Noncommutative Extension of Minkowski Spacetime and Its Primary Application
aa r X i v : . [ h e p - t h ] A p r Noncommutative Extension of Minkowski Spacetimeand Its Primary Application
Yan-Gang Miao ∗ Department of Physics, Nankai University, Tianjin 300071,People’s Republic of ChinaDepartment of Physics, University of Kaiserslautern, P.O. Box 3049,D-67653 Kaiserslautern, GermanyBethe Center for Theoretical Physics and Institute of Physics, University of Bonn,Nussallee 12, D-53115 Bonn, Germany
Abstract
We propose a noncommutative extension of the Minkowski spacetime by introducing awell-defined proper time from the κ -deformed Minkowski spacetime related to the standardbasis. The extended Minkowski spacetime is commutative, i.e. it is based on the standardHeisenberg commutation relations, but some information of noncommutativity is encodedthrough the proper time to it. Within this framework, by simply considering the Lorentzinvariance we can construct field theory models that comprise noncommutative effects nat-urally. In particular, we find a kind of temporal fuzziness related to noncommutativity inthe noncommutative extension of the Minkowski spacetime. As a primary application, weinvestigate three types of formulations of chiral bosons, deduce the lagrangian theories ofnoncommutative chiral bosons and quantize them consistently in accordance with Dirac’smethod, and further analyze the self-duality of the lagrangian theories in terms of the parentaction approach.PACS Number(s): 02.40.Gh; 11.10.Nx; 11.10.KkKeywords: extended Minkowski spacetime, κ -Minkowski spacetime, chiral boson ∗ e-mail address: [email protected] Introduction
In history, Snyder [1] published the first work on a noncommutative spacetime althoughthe idea might be traced back earlier to W. Heisenberg, R.E. Peierls, W. Pauli, and J.R.Oppenheimer. The quantized spacetime was introduced in order to remove the divergencetrouble caused by point interactions between matter and fields. In modern quantum fieldtheory, instead of a discrete spacetime, the well-developed renormalization is utilized toovercome this difficulty in the spacetime with the scale larger than Planck’s. However, theidea of noncommutative spacetimes has revived due to the intimate relationship betweenthe noncommutative field theory (NCFT) and string theory [2] and between the NCFT andquantum Hall effect [3], respectively. In the former relationship, some low-energy effectivetheory of open strings with a nontrivial background can be described by the NCFT andthus some relative features of the string theory may be clarified through the NCFT withina framework of the quantum field theory, and in the latter the quantum Hall effect canbe deduced from the abelian noncommutative Chern-Simons theory at level n shown to beexactly equivalent to the Laughlin theory at filling fraction 1 /n . Recently it is commonlyacceptable that the noncommutativity would occur in the spacetime with the Planck scaleand the NCFT [4] would play an important role in describing phenomena at planckianregimes.The mathematical background for the NCFT is the noncommutative geometry [5]. Ingeneral, the spacetime noncommutativity can be distinguished in accordance with the Hopf-algebraic classification by the three types, i.e. the canonical, Lie-algebraic and quadraticnoncommutativity, respectively. Among them, the κ -deformed Minkowski spacetime [6, 7]as a specific case of the Lie-algebraic type has recently been paid much attention because itis a natural candidate for the spacetime based on which the Doubly Special Relativity [8]has been established. The κ -Minkowski spacetime is defined by the following commutationrelations of the Lie-algebraic type,[ˆ x , ˆ x j ] = iκ ˆ x j , [ˆ x i , ˆ x j ] = 0 , i, j = 1 , , . (1)In addition, the vanishing momentum commutation relations taken from the κ -deformedPoincar´e algebra should be supplemented,[ˆ p µ , ˆ p ν ] = 0 , µ, ν = 0 , , , . (2)Therefore eqs. (1) and (2), together with the cross relations between ˆ x µ and ˆ p ν that shouldcoincide with the Jacobi identity, constitute a complete algebra of the noncommutativephase space. Here the noncommutative parameter κ with the mass dimension is considered to be real and positive. here we briefly recapitulate thetraditional scheme for the sake of presentation of a big difference between the two schemes.In the latter the noncommutativity can be described by the way of Weyl operators or forthe sake of practical applications by the way of normal functions with a suitable definitionof star-products. The relationship between the two ways is, as was stated in ref. [4], thatthe noncommutativity of spacetimes may be encoded through ordinary products in thenoncommutative C ∗ -algebra of Weyl operators, or equivalently through the deformation ofthe product of the commutative C ∗ -algebra of functions to a noncommutative star-product.For instance, in the canonical noncommutative spacetime the star-product is just the Moyal-product [10], while in the κ -deformed Minkowski spacetime the star-product requires a morecomplicated formula [11]. Some recent progress on star-products shows [12] that a local fieldtheory in the κ -deformed Minkowski spacetime can be described by a nonlocal relativisticfield theory in the Minkowski spacetime.The arrangement of this paper is as follows. In the next section, a new approachquite different from the traditional one mentioned above is proposed for disposing the non-commutativity of the κ -Minkowski spacetime, that is, a noncommutative extension of theMinkowski spacetime is introduced. As a byproduct, the approach makes it possible that a local field theory in the κ -deformed Minkowski spacetime can be reduced under some postu-lation of operator linearization to a still local relativistic field theory in the noncommutativeextension of the Minkowski spacetime. In section 3, the idea of the extended Minkowskispacetime is applied as an example to chiral bosons, and the lagrangian theories of noncom-mutative chiral bosons are established in such a framework of the Minkowski spacetime andthen quantized consistently in accordance with Dirac’s method. Furthermore, in section 4the self-duality of the lagrangian theories is analyzed in terms of the parent action method.Finally section 5 is devoted to the conclusion and perspective, in which a kind of temporalfuzziness related to noncommutativity is discussed in detail. In general the so-called noncommutativity is closely related to a noncommutative spacetimelike the κ -deformed Minkowski spacetime, and it has nothing to do with a commutativespacetime, such as the Minkowski spacetime. In this paper we encode enough informationof noncommutativity of the κ -Minkowski spacetime to a commutative spacetime, and pro-pose a noncommutative extension of the Minkowski spacetime. This extended Minkowski The concept of noncommutative fields [9] is different from that of the noncommutative spacetime andis not involved in this paper. κ -deformed Minkowski spacetime can be dealt with in some sense in thesame way as the Minkowski spacetime. This idea may be realized by connecting a commu-tative spacetime with the κ -Minkowski spacetime and by introducing a well-defined propertime. As a result, the noncommutative field theories defined in the κ -deformed Minkowskispacetime may be investigated somehow by the way of the ordinary (commutative) fieldtheories in the noncommutative extension of the Minkowski spacetime and thus the non-commutativity may be depicted within the framework of this commutative spacetime. Thatis, we may provide from the point of view of noncommutativity a simplified treatment to thenoncommutativity of the κ -Minkowski spacetime, which indeed unveils certain interestingfeatures, such as fuzziness in the temporal dimension.We start with an algebra of coordinate and momentum operators of a commutativespacetime, which is nothing but the standard Heisenberg commutation relations, [ ˆ X µ , ˆ X ν ] = 0 , [ ˆ X µ , ˆ P ν ] = iδ µν , [ ˆ P µ , ˆ P ν ] = 0 . (3)The connection between the commutative spacetime and the κ -deformed Minkowski space-time should be found because it is the basis for us to give the noncommutative extensionof the Minkowski spacetime. In accordance with ref. [13] the following relations may bechosen, ˆ x = ˆ X − κ [ ˆ X j , ˆ P j ] + , (4)ˆ x i = ˆ X i + Aη ij ˆ P j exp (cid:18) κ ˆ P (cid:19) , (5)where [ ˆ O , ˆ O ] + ≡ (cid:16) ˆ O ˆ O + ˆ O ˆ O (cid:17) , η µν ≡ diag(1 , − , − , − A is an arbitraryconstant. This choice is, though not unique, rational because we are able to guaranteeeq. (1) in terms of eqs. (3), (4) and (5).According to the Casimir operator of the κ -deformed Poincar´e algebra on the standardbasis [6], ˆ C = κ sinh ˆ p κ ! − ˆ p i , (6)we supplement the relations of momentum operators between the commutative spacetimeand the κ -Minkowski spacetime as follows:ˆ p = 2 κ sinh − ˆ P κ , (7)ˆ p i = ˆ P i , (8) The speed of light c and the Planck constant ¯ h are set to unit throughout this paper. x µ and ˆ p ν ,[ˆ x i , ˆ p j ] = iδ ij , [ˆ x , ˆ p ] = i cosh ˆ p κ ! − , [ˆ x , ˆ p i ] = − iκ ˆ p i , [ˆ x i , ˆ p ] = 0 . (9)Fortunately, the complete algebra of the noncommutative phase space composed of eqs. (1),(2) and (9) indeed satisfies the Jacobi identity, which is, from the point of view of consistency,crucial to the relationship, i.e. eqs. (4), (5), (7) and (8) established between the commutativespacetime and the κ -Minkowski spacetime. Moreover, such a relationship makes the Casimiroperator have the usual formula as expected,ˆ C = ˆ P − ˆ P i , (10)which is consistent with the standard Heisenberg commutation relations (eq. (3)).When ˆ p µ take the usual forms, ˆ p = − i ∂∂t , (11)ˆ p i = − i ∂∂x i , (12)which coincide with eq. (2), the operator ˆ P then readsˆ P = − i κ sin 12 κ ∂∂t ! . (13)Next we demonstrate the meaning of t and x i defined by eqs. (11) and (12). For simplicityand without contradiction to eqs. (11) and (12), we set A = 0 in eq. (5). Therefore, thecanonical coordinates x i that are the eigenvalues of operators ˆ X i may be regarded as theeigenvalues of operators ˆ x i , while the eigenvalue of operator ˆ x does not match the ordinarytime variable because operator ˆ x , different from operator ˆ X , does not satisfy the standardHeisenberg commutation relations. As a result, for the κ -deformed Minkowski spacetimewe have a certain degree of freedom to introduce a parameter which is of the temporaldimension. This parameter that is required to tend to the ordinary time variable in thelimit κ → + ∞ is used to describe the evolution of systems. Here t , introduced througheq. (11), is dealt with for the sake of simplicity as such a parameter, rather than theeigenvalue of operator ˆ x , to describe the dynamical evolution of fields. It obviously tendsto the ordinary time variable in the limit κ → + ∞ as well as the eigenvalue of operator ˆ x ,which guarantees the consistency of the choice of the parameter. The eigenvalue of operator ˆ x does not match the t -parameter either, but takes a more complicatedformula that should coincide with the algebra of the noncommutative phase space defined by eqs. (1), (2)and (9). τ by defining the operatorˆ P ≡ − i ∂∂τ . (14) τ may be treated as the eigenvalue of operator ˆ X , which, together with eq. (14), is inagreement with the standard Heisenberg commutation relations. At the present stage, noconnection between τ and t can be determined. Here a linear realization or representationof operator ˆ P is postulated, which gives rise to a well-defined proper time that satisfiesthe following differential equation in accordance with eqs. (13) and (14),2 κ sin 12 κ ddt ! τ = 1 . (15)Fortunately, this equation has an exact solution, τ = t + P + ∞ n = −∞ c n exp(2 κnπt ), where n isan integer and coefficients c n are arbitrary real constants. The consistency requires that τ should be convergent and tend to parameter t in the limit κ → + ∞ , which guarantees thatboth τ and t can regress to the ordinary time variable. This requirement nevertheless adds the constraints, c n = 0 for n ≥
1. Therefore, the final form of the solution reads τ = t + + ∞ X n =0 c − n exp( − κnπt ) . (16)The noncommutative extension of the Minkowski spacetime spanned by ( τ, x i ) coordi-nates is thus given, in which the Casimir operator and the line element have the same formsas that in the Minkowski spacetime. However, some information of noncommutativity hasbeen encoded through the proper time to the framework, which can be seen clearly whenthis kind of extended spacetimes is transformed into ( t, x i ) coordinates. This is one thingwith two sides. Eq. (16) plays a crucial role in connecting the two coordinate systems toeach other. The connection means that a noncommutative spacetime, i.e. the κ -Minkowskispacetime may be represented under the postulation of the operator linearization (eq. (15)) This terminology is adopted here only because of its temporal dimension. It has nothing to do withthat of the special relativity. This form is a natural choice that corresponds to a kind of minimal extensions of the Minkowski space-time from the point of view of noncommutativity. The idea of minimal extensions is basic and usuallyadopted in physics, in particular at the beginning stage to establish a theory, such as the minimal super-symmetry. When we introduce light-cone components in the coordinate system related to τ , this form indeedinduces models of noncommutative chiral bosons with interesting physical properties. For the details, seefurther discussions. For the usual notation of t ≥
0. When t <
0, the solution takes the form, τ = t + P + ∞ n =1 c ′ n exp(2 κnπt ),where c ′ n are constants and the convergence requires c ′ n = 0 for n < An additional constraint should be imposed upon the non-vanishing coefficients, c − n for n ≥
0, if theinitial value of τ is required to be finite, which is related to the so-called temporal fuzziness that will beanalyzed in detail in the last section of this paper.
6y a commutative spacetime, i.e. the extended Minkowski spacetime, but the price paidfor such a simplified treatment is the appearance of infinitely many unfixed coefficients inthe commutative spacetime. This would be understandable because it is just these unfixedcoefficients that reflect the information of noncommutativity in the commutative spacetime.See the last section of this paper for a detailed discussion. By making use of eq. (16) wethen reduce the Casimir operator to beˆ C ′ = − ∂ ∂τ + ∂∂x i ∂∂x i = − τ ∂∂t τ ∂∂t ! + ∂∂x i ∂∂x i , (17)and give the line element ds = dτ − ( dx i ) = ˙ τ dt − ( dx i ) , (18)where ˙ τ means dτ /dt , ˙ τ = 1 − κπ + ∞ X n =0 nc − n exp( − κnπt ) . (19)The extended Minkowski spacetime is, as expected, commutative, which is a merit for usto construct field theory models in this framework. That is, we simply consider the Lorentzinvariance in the extended Minkowski spacetime, and the constructed models contain non-commutative effects naturally. In mathematics, there exists a specific map (see eqs. (4),(5), (7) and (8)) from the κ -Minkowski spacetime to the noncommutative extension of theMinkowski spacetime. The Lorentz invariance in the extended Minkowski spacetime reflectsin fact some invariance in the κ -Minkowski spacetime that seems an unknown symmetry upto now. In this way we might have circumvented a relatively complicated procedure forsearching for models that should possess such an unknown symmetry.The line element gives the fact that the noncommutative extension of the Minkowskispacetime connects with a special flat spacetime corresponding to a twisted t -parameter. In consequence we may say that the κ -Minkowski spacetime, the source of the extendedMinkowski spacetime, is somehow reduced to the flat spacetime whose metric is given by g = ˙ τ = " − κπ + ∞ X n =0 nc − n exp( − κnπt ) , g = g = g = − . (20)Before the end of this section we emphasize that the Casimir operator defined by eqs. (6),(11) and (12) with infinite order in t -parameter derivative has now been reduced to the See, e.g. ref. [14] and the references therein. The curvature of this spacetime equals to zero. The author would like to thank the anonymous refereefor pointing it out. Here it only means that ˆ C is reduced to ˆ C ′ , while the inverse procedure is never implied in this paper. t -parameter derivative under the postulationof the operator linearization (eq. (15)). This realization or representation of the Casimiroperator (eq. (17)) fulfills in a way the aim that some information of noncommutativity canbe encoded to a commutative spacetime. The feature of the extended Minkowski spacetimespanned by ( t, x i ) is that the evolution parameter is twisted while the spaces are still flat(see, e.g. eq. (18)), which coincides with [6] the characters of the κ -Minkowski spacetime, i.e. with a “quantum time” and a three-dimensional euclidean space. This implies thatthe extended Minkowski spacetime does contain enough information of noncommutativitythat is able to reflect the characteristic of the κ -Minkowski spacetime although it does notrecover the whole information of noncommutativity due to the postulation of the operatorlinearization. We deviate from the discussion of noncommutativity temporarily and give a brief intro-duction of chiral bosons in the Minkowski spacetime. The main reason that chiral bosons have received much attention is that they appear in various theoretical models that relate tosuperstring theories, and reflect especially the existence of a variety of important dualitiesthat connect these theories among one another. One has to envisage two basic problemsin a lagrangian description of chiral bosons: the first is the consistent quantization and thesecond is the harmonic combination of the manifest duality and Lorentz covariance, sincethe equation of motion of a chiral boson, i.e. the self-duality condition is first order withrespect to the derivatives of space and time. In order to solve these problems, various typesof formulations of chiral bosons, each of which possesses its own advantages, have beenproposed [15, 16, 17]. It is remarkable that these models of chiral bosons have close rela-tionships among one another, especially various dualities that have been demonstrated indetail from the points of view of both the configuration [18, 19] and momentum [20] spaces.In this section we mainly propose noncommutative chiral bosons in the noncommu-tative extension of the Minkowski spacetime. The method is as follows: a lagrangian ofnoncommutative chiral bosons is given simply by the requirement of the Lorentz invari-ance in the extended Minkowski spacetime spanned by the coordinates ( τ, x ), and throughthe coordinate transformation eq. (16), it is then converted into its ( t, x )-coordinate for-mulation with explicit noncommutativity. As a result, we establish the lagrangian theory In general, one should mention chiral p -forms that include chiral bosons as the p = 0 case. A chiral0-form in the (1+1)-dimensional Minkowski spacetime is usually called a chiral boson which describes a left-or right-moving boson in one spatial dimension. We can also consider chiral p -forms ( p ≥
1) and their noncommutative generalizations. This is one ofthe further topics that will probably be reported elsewhere soon.
8f noncommutative chiral bosons in the extended framework of the Minkowski spacetime.Alternatively, we can also construct a lagrangian theory directly in the ( t, x )-coordinateframework in terms of the metric eq. (20). It should be noted that for a certain formulationof chiral bosons both procedures give rise to the noncommutative generalizations that havethe same physical spectrum.Let us begin from the light-cone coordinates and their derivatives defined in the (1 + 1)-dimensional noncommutative extension of the Minkowski spacetime, respectively, as follows: X ± ≡ √ ± τ + x ) , (21) D ± ≡ √ ± ∂∂τ + ∂∂x ! . (22)It is obvious that they satisfy D ± X ± = 1 and D ± X ∓ = 0. In the spacetime spanned bythe coordinates ( τ, x ) the equation of motion for a noncommutative chiral boson takes theusual form D ∓ φ = 0 , (23)and its solution thus reads φ = φ ( X ± ) , (24)where the upper sign corresponds to the left-moving while the lower the right-moving.Through the coordinate transformation we convert the equation of motion eq. (23) into itscorresponding formulation in the coordinate system spanned by ( t, x ), ∓ τ ∂φ∂t + ∂φ∂x = 0 . (25)The solution takes the same form as eq. (24) and can easily be expressed by the ( t, x )coordinates through the well-defined X ± , φ = φ √ " ± t + x ± + ∞ X n =0 c − n exp ( − κnπt ) . (26)Consequently, the equation of motion and its solution comprise in a natural way the noncom-mutative effects related to the finite noncommutative parameter κ . Incidentally, they be-come their ordinary forms correspondent to the Minkowski spacetime in the limit κ → + ∞ .The following task is straightforward, that is, to construct in the extended frameworkof the Minkowski spacetime such a lagrangian that yields the equation of motion (eq. (25))for noncommutative chiral bosons by the method mentioned in the second paragraph of thissection. Three typical formulations of chiral bosons are investigated as a primary applicationin this section, i.e. the non-manifestly Lorentz covariant version [15] and manifestly Lorentzcovariant versions with the linear self-duality constraint [16] and with the quadratic one [17],respectively. 9 .1 The non-manifestly Lorentz covariant formulation It is non-manifestly Lorentz covariant but indeed Lorentz invariant [15]. Simply consideringthe Lorentz invariance in the extended Minkowski spacetime ( τ, x ), we give the action S = Z dτ dx ∂φ∂τ ∂φ∂x − ∂φ∂x ! , (27)which is nothing but the formulation of Floreanini and Jackiw’s left-moving chiral bosons if τ is replaced by the ordinary time. After making the coordinate transformation, we obtainthe action written in terms of the coordinates ( t, x ), S = Z dtdx √− g τ ∂φ∂t ∂φ∂x − ∂φ∂x ! , (28)where √− g is the Jacobian and also the nontrivial measure of the flat spacetime eq. (20)connected with the κ -Minkowski spacetime, √− g = | ˙ τ | . (29)Therefore the lagrangian takes the form L = ± ˙ φφ ′ − √− gφ ′ , (30)where a dot and a prime stand for derivatives with respect to time t and space x , respec-tively. A plus and minus sign appears in front of the first term due to the ratio √− g/ ˙ τ , andthe choice depends on either ˙ τ > τ <
0. However, this does not cause any ambiguity. By making use of Dirac’s quantization [21] we can prove that the lagrangian only describesa left-moving noncommutative chiral boson in the extended framework of the Minkowskispacetime, which is independent of whether ˙ τ > τ <
0, and that the similar case alsohappens in the linear and quadratic self-duality constraint formulations of chiral bosons.Incidentally, this feature does not exist in the case of ordinary (commutative) chiral bosonsdue to the triviality √− g = ˙ τ = 1 in the limit κ → + ∞ .In terms of Dirac’s quantization [21] we can verify that the lagrangian L indeed de-scribes a noncommutative chiral boson which satisfies the equation of motion eq. (25) withthe choice of the upper sign correspondent to the left-handed chirality. To this end, at firstdefine the momentum conjugate to φ , π φ ≡ ∂ L ∂ ˙ φ = ± φ ′ , For the sake of convenience in description, time , here different from the ordinary time variable, standsonly for t-parameter in the three subsections of section 3. The same result can be obtained if we start with the action for right-moving chiral bosons, which isalso available in subsections 3.2 and 3.3. H = π φ ˙ φ − L = √− gφ ′ . Note that this hamiltoian explicitly contains time and that it is positive definite as itscounterpart in the Minkowski spacetime. The definition of momenta actually yields oneprimary constraint Ω( x ) ≡ π φ ∓ φ ′ ≈ , where “ ≈ ” stands for Dirac’s weak equality. Because of no further constraints and of thenon-vanishing equal-time Poisson bracket, { Ω( x ) , Ω( y ) } P B = ∓ ∂ x δ ( x − y ) , this constraint itself constitutes a second-class set. With the inverse of the Poisson bracket, { Ω( x ) , Ω( y ) } − P B = ∓ ε ( x − y ) , where ε ( x ) is the step function with the property dε ( x ) /dx = δ ( x ), we calculate the equal-time Dirac brackets: { φ ( x ) , φ ( y ) } DB = ∓ ε ( x − y ) , { φ ( x ) , π φ ( y ) } DB = 12 δ ( x − y ) , { π φ ( x ) , π φ ( y ) } DB = ± ∂ x δ ( x − y ) . In the sense of Dirac brackets weak constraints become strong conditions. As a consequence,we write the reduced hamiltonian H r = √− gφ ′ = √− gπ φ = 12 √− g (cid:16) φ ′ + π φ (cid:17) , and derive from the canonical hamiltonian equation˙ φ = Z dy { φ ( x ) , H r ( y ) } DB , the equation of motion for the noncommutative chiral boson,˙ φ = ±√− gφ ′ = ˙ τ φ ′ , which is nothing but eq. (25) with the upper sign corresponding to the left-handed chirality. The Dirac quantization is shown to be available in constrained systems whose hamiltoians contain timeexplicitly. For instance, see ref. [22]. .2 The manifestly Lorentz covariant formulation with linear self-duality constraint In this formulation the self-duality constraint is imposed upon a massless real scalar field [16].Although it has some defects [23], the linear formulation strictly describes a chiral bosonfrom the point of view of equations of motion at both the classical and quantum levels.Its generalization in the canonical noncommutative spacetime has been studied in detail,and in particular a kind of fuzziness on the left- and right-handed chiralities in the spatialdimension has been noticed [22]. Under the requirement of the Lorentz invariance in theextended Minkowski spacetime, we can write the action in a straightforward way, S = Z dτ dx (cid:16) − D + φD − φ − √ λ + D − φ (cid:17) , (31)where λ + ≡ √ ( λ + λ ), a light-cone component of the vector field λ µ , µ = 0 ,
1, introducedas a Lagrange multiplier. In terms of the method adopted in subsection 3.1, we then convertit into its formulation in the framework spanned by the coordinates ( t, x ), S = Z dtdx √− g
12 ˙ τ ∂φ∂t ! − ∂φ∂x ! + λ + τ ∂φ∂t − ∂φ∂x ! , (32)from which the lagrangian reads L = 12 √− g (cid:20) ˙ φ − (cid:16) √− gφ ′ (cid:17) (cid:21) + λ + (cid:16) ± ˙ φ − √− gφ ′ (cid:17) , (33)where a plus and minus sign also exists in front of ˙ φ as explained in subsection 3.1.As was done in the above subsection, we make the hamiltonian analysis by using Dirac’smethod and prove that L describes, as expected, a noncommutative chiral boson with theleft-handed chirality. Let us define momenta conjugate to φ and λ + , respectively, π φ ≡ ∂ L ∂ ˙ φ = 1 √− g ˙ φ ± λ + ,π λ + ≡ ∂ L ∂ ˙ λ + ≈ . The latter gives in fact one primary constraintΩ ( x ) ≡ π λ + ≈ . Making the Legendre transformation, we get the canonical hamiltonian H = π φ ˙ φ + π λ + ˙ λ + − L = √− g (cid:20) (cid:16) π φ + φ ′ (cid:17) + λ + ( ∓ π φ + φ ′ ) + 12 λ (cid:21) .
12s a basic consistency requirement in dynamics of constrained systems, Ω ( x ) should bepreserved in time, which yields one secondary constraintΩ ( x ) ≡ ± π φ − φ ′ − λ + ≈ . Because the preservation of Ω ( x ) does not give further constraints, the constraint set con-sists of Ω ( x ) and Ω ( x ). With the matrix elements of equal-time Poisson brackets of theconstraints, C ( x, y ) = 0 ,C ( x, y ) = − C ( x, y ) = δ ( x − y ) ,C ( x, y ) = ∓ ∂ x δ ( x − y ) , we can easily deduce their inverse elements C − ( x, y ) = ∓ ∂ x δ ( x − y ) ,C − ( x, y ) = − C − ( x, y ) = − δ ( x − y ) ,C − ( x, y ) = 0 , and then compute the non-vanishing equal-time Dirac brackets { φ ( x ) , π φ ( y ) } DB = δ ( x − y ) , { φ ( x ) , λ + ( y ) } DB = ± δ ( x − y ) , { π φ ( x ) , λ + ( y ) } DB = − ∂ x δ ( x − y ) , { λ + ( x ) , λ + ( y ) } DB = ∓ ∂ x δ ( x − y ) . After making Dirac’s weak constraints be strong conditions, we obtain the reduced hamil-tonian in terms of independent phase space variables, H r = ±√− gπ φ φ ′ . Note that the linear self-duality model of chiral bosons in the ordinary spacetime [16] isintrinsically non-positive definite, the non-positive definition here should not be inducedby its generalization in the extended Minkowski spacetime but emerges from the original(commutative) formulation. From the canonical hamiltonian equation,˙ φ = Z dy { φ ( x ) , H r ( y ) } DB , we thus arrive at the expected equation of motion˙ φ = ±√− gφ ′ = ˙ τ φ ′ . .3 The manifestly Lorentz covariant formulation with quadraticself-duality constraint In this formulation the square of the self-duality constraint, instead of the self-duality itself,is imposed upon a massless real scalar field [17]. In accordance with the Lorentz invariancewe firstly write the following action in the extended Minkowski spacetime related to thecoordinates ( τ, x ), S = Z dτ dx h − D + φD − φ − λ ++ ( D − φ ) i , (34)where λ ++ ≡ ( λ + λ + λ + λ ), a light-cone component of the tensor field λ µν , µ, ν = 0 ,
1, introduced as a Lagrange multiplier, and then rewrite it in the ( t, x )-coordinateframework in terms of eq. (16), S = Z dtdx √− g
12 ˙ τ ∂φ∂t ! − ∂φ∂x ! − λ ++ τ ∂φ∂t − ∂φ∂x ! , (35)which yields at last the lagrangian L = 12 √− g (cid:20) ˙ φ − (cid:16) √− gφ ′ (cid:17) (cid:21) − √− g λ ++ (cid:16) ± ˙ φ − √− gφ ′ (cid:17) , (36)where a plus and minus sign emerges in front of ˙ φ once again as occurred in subsections 3.1and 3.2.Briefly repeating the procedure gone through in the above two subsections, we can makethe conclusion that L also describes a noncommutative chiral boson with the left-handedchirality in the extended Minkowski spacetime. At first, through introducing canonicalmomenta conjugate to φ and λ ++ , respectively, π φ ≡ ∂ L ∂ ˙ φ = 1 √− g (1 − λ ++ ) ˙ φ ± λ ++ φ ′ ,π λ ++ ≡ ∂ L ∂ ˙ λ ++ ≈ , we get a primary constraint Ω ( x ) ≡ π λ ++ ≈ , and derive the canonical hamiltonian in terms of the Legendre transformation, H = π φ ˙ φ + π λ ++ ˙ λ ++ − L = √− g − λ ++ (cid:18) π φ ∓ λ ++ π φ φ ′ + 12 φ ′ (cid:19) . The consistency of time evolution of Ω ( x ) then gives rise to one secondary constraint˜Ω ( x ) ≡ ( π φ ∓ φ ′ ) ≈ , ( x ) by its linearized version, Ω ( x ) ≡ π φ ∓ φ ′ ≈ , which is second-class, as was dealt with [24] to the ordinary quadratic self-duality constraintformulation in the Minkowski spacetime under the consideration of the classical equivalencebetween the two constraints. According to Dirac’s method, a gauge fixing condition χ ( x ) ≡ λ ++ ( x ) − F ( x ) ≈ F ( x ) an arbitrary function in the extended framework of the Minkowski spacetime,should be added, and thus the constraint set, ( χ ( x ) , Ω ( x ) , Ω ( x )), becomes second-class.The remainder of the canonical analysis can be followed straightforwardly and the results,such as the non-vanishing equal-time Dirac brackets, the reduced hamiltonian, and the equa-tion of motion, are exactly same as that obtained in subsection 3.1. As a consequence, weverify that the quadratic formulation can be reduced to the noncommutative generalizationof Floreanini and Jackiw’s chiral bosons, which reveals the connection between the twoformulations in the noncommutative extension of the Minkowski spacetime. Based on the important role played by the duality and/or self-duality in the ordinary chiral p -forms and the related theories [18, 19, 20], it is curious to argue whether such a symme-try maintains or not in their various noncommutative generalizations. It was revealed [22]that the self-duality is not preserved when the chiral boson action with the linear chiralityconstraint [16] is generalized from the Minkowski spacetime to the canonical noncommu-tative spacetime. However, it will be verified in the following context that this kind ofsymmetries still exists in the generalizations performed in the noncommutative extension ofthe Minkowski spacetime. The situation implies somehow a complex relationship betweenduality/self-duality and noncommutativity, which may be worth notice.In this section a systematic approach, known as the parent action method [25], is usedto investigate the self-duality of the actions eqs. (28), (32) and (35). The approach, fromthe Legendre transformation and for the foundation of equivalence among theories at thelevel of actions instead of equations of motion, can be summarized briefly as follows: (i) tointroduce auxiliary fields and then to construct a parent or master action based on a sourceaction, and (ii) to make variation of the parent action with respect to each auxiliary field, As to the original proposal and recent development of the parent action method, see, for instance, thereferences cited by ref. [25].
15o solve one auxiliary field in terms of other fields and then to substitute the solution intothe parent action. Through making variations with respect to different auxiliary fields, wecan obtain different forms of an action. These forms are, of course, equivalent classically,and the relation between them is usually referred to duality. If the resulting forms are same,their relation is called self-duality.
According to the parent action approach [25], we introduce two auxiliary vector fields G µ and F µ , and write down the parent action correspondent to S , i.e. eq. (28), S p1 = Z dtdx h F F − √− gF + G µ ( F µ − ∂ µ φ ) i . (37)Now varying eq. (37) with respect to G µ gives F µ = ∂ µ φ , together with which eq. (37)regresses to the action eq. (28). This shows the classical equivalence between the parentaction S p1 and its source action S . However, varying eq. (37) with respect to F µ leads tothe expression of F µ in terms of G µ : F = − √− gG − G ,F = − G . (38)Substituting eq. (38) into eq. (37), we obtain a kind of dual versions for the action S ,˜ S dual1 = Z dtdx (cid:20) − G G − √− g (cid:16) G (cid:17) + φ∂ µ G µ (cid:21) , (39)where φ is dealt with at present as a Lagrangian multiplier. Further varying eq. (39) withrespect to φ gives ∂ µ G µ = 0, whose solution takes the form G µ = ǫ µν ∂ ν ϕ, (40)where ǫ = ǫ = 0, ǫ = − ǫ = +1, and ϕ is a scalar filed that is in general different from φ . Therefore the dual version eq. (39) can be reduced to its simpler formula described onlyby ϕ , S dual1 = Z dtdx h ˙ ϕϕ ′ − √− gϕ ′ i . (41)It has the same form as eq. (28) just with the replacement of φ by ϕ , that is, eq. (28) oreq. (41) is self-dual with respect to the chiral boson field.In accordance with the illustration utilized for duality in ref. [25], the self-duality of S and S dual1 may be illustrated by Fig. 1. For simplicity, we only consider case ˙ τ > τ <
0, the same result will be deduced. p1 ✡✡✡✡✡✡✡✢ ❏❏❏❏❏❏❏❫ δG µ δF µ + δφS S dual1 ✻ ✻ self-dual Figure 1:
Schematic relation of the actions: the parent action S p1 shows that S and S dual1 haveself-duality with respect to the chiral boson field. Following the procedure gone through in the above subsection, we can verify that S (eq. (32)) is self-dual with respect to the chiral boson field. To this end, let us at firstregard S as a source action and give its corresponding parent action S p2 = Z dtdx ( √− g (cid:20) F − (cid:16) √− gF (cid:17) (cid:21) + λ + (cid:16) F − √− gF (cid:17) + G µ ( F µ − ∂ µ φ ) ) , (42)where G µ and F µ stand for auxiliary vector fields as before. Next, through making thevariation of eq. (42) with respect to G µ we simply obtain F µ = ∂ µ φ , with which eq. (42)becomes the source action eq. (32). This implies that S p2 is classically equivalent to S .Thirdly, by making the variation of eq. (42) with respect to F µ we have F = −√− g (cid:16) G + λ + (cid:17) ,F = 1 √− g (cid:16) G − √− gλ + (cid:17) . (43)After substituting eq. (43) into eq. (42) we then deduce a dual version of S ,˜ S dual2 = Z dtdx ( − √− g (cid:20)(cid:16) √− gG (cid:17) − (cid:16) G (cid:17) (cid:21) − λ + (cid:16) √− gG + G (cid:17) + φ∂ µ G µ ) . (44)At last, varying eq. (44) with respect to φ which is now dealt with as a Lagrangian multiplier,we derive eq. (40) again and therefore simplify the above dual action to be S dual2 = Z dtdx ( √− g (cid:20) ˙ ϕ − (cid:16) √− gϕ ′ (cid:17) (cid:21) + λ + (cid:16) ˙ ϕ − √− gϕ ′ (cid:17)) , (45)which is nothing but eq. (32) if ϕ is replaced by φ . The illustration of self-duality of S and S dual2 can be seen in Fig. 2. 17 p2 ✡✡✡✡✡✡✡✢ ❏❏❏❏❏❏❏❫ δG µ δF µ + δφS S dual2 ✻ ✻ self-dual Figure 2:
Schematic relation of the actions: the parent action S p2 shows that S and S dual2 haveself-duality with respect to the chiral boson field. Simply repeating the procedure followed in the above two subsections, we can easily showthe self-duality of S (eq. (35)) with respect to the chiral boson field φ . The parent action,correspondent to S as a source, takes the form S p3 = Z dtdx ( √− g (cid:20) F − (cid:16) √− gF (cid:17) (cid:21) − √− g λ ++ (cid:16) F − √− gF (cid:17) + G µ ( F µ − ∂ µ φ ) ) , (46)where G µ and F µ are auxiliary vector fields introduced. Varying the parent action withrespect to G µ gives F µ = ∂ µ φ , which provides nothing new but just the classical equivalencebetween S p3 and S . However, varying it with respect to F µ instead leads to the usefulrelations between the two auxiliary fields, F = −√− g (1 + λ ++ ) G − λ ++ G ,F = − λ ++ G + 1 √− g (1 − λ ++ ) G . (47)Substituting the above equation into the parent action, we derive a dual action describedby G µ and φ ,˜ S dual3 = Z dtdx ( − √− g (cid:20)(cid:16) √− gG (cid:17) − (cid:16) G (cid:17) (cid:21) − √− g λ ++ (cid:16) √− gG + G (cid:17) + φ∂ µ G µ ) . (48)Now treating φ as a Lagrangian multiplier, we arrive at eq. (40) once more and thus deducethe formula of the dual action with the obvious self-duality, S dual3 = Z dtdx ( √− g (cid:20) ˙ ϕ − (cid:16) √− gϕ ′ (cid:17) (cid:21) − √− g λ ++ (cid:16) ˙ ϕ − √− gϕ ′ (cid:17) ) . (49)As done in the above two subsections, the self-duality of S and S dual3 may be illustrated byFig. 3. 18 p3 ✡✡✡✡✡✡✡✢ ❏❏❏❏❏❏❏❫ δG µ δF µ + δφS S dual3 ✻ ✻ self-dual Figure 3:
Schematic relation of the actions: the parent action S p3 shows that S and S dual3 haveself-duality with respect to the chiral boson field. In conclusion we emphasize the key point of this paper, that is, the proposal of the non-commutative extension of the Minkowski spacetime. This newly proposed spacetime isfounded by introducing a proper time from the κ -deformed Minkowski spacetime related tothe standard basis. It is a commutative spacetime, but contains some information of non-commutativity encoded from the κ -deformed Minkowski spacetime. Our performance thusgives the possibility to deal with noncommutativity in a simplified way within a commuta-tive framework. As a byproduct, a local field theory in the κ -deformed Minkowski spacetimecan be reduced to a still local relativistic field theory in the noncommutative extension ofthe Minkowski spacetime. Due to the postulation of the operator linearization (eq. (15))the new spacetime may be regarded as a minimal extension of the Minkowski spacetimeto which the noncommutativity is encoded. Mathematically, we give a specific map (seeeqs. (4), (5), (7) and (8)): (ˆ x µ , ˆ p ν ) → ( ˆ X µ , ˆ P ν ), i.e. from the κ -Minkowski spacetime to theextended Minkowski spacetime, and nevertheless the latter spacetime contains intrinsicallythe primordial information of the former, i.e. the noncommutativity expressed by the finitenoncommutative parameter κ . In applications one keeps a model Lorentz invariant in the ex-tended Minkowski spacetime, which actually reflects it some invariance in the κ -Minkowskispacetime related to the standard basis. This is the method that makes the model com-prise noncommutative effects naturally. Moreover, we notice the connection between the κ -Minkowski spacetime and the flat spacetime with the nontrivial metric eq. (20), whichprovides an alternative way to fulfil noncommutative generalizations for field theory models.As a primary application of this extended Minkowski spacetime, three types of formulationsof chiral bosons are generalized in terms of this method to the corresponding noncommuta-tive versions and the lagrangian theories of noncommutative chiral bosons are acquired andthen quantized consistently by the use of Dirac’s method. In addition, the self-duality ofthe lagrangian theories of noncommutative chiral bosons is verified in terms of the parentaction method. Here we should mention that the self-duality is not preserved [22] when19he chiral boson with the linear chirality constraint is generalized to the canonical noncom-mutative spacetime. This implies that the noncommutativity of spacetimes has a complexrelationship with the self-duality of lagrangian theories as has been stated in ref. [22].In particular, we note that the proper time τ (eq. (16)) contains infinitely many realcoefficients, c − n for n ≥
0, which can not be fixed within the framework of the noncommu-tative extension of the Minkowski spacetime. An interesting feature caused by the indefinitecoefficients is that the initial value of the proper time, τ | t =0 = + ∞ X n =0 c − n , (50)is uncertain even if the convergence requirement of the series of constant terms is added, andthe uncertainty can further extend to any value of the proper time. This may be interpretedto be a kind of temporal fuzziness that exists in the extended Minkowski spacetime. Such afuzziness is compatible with the κ -Minkowski spacetime in which a time and a space opera-tors commute to the space operator, that is, it originates from the special noncommutativitydescribed in mathematics by the κ -deformed Poincar´e algebra. For noncommutative chiralbosons, nevertheless, we point out that the fuzziness is different from that discovered [22]in the canonical noncommutativity where the similar phenomenon presents ambiguous left-and right-handed chiralities in the spatial dimension. In spite of the distinction mentioned,the existence of fuzziness that is closely related to noncommutative spacetimes might beinevitable although it is not clear how the noncommutativity brings about fuzziness in thetemporal or spatial dimension, or even probably in multiple temporal and spatial dimen-sions.Further considerations focus on the following two aspects. The first is whether we cangive a map from the κ -deformed Minkowski spacetime related to the bicrossproduct basis [7]to a commutative spacetime. This is worth noticing because the κ -Poincar´e algebra in thebicrossproduct basis contains the undeformed (classical) Lorentz subalgebra. Different fromthe case of the standard basis, now the difficulty lies in the appearance of entangled termsof ˆ p and ˆ p i in the Casimir operator related to the bicrossproduct basis. Namely, we try tofind, by following the way adopted for the standard basis or by setting up some new way,such a proper time and such “proper” spatial coordinates as well that the entanglementcould be removed. The second aspect is to enlarge the application of the noncommutativeextension of the Minkowski spacetime by considering any models that are interesting inthe ordinary (commutative) spacetime, in particular, the models whose noncommutativegeneralizations connect with phenomena probably tested in experiments, such as in ref. [26]where the canonical noncommutativity is involved. In this way we may have opportunitiesto test the extended Minkowski spacetime and/or to determine the value of the noncommu-tative parameter by comparing theoretical results with available experimental data. A quick20nd direct consideration is to investigate chiral p -forms [18, 19] and the interacting theoryof chiral bosons and gauge fields [27] in the noncommutative extension of the Minkowskispacetime, and results will be given separately. Acknowledgments
The author would like to thank H.J.W. M¨uller-Kirsten of the University of Kaiserslautern forhelpful discussions and H.P. Nilles of the University of Bonn for warm hospitality. This workwas supported in part by the DFG (Deutsche Forschungsgemeinschaft), by the NationalNatural Science Foundation of China under grant No.10675061, and by the Ministry ofEducation of China under grant No.20060055006.21 eferences [1] H.S. Snyder,
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