Domain wall solitons and Hopf algebraic translational symmetries in noncommutative field theories
aa r X i v : . [ h e p - t h ] N ov YITP-07-78
Domain wall solitons and Hopf algebraic translationalsymmetries in noncommutative field theories
Yuya Sasai ∗ and Naoki Sasakura † Yukawa Institute for Theoretical Physics, Kyoto University,Kyoto 606-8502, Japan
Abstract
Domain wall solitons are the simplest topological objects in field theories.The conventional translational symmetry in a field theory is the generator of aone-parameter family of domain wall solutions, and induces a massless modulifield which propagates along a domain wall. We study similar issues in braidednoncommutative field theories possessing Hopf algebraic translational symme-tries. As a concrete example, we discuss a domain wall soliton in the scalar φ braided noncommutative field theory in Lie-algebraic noncommutative spacetime,[ x i , x j ] = 2 iκǫ ijk x k ( i, j, k = 1 , , κ . We then find the massless modulifield which propagates on the domain wall soliton. We further extend our analysisto the general Hopf algebraic translational symmetry. ∗ e-mail: [email protected] † e-mail: [email protected] Introduction
Noncommutative field theories [1, 2, 3, 4] are important subjects for studying the Planckscale physics. The most well-studied are the noncommutative field theories in Moyalspacetime, whose coordinate commutation relation is given by [ x µ , x ν ] = iθ µν with anantisymmetric constant θ µν . Field theories in Moyal spacetime are also known to appearas effective field theories of open string theory with a constant background B µν field[5, 6]. Thus, not only as the simplest field theories in quantum spacetime but also astoy models of string theory, various perturbative and non-perturbative aspects such asunitarity [7, 8, 9], causality [10], UV-IR mixing [11, 12, 13], renormalizability [12], scalarsolitons [14, 15, 16, 17], instantons [18, 19, 20, 21, 22, 23], monopoles [24, 25, 23], andother solitonic solutions [26, 27, 28, 29, 30] have extensively been analyzed. Recentlyit has been pointed out that Moyal spacetime is invariant under the twisted Poincar´esymmetry, which is a kind of Hopf algebraic symmetry [31, 32, 33]. There have beenvarious proposals to implement the twisted Poincar´e invariance in quantum field theories[34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48]. Gravity in fuzzy spacetimes hasalso been discussed in [49, 50, 51, 52, 53, 54, 55, 56].A prominent feature of Hopf algebraic symmetries is the general requirement of non-trivial statistics, which is called braiding, of fields to keep the symmetries at quantumlevel. In our previous paper [47], it has been shown that symmetry relations amongcorrelation functions can systematically be derived from Hopf algebraic symmetries inthe framework of braided quantum field theories [57], if appropriate braiding of fieldscan be chosen. This feature is in parallel with the existence of similar relations, such asWard-Takahashi identities, in field theories possessing conventional symmetries.The main motivation of this paper is to understand better the physical roles ofHopf algebraic symmetries in another setting. In this paper we study a domain wallsoliton in the three-dimensional noncommutative scalar field theory in Lie-algebraicnoncommutative space-time [ x i , x j ] = 2 iκǫ ijk x k ( i, j, k = 0 , ,
2) [58, 59, 60, 61, 62]. Thisnoncommutative spacetime has also a Hopf algebraic Poincar´e symmetry [61, 62, 47], butthe difference from Moyal spacetime is that its translational symmetry is Hopf algebraic,while the rotation-boost symmetry is Hopf algebraic in Moyal spacetime. Therefore thisnoncommutative field theory provides an interesting stage for investigating the physicalroles of the braiding and the Hopf algebraic translational symmetry on a domain wallsoliton, since the conventional translational symmetry in a field theory is the generatorof a one-parameter family of domain wall solutions, and induces a massless moduli fieldwhich propagates along a domain wall.This paper is organized as follows. In section 2.1, we review the three dimensionalnoncommutative φ theory in the Lie-algebraic noncommutative space-time [ x i , x j ] =2 iκǫ ijk x k . In section 2.2, we apply the criterion of Derrick’s theorem [63] to the φ theory and conclude that a domain wall solution is possible at least perturbatively in κ .In section 2.3, we solve the equation of motion to obtain a one-parameter family of the1ink solutions in perturbation of the noncommutativity parameter κ . In section 2.4, wediscuss the moduli space. In section 2.5, we analyze the moduli field, which propagatesalong the domain wall soliton, and conclude that the moduli field is massless. In section3, we study the general Hopf algebraic translational symmetry. The final section isdevoted to the summary and comments. φ theory in Lie-algebraic space-time and the domain wall solutions φ theory in Lie-algebraic spacetime In this subsection, we review the noncommutative φ theory in Lie-algebraic noncom-mutative space-time whose commutation relation is given by[ˆ x i , ˆ x j ] = 2 iκǫ ijk ˆ x k , (1)where i, j, k = 0 , , [58, 59], following the constructions of [60, 61, 47]. Imposing theJacobi identity and Lorentz invariance, we can determine the commutation relationsbetween the coordinates and momenta as follows [58]:[ ˆ P i , ˆ x j ] = − iη ij q κ ˆ P + iκǫ ijk ˆ P k , (2)where we have also imposed [ ˆ P i , ˆ P j ] = 0. We can identify these operators with the Liealgebra of ISO (2 ,
2) as follows: ˆ x i = κ ( ˆ J − ,i − ǫ ijk ˆ J jk ) , (3)ˆ P i = ˆ P µ = i , (4)1 + κ ˆ P µ ˆ P µ = 0 , (5)where the commutation relations of the Lie algebra of ISO (2 ,
2) are given by[ ˆ J µν , ˆ J ρσ ] = − i ( η µρ ˆ J νσ − η µσ ˆ J νρ − η νρ ˆ J µσ + η νσ ˆ J µρ ) , (6)[ ˆ J µν , ˆ P ρ ] = − i ( η µρ ˆ P ν − η νρ ˆ P µ ) , (7)[ ˆ P µ , ˆ P ν ] = 0 , (8)and the Greek indices run through − SL (2 , R ). The signature of the metric η ij is ( − , , η µν is ( − , − , , φ ( x ) be a scalar field in the three-dimensional spacetime. Its Fourier transfor-mation is given by φ ( x ) = Z dg ˜ φ ( g ) e iP ( g ) · x , (9)where P i ( g ) are determined by g = P − − iκP i ˜ σ i ∈ SL (2 , R ) and R dg is the Haarmeasure of SL (2 , R ). This P − can take two values, P − = ± p κ P i P i , (10)for each P i . This unphysical two-fold degeneracy can be deleted by imposing˜ φ ( g ) = ˜ φ ( − g ) . (11)The definition of the star product is given by e iP ( g ) · x ⋆ e iP ( g ) · x = e iP ( g g ) · x . (12)This determines the coproducts of P i and P − via the group product g g as∆ P i = P i ⊗ P − + P − ⊗ P i − κǫ ijk P j ⊗ P k , (13)∆ P − = P − ⊗ P − + κ P i ⊗ P i . (14)We consider the φ theory in the Lie-algebraic noncommutative space-time. We givethe action as follows: S = Z d x (cid:20) −
12 ( ∂ i φ ⋆ ∂ i φ )( x ) + 12 m ( φ ⋆ φ )( x ) − λ φ ⋆ φ ⋆ φ ⋆ φ )( x ) − m λ (cid:21) , (15)where we have chosen the constant term so that the minima of the potential vanish when κ = 0.Carrying out the coordinate integration, one finds a modified energy-momentumconservation: P i ( g g · · · ) = P + P + · · · + O ( κ ) = 0 at the classical level. Thisshould be regarded as a consequence of the Hopf algebraic translational symmetry. Anaive construction of noncommutative quantum field theory in this space-time leads todisastrous violations of the energy-momentum conservation in the non-planar diagrams The definition of ˜ σ i is given by ˜ σ = σ , ˜ σ = iσ , ˜ σ = iσ , with Pauli matrices ( σ , σ , σ ). We have also changed the normalization of P − by κ from (5). In fact, one can produce the commutation relation between the coordinates (1) by differentiatingboth hand sides of (12) with respect to P i ≡ P i ( g ) and P j ≡ P j ( g ) and then taking the limit P i , P i → ψ ( ˜ φ ( g ) ˜ φ ( g )) = ˜ φ ( g ) ˜ φ ( g − g g ) , (16)where ψ is an exchanging map. This is denoted by braiding. This braiding was firstderived from three dimensional quantum gravity with scalar particles [61]. With thisbraiding, correlation functions respect the Hopf algebraic symmetry at the full quantumlevel [47]. φ theory We consider a domain wall soliton in the noncommutative φ theory. At first we considerwhether the domain wall solution may exist or not by applying the criterion of Derrick’stheorem [63].Varying the action (15) with respect to φ ( x ), we obtain the equation of motion, ∂ φ ( x ) + m φ ( x ) − λ ( φ ⋆ φ ⋆ φ )( x ) = 0 . (17)Since our interest is in a domain wall, we consider only one spatial direction of thecoordinates . Let us change the variables P, P − as follows : P = 1 κ sinh( κθ ) P − = cosh( κθ ) , (18)where −∞ < θ < ∞ . Then the field φ ( x ) is given by φ ( x ) = Z dθ π ˜ φ ( θ ) e iκ sinh( κθ ) x . (19)The star product simply becomes e iκ sinh( κθ ) x ⋆ e iκ sinh( κθ ) x = e iκ sinh( κ ( θ + θ )) x . (20)Here we notice that the nontrivial momentum sum, which comes from the star prod-uct, can be described by the usual sum of θ . In fact, from (2), we can find that thecommutation relation between ˆ θ = κ sinh − ( κ ˆ P ) and ˆ x becomes[ˆ θ, ˆ x ] = − i, (21) In one dimension, there is no non-trivial noncommutativity of coordinates, but the coordinateand the momentum are noncommutative as in (2). Thus a soliton solution is not the same as thecommutative case. When one considers only spatial directions, one can safely take only the positive branch of P − in(10). θ becomes∆ˆ θ = ˆ θ ⊗ ⊗ ˆ θ, (22)which is the usual Leibnitz rule.Using (19) and (20), the equation of motion (17) becomes Z dθ π (cid:18) − κ sinh ( κθ ) ˜ φ ( θ ) + m ˜ φ ( θ ) − λ Z dθ π dθ π dθ π (2 π ) δ ( θ − θ − θ − θ ) ˜ φ ( θ ) ˜ φ ( θ ) ˜ φ ( θ ) (cid:19) e iκ sinh( κθ ) x = 0 . (23)Thus we find that (cid:18) − κ sinh ( κθ )+ m (cid:19) ˜ φ ( θ )2 π − λ Z dθ π dθ π dθ π δ ( θ − θ − θ − θ ) ˜ φ ( θ ) ˜ φ ( θ ) ˜ φ ( θ ) = 0 . (24)Next we define h ( x ) = Z dθ π ˜ φ ( θ ) e iθx . (25)Rewriting (24) with h ( x ), we obtain an equation of motion for h ( x ):1 κ sin ( κ∂ ) h ( x ) + m h ( x ) − λh ( x ) = 0 . (26)Now the equation has a familiar local interaction term, but has infinite higher derivativeterms. Another very important feature is that, though the star product (20) and hence(17) are not invariant under the simple translation x → x + a , the equation (26) has theobvious translational symmetry.To analyze (26), we may consider an action for h ( x ), which is given by S h = Z dx (cid:20) − κ sin( κ∂ ) h ( x ) sin( κ∂ ) h ( x ) + 12 m h ( x ) − λ h ( x ) − m λ (cid:21) . (27)Then the problem becomes to find the minimum of the energy E h = − S h with anappropriate boundary condition at the infinities x → ±∞ , where the field takes thevacuum values h = ± m/ √ λ .In this regard, we will consider perturbation in κ . The energy can be expanded inthe form, E h = − S h = Z dx " ∞ X n =1 κ n − C n ∂ n h ( x ) ∂ n h ( x ) ! + V ( h ( x )) , (28)where C n = 2 n − / ( n !(2 n − V ( h ( x )) = − m h ( x ) + λ h ( x ) + m λ ≥
0. Thepositivity of all the coefficients C n will play an essential role in the following discussions.5et us rescale x i → x ′ i = µ x i (0 < µ < ∞ ) and define h ( µ ) ( x ) = h ( µx ). Derrick’stheorem [63] tells us that if the energy for the rescaled field does not have any stationarypoints with respect to µ , there exist no soliton solutions. In our case, the energy for h ( µ ) ( x ) is given by E h ( µ ) = Z dx " ∞ X n =1 κ n − C n ∂ n h ( µ ) ( x ) ∂ n h ( µ ) ( x ) ! + V (cid:0) h ( µ ) ( x ) (cid:1) (29)= Z dx ′ µ " ∞ X n =1 µ n κ n − C n ∂ ′ n h ( x ′ ) ∂ ′ n h ( µ ) ( x ′ ) ! + V (cid:0) h ( µ ) ( x ′ ) (cid:1) (30)= 1 µ E + ∞ X n =1 µ n − E n , (31)where E = Z dx V ( h ( x )) ,E n = C n Z dx ( ∂ n h ( x )) . (32)All the E and E n are non-negative in general. For an h ( x ) connecting the distinctvacua, E and at least some of the E n are positive. Therefore (31) diverges at µ → +0 , + ∞ (or a finite µ c ) , and takes a minimum value at a positive finite µ . Thus weconclude that a domain wall solution in this noncommutative field theory is possible. h ( x ) Next we consider the perturbative solution of h ( x ). We write the perturbation series as h ( x ) = h ( x ) + κ h ( x ) + κ h ( x ) + · · · . Inserting this into the equation of motion (26),we obtain for each order of κ , ∂ h ( x ) + 2 h ( x ) − h ( x ) = 0 , (33) ∂ h ( x ) + 2 h ( x ) − h ( x ) h ( x ) − ∂ h ( x ) = 0 , (34) ∂ h ( x ) + 2 h ( x ) − h ( x ) h ( x ) − ∂ h ( x ) + 245 ∂ h ( x ) − h ( x ) h ( x ) = 0 , (35)...where we have set m = 2 , λ = 2 for simplicity. For example, the convergence radius of the infinite sum is | µ | < µ c = π/ κ for h ( x ) = tanh( x ). h ( x = ±∞ ) = ±
1. The equation (33) is the same as the equation of motion in the commutativecase. The general solution of (33) has two integration constants. One is interpreted asthe translation of the solution, and the other can be determined by the behavior at x = −∞ or ∞ . If one assumes h ( x = ±∞ ) = ± h ( x ) diverges or oscillates at x = ±∞ . For such an h ( x ), the solutions of h n ( x ) ( n = 1 , , · · · ) diverge at x = ±∞ ,unless h n ( x = ±∞ ) = 0. Thus the boundary condition h ( x = ±∞ ) = ± h ( x = ±∞ ) = ± h ( x = ±∞ ) = ±
1, the solution to the equation (33) iswell known and given by h ( x ) = tanh( x + a ) , (36)where a ∈ R . The arbitrary parameter a results from the translational invariance of theequation (33).Next we will solve the equation (34) for a = 0. Let us put h ( x ) = f ( x )cosh x . (37)Inserting this and (36) for a = 0 into (34), we obtain f ′′ ( x ) − xf ′ ( x ) − (cid:18) x cosh x − tanh x (cid:19) = 0 . (38)Then let us put f ′ ( x ) = cosh ( x ) g ( x ) , (39)and insert this into (38). The equation becomes g ′ ( x ) = 83 cosh x (cid:18) x cosh x − tanh x (cid:19) . (40)Integrating (40) over x , we obtain g ( x ) = 23 cosh x −
43 cosh x + A , (41)where A is an integration constant. Thus the differential equation of f ( x ) becomes f ′ ( x ) = 23 −
43 cosh x + A cosh x. (42)Integrating this over x and using (37), we obtain h ( x ) = 2 x x − x x + A (cid:18) x x + 38 tanh x + 14 cosh x tanh x (cid:19) + A cosh x , (43)7here A is an integration constant.Since the term with A is divergent at x = ±∞ , we have to put A = 0 from theboundary condition. The A term is allowed but can just be absorbed into the parameter a in (36), because tanh( x + κ A ) = tanh( x ) + κ A / cosh ( x ) + · · · . To systematicallykill such redundant integration constants, we impose the oddness condition, h n ( x ) = − h n ( − x ) for a = 0. Then A = 0 is also required. Finally, recovering the parameter a ,we obtain h ( x ) = 2( x + a )3 cosh ( x + a ) − x + a )3 cosh ( x + a ) . (44)In the same way, we can obtain the solution to the equation (35), which is given by h ( x ) = 134( x + a )45 cosh ( x + a ) − x + a )3 cosh ( x + a ) −
40 tanh( x + a )9 cosh ( x + a ) − x + a ) tanh( x + a )9 cosh ( x + a ) + 52 tanh( x + a )9 cosh ( x + a ) . (45)This procedure will be able to be repeated to a required order. φ ( x ) and the moduli space In the preceding subsection, we have obtained the perturbative solution of h ( x ). Then weformally know the perturbative soliton solution of φ ( x ) through ˜ φ ( θ ), which are relatedto φ ( x ) and h ( x ) by (19) and (25), respectively.In the following let us discuss the moduli space of the domain wall solution. In h ( x ),the moduli parameter is just the translation parameter a . This translation correspondsto the phase rotation ˜ φ ( θ ) → e iaθ ˜ φ ( θ ), as can be seen in (25). Therefore the translationon φ ( x ) is given by T a φ ( x ) = Z dθ π ˜ φ ( θ ) e i ( θa + κ sinh( κθ ) x ) (46)= e ia ˆ θ φ ( x ) . (47)This last expression shows that the operator ˆ θ , which is a non-linear function of ˆ P , isthe generator of the translational moduli. In fact, by using the Leibnitz rule (22) andfollowing the same procedure as a conventional symmetry, one can directly show that,if φ ( x ) is a solution to the equation of motion (17), e ia ˆ θ φ ( x ) is also a solution. Thegeneralization of this fact to the general Hopf algebraic translational symmetry will bediscussed in section 3. 8 .5 The moduli field from the Hopf algebraic translationalsymmetry Another interesting consequence of the conventional translational symmetry in a fieldtheory is the existence of a massless propagating field along a domain wall. This field canbe obtained by generalizing the constant moduli parameter a to a field a ( x k ) dependingon the coordinates along a domain wall. In this subsection, we will study this aspect inour noncommutative field theory.We go back to the three dimensional case. For simplicity, we set κ = 1. We changethe variable P i ( g ) as follows: P i = sinh( √ k ) k i √ k . (48)This k i is the three-dimensional analog of θ in the previous subsections. The field φ ( x )can be rewritten as φ ( x ) = Z d P (2 π ) √ P ˜ φ ( P ) e iP · x = Z d k (2 π ) sinh ( √ k ) k ˜ φ ( k ) e i sinh( √ k ) ki √ k x i ≡ Z d k (2 π ) ˜ ϕ ( k ) e i sinh( √ k ) ki √ k x i . (49)Let us define h (ˆ x ) = Z d k (2 π ) ˜ ϕ ( k ) e ik · ˆ x (50)as in (25). Then it can be shown that the action (15) is equivalent to the followingaction [58]: S = h | (cid:18) − h (ˆ x )[ ˆ P i , [ ˆ P i , h (ˆ x )]] + 12 m h (ˆ x ) − λ h (ˆ x ) (cid:19) | i , (51)where | i denotes the momentum zero eigenstate ˆ P i | i = 0, and[ ˆ P i , ˆ x j ] = − iη ij q P + iǫ ijk ˆ P k , (52)[ ˆ P i , ˆ P j ] = 0 . (53)From the commutation relation, the following relation is satisfied [58]:ˆ P i e ik · ˆ x | i = sinh( √ k ) k i √ k e ik · ˆ x | i = P i e ik · ˆ x | i . (54)9hus e ik · ˆ x | i is the eigenstate of ˆ P i with an eigenvalue P i . In the following discussions,we use the notation | P i i ≡ e ik · ˆ x | i .The equation of motion from (51) is (cid:16) − [ ˆ P , h (ˆ x )] + m h (ˆ x ) − λh (ˆ x ) (cid:17) | i = 0 . (55)As has been discussed in the preceding subsections, there exists a one-parameter familyof domain wall solutions h asol (ˆ x ) to (55), where a is the translational parameter. Onemay expand the solution with respect to a as h asol (ˆ x ) = h sol (ˆ x ) + af (ˆ x ) + · · · , wherewe have chosen ˆ x as the spatial direction perpendicular to the domain wall . Then,putting this expansion into (55) and taking the first order of a , f (ˆ x ) is shown to satisfy (cid:16) − [ ˆ P , f (ˆ x )] + m f (ˆ x ) − λ ( h sol (ˆ x )) f (ˆ x ) (cid:17) | i = 0 . (56)To study the property of the moduli field, we will replace a to a ( ˆ x , ˆ x ). In doing so,the braiding property (16) plays essential roles. For general h (ˆ x ) , h (ˆ x ), we have thefollowing commuting property, h (ˆ x ) h (ˆ x ) = Z dg Z dg ˜ φ ( g ) ˜ φ ( g ) e ik ( g ) · ˆ x e ik ( g ) · ˆ x = Z dg Z dg ˜ φ ( g ) ˜ φ ( g − g g ) e ik ( g ) · ˆ x e ik ( g ) · ˆ x = Z dg Z dg ˜ φ ( g ) ˜ φ ( g ) e ik ( g g g − ) · ˆ x e ik ( g ) · ˆ x = Z dg Z dg ˜ φ ( g ) ˜ φ ( g ) e ik ( g g ) · ˆ x = h (ˆ x ) h (ˆ x ) , (57)where we have used the invariance of the Haar measure. Inserting h (ˆ x ) = h sol (ˆ x ) + a ( ˆ x , ˆ x ) f (ˆ x ) into the equation of motion (55) and taking the first order of a ( ˆ x , ˆ x ),we obtain (cid:16) − [ ˆ P , a ( ˆ x , ˆ x ) f (ˆ x )] + m a ( ˆ x , ˆ x ) f (ˆ x ) − λa ( ˆ x , ˆ x )( h sol (ˆ x )) f (ˆ x ) (cid:17) | i = 0 . (58)Then, from (56), we obtain [ ˆ P , a ( ˆ x , ˆ x )] f (ˆ x ) | i = 0 . (59)After the Fourier transformation, we find Z P Z P ˜ a ( P ) ˜ f ( P )( P ( g g ) − P ) | P ( g g ) i = 0 , (60) The following discussions do not depend on the value of a where the expansion with respect to a iscarried out. P i = ( P , , P ) and P i = (0 , P , P ,which is given by ∆( P ) = P ⊗ ⊗ P + P ⊗ P + 2 √ P P i ⊗ √ P P i + P i P j ⊗ P i P j , (61)the equation (60) becomes Z P Z P P ˜ a ( P )(1 + P ) ˜ f ( P ) | P ( g g ) i = 0 . (62)Operating h x | from the left, we find Z P Z P P ˜ a ( P )(1 + P ) ˜ f ( P ) e iP ( g g ) · x = Z P Z P P ˜ a ( P )(1 + P ) ˜ f ( P ) e iP · x ⋆ e iP · x = − ∂ a ( x , x )(1 − ∂ ) f ( x )=0 . (63)Thus, since (1 − ∂ ) f ( x ) does not vanish constantly, we obtain ∂ a ( x , x ) = 0 . (64)Thus we conclude that the moduli field is massless.The preceding discussions in the operator formalism can be repeated with the starproduct. Putting the expansion φ asol ( x ) = φ sol ( x ) + a g ( x ) + · · · into the equation ofmotion (17), one obtains ∂ g ( x ) + m g ( x ) − λφ sol ( x ) ⋆ φ sol ( x ) ⋆ g ( x ) = 0 . (65)Next we define the moduli field a ( x , x ), and consider φ ( x ) = φ sol ( x ) + a ( x , x ) ⋆ g ( x ).Putting this into the equation of motion and taking the first order of a ( x , x ), we obtain ∂ ( a ( x , x ) ⋆ g ( x )) + m ( a ( x , x ) ⋆ g ( x )) − λφ sol ( x ) ⋆ φ sol ( x ) ⋆ a ( x , x ) ⋆ g ( x ) = 0 , (66)where we have used the property similar to (16) for the star product. The first term of(66) can easily be computed by using the coproduct of P . Using (61) and (10), ∂ ( a ⋆ g )becomes ∂ ( a ( x , x ) ⋆g ( x )) = a ( x , x ) ⋆∂ g ( x )+ ∂ a ( x , x ) ⋆g ( x ) − ∂ a ( x , x ) ⋆∂ g ( x ) . (67)Thus (66) becomes0 = a ( x , x ) ⋆ ( ∂ g ( x ) + m g ( x ) − λg ( x ) ⋆ φ sol ( x ) ⋆ φ sol ( x ))+ ∂ a ( x , x ) ⋆ g ( x ) − ∂ a ( x , x ) ⋆ ∂ g ( x )= ( g ( x ) − ∂ g ( x )) ⋆ ∂ a ( x , x ) , (68)where we have used (65). Thus we obtain the same conclusion as above.11 The general Hopf algebraic translationalsymmetry
In the preceding section, the discussions are restricted to the specific noncommutativefield theory. However, it is interesting to know what holds for the general Hopf algebraictranslational symmetry. In this section, we will show that the results in the precedingsection are the general consequence of a Hopf algebraic translational symmetry.We first assume that, in considering domain wall solutions, only one direction ofmomentum is relevant. Then the (associative) coproduct of the momentum may bewritten as ∆( ˆ P ) = X i a i ( ˆ P ) ⊗ b i ( ˆ P ) . (69)This defines the associative sum of two momenta ⊕ .Let us consider a small momentum P ε . One may consider its n sum, P n ≡ n z }| { P ε ⊕ P ε ⊕ · · · ⊕ P ε . (70)For such P n , let us define θ ( P n ) = nP ε . (71)Then θ ( P ) can be shown to define an additive quantity for ⊕ as θ ( P n ⊕ P m ) = θ ( P n + m ) (72)= ( n + m ) P ε (73)= θ ( P n ) + θ ( P m ) , (74)where we have used the associativity of ⊕ . This shows the usual Leibnitz rule for thecoproduct of ˆ θ , ∆(ˆ θ ) = ˆ θ ⊗ ⊗ ˆ θ. (75)The above discussions may be generalized to negative n ’s, and further to a continuousmomentum by considering the limit P ε → θ ( P ) as dθ ( P ) dP = lim P ε → θ ( P ε ⊕ P ) − θ ( P ) P ε ⊕ P − P = lim P ε → P ε P ε ⊕ P − P . (76)The last limit can be computed from a given coproduct of momentum . The initialcondition should be taken as θ (0) = 0. For the limit to have a finite value, 0 ⊕ P = P is necessary. This is mathematically obtained fromthe axiom (id ⊗ ǫ )∆ = ( ǫ ⊗ id)∆ = 1 with ǫ ( P ) = 0, where ǫ is the counit map. dθ ( P ) dP = lim P ε → P ε √ κ P P ε + p κ P ε P − P = 1 √ κ P . (77)With the initial condition θ (0) = 0, the solution is actually given by P = κ sinh( κθ ),which agrees with (18).As explained in section 2.5, the usual Leibnitz rule (75) for ˆ θ implies that e ia ˆ θ φ ( x )forms a one-parameter family of domain wall solutions, provided that φ ( x ) is such asolution. It would be physically reasonable to assume that there exists at least onedomain wall solution which connects distinct vacua with the same energy, if a theoryhas multiple vacua and is physically sensible. Therefore a noncommutative field theorypossessing a Hopf algebraic translational symmetry will have a one-parameter family ofdomain wall solutions, if it has multiple vacua with the same energy. The associatedmoduli field will also have a vanishing mass, since the zero mode of the moduli field isthe parameter itself, and its potential should be flat in this direction. We have studied the domain wall soliton and its moduli field in the braided φ noncom-mutative field theory in the three dimensional Lie algebraic noncommutative spacetime[ x i , x j ] = 2 iκǫ ijk x k . This noncommutative spacetime is known to have a Hopf algebraictranslational symmetry, and provides an interesting stage for investigating the physicalroles of a Hopf algebraic translational symmetry on domain walls. We have found thatthere exists a one-parameter family of the solutions, and the mass of the moduli fieldpropagating along the domain wall vanishes. We have also argued that the results shouldalso hold in the general noncommutative field theory with a Hopf algebraic translationalsymmetry. This conclusion agrees with what can be obtained from the conventionaltranslational symmetry of the usual field theory. Therefore our results show another ev-idence for the physical importance of Hopf algebraic symmetries as much as the standardLie-algebraic symmetries.Two comments are in order. Firstly, we have used the braiding property when weanalyze the equation of motion for the moduli field. Therefore the non-trivial statistics ofboth the domain wall and the moduli field seem to play significant roles in their dynamics.Secondly, in our discussions, the operator ˆ θ , which has a Lie-algebraic coproduct, playsessential roles. 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