Some aspects of quantum mechanics and field theory in a Lorentz invariant noncommutative space
aa r X i v : . [ h e p - t h ] D ec Some aspects of quantum mechanics and field theory in a Lorentzinvariant noncommutative space
Everton M. C. Abreu a,b,c ∗ and M. J. Neves a † a Grupo de F´ısica Te´orica e Matem´atica F´ısica,Departamento de F´ısica,Universidade Federal Rural do Rio de JaneiroBR 465-07, 23890-971,Serop´edica, Rio de Janeiro, Brazil b LAFEX, Centro Brasileiro de Pesquisas F´ısicas (CBPF),Rua Xavier Sigaud 150,Urca, 22290-180, RJ, Brazil c Departamento de F´ısica, ICE,Universidade Federal de Juiz de Fora,36036-330, Juiz de Fora, MG, BrazilJuly 31, 2018
Abstract
We obtained the Feynman propagators for a noncommutative (NC) quantum mechanics definedin the recently developed Doplicher-Fredenhagen-Roberts-Amorim (DFRA) NC background thatcan be considered as an alternative framework for the NC spacetime of the early Universe. Theoperators formalism was revisited and we applied its properties to obtain a NC transition am-plitude representation. Two examples of DFRA’s systems were discussed, namely, the NC freeparticle and NC harmonic oscillator. The spectral representation of the propagator gave us theNC wave function and energy spectrum. We calculated the partition function of the NC harmonicoscillator and the distribution function. Besides, the extension to NC DFRA quantum field theoryis straightforward and we used it in a massive scalar field. We had written the scalar action withself-interaction φ using the Weyl-Moyal product to obtain the propagator and vertex of this modelneeded to perturbation theory. It is important to emphasize from the outset is that the formalismdemonstrated here will not be constructed introducing a NC parameter in the system, as usual.It will be generated naturally from an already NC space. In this extra dimensional NC space, wepresented also the idea of dimensional reduction to recover commutativity. PACS numbers: 11.15.-q; 11.10.Ef; 11.10.NxKeywords: Noncommutativity; quantum mechanics; field theory ∗ Electronic address: [email protected] † Electronic address: [email protected] uantum mechanics and field theory in a Lorentz invariant noncommutative space I. INTRODUCTION
There are theoretical evidences that make us to expect that at very small scales thespacetime acquires a foam-like (fuzzy) structure [1]. This foam-like framework, at thattime, was supposed to eliminate the problem of infinities in quantum field theory. However,it is well known now that noncommutativity, due to its Planck scale feature, introducesthis foam-like structure in spacetime, which is characteristic of quantum gravity, where oneshould expect a very large fluctuation of the metric and even of the topology of the spacetimemanifolds on short length scales [1]. Hence, to construct a NC spacetime implies directly toobtain this foam-like structure.However, a long time before the foam-like space ideas of Hawking and Wheeler, the beliefthat a noncommutative (NC) spacetime, instead of a continuous one, could free quantumfield theory (QFT) from the divergences that habit within it. This concept evolved throughthe years and the first published work concerning a NC concept of spacetime was carried outin 1947 by Snyder in his seminal paper [2]. The need to control the ultraviolet divergencesin quantum field theory (QFT) was the first motivation to consider a NC spacetime. Themain NC idea is that the spacetime coordinates x µ ( µ = 0 , , ,
3) are promoted to operatorsˆ x µ in order to satisfy the basic commutation relation[ ˆ x µ , ˆ x ν ] = i ℓθ µν , (1.1)where θ µν is an antisymmetric constant matrix, and ℓ is a length scale. The alternativewould be to construct a discrete spacetime with a NC algebra. Consequently, the coordinatesoperators are quantum observable that satisfy the uncertainty relation∆ˆ x µ ∆ˆ x ν ≃ ℓ θ µν , (1.2)it leads to the interpretation that noncommutativity of spacetime must emerge in a funda-mental length scale ℓ , the Planck scale , for example.However, Yang in [3], a little time later, demonstrated that Snyder’s hopes in cutting offthe infinities in QFT were not obtained by noncommutativity. This fact doomed Snyder’sNC theory to years of ostracism. After the important result that the algebra obtained witha string theory embedded in a magnetic background is NC, a new perspective concerningnoncommutativity was rekindle [4]. Nowadays the NC quantum field theory (NCQFT) isone of the most investigated subjects about the description of a physics at a fundamentallength scale of quantum gravity [5].The most popular noncommutativity formalism consider θ µν as a constant matrix, aswe said before. Although it maintains the translational invariance, the Lorentz symmetryis not preserved [6]. To heal this disease a recent approach was introduced by Doplicher,Fredenhagen and Roberts (DFR) [7]. It considers θ µν as an ordinary coordinate of thesystem in which the Lorentz symmetry is preserved. Recently, it has emerged the idea[8] of constructing an extension of the DFR spacetime introducing the conjugate canonicalmomenta associated with θ µν [9] (for a review the reader can see [10]). This extended NCspacetime has ten dimensions: four relative to Minkowski spacetime and six relative to θ -space. This new framework is characterized by a field theory constructed in a spacetime withextra-dimensions (4 + 6), and which does not need necessarily the presence of a length scale ℓ localized into the six dimensions of the θ -space, where θ µν has dimension of length-square. uantum mechanics and field theory in a Lorentz invariant noncommutative space x + θ )-spacetime must also be preserved [11]. The quantum field theorydefined in DFR space is well known in the literature [8, 12–15], but it does not take intoaccount any propagation of fields in the extra θ -dimension. In this paper we are interestedin analyzing the consequences of the propagation of fields in this θ -direction.By following this framework, a new version of NC quantum mechanics (NCQM) was in-troduced. In this formalism not only the coordinates x µ and their canonical momenta p µ areconsidered as operators ˆ x µ and ˆ p µ in a Hilbert space H , but the operators of noncommuta-tivity ˆ θ µν also have their canonical conjugate momenta operators ˆ π µν [9, 16–18]. All theseoperators belong to the same algebra and have the same hierarchical level, introducing aminimal canonical extension of DFR algebra, the so-called Doplicher-Fredenhagen-Roberts-Amorim (DFRA) formalism. This enlargement of the standard set of Hilbert space operatorspermits us to consider the theory to be invariant under the rotation group SO(D) treating itnonrelativisticaly. Rotation invariance is underlying when we are treating with nonrelativis-tic theories in order to depict consistently any physical system. In [16, 17] the relativistictreatment is introduced, which allows one to deal with the Poncar`e invariance as a dynamicalsymmetry [19] in NCQM [9].If the θ parameter is treated as a constant matrix, the non relativistic theory is notinvariant under rotation symmetry. This enlargement of the usual set of Hilbert spaceoperators allows the theory to be invariant under the rotation group SO ( D ), as showed indetail in [9, 18], where the treatment is a non relativistic one. Rotation invariance in a nonrelativistic theory is the main ingredient if one intends to describe any physical system in aconsistent way. In the first papers that treated the DFRA formalism, the main motivation,as we said before, was to construct a NC standard QM.The paper is organized as: the next section is dedicated (for self-containment of this work)to a review of the basics of QM in this NC DFRA framework, namely, the DFRA algebra.In the third section we apply the basics of DFRA NCQM to calculate the propagator viaequation for the time evolution of a quantum state. In the fourth section we discussed theexamples the free particle and the harmonic oscillator. We calculated the partition functionand the energy spectrum of the isotropic harmonic oscillator. The fifth section is dedicatedto field theory using the DFRA formalism in a well known standard model using the Weyl-Moyal product. To finish, we discussed the results obtained and we made the final remarksand conclusions. II. QUANTUM MECHANICS IN
DF RA
NC SPACE
In this section, in order to maintain this work self-contained, we will review the mainsteps published in [9, 16–18]. Namely, we will revisit the basics of the NCQM defined inthe DFRA space. We assume that this space has D ≥ x i ( i = 1 , , ..., D ) and ˆ p i are the position operator and its conjugated momentum, respectively.They satisfy the usual commutation relation[ˆ x i , ˆ p j ] = iδ ij , (2.1)where we has adopted the Natural units ( ~ = c = 1) and ℓ = 1. Having said that, in NCQMwe can rewrite the commutation relation between the position operator, that is[ˆ x i , ˆ x j ] = iθ ij , (2.2) uantum mechanics and field theory in a Lorentz invariant noncommutative space θ ij is an antisymmetric matrix. In DFR formalism θ ij is considered as a space co-ordinate, and consequently it is promoted to a position operator ˆ θ ij in θ -space. Thereforethis assumption leads us to a space with coordinates of position (cid:16) ˆ x i , ˆ θ ij (cid:17) , in which θ ij has D ( D − / θ ij are coordinates, the commutation relationsare assumed to be [ˆ x i , ˆ x j ] = i ˆ θ ij , h ˆ x i , ˆ θ jk i = 0 and h ˆ θ ij , ˆ θ kℓ i = 0 . (2.3)Moreover exist the canonical conjugate momenta operator ˆ π ij associated with the operatorˆ θ ij , and they must satisfy the commutation relation h ˆ θ ij , ˆ π kℓ i = iδ ijkℓ , (2.4)where δ ijkℓ = δ ik δ jℓ − δ iℓ δ jk . In order to obtain consistency we can write that [9][ˆ p i , ˆ p j ] = 0 , [ˆ p i , ˆ θ jk ] = 0 , [ˆ p i , ˆ π jk ] = 0 , (2.5)and this completes the DFRA algebra.The Jacobi identity formed by the operators ˆ x i , ˆ x j and ˆ π kl leads to the nontrivial relation[[ˆ x i , ˆ π kl ] , ˆ x j ] − [[ˆ x j , ˆ π kl ] , ˆ x i ] = − δ ijkl , (2.6)which solution, not considering trivial terms, is given by[ˆ x i , ˆ π jk ] = − i δ jkil ˆ p l . (2.7)It is possible to verify that the whole set of commutation relations listed above is indeedconsistent under all possible Jacobi identities and the CCR algebras [11]. Expression (2.7)suggests that the shifted coordinate operator [37–41]ˆ X i = ˆ x i + i θ ij ˆ p j , (2.8)commutes with ˆ π kl . The relation (2.8) is also known as Bopp shift in the literature. Thecommutation relation (2.7) also commutes with ˆ θ kl and ˆ X j , and satisfies a non trivial com-mutation relation with ˆ p i dependent objects, which could be derived from[ ˆ X i , ˆ p j ] = iδ ij , [ ˆ X i , ˆ X j ] = 0 and [ ˆ P i , ˆ P j ] = 0 , (2.9)where ˆ P i = ˆ p i and the property ˆ p i ˆ X i = ˆ p i ˆ x i is easily verified. Hence, we see from these bothequations that the shifted coordinated operator (2.8) allows us to recover the commutativity.The shifted coordinate operator ˆ X i plays a fundamental role in NC quantum mechanicsdefined in the ( x + θ )-space, since it is possible to form a basis with its eigenvalues. Thispossibility is forbidden for the usual coordinate operator ˆ x i since its components satisfynontrivial commutation relations among themselves (2.1). So, differently from ˆ x i , we cansay that ˆ X i forms a basis in Hilbert space. This fact will be important very soon. uantum mechanics and field theory in a Lorentz invariant noncommutative space SO ( D ). It is a fact that the usual orbital angular momentum operatorˆ ℓ ij = ˆ x i ˆ p j − ˆ x j ˆ p i , (2.10)does not closes in an algebra due to (2.2), that is h ˆ ℓ ij , ˆ ℓ kl i = iδ il ˆ ℓ kj − iδ jl ˆ ℓ ki − iδ ik ˆ ℓ lj + iδ jk ˆ ℓ li + i ˆ θ il ˆ p k ˆ p j − i ˆ θ jl ˆ p k ˆ p i − i ˆ θ ik ˆ p l ˆ p j + i ˆ θ jk ˆ p l ˆ p i , (2.11)and so their components cannot be SO ( D ) generators in this extended Hilbert space. It iseasy to see that the operator ˆ L ij = ˆ X i ˆ p j − ˆ X j ˆ p i , (2.12)closes in the SO ( D ) algebra. Besides, this result can be generalized to the total angularmomentum operator ˆ J ij = ˆ X i ˆ p j − ˆ X j ˆ p i + ˆ θ jl ˆ π li − ˆ θ il ˆ π lj , (2.13)that closes the algebra h ˆ J ij , ˆ J kl i = δ il ˆ J kj − δ jl ˆ J ki − δ ik ˆ J lj + δ jk ˆ J li , (2.14)and ˆ J ij generates rotation in Hilbert space.Now we return to the discussion about the basis in this NCQM. It is possible to introducea continuous basis for a general Hilbert space searching by a maximal set of commutatingoperators. The physical coordinates represented by the positions operators ˆ x i do not com-mute and their eigenvalues cannot be used to form a basis in the Hilbert space H . This doesnot occur with the shifted operators ˆ X i (2.8), and consequently, their eigenvalues are usedin the construction of such basis. Therefore one can use the shifted position operators ˆ X i as coordinate basis, although ˆ x i be the physical position operator. The noncommutativityof this space stays registered by the presence of the operator θ as a spatial coordinate ofthe system. A coordinate basis formed by the eigenvectors of ( ˆ X, ˆ θ ) can be introduced, andfor the momentum basis one chooses the eigenvectors of (ˆ p, ˆ π ). Let | X ′ , θ ′ i = | X ′ i ⊗ | θ ′ i and | p ′ , k ′ i = | p ′ i ⊗ | k ′ i be the position and momenta states in this ( x + θ )-space where thefundamental relations involving each basis areˆ X i | X ′ , θ ′ i = X ′ i | X ′ , θ ′ i , ˆ θ ij | X ′ , θ ′ i = θ ′ ij | X ′ , θ ′ i (2.15)ˆ p i | p ′ , k ′ i = p ′ i | p ′ , k ′ i , ˆ π ij | p ′ , k ′ i = k ′ ij | p ′ , k ′ i , (2.16) Z d D X ′ d D ( D − / θ ′ | X ′ , θ ′ ih X ′ , θ ′ | = , (2.17) Z d D p ′ (2 π ) D d D ( D − / k ′ (2 π ) D ( D − / | p ′ , k ′ ih p ′ , k ′ | = , (2.18) uantum mechanics and field theory in a Lorentz invariant noncommutative space h X ′ , θ ′ | X ′′ , θ ′′ i = δ ( D ) ( x ′ − x ′′ ) δ D ( D − / ( θ ′ − θ ′′ ) , (2.19) h p ′ , k ′ | p ′′ , k ′′ i = (2 π ) D δ ( D ) ( p ′ − p ′′ ) (2 π ) D ( D − / δ D ( D − / (cid:0) k ij − k ′ ij (cid:1) , (2.20)that are the eigenvalues equations, completeness relations and orthogonality, respectively.Concerning the last relations and the operators representation we can write that [9] h X ′ , θ ′ | ˆ p i | X ′′ , θ ′′ i = − i ∂∂ x ′ i δ ( D ) ( x ′ − x ′′ ) δ D ( D − / ( θ ′ − θ ′′ ) , (2.21)and h X ′ , θ ′ | ˆ π ij | X ′′ , θ ′′ i = δ ( D ) ( x ′ − x ′′ ) ( − i ) ∂∂θ ′ ij δ D ( D − / ( θ ′ − θ ′′ ) . (2.22)It is important to pay attention to the notation. It is obvious that the prime and doubleprime notation indicates two different points in ( x + θ )-space. But the meaning is the same,i.e., two different points in ( x + θ )-space.The transformations (2.22) from one basis to the other are constructed using extendedFourier transforms. The wave plane defined in this ( x + θ )-space is obtained by internalproduct between position and momentum states h X ′ , θ ′ | p ′′ , k ′′ i = e i ( p ′′ · x ′ + k ′′ · θ ′ ) , (2.23)where p ′′ · x ′ + k ′′ · θ ′ = p ′′ i x i ′ + k ′′ ij θ ij ′ /
2, and we have used the property of scalar product p · X = p · x . Others properties of this NCQM are explored in more details in [9]. III. FEYNMAN PROPAGATOR IN NC QUANTUM MECHANICS
To apply the basics of DFRA NCQM to a path integral formalism, for simplicity weconsider a space of D = 3, i.e. , we have three independent coordinates associated to θ ij plus three usual position coordinates ˆ x i ( i = 1 , , ij ) in θ ij indicate onepoint in θ -space formed by ( θ , θ , θ ) coordinates. The same notation will be used for themomentum k ij , say ( k , k , k ). Thus we will work in a six dimensional space.The time evolution of a quantum state | ψ i is governed by the dynamical equation i ddt | ψ ( t ) i = ˆ H | ψ ( t ) i , (3.1)where ˆ H is the Hamiltonian operator. The solution of equation (3.1) is the expression | ψ ( t ) i = e − i ˆ H ( t − t ′ ) | ψ ( t ′ ) i , (3.2)that represents the time evolution of a state | ψ i in a time interval t − t ′ > X , θ ) and ( X ′ , θ ′ ) in this ( x + θ ) − space, and the Hamiltonian operator ˆ H is consideredtime independent. The coordinates like θ and θ ′ indicate that we are considering two differentpoints in space, for instance, It is like the prime and double prime in Eqs. (2.19)-(2.23) andso on. uantum mechanics and field theory in a Lorentz invariant noncommutative space | X , θ i be a position quantum state in ( X , θ ), we operate it in (3.2) to obtain h X , θ | ψ ( t ) i = h X , θ | e − i ˆ H ( t − t ′ ) | ψ ( t ′ ) i , (3.3)and introducing the identity (2.17) we have ψ ( X , θ ; t ) = Z d X ′ d θ ′ K ( X , θ ; X ′ , θ ′ ; t − t ′ ) ψ ( X ′ , θ ′ ; t ′ ) , (3.4)which provides the transition of the particle wave-function ψ between the points ( X , θ ) and( X ′ , θ ′ ) in the ( x + θ ) space, and K is the Feynman propagator of this transition K ( X , θ ; X ′ , θ ′ ; t − t ′ ) := h X , θ | e − i ˆ H ( t − t ′ ) | X ′ , θ ′ i . (3.5)Hence, the propagator (3.5) satisfies the Green equation (cid:26) i ∂∂τ − ˆ H (cid:16) ˆ X i , ˆ p i ; ˆ θ ij , ˆ π ij (cid:17)(cid:27) K ( X, θ ; X ′ , θ ′ ; τ ) = δ (3) ( x − x ′ ) δ (3) ( θ − θ ′ ) δ ( τ ) , (3.6)where we assumed the condition K ( X, θ ; X ′ , θ ′ ; τ ) = 0 when τ = t − t ′ < . (3.7)The Hamiltonian operator ˆ H is a function of the position in the six-dimensional space ( ˆ X i , ˆ θ ij )and of the momenta operators (ˆ p i , ˆ π ij ) discussed in the last section. The spectral represen-tation of the propagator is obtained by inserting the complete set = X n i , e n i | n i ; e n i ih n i ; e n i | = X n i | n i ih n i | X e n i | e n i ih e n i | (3.8)in the definition (3.5), so we can write that K ( X , θ ; X ′ , θ ′ ; τ ) := Θ( τ ) X n i Φ n i , e n i ( X , θ )Φ ∗ n i , e n i ( X , θ ) e − iE ni, e ni τ , (3.9)where Φ n i , e n i ( X , θ ) := φ n i ( X ) ξ e n i ( θ ), and φ n i ( X ) := h X | n i i , ξ e n i ( θ ) := h θ | e n i i are the wavefunctions of the system, and E n i , e n i the energy spectrum. The eigenvectors | n i ; e n i i = | n i i⊗| e n i i are the eigenstates of spectrum energy, say ˆ H | n i ; e n i i = E n i , e n i | n i ; e n i i . It is important to noticethat we are constructing these eigenvectors and its respective eigenstates in the ( X, θ ) NCspace. Namely, in the extended NC Hilbert space H ′ so that the general Hilbert space isgiven by H ( X, θ ) = H ⊕ H ′ .By using the definitions of operators of the DFRA algebra and basis on the Hilbert space H , the representation of the Feynman as a functional integral over configuration space is h X , θ | e − i ˆ H ( t − t ′ ) | X ′ , θ ′ i = N Z D X (2 π ) D θ (2 π ) e iS ( X ,θ ) , (3.10)where we sum of all possible transition amplitudes between points ( X , θ ) and ( X ′ , θ ′ ) of thespace X + θ . Here N is just a normalization constant, S ( X , θ ) is the action integral S ( X , θ ) = Z t ′ t dt ′′ L ( ˙ X , ˙ θ ) , (3.11) uantum mechanics and field theory in a Lorentz invariant noncommutative space L is the Lagrangian function of the system L ( X , ˙ X ; θ, ˙ θ ) = 12 m ˙ X + 12 Λ ˙ θ − V ( X , θ ) . (3.12)Here the parameter Λ has dimension of mass [9]. As we expected, the representation ofthe path integral is given by the functional integration over the configuration space-( X + θ )of the exponential function of the action integral. The function W is a measure in the θ -integration, and consequently, it attenuates the functional integral on the θ -space. Naturally,what emerges in this result is the DFRA-Lagrangian function of the system. In the nextsection we apply it to some simple examples, as free particle and the isotropic harmonicoscillator. IV. EXAMPLES
In this section we will exemplify the formalism, developed before, using two simple sys-tems: the NC free particle and the NC harmonic oscillator. For the first case we have thepropagator as the matrix element (3.5), where we insert the completeness relation in themomentum space (2.18) to obtain the integrals K ( x , θ ; x ′ , θ ′ ; τ ) = h X , θ | e − i ˆ H τ | X ′ , θ ′ i = Z d p (2 π ) e i p · ( x − x ′ ) − iτ m p Z d k (2 π ) e i k · ( θ − θ ′ ) − iτ k , (4.1)where ˆ H is the Hamiltonian operator of the free particle in ( x + θ )-space (for simplicity wewill use from now on x meaning X )ˆ H (ˆ p i , ˆ π ij ) = ˆ p i m + ˆ π ij . (4.2)Using the Gaussian integrals, the free particle propagator is K ( x − x ′ , θ − θ ′ ; t − t ′ ) = i ( m Λ) / ( t − t ′ ) exp " im ( x − x ′ ) + i Λ ( θ − θ ′ ) t − t ′ ) , (4.3)with the condition t − t ′ > X and thefree propagator in the space- θ .As a second example we have the Hamiltonian of the NC isotropic harmonic oscillator(IHO) [9] ˆ H ( IHO ) ( ˆ X i , ˆ p i ; ˆ θ ij , ˆ π ij ) = ˆ p i m + 12 mω ˆ X i + ˆ π ij
2Λ + 12 ΛΩ ˆ θ ij , (4.4)where ω and Ω are the oscillation frequencies in the spaces ( X , θ ), respectively. The operatorHamiltonian (4.4) is the sum of usual operator Hamiltonian of the harmonic oscillator and NCHamiltonian, and consequently, the propagator is the product of usual harmonic oscillator uantum mechanics and field theory in a Lorentz invariant noncommutative space K ( IHO ) ( X , θ ; X ′ , θ ′ ; τ ) = (cid:18) mω πi sin( ωτ ) (cid:19) / exp (cid:20) imω ωτ ) (cid:0) cos( ωτ ) (cid:0) X + X ′ (cid:1) − X · X ′ (cid:1)(cid:21) × (cid:18) ΛΩ2 πi sin(Ω τ ) (cid:19) / exp (cid:20) i ΛΩ2 sin(Ω τ ) (cid:0) cos(Ω τ ) (cid:0) θ + θ ′ (cid:1) − θ · θ ′ (cid:1)(cid:21) , (4.5)like in the free propagator case, where τ is defined as τ := t − t ′ . It is easy to write thisexpression of the propagator in the spectral form (3.9), and so we obtain the wave functionΦ ( n n n ; e n e n e n ) ( X , θ ) = (cid:16) mωπ (cid:17) / Y i =1 e − mω X i H n i ( √ mωX i ) √ n i n i ! ×× (cid:18) ΛΩ π (cid:19) / Y j =1 e − ΛΩ4 θ j H n j (cid:16) √ ΛΩ θ j (cid:17)p n j n j ! , (4.6)where H n is the Hermite function, and the energy spectrum E ( n n n ; e n e n e n ) = X i =1 (cid:18) n i + 12 (cid:19) ω + X i =1 (cid:18)e n i + 12 (cid:19) Ω (4.7)for the NC isotropic harmonic oscillator. For the ground state, we have the wave functionΦ (0;0) ( X , θ ) = (cid:16) mωπ (cid:17) / e − mω X (cid:18) ΛΩ π (cid:19) / e − ΛΩ4 θ ij θ ij . (4.8)An important point concerning (4.8) is the natural introduction of a weight function W inthe θ -sector that appears in the context of NC QFT [8, 12, 14]. It must be connected to θ -integral as an integration measure. We observe explicitly by calculating the expected valueof any function f over the fundamental state h f ( X , θ ) i = Z d X d θ Φ ∗ ( X , θ ) f ( X , θ )Φ( X , θ ) == (cid:16) mωπ (cid:17) / Z d X e − mω X Z d θ W ( θ ) f ( X , θ ) , (4.9)where W ( θ ) is W ( θ ) = (cid:18) ΛΩ π (cid:19) / e − ΛΩ2 θ ij θ ij . (4.10)We have some properties of the function W [8,12,14,15] h i = 1 , h θ ij i = 0 , h θ ij θ kl i = h θ i δ [ ij,kl ] , (4.11)with h θ i = (2ΛΩ) − . More details about the W -function will be discussed in the nextsection. uantum mechanics and field theory in a Lorentz invariant noncommutative space Z ( β ) := T r (cid:16) e − β ˆ H (cid:17) , (4.12)in which the trace operation is taken over the continuous set of eigenstates of the positionoperators | X , θ i Z ( β ) = Z d X d θ W ( θ ) h X , θ | e − β ˆ H | X , θ i , (4.13)where the terms in the propagator we have written X ′ = X , θ ′ = θ and τ = − iβ as theimaginary time interval, so we have Z ( IHO ) ( β ) = Z d X d θ W ( θ ) K ( IHO ) ( X , θ ; X , θ, − iβ ) . (4.14)Substituting the propagator (4.5) and the W function (4.10) into (4.14), after a trivialGaussian integration, we obtain that Z ( IHO ) ( β ) = 164 csch (cid:18) ωβ (cid:19) csch (cid:18) Ω β (cid:19) (cid:20) (cid:18) Ω β (cid:19)(cid:21) − / . (4.15)We can see clearly that the NC space contribution is through the Ω frequency. The meanenergy computed using the partition function is given by the equation h E i = − ∂∂β ln Z ( β ) (cid:12)(cid:12)(cid:12)(cid:12) β = T , (4.16)where the β parameter is identified as the inverse of the temperature T , and we have theresult h E i = 3 ω (cid:16) ω T (cid:17) + 3Ω2 " (cid:18) Ω2 T (cid:19) − /
41 + coth (cid:0) Ω2 T (cid:1) . (4.17)This is the Planck’s formula for the average energy of the oscillator. At very low temperatures( T ≪ ω ) and ( T ≪ Ω), we have the ground state h E i ≈
32 ( ω + Ω) , (4.18)and at high temperature we obtain the classical Boltzman statistics h E i ≈ T , (4.19)in which the extra dimension- θ contributes with a factor of 3 /
2. In the next section we applythe path integral (3.10) to the framework of quantum field theory in the NC
DF RA spacetimediscussing the generating functional, perturbation theory and n -points Green functions. uantum mechanics and field theory in a Lorentz invariant noncommutative space V. FIELD THEORY, GREEN FUNCTIONS AND φ ⋆ INTERACTION NC IN THEDFRA SPACE
Concerning the extension of a NCQFT in DFRA spacetime, the spacetime coordinates x µ = ( t, x ) do not commute with itself satisfying the commutation relation (1.1). Theparameter θ µν is promoted to be a coordinate of this spacetime. So, in D = 4, we havesix independents spatial coordinates associated with θ µν . The commutation relation of theDFRA algebra in Eqs. (2.1)-(2.5) can be easily extended to this NC space[ˆ x µ , ˆ x ν ] = i ˆ θ µν , h ˆ x µ , ˆ θ να i = 0 , h ˆ θ µν , ˆ θ αβ i = 0 , [ˆ x µ , ˆ p ν ] = iη µν , [ˆ p µ , ˆ p ν ] = 0 , h ˆ p µ , ˆ θ να i = 0 , [ ˆ p µ , ˆ π να ] = 0 , [ˆ x µ , ˆ π νρ ] = − i δ µσνρ ˆ p σ , (5.1)where (ˆ p µ , ˆ π µν ) are the momenta operators associated with the coordinates (ˆ x µ , ˆ θ µν ), respec-tively. The θ µν coordinates are constrained by quantum conditions θ µν θ µν = 0 and (cid:18) ⋆ θ µν θ µν (cid:19) = λ P , (5.2)where ⋆θ µν = ε µνρσ θ ρσ and λ P is the Planck length. In analogy to (2.13), the Lorentz groupgenerator is ˆ M µν = ˆ X µ ˆ p ν − ˆ X ν ˆ p µ + ˆ θ νρ ˆ π ρµ − ˆ θ µρ ˆ π ρν , (5.3)and from (5.1) we can write the translations generators as ˆ p µ = − i∂ µ . The shifted coordinateoperator ˆ X µ has the analogous definition of (2.8), and it satisfies the commutation relations[ ˆ X µ , ˆ p ν ] = iη µν and [ ˆ X µ , ˆ X ν ] = 0 . (5.4)With these ingredients it is easy to construct the commutation relations[ˆ p µ , ˆ p ν ] = 0 , h ˆ M µν , ˆ p ρ i = i (cid:0) η µρ ˆ p ν − η µν ˆ p ρ (cid:1) , h ˆ M µν , ˆ M ρσ i = i (cid:16) η µσ ˆ M ρν − η νσ ˆ M ρµ − η µρ ˆ M σν + η νρ ˆ M σµ (cid:17) , (5.5)and it closes the proper algebra. We can say that ˆ p µ and ˆ M µν are the generators for theextended DFR algebra.Analyzing the Lorentz symmetry in NCQM following the lines above, we can introducea proper theory, for instance, given by a scalar action. We know, however, that elementaryparticles are classified according to the eigenvalues of the Casimir operators of the inho-mogeneous Lorentz group. Hence, let us extend this approach to the Poincar´e group P .Considering the operators presented here, we can in principle consider thatˆ G = 12 ω µν ˆ M µν − a µ ˆ p µ + 12 b µν ˆ π µν , (5.6) uantum mechanics and field theory in a Lorentz invariant noncommutative space P ′ , which has the Poincar´e group as a subgroup. By definingthe dynamical transformation of an arbitrary operator ˆ A in H in such a way that δ ˆ A = i [ ˆ A, ˆ G ] we arrive at the set of transformations, δ ˆ x µ = ω µν ˆ x ν + a µ δ ˆ p µ = ω νµ ˆ p ν δ ˆ θ µν = ω µρ ˆ θ ρν + ω νρ ˆ θ µρ + b µν δ ˆ π µν = ω ρµ ˆ π ρν + ω ρν ˆ π µρ δ ˆ M µν = ω µρ ˆ M ρν + ω νρ ˆ M µρ + a µ ˆ p ν − a ν ˆ p µ δ ˆ M µν = ω µρ ˆ M ρν + ω νρ ˆ M µρ + b µρ ˆ π νρ + b νρ ˆ π µρ δ ˆ x µ = ω µν ˆ x ν + a µ + 12 b µν ˆ p ν . (5.7)We observe that there is an unexpected term in the last equation of (5.7). This is aconsequence of the coordinate operator in (2.8), which is a nonlinear combination of operatorsthat act on different Hilbert spaces.The action of P ′ upon Hilbert space operators is in some sense equal to the action ofthe Poincar´e group with an additional translation operation on the (ˆ θ µν ) sector. Its gen-erators, all of them, close in a commutation algebra. Hence, P ′ is a well defined group oftransformations. As a matter of fact, the commutation of two transformations closes in thealgebra [ δ , δ ] ˆ y = δ ˆ y , (5.8)where y represents any one of the operators appearing in (5.7). The parameters compositionrule is given by ω µ ν = ω µ α ω α ν − ω µ α ω α ν a µ = ω µ ν a ν − ω µ ν a ν b µν = ω µ ρ b ρν − ω µ ρ b ρν − ω ν ρ b ρµ + ω ν ρ b ρµ . (5.9)To sum up, the framework showed above demonstrated that in NCQM, the physicalcoordinates do not commute and the respective eigenvectors cannot be used to form a basisin H = H ⊕ H [16]. This can be accomplished with the Bopp shift defined in (2.8) with(5.4) as consequence. So, we can introduce a coordinate basis | X ′ , θ ′ i = | X ′ i ⊗ | θ ′ i and | p ′ , k ′ i = | p ′ i ⊗ | k ′ i , in such a way thatˆ X µ | X ′ , θ ′ i = X ′ µ | X ′ , θ ′ i and ˆ θ µν | X ′ , θ ′ i = θ ′ µν | X ′ , θ ′ i , (5.10)and ˆ p µ | p ′ , k ′ i = p ′ µ | p ′ , k ′ i , ˆ π µν | p ′ , k ′ i = k ′ µν | p ′ , k ′ i . (5.11)The wave function φ ( X ′ , θ ′ ) = h X ′ , θ ′ | φ i represents the physical state | φ i in the coordinatebasis defined above. This wave function satisfies some wave equation that can be derivedfrom an action, through a variational principle, as usual. In [16], the author constructed uantum mechanics and field theory in a Lorentz invariant noncommutative space (cid:0) ˆ p µ ˆ p µ − m (cid:1) | φ i = 0 , (5.12)demonstrated through the Casimir operator C = ˆ p µ ˆ p µ (for more algebraic details see [16]).It is easy to see that in the coordinate representation, this originates the NC KG equation.Condition (5.12) selects the physical states that must be invariant under gauge transforma-tions. To treat the NC case, let us assume that the second mass-shell condition(ˆ π µν ˆ π µν − ∆) | φ i = 0 , (5.13)and must be imposed on the physical states, where ∆ is some constant with dimension M ,which sign and value can be defined if π is spacelike, timelike or null. Analogously theCasimir invariant is C = ˆ π µν ˆ π µν , demonstrated the validity of (5.12) (see [16] for details).Both equations (5.12) and (5.13) permit us to construct a general expression for the planewave solution such as [16] φ ( x ′ , θ ′ ) := h X ′ , θ ′ | φ i = Z d p (2 π ) d k (2 πλ − ) e φ ( p, k µν ) exp (cid:18) ip µ x ′ µ + i k µν θ ′ µν (cid:19) , (5.14)where p − m = 0 and k − ∆ = 0, and we have used that p · X = p · x . The length λ − is introduced conveniently in the k -integration in order to keep the usual dimensionsof the fields since the action S must have null dimension in natural units. Or an evenweight function that will be defined in a few moments, used to make the bridge between theformalism in D = 4 + 6 and the standard one in D = 4. Or finally it can be understood asa distribution used to impose further conditions [7].In coordinate representation, the operators (ˆ p, ˆ π ) are written in terms of the derivativesˆ p µ → − i∂ µ and ˆ π µν → − i ∂∂θ µν , (5.15)and consequently, both (5.12) and (5.13) are just the Klein-Gordon equations (cid:0) ✷ + M (cid:1) φ ( x, θ ) = 0 (5.16)and ( ✷ θ + ∆ ) φ ( x, θ ) = 0 , (5.17)respectively, where we have defined ✷ θ = ∂ µν ∂ µν and ∂ µν = ∂∂θ ′ µν , with η µν =diag(1 , − , − , − φ (cid:0) ✷ + λ ✷ θ + m (cid:1) φ ( x, θ ) = 0 , (5.18)which is the Klein-Gordon equation in DFRA space. The parameters M and ∆ have massdimension, and it is related with the mass scalar field, that is, m = M + λ ∆ . Substitutingthe wave plane solution (5.14), we obtain the mass invariant p + λ k µν k µν = m . (5.19) uantum mechanics and field theory in a Lorentz invariant noncommutative space λ is a parameter with dimension of length defined before, as the Planck length. Wedefine the components of the k -momentum k µν = ( − k , − e k ) and k µν = ( k , e k ), to get theDFRA dispersion relation ω ( p , k , e k ) = r p + λ (cid:16) k + e k (cid:17) + m , (5.20)in which e k i is the dual vector of the components k ij , that is, k ij = ǫ ijk e k k ( i, j, k = 1 , , , ).To propose the action of a scalar field we need to define the Weyl representation for DFRAoperators. It is given by the mappingˆ W ( f )(ˆ x, ˆ θ ) = Z d p (2 π ) d k (2 πλ − ) e f ( p, k µν ) e ip · ˆ x + i k · ˆ θ , (5.21)where (ˆ x, ˆ θ ) are position operators satisfying the DFRA algebra, p µ and k µν are the conju-gated momentum of the coordinates x µ and θ µν , respectively. The Weyl symbol provides amap from the operator algebra to the algebra of functions equipped with a star-product, viathe Weyl-Moyal correspondenceˆ f (ˆ x, ˆ θ ) ˆ g (ˆ x, ˆ θ ) ↔ f ( x, θ ) ⋆ g ( x, θ ) , (5.22)and the star-product turns out to be the same as in the usual NC case f ( x, θ ) ⋆ g ( x, θ ) = e i θ µν ∂ µ ∂ ′ ν f ( x, θ ) g ( x ′ , θ ) (cid:12)(cid:12)(cid:12) x ′ = x , (5.23)for any functions f and g . The Weyl operator (5.21) has the following trace propertiesTr h ˆ W ( f ) i = Z d x d θ W ( θ ) f ( x, θ ) , (5.24)and for a product of n functions ( f , ..., f n )Tr h ˆ W ( f ) ... ˆ W ( f n ) i = Z d x d θ W ( θ ) f ( x, θ ) ⋆ ... ⋆ f n ( x, θ ) . (5.25)The function W is a Lorentz invariant integration- θ measure. This weight function isintroduced in the context of NC field theory to control divergences of the integration in the θ -space [8, 12, 14]. It will permits us to work with series expansions in θ , i.e. , with truncatedpower series expansion of functions of θ . For any large θ µν it falls to zero quickly so that allintegrals are well defined, in that it is assumed the normalization condition h i = Z d θ W ( θ ) = 1 . (5.26)The function W should be a even function of θ , that is, W ( − θ ) = W ( θ ), and consequentlyit implies that h θ µν i = Z d θ W ( θ ) θ µν = 0 . (5.27) uantum mechanics and field theory in a Lorentz invariant noncommutative space h θ n i = Z d θ W ( θ ) ( θ µν θ µν ) n , with n ∈ Z + , (5.28)in which the normalization condition corresponding to the case n = 0. For n = 1, we have Z d θ W ( θ ) θ µν θ ρλ = h θ i [ µν,ρλ ] , (5.29)where [ µν,ρλ ] := ( g µρ g νλ − g µλ g νρ ) / µν ) and ( ρλ ).For n = 2, we have Z d θ W ( θ ) θ µν θ ρλ θ αβ θ γσ = h θ i (cid:0) [ µν,ρλ ] [ αβ,γσ ] + [ µν,αβ ] [ ρλ,γσ ] + [ µν,γσ ] [ ρλ,αβ ] (cid:1) . (5.30)A explicit form for the weight function W that satisfies all this previous properties is W ( θ ) = (cid:18) πλ (cid:19) e − | θµν θµν | λ , (5.31)and the absolute value has been introduced to assure that there is not directions in which W -function blows up to infinity. In the limit λ → W -function tends to a Dirac’s deltafunction, that when integrated in d θ , it has the interpretation of spatial volume of the extradimension- θ . Using the previous properties we have an important integration for us Z d θ W ( θ ) e i k µν θ µν = e − λ | k µν k µν | , (5.32)With the definition of the Moyal product (5.23) it is trivial to obtain the property Z d x d θ W ( θ ) f ( x, θ ) ⋆ g ( x, θ ) = Z d x d θ W ( θ ) f ( x, θ ) g ( x, θ ) . (5.33)The physical interpretation of the average of the components of θ µν , i.e. h θ i , is the definitionof the NC energy scale [12] Λ NC = (cid:18) h θ i (cid:19) / =: 1 λ , (5.34)in which λ is the fundamental length scale which appeared in the KG equation (5.18), and inthe dispersion relation (5.20). This approach has the advantage of being independent of theform of the function W , at least for lowest-order processes. The study of Lorentz-invariantNC QED, as Bhabha scattering, dilepton and diphoton production to LEP data led theauthors of [13, 14] to the boundΛ NC > GeV
C.L. . (5.35)After the discussion of the W -function it can be postulated the completeness relations Z d x ′ d θ ′ W ( θ ′ ) | X ′ , θ ′ ih X ′ , θ ′ | = , (5.36) uantum mechanics and field theory in a Lorentz invariant noncommutative space Z d p ′ (2 π ) d k ′ (2 πλ − ) | p ′ , k ′ ih p ′ , k ′ | = . (5.37)Using the previous completeness relations and the integral (5.32), we obtain h X ′ , θ ′ | X ′′ , θ ′′ i = δ (4) ( x ′ − x ′′ ) W − ( θ ′ ) δ (6) ( θ ′ − θ ′′ ) , (5.38)and h p, k | p ′ , k ′ i = (2 π ) δ (4) ( p − p ′ ) e − λ (cid:12)(cid:12)(cid:12) ( k µν − k ′ µν ) (cid:12)(cid:12)(cid:12) , (5.39)where we have used h θ i = 12 λ of (5.34), and the matrix elements h X ′ , θ ′ | ˆ p µ | X ′′ , θ ′′ i = − i ∂ ′ µ δ (4) ( x ′ − x ′′ ) W − ( θ ′ ) δ (6) ( θ ′ − θ ′′ ) , (5.40)and h X ′ , θ ′ | ˆ π µν | X ′′ , θ ′′ i = δ (4) ( x ′ − x ′′ )( − i ) ∂∂θ ′ µν (cid:0) W − ( θ ′ ) δ (6) ( θ ′ − θ ′′ ) (cid:1) , (5.41)that confirms the differential representation of (5.15). The result (5.39) reveals that thecanonical momentum k µν associated to θ µν is not conserved due to introduction of the W -function.Since we constructed the NC KG equation, we will now provide its correspondent action.We use the definition of Moyal-product (5.23) to write the action with a φ ⋆ interaction term S ( φ ) = Z d x d θ W ( θ ) (cid:18) ∂ µ φ ⋆ ∂ µ φ + λ ∂ µν φ ⋆ ∂ µν φ − m φ ⋆ φ − g φ ⋆ φ ⋆ φ ⋆ φ (cid:19) , (5.42)where g is a constant coupling, and using the identity (5.33), the free part of the action isreduced to the usual product one S ( φ ) = Z d x d θ W ( θ ) (cid:20)
12 ( ∂ µ φ ) + λ ∂ µν φ ) − m φ − g
4! ( φ ⋆ φ ) (cid:21) , (5.43)where W is the previous weight function in the measure- θ , that can be used to connect D = 4 + 6 and D = 4 DFRA formalisms. Firstly, we study the free part of (5.43) byrevisiting the retarded, advanced and causal free Green functions [11].The NC KG equation DFRA in the presence of an external source J admits two solutionsof the type φ ( x, θ ) = φ in ( x, θ ) + Z d x ′ d θ ′ W ( θ ′ ) ∆ ( − ) ( x − x ′ , θ − θ ′ ) J ( x ′ , θ ′ ) , (5.44)and φ ( x, θ ) = φ out ( x, θ ) + Z d x ′ d θ ′ W ( θ ′ ) ∆ (+) ( x − x ′ , θ − θ ′ ) J ( x ′ , θ ′ ) , (5.45) uantum mechanics and field theory in a Lorentz invariant noncommutative space φ in and φ out are asymptotic fields and solutions of the NC KG free equation. The∆ ( − ) and ∆ (+) are the retarded and advanced Green functions in the NC DFRA, respectively.It is obtained inverting the Green equation (cid:0) ✷ + λ ✷ θ + m (cid:1) ∆ ( ± ) ( x − x ′ ; θ − θ ′ ) = δ (4) ( x − x ′ ) W − ( θ ) δ (6) ( θ − θ ′ ) , (5.46)by the traditional Fourier method, we have∆ ( ∓ ) ( x − x ′ ; θ − θ ′ ) = Z ( ∓ ) dp π d p (2 π ) Z d k (2 πλ − ) e ip · ( x − x ′ )+ i k µν ( θ − θ ′ ) µν ( p ± iε ) − p + λ k µν k µν − m . (5.47)Here we have added the prescription p ω + iε (retarded Green function), and p ω − iε (advanced Green function). The symbols ( ∓ ) in the p -integration denotes the convenientcontour associated to those Green functions. For more details on these NC Green functions,see [11].For the formalism of perturbative field theory we need the Causal Green function, whichhas a different prescription comparing with (5.47). To simplify our task we write the freepart of (5.43) in the momentum space by using the Fourier representation (5.14) and theintegral (5.32), so we obtain S ( e φ ) = Z d p (2 π ) d k (2 πλ − ) d k ′ (2 πλ − ) e − λ (cid:12)(cid:12)(cid:12) ( k µν + k ′ µν ) (cid:12)(cid:12)(cid:12) ×× e φ ( − p, k ′ µν ) 12 (cid:18) p + λ k ′ µν k µν − m (cid:19) e φ ( p, k µν ) . (5.48)Therefore we postulate the free generating functional defined with configurations in themomentum space Z ( e J ) = Z D e φ exp h iS ( e φ ; − iε )++ i Z d p (2 π ) d k (2 πλ − ) d k ′ (2 πλ − ) e − λ (cid:12)(cid:12)(cid:12) ( k µν + k ′ µν ) (cid:12)(cid:12)(cid:12) e J ( − p, k ′ µν ) e φ ( p, k µν ) (cid:21) , (5.49)where the prescription m m − iε has been added to the mass term in the free action(5.48) to obtain a functional integration well defined. This prescription gives rise the causalGreen function. To obtain the Feynman propagator (causal Green function) we make thetransformation e φ e φ − ( p + λ k ′ µν k µν − m + iε ) − e J , and the Z functional is Z ( e J ) = exp − Z d p (2 π ) d k (2 πλ − ) d k ′ (2 πλ − ) e J ( − p, k ′ µν ) i e − λ (cid:12)(cid:12)(cid:12) ( k µν + k ′ µν ) (cid:12)(cid:12)(cid:12) p + λ k ′ µν k µν − m + iε e J ( p, k µν ) . (5.50)The free 2-points Green function in the momentum space is defined by derivatives of Z inrelation to e J , so we have e ∆ (2) F ( p, p ′ ; k µν , k ′ µν ) = (2 π ) δ (4) ( p + p ′ ) ie − λ (cid:12)(cid:12)(cid:12) ( k µν + k ′ µν ) (cid:12)(cid:12)(cid:12) p + λ k ′ µν k µν − m + iε , (5.51) uantum mechanics and field theory in a Lorentz invariant noncommutative space x + θ )-space is the Fourier transform∆ F ( x − x ′ ; θ, θ ′ ) = Z ( F ) dp π d p (2 π ) e ip · ( x − x ′ ) ×× Z d k (2 πλ − ) d k ′ (2 πλ − ) e i ( k µν θ µν − k ′ µν θ ′ µν ) ie − λ (cid:12)(cid:12)(cid:12) ( k µν + k ′ µν ) (cid:12)(cid:12)(cid:12) p − p + λ k ′ µν k µν − m + iε , (5.52)where the symbol ( F ) indicates the integration contour correspondent to Causal Green func-tion.For the interaction part φ ⋆ of (5.43) we also write it in the momentum space by usingthe Fourier transform (5.14), the interaction is S int ( e φ ) = − g Z Y i =1 d p i (2 π ) d k i (2 πλ − ) e φ ( p i , k iµν )(2 π ) δ (4) ( p + p + p + p ) ×× Z d θ W ( θ ) F ( p , p , p , p ) e i ( k µν + k µν + k µν + k µν ) θ µν , (5.53)where F is a function totally symmetric exchanging the momenta ( p , p , p , p ) F ( p , p , p , p ) = 13 h cos (cid:16) p ∧ p (cid:17) cos (cid:16) p ∧ p (cid:17) + cos (cid:16) p ∧ p (cid:17) cos (cid:16) p ∧ p (cid:17) + cos (cid:16) p ∧ p (cid:17) cos (cid:16) p ∧ p (cid:17)i , (5.54)and the symbol ∧ means the product p i ∧ p j = θ µν p µi p νj if i = j ( i, j = 1 , , , p i ∧ p j = 0, if i = j . The θ -integral of (5.53) can be calculated with the help of (5.32) which,after some manipulations we obtain S int ( e φ ) = − g × Z Y i =1 d p i (2 π ) d k i (2 πλ − ) e φ ( p i , k iµν )(2 π ) δ (4) ( p + p + p + p ) × h e − λ ( k sµν + p µ p ν + p µ p ν ) + e − λ ( k sµν − p µ p ν − p µ p ν ) + e − λ ( k sµν + p µ p ν − p µ p ν ) + e − λ ( k sµν − p µ p ν + p µ p ν ) + e − λ ( k sµν + p µ p ν + p µ p ν ) + e − λ ( k sµν − p µ p ν − p µ p ν ) + e − λ ( k sµν + p µ p ν − p µ p ν ) + e − λ ( k sµν − p µ p ν + p µ p ν ) + e − λ ( k sµν + p µ p ν + p µ p ν ) + e − λ ( k sµν − p µ p ν − p µ p ν ) + e − λ ( k sµν + p µ p ν − p µ p ν ) + e − λ ( k sµν − p µ p ν + p µ p ν ) i . (5.55) uantum mechanics and field theory in a Lorentz invariant noncommutative space V (4) of the φ ⋆ interaction in the momentum space is V (4) ( p , ..., p ; k µν , ..., k µν ) = − g π ) δ (4) ( p + p + p + p ) ×× e − λ | k sµν k µνs | (cid:26) e − λ ( p µ p ν + p µ p ν ) cosh (cid:20) λ k sµν ( p µ p ν + p µ p ν ) (cid:21) + e − λ ( p µ p ν − p µ p ν ) cosh (cid:20) λ k sµν ( p µ p ν − p µ p ν ) (cid:21) + e − λ ( p µ p ν + p µ p ν ) cosh (cid:20) λ k sµν ( p µ p ν + p µ p ν ) (cid:21) + e − λ ( p µ p ν − p µ p ν ) cosh (cid:20) λ k sµν ( p µ p ν − p µ p ν ) (cid:21) + e − λ ( p µ p ν + p µ p ν ) cosh (cid:20) λ k sµν ( p µ p ν + p µ p ν ) (cid:21) + e − λ ( p µ p ν − p µ p ν ) cosh (cid:20) λ k sµν ( p µ p ν − p µ p ν ) (cid:21)(cid:27) , (5.56)where we have defined k sµν := k µν + k µν + k µν + k µν . (5.57)The expressions (5.51) and (5.56) are the Feynman rules of the φ ⋆ DFRA model. Forradiative corrections of the perturbative series, lines and vertex are represented in the mo-mentum space by those expressions. Clearly, the vertex expression shows that the externaltotal momentum associated to extra dimension θ is not conserved, that is, k sµν = 0, whilethe total usual momentum p µ is conserved. VI. CONCLUSIONS
The idea of noncommutativity brings hope to the elimination of divergences that plagueQFT. Snyder was the first one who published a way to deal with these ideas but Yangshowed that the divergences were still there. This result provoke an hibernation of Snyderwork in particular and of the noncommutativity concepts in general for more than forty years.Calculations concerning string theory algebra demonstrated that nature can be NC. Sincestring theory is one of the candidates to unify QM with general relativity, noncommutativityconcepts were vigorously reborn through a huge and dynamical literature.To describe some NC aspects, one way that can be used is to analyze noncommutativitythrough the Moyal-Weyl product where the standard product of two or more fields is sub-stituted by a star product. In this case, the mathematical consistency of this star productis guaranteed because the NC parameter is constant ([42] and references therein). However,there are NC versions, not using the Moyal-Weyl product, where the NC parameter is not aconstant. We can also introduce noncommutativity through the so-called Bopp shift. Theseones are the most popular realizations of the NC concepts.In this paper we work with an alternative and ingenuous formulation of NC theory devel-oped recently, motivated by the ideas that in the early Universe, the spacetime may be NC. uantum mechanics and field theory in a Lorentz invariant noncommutative space θ µν coordinates, namely D ( D − / θ µν -coordinate has its conjugated momenta k µν (DFRA),where we are interested in studying the consequences of the propagation in the θ -direction.Besides, to work with a ten dimensional NC spacetime can disclose new physics beyond theStandard Model.On the other hand, to work with NCQM we need only three space coordinates andconsequently the NC sector has three coordinates also. These both sets combined are theso-called DFRA space which has been developed through these last few years using thestandard concepts of QM and constructing an extension of the Hilbert space.Here we showed that the alternative Feynman vision for QM can be treated in this DFRAspace. We used this new formalism to quantize the NC free particle and the NC harmonicoscillator. After that we also constructed the DFRA action of a scalar field with a self-interaction φ ⋆ . From the action we have obtained the propagators and vertex φ ⋆ in themomentum space. The detailed analysis of the radiative corrections of this model is themain motivation of the next paper.As a final remark we would like to say that the formalism developed here is very unusualconcerning the fact that it was totally constructed within a NC space. We did not introduce(by hand) any NC parameter such that its elimination recover the commutative theory.To recover its commutative behavior, we have to perform a dimensional reduction in this( x + θ )-space which has nine dimensions. Since some thermodynamics were considered andsome objects like transition amplitude and partition function with NC coordinates werecalculated, we believe that the construction of a NC thermofield dynamics can be the nextmove in this direction. Some cosmological models like black holes and wormholes can be atarget for this analysis. The idea would be, instead of introducing noncommutativity in suchmodels existing in commutative spacetime, to construct black holes and wormholes in a NCspacetime like the one discussed here. [1] S. W. Hawking, Nucl. Phys. B 144 (1978) 349, and references therein.[2] H. S. Snyder, Phys. Rev. (1947) 38.[3] C. N. Yang, Phys. Rev. (1947) 874.[4] N. Seiberg and E. Witten, JHEP (1999) 032.[5] L. Alvarez-Gaum´e, F. Meyer and M. A. Vazquez-Mozo,
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