Some considerations on duality concerning kappa-Minkowski spacetime theories
aa r X i v : . [ h e p - t h ] D ec Some considerations on duality concerningkappa-Minkowski spacetime theories
Vahid Nikoofard a,c and Everton M. C. Abreu b,c a LAFEX, Centro Brasileiro de Pesquisas F´ısicas, Rua Xavier Sigaud 150, 22290-180,Rio de Janeiro, RJ, Brazil b Grupo de F´ısica Te´orica e Matem´atica F´ısica, Departamento de F´ısica, Universidade Federal Ruraldo Rio de Janeiro, 23890-971, Serop´edica, RJ, Brazil c Departamento de F´ısica, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora, MG,Brazil
Abstract
In this paper we have analyzed the κ -deformed Minkowski spacetime through the light ofthe interference phenomena in QFT where two opposite chiral fields are put together inthe same multiplet and its consequences are discussed. The chiral models analyzed hereare the chiral Schwinger model, its generalized version and its gauge invariant version,where a Wess-Zumino term were added. We will see that the final actions obtained hereare in fact related to the original ones via duality transformations. [email protected] [email protected] Introduction
The fermion-boson mapping is one of the most investigated topics in theoretical physics duringthe last three decades due to its importance in the quantization of strings and also the Hallquantum effect. The possibility of mapping a complicated fermionic model into a scalar bosonicone was really attractive. This mapping is called bosonization and the chiral bosons can beobtained from the restriction of a scalar field to move in one direction only, as done by Siegel[1], or by a first-order Lagrangian theory, as proposed by Floreanini and Jackiw [2].In two dimensions (2D), scalar fields can be viewed as bosonized versions of Dirac fermionsand chiral bosons can be seen to correspond to two-dimensional versions of Weyl fermions. Asa generalization, in supergravity models, the extension of the chiral boson to higher dimensionshas naturally introduced the concept of the chiral p-forms. Harada, in [4], has investigatedthe chiral Schwinger model via chiral bosonization and he has analyzed its spectrum. He hasshowed how to obtain a consistent coupling of FJ chiral bosons with a U(1) gauge field, startingfrom the chiral Schwinger model and discarding the right-handed degrees of freedom by meansof a restriction in the phase space implemented by imposing the chiral constraint π = φ ′ . In[3], Bellucci, Golterman and Petcher introduced an O(N) generalization of SiegelâĂŹs modelfor chiral bosons coupled to Abelian and non-Abelian gauge fields. The physical spectrum ofthe resulting Abelian theory is that of a (massless) chiral boson and a free massive scalar field.Initially, several models were suggested for chiral bosons but latter it was shown that thereare some relations between these models [5]. For instance, the Floreanini-Jackiw (FJ) modelis the chiral dynamical sector of the more general model proposed by Siegel. The Siegel modes(rightons and leftons) carry not only chiral dynamics but also symmetry information. Thesymmetry content of the theory is described by the Siegel algebra, a truncate diffeomorphism,that disappears at the quantum level. As another application, chiral bosons appear in theanalysis of quantum Hall effect [6]. The introduction of a soliton field as a charge-creatingfield obeying one additional equation of motion leads to a bosonization rule [7].The direct sum of two chiral fermions in 2D gives rise to a full Dirac fermion, however thisis not true for their bosonized versions as noticed in [8], see also [9]. Besides, the fermionicdeterminant of a Dirac fermion interacting with a vector gauge field in D = 1 + 1 factorizesinto the product of two chiral determinants but the full bosonic effective action is not thedirect sum of the naive chiral effective actions as discussed in [11]. Stated differently, theaction of a bosonized Dirac fermion is not simply the sum of the actions of two bosonizedWeyl fermions, or chiral bosons. Physically, this is connected with the necessity to abandonthe separated right and left symmetries, and accept that vector gauge symmetry should bepreserved at all times. This restriction will force the two independent chiral bosons to belongto the same multiplet , effectively soldering them together. In both cases it turns out that aninterference term between the opposite chiral bosonic actions is needed to achieve the expectedresult, such term is provided by the so called soldering procedure.The concept of soldering has proved extremely useful in different contexts [10]. This for-malism essentially combines two distinct Lagrangians carrying dual aspects of some symmetryto yield a new Lagrangian which is exposed of, or rather hides, that symmetry. These so-calledquantum interference effects, whether constructive or destructive, among the dual aspects ofsymmetry, are thereby captured through this mechanism [11]. The formalism introduced byM. Stone [8] could actually be interpreted as a new method of dynamical mass generation2hrough the result obtained in [11]. This is possible by considering the interference of rightand left gauged FJ chiral bosons. The result of the chiral interference shows the presenceof a massive vectorial mode for the special case where the Bose symmetry fixed the Jackiw-Rajaraman regularization parameter as a = 1 [12], which is the value where the chiral theorieshave only one massless excitation in their spectra. This clearly shows that the massive vectormode results from the interference between two massless modes.It was shown lately [13], that in the soldering process of two opposite chiral fields, alefton and a righton, coupled to a gauge field, the gauge field decouples from the physicalfield. The final action describes a non-mover field (a noton) at the classical level. The notonacquires dynamics upon quantization. This field was introduced by Hull [14] to cancel out theSiegel anomaly. It carries a representation of the full diffeomorphism group, while its chiralcomponents carry the representation of the chiral diffeomorphism.The same procedure works in D = 2 + 1 if we substitute chirality by helicity. For instance,by fusing together two topologically massive modes generated by the bosonization of twomassive Thirring models with opposite mass signatures in the long wave-length limit. Thebosonized modes, which are described by self and anti-self dual Chern-Simons models [15, 16],were then soldered into two massive modes of the 3 D Proca model [17]. More generally, the ± α witha sign freedom which plays a role whenever interactions are present. In the soldering of twochiral Schwinger models that results either to an axial ( α = âĹŠ1) or to a vector ( α = +1)Schwinger model which are dual do each other. In the case of the two Maxwell-Chern-Simonstheories, the choice of the α -parameter with opposite sign leads to dual interaction terms.We can have either a derivative coupling or a minimal coupling plus a Thirring term. Afterintegration over the soldering field the dependence on the sign of α disappears which provesthat they correspond to dual forms of the same interacting theory. Recently, a new ideaconcerning the construction of the so-called Noether vector, the concept of can be directlyanalyzed from an initial master action [ ? ]. We will discuss this issue here in the future.Recently the soldering formalism was used to investigate the self-dual theories with spin s ≥ ⋆ -algebra of Weyl operators, or equivalently through the defor-mation of the product of the commutative C ⋆ -algebra of functions to a NC star-product. Forinstance, in the canonical NC spacetime the star-product is simply the Moyal-product [25],while on the κ -deformed Minkowski spacetime the star-product requires a more complicatedformula [26].In order to treat the κ -deformed Minkowski spacetime in a very similar way to the usual3inkowski spacetime, the authors in [27, 29] have proposed a quite different approach tothe implementation of noncommutativity. To this aim a well-defined proper time from the κ -deformed Minkowski spacetime has been defined that corresponds to the standard basis.In this way we encode enough information of noncommutativity of the κ -Minkowski space-time to a commutative spacetime in this new parameter, and then set up a NC extensionof the Minkowski spacetime. This extended Minkowski spacetime is as commutative as theMinkowski spacetime, but it contains noncommutativity already. Therefore, one can somehowinvestigate the NC field theories defined on the κ -deformed Minkowski spacetime by followingthe way of the ordinary (commutative) field theories on the NC extension of the Minkowskispacetime, and thus depict the noncommutativity within the framework of this commuta-tive spacetime. With this simplified treatment of the noncommutativity of the κ -Minkowskispacetime, we unveil the fuzziness in the temporal dimension and build noncommutative chiralboson models in [27].The organization of the issues through this paper obeys the following sequence: in section2 we have written a review of the κ -Minkowski noncommutativity and in section 3, a review ofthe essentials of the soldering formalism. In section 4, we have analyzed the soldering of theNC chiral Scwinger model (CSM) and in section 5, the NC version of the generalized CSM. Insection 6 we have discussed the gauge invariant CSM. The (anti)self-dual model in D = 2 + 1was analyzed in section 7. As usual, the conclusions and perspectives were described in thelast section. The commutative spacetime is characterized by the canonical Heisenberg commutation rela-tions h ˆ X µ , ˆ X ν i = 0 , h ˆ X µ , ˆ P ν i = iδ µν , h ˆ P µ , ˆ P ν i = 0 (1)where µ, ν = 0 , , ,
3. In order to introduce the κ -deformed Minkowski spacetime we have[27] ˆ x = ˆ X − k h ˆ X i , ˆ P j i + , ˆ x i = ˆ X i + A η ij ˆ P j exp ( 2 k ˆ P ) (2)where h ˆ O , ˆ O i + ≡ ( ˆ O ˆ O + ˆ O ˆ O ), η µν ≡ diag (1 , − , − , − i, j = 1 , , A is anarbitrary constant. The NC parameter κ has mass dimension and it is real and positive. TheCasimir operator related to the κ -deformed PoincarÃľ’s algebra isˆ C = ksinh ˆ p k ! − ˆ p i , (3)and for the momentum operators we haveˆ p = 2 k sinh − ˆ P k , ˆ p i = ˆ P i . (4)4ith these last results we can construct our NC phase-space (ˆ x µ , ˆ p ν ) h ˆ x , ˆ x j i = ik ˆ x j , h ˆ x i , ˆ x j i = 0 , [ ˆ p µ , ˆ p ν ] = 0 , h ˆ x i , ˆ p j i = iδ ij (5) h ˆ x , ˆ p i = i cosh ˆ p k ! − , h ˆ x , ˆ p i i = − ik ˆ p i , h ˆ x i , ˆ p i = 0 (6)which satisfies the Jacobi identity. It is easy to see that when k → ∞ we recover the commu-tative phase-space in Eq.(1).The Casimir operator described above in Eq.(3) can now be written in the standard wayˆ C = ˆ P − ˆ P i (7)where it is easy to see that this selection coincides with the ones in Eq.(1). In the case thatˆ p µ has standard forms like ˆ p = − i ∂∂t , ˆ p i = − i ∂∂x i , (8)so that the operator ˆ P then reads ˆ P = − ik sin k ∂∂t ! . (9)In [27] the author has introduced a proper time τ through the operatorˆ P ≡ − i ∂∂τ (10)and using Eqs. (9) and (10) we have that2 k sin k ddt ! τ = 1 (11)which solution is τ = t + + ∞ X n =0 c − n exp ( − knπt ) (12)where n ≥ n ∈ N . The coefficients c − n are arbitrary real constants. This property impliesa kind of temporal fuzziness coherent in the κ -Minkowski spacetime. Notice that as k → ∞ ,the proper time turns back to the ordinary time variable.To construct a NC extension of Minkowski spacetime ( τ, x i )(where the NC feature is insidethe proper time), let us define a twisted t -coordinate, such that the metric is g = ˙ τ = " − kπ + ∞ X n =0 nc − n exp ( − knπt ) g = g = g = − . (13)So, we can use Eq.(13), construct NC models in the commutative framework. Namely, weconstruct a Lagrangian theory for NC model in the extended framework of the Minkowskispacetime. 5 The canonical soldering formalism
The basic idea of the soldering procedure is to raise a global Noether symmetry of the self andanti-self dual constituents into a local one, but for an effective composite system, consistingof the dual components and an interference term. The objective in [28] is to systematize theprocedure like an algorithm and, consequently, to define the soldered action. The physicsconsiderations will be taken based on the resulting action. For example, in [5], one of us haveobtained a mass generation in the process.An iterative Noether procedure was adopted to lift the global symmetries to local ones.Therefore, we will assume that the symmetries in question are being described by the localactions S ± ( φ η ± ), invariant under a global multi-parametric transformation δφ η ± = α η , (14)where η represents the tensorial character of the basic fields in the dual actions S ± and, fornotational simplicity, will be dropped from now on. Here the ± subscript is referring to theopposite/complementary aspects of two models at hand, for instance, φ + may refer to a leftchiral field and φ − to a field with right chirality. As it is well known, we can write, δS ± = J ± ∂ ± α , (15)where J ± are the Noether currents.Now, under local transformations these actions will not remain invariant, and Noethercounterterms become necessary to reestablish the invariance, along with appropriate auxiliaryfields B ( N ) , the so-called soldering fields which have no dynamics where the N superscript isreferring to the level of the iteration. This makes a wider range of gauge-fixing conditionsavailable. In this way, the N -action can be written as, S ± ( φ ± ) (0) → S ± ( φ ± ) ( N ) = S ± ( φ ± ) ( N − − B ( N ) J ( N ) ± . (16)Here J ( N ) ± are the N − iteration Noether currents. For the self and anti-self dual systemswe have in mind that this iterative gauging procedure is (intentionally) constructed not toproduce invariant actions for any finite number of steps. However, if after N repetitions, thenon-invariant piece ends up being only dependent on the gauging parameters, but not on theoriginal fields, there will exist the possibility of mutual cancellation if both gauged version ofself and anti-self dual systems are put together. Then, suppose that after N repetitions wearrive at the following simultaneous conditions, δS ± ( φ ± ) ( N ) = 0 δS B ( φ ± ) = 0 , (17)with S B being the so-called soldered action S B ( φ ± ) = S ( N )+ ( φ + ) + S ( N ) − ( φ − ) + Contact Terms , (18)and the ”Contact Terms” being generally quadratic functions of the soldering fields. Thenwe can immediately identify the (soldering) interference term as, S int = Contact Terms − X N B ( N ) J ( N ) ± . (19)6ncidentally, these auxiliary fields B ( N ) may be eliminated, for instance, through theirs equa-tions of motion, from the resulting effective action, in favor of the physically relevant degreesof freedom. It is important to notice that after the elimination of the soldering fields, theresulting effective action will not depend on either self or anti-self dual fields φ ± but only insome collective field, say Φ, defined in terms of the original ones in a (Noether) invariant way S B ( φ ± ) → S eff (Φ) . (20)Analyzing in terms of the classical degrees of freedom, it is obvious that we have now a theorywith bigger symmetry groups. Once such effective action has been established, the physicalconsequences of the soldering are readily obtained by simple inspection. The Chiral Schwinger model (CSM) is a 2D (1 spatial dimension + 1 time dimension) Eu-clidean quantum electrodynamics for a Dirac fermion. This model exhibits a spontaneoussymmetry breaking of the U(1) group due to a chiral condensate from a pool of instantons[36]. The photon in this model becomes a massive particle at low temperatures. This modelcan be solved exactly and it is used as a toy model for other complex theories. The bosoniza-tion of this theory can be done in several ways that apparently leads to different bosonizedmodels. But these apparently inequivalent models are related by some gauge transformations[4]. Here we shall not enter into the details of this equivalence. We will discuss the applicationof the soldering mechanism in the different forms concerning these chiral models.The CSM is described by the Lagrangian density L ch = ˙ φφ ′ − ( φ ′ ) + 2 eφ ′ ( A − A ) − e ( A − A ) + 12 e aA µ A µ , (21)where the last term is the CSM mass term for the gauge field A µ .In fact, this Lagrangian is the gauged version of the FJ’s Lagrangian, L = ˙ φφ ′ − ( φ ′ ) [2].On the 2D extended Minkowski spacetime ( τ, x ) the Lagrangian (21) takes the following actionform ˆ S = Z dτ dx ∂φ∂τ ∂φ∂x − ( ∂φ∂x ) + 2 e ∂φ∂x ( A − A ) − e ( A − A ) (22)+ 12 e aη µν A µ A ν − F µν F µν , where η µν = diag (1 , −
1) is the flat metric of the extended Minkowski spacetime ( τ, x ) and a is a real parameter ( a > t, x )with explicit noncommutativity,ˆ S = Z dtdx √− g τ ∂φ∂t ∂φ∂x − ( ∂φ∂x ) + 2 e ∂φ∂x ( A − A ) − e ( A − A ) + 12 e aη µν A µ A ν + 12 τ ∂A ∂t − ∂A ∂x ! , (23)7here √− g is the Jacobian of the transformation and also the non-trivial measure of the k -deformed Minkowski spacetime. Note that always √− g = | ˙ τ | but here we only focus onthe case ˙ τ > L + = ˙ φφ ′ − √− g ( φ ′ ) + √− g { eφ ′ ( A − A ) − e ( A − A ) + 12 e a [( A ) − ( A ) ] } + 12 √− g ( ˙ A − √− gA ′ ) (24)ˆ L − = − ˙ ρρ ′ − √− g ( ρ ′ ) + √− g { eρ ′ ( A − A ) − e ( A − A ) + 12 e b [( A ) − ( A ) ] } + 12 √− g ( ˙ A − √− gA ′ ) . (25)Note that + and − signs are associated to left and right moving chiral bosons, respectively.These models contain noncommutativity through the proper time τ with the finite NC param-eter k . In the limit k → + ∞ , √− g = ˙ τ = 1 these Lagrangians turn back to theirs ordinaryforms on the Minkowski spacetime.Now we are ready to sold these two chiral Lagrangians. To accomplish the task we calculatethe variations of Eqs. (24) and (25) under the following local variations δφ = η ( t, x ) = δρ. (26)In fact we are imposing this local symmetry into these models in order to obtain a gaugeinvariant Lagrangian. Under this variation we have, after some algebra, that δ ( ˆ L + + ˆ L − ) = ( J + + J − ) δB (27)where J + = 2 ˙ φ − √− gφ ′ + 2 e √− g ( A − A ) (28)and J − = − ρ − √− gρ ′ + 2 e √− g ( A − A ) (29)where B (mentioned in the previous section) and B (which will be necessary) are auxiliaryfields which variations can be defined as δB = ∂ x η and δB = ∂ t η. (30)So we must add a counterterm to both original Lagrangians (24) and (25) to cover theabove extra terms. So ˆ L +1 = ˆ L + − J + B (31)ˆ L − = ˆ L − − J − B (32)8ow let us check the variation of the above Lagrangians δ ˆ L +1 = − ( δJ + ) B = − (2 ˙ η − √− gη ′ ) B = − B ( δB ) + 2 √− gB ( δB ) (33) δ ˆ L − = − ( δJ − ) B = (2 ˙ η − √− gη ′ ) B = 2 B ( δB ) + 2 √− gB ( δB ) . (34)As we can see, it is not zero but the extra terms are independent of original fields. So theiteration will finish in this second step by adding another counterterm.Finally we can sold these two Lagrangians in order to construct an invariant one.ˆ W = ˆ L + + ˆ L − − ( J + + J − ) B − √− g ( B ) . (35)where the B field were eliminated algebraically. On the other hand, we can eliminate theauxiliary field B by its equation of motion δ W δB = 0 = ⇒ − ( J + + J − ) − √− gB = 0 ⇒ B = − √− g ( J + + J − ) (36)By substituting Eq. (36) into W we findˆ W = ˆ L + + ˆ L − + 18 √− g ( J + + J − ) . (37)Here we define a new field the soldering field Ψ = φ − ρ . By this definition we can rewrite W in a compact and nice formˆ W = − √− g ′ + 12 √− g ˙Ψ + 2 e ˙Ψ( A − A ) + 2 ξ (38)where ξ is ξ = √− g { e ( A − A ) + 14 e ( a + b )[( A ) − ( A ) ] } + 12 √− g ( ˙ A − √− gA ′ ) (39)As the final result, the action (38) is not “chiral” theory anymore and it has a biggersymmetry group than the two initial models. To this aim, we have soldered the two chiralmodels and as a consequence we have gained an additional term in the final Lagrangian thatwas absent initially. One of the peculiar consequences of this action is that the electromagneticfield interacts just with the temporal derivative of the soldered field. This peculiarity has itsorigin in the noncovariant initial Jackiw-Floreanini Lagrangian. In fact one can decomposethe above action into two distinct ones using the dual projection approach []. The result is aself-dual and a free massive scalar fields.This mechanism in some sense is analogous to adding a mass term into the Dirac action.Without this mass term the Dirac equation describes two chiral electrons and by adding themass, we have merged these two chiral electrons to obtain the real electron.9 The soldering of the generalized bosonized CSM
Bassetto et al. [30] have suggested the generalized chiral Schwinger model (GCSM), i.e., avector and axial-vector theory characterized by a parameter which interpolates between purevector and chiral Schwinger models. This 2D model is given by the actionˆ S = Z d t d x (cid:20)
12 ( ∂ µ φ ) ( ∂ µ φ ) + eA µ ( ǫ µν − rη µν ) ∂ ν φ + 12 e aA µ A µ − F µν F µν (cid:21) . (40)The quantity r is a real interpolating parameter between the vector ( r = 0) and the chiralSchwinger models ( r = ± L = 12 √− g ˙ φ − √− g φ ′ − k ˙ φ + k φ ′ + ξ (41)where k = e ( rA + A ) (42) k = e √− g ( A + rA ) (43) ξ = 12 √− g (cid:16) ˙ A − √− gA ′ (cid:17) + 12 e a √− g h ( A ) − ( A ) i . (44)By defining the value of the parameter r in two extreme points ± L + = 12 √− g ˙ φ − √− g φ ′ − e ( A + A ) ˙ φ + e √− g ( A + A ) φ ′ + 12 √− g (cid:16) ˙ A − √− gA ′ (cid:17) + 12 ae √− g h ( A ) − ( A ) i (45)ˆ L − = 12 √− g ˙ ρ − √− g ρ ′ − e ( − A + A ) ˙ ρ + e √− g ( A − A ) ρ ′ + 12 √− g (cid:16) ˙ A − √− gA ′ (cid:17) + 12 be √− g h ( A ) − ( A ) i (46)where a and b are the Jackiw-Rajaraman coefficients for each chirality, respectively. Here,through the iterative Noether embedding procedure, we will transform both Lagrangians (45)and (46) into two embedded Lagrangians which are invariant under transformations δφ = η ( x )and δρ = η ( x ). After that, we will be able to sold these new Lagrangians in order to yield aninvariant one that describes a fermionic system. By varying the Lagrangians with respect tothe variables ∂ t Φ and ∂ x Φ, (Φ = ( φ, ρ )), we obtain the following Noether currents J = 1 √− g ˙ φ − e ( A + A ) (47) J = −√− g [ φ ′ − e ( A + A )] (48) J − = 1 √− g ˙ ρ + e ( A − A ) (49) J − = −√− g [ ρ ′ − e ( A − A )] . (50)10fter two iterations and by adding the counterterms to the original Lagrangians, we can findthat ˆ L (2)+ = L + − J B − J B + √− g ( B ) − √− g ( B ) + ξ + (51)ˆ L (2) − = L − − J − B − J − B + √− g ( B ) − √− g ( B ) + ξ − (52)where ξ ± are non-dynamical terms of L ± . The embedding process ends after these two stepsand these Lagrangians are invariant under the desired transformation δφ = η ( x ) δρ . Now wecan solder them by adding up two Lagrangians Eqs. (51) and (52)ˆ W = ˆ L (2)+ + ˆ L (2) − (53)= ˆ L + + ˆ L − − ( J + J − ) B − ( J + J − ) B + 1 √− g ( B ) − √− g ( B ) To express this Lagrangian just in terms of the original fields, we can eliminate B and B easily by using their equation of motions, which reads B = √− g ( J + J − ) (54) B = − √− g ( J + J − ) . (55)After substituting these results into W , defining a new field Ψ = φ − ρ and fixing the Jackiw-Rajaraman coefficients a = b = 1, for simplicity, we can write thatˆ W = 14 √− g ˙Ψ − √− g ′ − eA ˙Ψ + eA √− g Ψ ′ + 12 √− g (cid:16) ˙ A − √− gA ′ (cid:17) + e √− g h ( A ) − ( A ) i . (56)This Lagrangian describes a 2D fermionic system and has a larger symmetry group than theinitial Lagrangians (45) and (46). As the previous case, the soldering process included anextra noton term into the original Lagrangians to fuse the chiral states. This non-dynamicalterm can acquire dynamics upon quantization [28]. In [31], the authors have introduced the Wess-Zumino (WZ) term for the GCSM and con-structed its gauge invariant formulation by adding the WZ term into the Lagrangian of themodel. This gauge invariant model is described byˆ S = Z d t d x (cid:26)
12 ( ∂ µ φ ) ( ∂ µ φ ) + eA µ ( ǫ µν − rη µν ) ∂ ν φ + 12 e aA µ A µ − F µν F µν + 12 (cid:16) a − r (cid:17) ( ∂ µ θ ) ( ∂ µ θ ) + eA µ h rǫ µν + (cid:16) a − r (cid:17) η µν i ∂ ν θ (cid:27) , (57)11here θ ( x ) is the WZ field. The Lagrangians of left/right moving bosons are given by definingthe parameter r at its two opposite points ± L + = 12 √− g ( ˙ φ ) − √− g φ ′ ) − b ˙ φ + b √− gφ ′ + b √− g ( ˙ θ ) − b √− g ( θ ′ ) + b ˙ θ + b θ ′ + ξ + ˆ L − = 12 √− g ( ˙ ρ ) − √− g ρ ′ ) − b ˙ ρ + b √− gρ ′ + b ′ √− g ( ˙ η ) − b ′ √− g ( η ′ ) + b ˙ η + b η ′ + ξ − (58)where η is also another WZ field and b ≡ e ( A + A ) , b ≡ a − , b ′ ≡ b − , b ≡ e [ A ( a − − A ] b ≡ e √− g [ A − A ( a − , b ≡ e ( A − A ) ,b ≡ e [ A ( b −
1) + A ] , b ≡ e √− g [ − A − A ( b − ξ ± ≡ √− g ( ˙ A ) − √− g A ′ ) + √− g e ( ab ) h ( A ) − ( A ) i − ˙ A A ′ . (59)The goal here is to gauge these Lagrangians under the following transformations δφ = δρ = α ( x ) δθ = δη = β ( x ) . (60)The Noether currents under these transformations are J = 1 √− g ˙ φ − b , J − = 1 √− g ˙ ρ − b ,J = −√− gφ ′ + b √− g, J − = −√− gρ ′ + b √− g,J = 2 b √− g ˙ θ + b , J − = 2 b ′ √− g ˙ η + b ,J = − b √− gθ ′ + b , J − = − b ′ √− gη ′ + b . (61)The first iteration Lagrangians readˆ L (1)+ = ˆ L + − J B − J B − J B − J B ˆ L (1) − = ˆ L − − J − B − J − B − J − B − J − B (62)where B , B , B and B are new auxiliaries fields which have the following variations δB = ∂ t α, δB = ∂ x α, δB = ∂ t β, δB = ∂ x β. (63)The variation of the first iterated Lagrangians are given by δ ˆ L (1)+ = − √− g ( δB ) B + √− g ( δB ) B − b √− g ( δB ) B + 2 b √− g ( δB ) B (64) δ ˆ L (1) − = − √− g ( δB ) B + √− g ( δB ) B − b √− g ( δB ) B + 2 b √− g ( δB ) B . (65)As we can see, these variations are completely independent of the original fields. Thereforethe embedding process finished here and by adding the counterterms associated with these12ariations we can obtain our desired invariant Lagrangian. Now we are ready to fuse bothLagrangians in Eqs. (62) by adding them up and introducing a countertermˆ W = ˆ L + + ˆ L − − J B − J B − J B − J B − J − B − J − B − J − B − J − B + 1 √− g ( B ) − √− g ( B ) + 2 b √− g ( B ) − b √− g ( B ) , (66)where we have fixed the Jackiw-Rajaraman coefficients a = b for simplicity. To express thefinal result only in terms of the original fields, one can eliminate the auxiliary fields by usingtheir equations of motions B = √− g J + J − ) (67) B = − √− g ( J + J − ) B = √− g b ( J + J − ) B = − b √− g ( J + J − ) . By substituting these results into Eq. (66) and introducing two soldering fields Ψ = φ − ρ andΩ = θ − η we obtain an effective actionˆ W eff = 14 √− g ( ˙Ψ) − √− g ′ ) − eA ˙Ψ + e √− gA Ψ ′ + b √− g ( ˙Ω) (68) − b √− g ′ ) − eA ˙Ω + 12 h eA + e √− g ( A − A b ) + 2 eA b i Ω ′ − e b √− g ( A ) + e √− g b ( A − A b ) − e A b ( A − A b )+ e √− gb ( A ) + e √− g A A + e b √− g ( A ) − e A ( A − A b ) + 2 ξ where ξ = ξ − + ξ + . The initial Lagrangians were invariant under a semilocal gauge group, butthis effective Lagrangian is invariant under the local version of the initial gauge group andmoreover it is invariant under gauge transformations (60).One can ask about the counterpart of this model in the commutative spacetime. We canfind it just by putting √− g = 1. It reads W eff = 14 ∂ µ Ψ ∂ µ Ψ + eǫ µν A µ ∂ ν Ψ + ( a − ∂ µ Ω ∂ µ Ω − eA µ ǫ µν ∂ ν Ω (69)+ 12 e aA µ A µ − F µν F µν where ξ ′ = ξ | √− g =1 . We have succeeded in including the effects of interference between rightonsand leftons (right/left moving scalar). Consequently, these components have lost their indi-viduality in favor of a new, gauge invariant, collective field that does not depend on φ or ρ separately. 13s it can be seen, this Lagrangian is apparently different from the initial ones and the newfields Ψ and Ω are not chiral anymore. If we fix the Jackiw-Rajaraman coefficients a = b = 1,the field Ω becomes non-dynamical and it will just interact with electromagnetic field. Thecombination of the massless modes led us to a massive vectorial mode as a consequence of thechiral interference. The noton field, that was defined before, propagates neither to the left norto the right directions. The Thirring model is an exactly solvable QFT that describes the self-interactions of a Diractheory in (2+1) dimensions. For the first time S. Coleman has discovered an equivalencebetween this model and the Sine-Gordon one which is a bosonic theory [37].In D = 1 + 1, the starting point was to consider two distinct fermionic theories withopposite chiralities. The analogous thing is to take two independent Thirring models withidentical coupling strengths but opposite mass signatures, L + = ¯ ψ ( i∂/ + m ) ψ − λ (cid:16) ¯ ψγ µ ψ (cid:17) L − = ¯ ξ ( i∂/ − m ′ ) ξ − λ (cid:16) ¯ ξγ µ ξ (cid:17) , (70)where the bosonized Lagrangians are, respectively, L + = 12 M ǫ µνλ f µ ∂ ν f λ + 12 f µ f µ L − = − M ǫ µνλ g µ ∂ ν g λ + 12 g µ g µ , (71)where f µ and g µ are the distinct bosonic vector fields. The current bosonization formula inboth cases are given by j + µ = ¯ ψγ µ ψ = λ π ǫ µνρ ∂ ν f ρ j − µ = ¯ ξγ µ ξ = − λ π ǫ µνρ ∂ ν g ρ . (72)These models are known as the self and anti-self dual models [32, 33, 34].On the extended Minkowski spacetime ( τ, x ) the Lagrangian (71) takes the following actionform ˆ S ± = Z dτ d x " h µ h µ ± M ǫ µ λ h µ ∂h λ ∂τ + ǫ µiλ h µ ∂ i h λ ! (73)where h µ = f µ , g µ .After making the coordinate transformation, we obtain the action written in terms of thecoordinates ( t, x ),ˆ S ± = Z dtd x √− g (cid:20) h µ h µ ± M ǫ µiλ h µ ∂ i h λ (cid:21) ± M ǫ µ λ h µ ∂h λ ∂t . (74)14aking a hint from the two dimensional case, let us consider the gauging of the followingsymmetry δf µ = δg µ = ǫ µρσ ∂ ρ α σ . (75)Under these transformations the bosonized Lagrangians change as δ ˆ S ± = Z dtd x (cid:20) √− g (cid:26) ǫ µρσ h µ ± M ǫ µiλ ǫ µρσ ∂ i h λ (cid:27) ± M ǫ µ λ ǫ µρσ ∂ h λ (cid:21) ∂ ρ α σ . (76)We can identify the Noether currents J ρσ ± ( h µ ) = √− g (cid:26) ǫ µρσ h µ ± M ǫ µiλ ǫ µρσ ∂ i h λ (cid:27) ± M ǫ µ λ ǫ µρσ ∂ h λ . (77)As a comment about the form of the gauge transformation in Eq. (75) we can say thatthe simpler form such as the one we have assumed in 2D case, will not be suitable and thevariations cannot be combined to give a single structure like the (77) one. Now we introducethe auxiliary field coupled to the antisymmetric currents. In the two dimensional case, thisfield was a vector. In the three dimensional case, as a natural generalization, we adopt anantisymmetric second rank Kalb-Ramond tensor field B ρσ where its transformation is givenby δB ρσ = ∂ ρ α σ − ∂ σ α ρ (78)It is worthwhile to mention that in the canonical NC approach, one must include thevariation of the current associated with the NC field/parameter concerning the transformationof the auxiliary tensor field in order to obtain an effective Lagrangian after the solderingprocedure [35].To eliminate the non-vanishing change (76), we add a counter-term to the original La-grangian. So, the first iterated Lagrangians are L (1) ± = L ± − J ρσ ± ( h µ ) B ρσ (79)which transforms as, δ L (1) ± = − δJ ρσ ± B ρσ . (80)The variation of the currents coupled to the auxiliary field is δJ ρσ ± B ρσ = √− g (cid:20) δB ρσ B ρσ ∓ M ǫ λγθ ( ∂ i ∂ γ α θ ) B iλ (cid:21) ∓ M ǫ λγθ ( ∂ ∂ γ α θ ) B λ . (81)As we can see, the above Lagrangians also are not invariant under the transformations (75),hence we must go further and add another counter term. As a key point in the solderingformalism, the invariance of one Lagrangian alone is not desired. We are looking for a com-bination of both Lagrangians that are gauge invariant. To this aim, the second iterationLagrangians is defined by L (2) ± = L (1) ± + √− g B ρσ B ρσ . (82)15y this definition, a straightforward algebra shows that the following combination is invariantunder transformation (75) and (78). So, L S = L (2)+ + L (2) − = L + + L − − B ρσ (cid:16) J + ρσ ( f ) + J − ρσ ( g ) (cid:17) + √− g B ρσ B ρσ . (83)The gauging procedure of the symmetry is therefore complete now. But the final result wouldbe more interesting if we express the above Lagrangian in terms of the original fields. By usingthe equation of motion for B ρσ we can eliminate this auxiliary field B ρσ = 12 √− g (cid:16) J + ρσ ( f ) + J − ρσ ( g ) (cid:17) . (84)Including this solution into (83) the final soldered Lagrangian is expressed only in terms ofthe original fields, L S = L + + L − − √− g (cid:16) J + ρσ ( f ) + J − ρσ ( g ) (cid:17) (cid:16) J + ρσ ( f ) + J − ρσ ( g ) (cid:17) . (85)The crucial point of soldering formalism becomes clear now. By using the explicit structuresfor the currents, the above Lagrangian is no longer a function of f µ and g µ separately, butsolely on the combination A µ = 1 √ M ( f µ − g µ ) . (86)By this field redefinition we can obtain the final effective action as L S = M √− g A µ A µ + ∂ i A ∂ A i − √− g ∂ A i ∂ A i − √− g ∂ i A ∂ i A + ∂ i A j ∂ i A j − ∂ j A i ∂ i A j ) . (87)In the usual commutative Minkowski spacetime we yield the Proca theory by soldering two(anti)self-dual theories [34]. As a generalization, we claim that the Lagrangian (87) are the NCversion of the Abelian Proca theory in the κ -deformed (2+1)D Minkowski spacetime. In orderto check that our calculation is correct we can obtain directly this Lagrangian by applying thecoordinate transformation ( τ, x ) → ( t, x ) in Proca theory. The Abelian Proca model on theextended Minkowski spacetime ( τ, x ) isˆ S = Z dτ d x (cid:20) − F µν F µν + M A µ A µ i = Z dτ d x − h ∂A i ∂τ ( ∂A i ∂τ − ∂A ∂x i ) + ∂A ∂x i ( ∂A ∂x i − ∂A i ∂τ ) + ∂A j ∂x i ( ∂A j ∂x i − ∂A i ∂x j ) i + M A µ A µ ! (88)where F µν = ∂ µ A ν − ∂ ν A µ . By a coordinate transformation (12) we can rewrite the aboveactions in terms of ( t, x ) with explicit noncommutativity,ˆ S = Z dtd x √− g (cid:18) − (cid:20) √− g ∂A i ∂t ( 1 √− g ∂A i ∂t − ∂A ∂x i ) + ∂A ∂x i ( ∂A ∂x i − √− g ∂A i ∂t )+ ∂A j ∂x i ( ∂A j ∂x i − ∂A i ∂x j ) (cid:21) + M A µ A µ (cid:19) . (89)16ere we have assumed that ˙ τ = √− g >
0. After some straightforward manipulation we findthat ˆ S = 12 Z dtd x (cid:18) ∂ A i ∂ i A + √− g∂ i A j ∂ j A i − √− g∂ i A j ∂ i A j − √− g∂ i A ∂ i A − √− g ∂ A i ∂ A i + M √− gA µ A µ (cid:19) . (90)As we have expected, this action is equal to the model described by the Lagrangian (87).Notice that this NC version is that, besides the modification of the field dynamics in thisnew spacetime, the mass term has also changed and it is not equal to the usual Minkowskispacetime so, the particle associated with this field must have a different mass in this spacetime.It is noteworthy that the transformations (75) are not the unique ones that lead to thisresult. We can also use the transformation δf µ = − δg µ = ǫ µρσ ∂ ρ α σ . (91)By assuming the above transformation and defining the final soldered field A µ = 1 √ M ( f µ − g µ ) (92)we can arrive at the same Lagrangian as in (87). This result led the authors of [19] to theidea of generalizing the soldering formalism. As it was mentioned before, the basic idea ofsoldering was that adding two independent dual Lagrangians does not give us new informationand for obtaining a gauge invariant model we have to fuse two Lagrangians via the Noetherprocedure. This idea was successfully applied to different models in various dimensions suchas chiral Schwinger model with opposite chiralities.Some years after proposing this idea it was shown that the usual sum of opposite chiralbosons models is, in fact, gauge invariant and it corresponds to a composite model, where thecomponent models are the vector and axial Schwinger models [19]. As a consequence, we canreinterpret the soldering formalism as a kind of degree of freedom reduction mechanism.In the case at hand, two transformations (75) and (91) result in the same effective actionbut in a general case we may obtain two apparently different actions. For example, if we add aninteraction term to the Lagrangians (71), the final result will be different. This property is thesubject of the generalized soldering formalism [19]. Now this question may arise whether thesetwo actions are describing two distinct phenomena. However, by calculating the generatingfunctional of these two Lagrangians we have the same result. This shows that we are dealingwith the same physics but described by different Lagrangians. The idea that we can construct a bosonized version for some fermionic models in order tostudy the properties of the target model through a theoretically easier version, the bosonizedone, has dwelled in the theoretical physicists mind during the 80’s and 90’s. In two spacetimedimensions, the concept of chirality together with the bosonization one were discussed afterthe influence of the chiral boson version in string theories.17olding that thought, M. Stone provided a method which objective was to put togetherin the same multiplet, two chiral versions of bosonized model in such a way that an effectivefinal model was obtained and the target was to analyze physically the properties of this lastone. Another result obtained in the soldering technique is to discuss the fact that the finalaction is connected to the first ones through duality properties.There was a relevant production of papers considering several models but none of themhave considered NC models, which in fact was our objective here. We have analyzed the κ -Minkowski noncommutativity where some variations of the CSM were soldered and thesoldered (final) action yielded were discussed in the aftermath.However, as a perspective, we can provide a constraint analysis via Dirac and symplecticformalisms, in order to compare the before and after soldering. The comparison can alsobe made together with commutative models, namely, what is new in the NC introduction.Another path is to investigate soldering in the light of the canonical noncommutativity, wherethe NC parameter is constant.The conversion of NC second-class constraints into first-class ones concerning the solderedactions can reveal interesting properties evolving gauge invariance of NC models. These ideasare, as a matter of fact, ongoing research that will be published elsewhere. V.N. would like to thank Prof. J. A. Helay¨el-Neto for valuable and insightful discussions.E.M.C.A. thanks CNPq (Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico)through Grants No. 301030/2012-0 and No. 442369/2014-0, for partial financial support,CNPq is a Brazilian scientific research support federal agency, and the hospitality of Theoret-ical Physics Department at Federal University of Rio de Janeiro (UFRJ), where part of thiswork was carried out.
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