ + bνρπµ ρ, (6.1f) generalizing the action of the Poincaré group P in order to include θ and π transformations, i.e., P ′. The P ′ transformations close in an algebra, such that [δ2, δ1]A = δ3A, 28 E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. Zangirolami and the parameters composition rule is given by ωµ 3 ν = ωµ 1 αω α 2 ν − ω µ 2 αω α 1 ν , aµ 3 = ωµ 1 νa ν 2 − ω µ 2 νa ν 1 , bµν 3 = ωµ 1 ρb ρν 2 − ω µ 2 ρb ρν 1 − ω ν 1 ρb ρµ 2 + ων 2 ρb ρµ 1 . 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 way for 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 = − ∫ d4 x d6θ { ∂µφ∗∂µφ+ λ2 4 ∂µνφ∗∂µνφ+m2φ∗φ } , (6.2) where λ is a parameter with dimension of length, as the Planck length, which is introduced due to 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 δS δφ = ( 2 + λ22θ −m2 ) φ∗ = 0, (6.3) with a similar equation of motion for φ. The action (6.2) is invariant under the transformation δφ = −(aµ + ωµ νx ν) ∂µφ− 1 2 (bµν + 2ωµ ρθ ρν) ∂µνφ, (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. We observe that (6.1c) closes in an algebra, as in (4.7), with the same composition rule defined in (4.8). That equation defines how a complex scalar field transforms in the x + θ space under P ′. The transformation subalgebra generated by (6.1d) is of course Abelian, although it could be directly generalized to a more general setting. Associated with those symmetry transformations, we can define the conserved currents [47] jµ = ∂L ∂∂µφ δφ+ δφ∗ ∂L ∂∂µφ∗ + Lδxµ, jµν = ∂L ∂∂µνφ δφ+ δφ∗ ∂L ∂∂µνφ∗ + Lδθµν . Actually, by using (6.1c) and (6.1d), as well as (6.1b) and (6.1d), we can show, after some algebra, that ∂µj µ + ∂µνj µν = −δS δφ δφ− δφ∗ δS δφ∗ . The expressions above, as seen before, allow us to derive a specific charge Q = − ∫ d3xd6θ j0, for each kind of conserved symmetry encoded in (6.1c) and (6.1d), since Q̇ = ∫ d3xd6θ ( ∂ij i + 1 2 ∂µνj µν ) The Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 29 vanishes as a consequence of the divergence theorem in this (x, θ) extended space. Let us consider each specific symmetry in (6.1c) and (6.1d). For usual x-translations, we can write j0 = j0µa µ, and so we can define the total momentum Pµ = − ∫ d3xd6θ j0µ = ∫ d3xd6θ ( φ̇∗∂µφ+ φ̇∂µφ ∗ − Lδ0µ ) . (6.6) For θ-translations, we can write that j0 = j0µνb µν , and consequently, giving Pµν = − ∫ d3xd6θ j0µν = 1 2 ∫ d3xd6θ ( φ̇∗∂µνφ+ φ̇∂µνφ ∗). (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 j0 = j̄0µνω µν , we can write Mµν = − ∫ d3xd6θ j̄0µν = ∫ d3xd6θ ( φ̇∗∆νµφ+ φ̇∆νµφ ∗ − Lδ0[µxν] ) . (6.9) At last, for the symmetry given by (6.1d), we can write the U(1) charge as Q = i ∫ d3xd6θ (φ̇∗φ− φ̇φ∗). (6.10) Now let us show that these charges generate the appropriate field transformations (and dynamics) in a quantum scenario, as generalized Heisenberg relations. To start the quantization of such theory, we can define as usual the field momenta π = ∂L ∂φ̇ = φ̇∗, π∗ = ∂L ∂φ̇∗ = φ̇, (6.11) satisfying the non vanishing equal time commutators (in what follows the commutators are to be understood as equal time commutators) [π(x, θ), φ(x′, θ′)] = −iδ3(x− x′)δ6(θ − θ′), [π∗(x, θ), φ∗(x′, θ′)] = −iδ3(x− x′)δ6(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. After that we use (6.12) to dynamically generate the symmetry operations. In this spirit, accordingly to (6.6) and (6.11), the spatial translation is generated by Pi = ∫ d3xd6θ ( π∂iφ+ π∗∂iφ ∗), and it is trivial to verify, by using (6.12), that [Pi,Y(x, θ)] = −i∂iY(x, θ), where Y represents φ, φ∗, π or π∗. The dynamics is generated by P0 = ∫ d3xd6θ ( π∗π + ∂iφ∗∂iφ+ λ2 4 ∂µνφ∗∂µνφ+m2φ∗φ ) accordingly to (6.6) and (6.11). 30 E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. Zangirolami With the Lagrangian given in (6.2) we can have that πµν = ∂L ∂(∂µνφ) = λ2 4 ∂µνφ ∗, π∗µν = ∂L ∂(∂µνφ∗) = λ2 4 ∂µνφ 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 ∂µνφ∗∂µνφ ≥ 0 to assure that the Hamiltonian H = P0 is positive definite. This condition can also be written as πµνπ∗µν ≥ 0, 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 Heisenberg relation [P0,Y(x, θ)] = −i∂0Y(x, θ). The θ-translations, accordingly to (6.7) and (6.11), are generated by Pµν = 1 2 ∫ d3xd6θ ( π∂µνφ+ π∗∂µνφ ∗), and one obtains trivially by (6.12) that [Pµν ,Y(x, θ)] = −i∂µνY(x, θ). Lorentz transformations are generated by (6.9) in a similar way. The spatial rotations gen- erator is given by Mij = ∫ d3xd6θ ( π∆jiφ+ π∗∆jiφ ∗), (6.13) while the boosts are generated by M0i = 1 2 ∫ d3xd6θ { π∗πxi − x0 ( π∂iφ+ π∗∂iφ ∗)+ π ( 2θ γ [i ∂0]γ − x0∂i ) φ + π∗ ( 2θ γ [i ∂0]γ − x0∂i ) φ∗ + ( ∂jφ ∗∂jφ+ λ2 4 ∂µνφ∗∂µνφ+m2φ∗φ ) xi } . (6.14) As can be verified in a direct way for (6.13) and in a little more indirect way for (6.14) [Mµν ,Y(x, θ)] = i∆µνY(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 ∫ d3xd6θ ( πφ− π∗φ∗ ) , 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 = 1 2 ωµνM µν − aµPµ + 1 2 bµνPµν − αQ The Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 31 over 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 are also four Casimir operators defined with the operators given above, with the same form as those previously defined at a first quantized perspective. So, the structure displayed above is very similar to the usual one found in ordinary quantum complex scalar fields. We can go one step further, 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 related to free bosonic fields. In what follows we consider some of these issues and postpone others for forthcoming works. 6.2 Plane waves and Green’s functions In order to evaluate a little more the framework described in the last sections, let us rewrite the generalized charged Klein–Gordon action (6.2) with source terms as S = − ∫ d4xd6θ { ∂µφ∗∂µφ+ λ2 4 ∂µνφ∗∂µνφ+m2φ∗φ+ J∗φ+ Jφ∗ } . (6.15) The corresponding equations of motion are( 2 + λ22θ −m2 ) φ(x, θ) = J(x, θ) (6.16) as well as its complex conjugate one. We have the following formal solution φ(x, θ) = φJ=0(x, θ) + φJ(x, θ), where, clearly, φJ=0(x, θ) is the source free solution and φJ(x, θ) is the solution with J 6= 0. The Green’s function for (6.16) satisfies( 2 + λ22θ −m2 ) G(x− x′, θ − θ′) = δ4(x− x′)δ6(θ − θ′), (6.17) where δ4(x− x′) and δ6(θ − θ′) are the Dirac’s delta functions δ4(x− x′) = 1 (2π)4 ∫ d4K(1)e iK(1)·(x−x′), (6.18) δ6(θ − θ′) = 1 (2π)6 ∫ d6K(2)e iK(2)·(θ−θ′). (6.19) Now let us define in D = 10 X = ( xµ, 1 λ θµν ) (6.20) and K = ( Kµ (1), λK µν (2) ) , (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)µx µ + 1 2 K(2)µνθ µν . 32 E.M.C. Abreu, A.C.R. Mendes, W. Oliveira and A.O. Zangirolami The factor 1 2 is introduced in order to eliminate repeated terms. In what follows it will also be considered that d10K = d4K(1)d 6K(2) and d10X = d4xd6θ. So, from (6.16) and (6.17) we formally have that φJ(X) = ∫ d10X ′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 of plane waves. Hence, we can write that G(X −X ′) = 1 (2π)10 ∫ d10K G̃(K)eiK·(X−X′). (6.23) Now, from (6.17), (6.18), (6.19) and (6.23) we obtain that, ( 2 + λ22θ −m2 ) ∫ d10K (2π)10 G̃(K)eiK·(x−x′) = ∫ d10K (2π)10 eiK·(x−x′) giving the solution for G̃(K) as G̃(K) = − 1 K2 +m2 (6.24) where, from (6.21), K2 = K(1)µK µ (1) + λ2 2 K(2)µνK µν (2). Substituting (6.24) in (6.23) we obtain G(x− x′, θ − θ′) = 1 (2π)10 ∫ d9K ∫ dK0 1 (K0)2 − ω2 eiK·(x−x′), (6.25) where the “frequency” in the (x+ θ) space is defined to be ω = ω( ~K(1),K(2)) = √ ~K(1) · ~K(1) + λ2 2 K(2)µνK µν (2) +m2, (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 K0 = ±ω in this framework. Of course we can construct an analogous solution for φ∗J(x, θ). In general, the poles of the Green’s function can be interpreted as masses for the stable particles described by the theory. We can see directly from equation (6.26) that the plane waves in the (x+ θ) space establish the interaction between the currents in this space and have energy given by ω( ~K(1),K(2)) since ω2 = ~K2 (1) + λ2 2 K2 (2) +m2 = K2 (1,2) +m2, The Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space 33 where K2 (1,2) = ~K2 (1) + λ2 2 K2 (2). So, one can say that the plane waves that mediate the interaction describe the propagation of particles in a x+ θ spacetime with a mass equal to m. We ask if we can use the Cauchy residue theorem in this new space to investigate the contributions of the poles in (6.25). Accordingly to the point described above, we can assume that the Hamiltonian is positive definite and it is directly related to the hypothesis that K2 (1,2) = −m2 < 0. However if the observables are constrained to a four dimensional spacetime, due to some kind of compactification, the physical mass can have contributions from the NC sector. This point is left for a forthcoming work [53], when we will consider the Fock space structure of the theory and possible schemes for compactification. For completeness, let us note that substituting (6.22) and (6.25) into (6.15), we arrive at the effective action Seff = − ∫ d4xd6θd4x′d6θ′ J∗(X) ∫ d9K (2π)10 ∫ dK0 1 (K0)2 − ω2 + iε eiK·(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 takes into account noncommutativity without destroying the symmetry content of the corresponding commutative 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 space sector associated with noncommutativity can not assessed, ground state effects could in principle be detectable. To end this revision work we have considered in this section, complex scalar fields using a new framework where the object of noncommutativity θµν represents independent degrees of freedom.We have started from a first quantized formalism, where θµν and its canonical mo- mentum πµν are considered as operators living in some Hilbert space. This structure, which is compatible with the minimal canonical extension of the Doplicher–Fredenhagen–Roberts– Amorim (DFRA) algebra, is also invariant under an extended Poincaré group of symmetry, but keeping, among others, the usual Casimir invariant operators. After that, in a second quantized formalism perspective, we explained an explicit form for the extended Poincaré generators and the same algebra of the first quantized description has been generated via generalized Heisenberg relations. 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The Noncommutative Doplicher-Fredenhagen-Roberts-Amorim Space

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