Reply to ‘‘Comment on ‘Noncommutative gauge theories and Lorentz symmetry’’’

Rabin Banerjee* and Biswajit Chakraborty†

S. N. Bose National Centre for Basic Sciences, JD Block, Sector 3, Salt Lake, Kolkata 700098, India

Kuldeep Kumar‡

Department of Physics, Panjab University, Chandigarh 160014, India(Received 11 September 2007; published 19 February 2008)

This is a reply to the preceding ‘‘Comment on ‘Noncommutative gauge theories and Lorentzsymmetry,’’’ Phys. Rev. D 77, 048701 (2008) by Alfredo Iorio.

DOI: 10.1103/PhysRevD.77.048702 PACS numbers: 11.30.Cp, 11.10.Nx, 11.15.�q

This is a reply to the preceding Comment [1] on ourpaper . The author has criticized our earlier work withoutprecisely pointing out the error. However we remain con-tent with our analysis for the following reasons.

The main confusion of the author concerns a textbookapplication of Noether theorem to the problem at hand.There are two distinctions to be noted. First, such applica-tions involve only scalar parameters rather than vector ortensor ones as is the case here. Second, and more impor-tantly, textbook applications do not discuss actions thathave parameters that are not included in the configurationspace. In that case the Noether procedure gets nontriviallymodified as was shown in our paper [2]. We shall herereinforce this point by first discussing a simple examplefrom particle mechanics and finally make the connectionwith the field theoretic models considered earlier [2].

Consider a nonrelativistic particle of mass m moving inthree dimensions, subjected to a constant external force ~F:

m �xi � Fi: (1)

It follows immediately that the rate of change of its angularmomentum ~J � ~x� ~p, with ~p � m _~x being the linear mo-mentum, is precisely the applied torque ~� � ~x� ~F:

d ~Jdt� ~�: (2)

Now observe that Eq. (1) follows from the followingaction:

S �ZLdt �

Z �1

2m _x2 � ~F � ~x

�dt: (3)

It is natural to expect that the force Fi transforms cova-riantly under SO�3� rotation, so that Eq. (1) has a covariantform. Correspondingly, the action (3) is invariant underSO�3� rotation. But despite its rotational invariance, thisaction does not yield a conserved charge, which is theangular momentum ~J in this case; it rather satisfiesEq. (2), as mentioned above. Let us derive this from the

action S using Noether’s approach and make some impor-tant and relevant observations on the way, by consideringits response to an infinitesimal SO�3� rotation,

xi ! x0i � xi � �xi; (4)

where �xi � !ijxj, j!ijj � 1, !ij � �!ji. Under thistransformation Fi also undergoes the transformation

Fi ! F0i � Fi � �Fi; �Fi � !ijFj: (5)

The invariance of the action S under this transformationimplies

0 � �S �Z

dt�m _xi� _xi � Fi�xi � xi�Fi�: (6)

Note that it is important to consider the xi�Fi term here.We shall see later that this will play an important role.

Equation (6) can be rewritten, using the fact that� _xi �

ddt ��xi�, as

0 � �S �Z

dt�

d

dt�m _xi�xi� � �m �xi � Fi��xi � xi�Fi

�:

(7)

We can now get rid of the central term involving �xi, usingthe equation of motion (1), so that the on-shell version ofEq. (7) becomes

0 � �S �Z

dt�

d

dt�m _xi�xi� � xi�Fi

�: (8)

Before proceeding further, let us note at this stage that wecannot regard the force Fi as an auxiliary variable belong-ing to the configuration space. Variables like Fi should beinterpreted as coordinates in an extended space hiddenbehind the external dynamics. Correspondingly, any ofthe transformations in Fi, in particular, the rotation, cannotbe generated by a naive Poisson bracketing with the angu-lar momentum generator Ji:

fFi; Jjg � "ijkFk (9)

unlike

fxi; Jjg � fxi; "jkmxkpmg � "ijkxk: (10)

*rabin@bose.res.in†biswajit@bose.res.in‡kuldeepk@pu.ac.in

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Poisson brackets involving variables like Fi can be defined,but only in the extended space. Therefore, for a nondy-namical variable like Fi, forcing a requirement of ‘‘dy-namical consistency’’ on it and thereby saying that ~F doesnot transform under rotation does not make sense. Forother phase-space variables, of course, the requirement ofdynamical consistency

���xi; pi� � f��xi; pi�; Gg (11)

must be satisfied for any symmetry transformation gener-ated by G, where the �� on the left-hand side is the‘‘algebraic transformation’’ defined in the precedingComment.

Coming back to Eq. (8) we can now introduce a vector~! � f!kg, dual to the antisymmetric tensor !ij, as

!k �12"ijk!ij: (12)

This enables us to cast Eq. (8) in the form

0 �Z

dt!k

��

d

dt�"ijkxipj� � "kijxiFj

�: (13)

Now the arbitrariness of !k readily yields

d

dt�"ijkxipj� � "kijxiFj (14)

which is nothing but Eq. (2) in component form. We nowmake few comments.

(1) The covariant forms of the Eqs. (1) and (2) wereensured by the fact that we started with an invariantaction S in Eq. (3).

(2) This exercise demonstrates that SO�3� invariance ofS does not yield a conserved angular momentum ~Janymore. This is clearly in contrast with the trans-lational invariance of S, as the Lagrangian Lchanges only by a total time-derivative:

~x! ~x0 � ~x� ~a; (15)

L! L0 � L� ~F � ~a � L�d

dt� ~Ft� � ~a: (16)

(Note that the translational symmetry is not pre-served in presence of a nonconstant force ~F.) Thecorresponding conserved charge is (m _~x� ~Ft), asfollows from Eq. (1) by simple inspection.

(3) The nonconservation of angular momentum ~J, de-spite having an SO�3� symmetry in the action (3), isentirely due to the ‘‘transforming’’ ~F coming fromthe xi�Fi term in Eq. (6) which gives rise to thetorque ~� � ~x� ~F. One can therefore identifyEq. (2) or Eq. (14) as the criterion for SO�3� sym-metry. Further, ~J still generates rotation on all thephase-space variables � ~x; ~p� and functions thereofbut not on Fi, as mentioned earlier.

(4) If Fi were not to transform, then S is no longer

invariant. And even if we insist on �S � 0 underthe transformation (4), then we shall meet with acontradiction, To see that, set �Fi � 0 in Eq. (8).Then in place of Eq. (13) we have

Zdt!k

d

dt�"kijxipj� � 0 (17)

yielding ddt � ~x� ~p� � 0 as the equation, which ap-

pears to be a modified criterion for SO�3� invarianceof S in presence of a nontransforming ~F. But this isclearly in conflict with Eq. (2). Besides, note thatone cannot actually set �S � 0 in the left-hand sideof Eq. (8) to begin with as S in Eq. (3) is no longerinvariant if �Fi � 0. This indicates that the system(3) respects SO�3� symmetry only in presence of atransforming Fi as in Eq. (5).

Having studied the symmetry aspects of this particlemodel, we now turn our attention to our field-theory modeland try to reassess the comments made by Iorio in thepreceding Comment.

First of all, the existing similarity between this simplemodel from particle mechanics, Eq. (3), with the toy model(Eq. (23) in our paper [2])

S �Z

d4xL � �Z

d4x�1

4F��F�� � j�A�

�(18)

or the first-order (in ���) terminated effective commutativetheory (Eq. (76) in our paper [2])

S � �Z

d4x�

1

4F��F��

� ����1

2F��F�� �

1

8F��F��

�F��

�(19)

should be obvious. The constant background fields like thevector field j� and the tensor field ��� transforming under(homogeneous) Lorentz transformations ensure Lorentzinvariance of the actions. Both j� and the noncommuta-tivity parameter ��� here are the counterparts of the con-stant force Fi introduced in Eq. (3). Sheer presence of these‘‘transforming’’ external parameters gives rise to noncon-serving angular momentum tensors in these two respectivetheories (Eqs. (38) and (82) in our paper [2]):

@�M� � Aj � Aj � 0; (20)

@�M� � ��F��

�F��F� � 1

4F��F�

�

� ��F��

�F��F� � 1

4F��F�

�� 0; (21)

as happens in the particle model, as we have seen already.These two results were obtained in an ab initio computa-tion carried out in [2]. The main point of derivation is thatwe had to consider the terms involving �j� and ���� inthese respective analyses [2] as we did earlier in Eq. (6) for

COMMENTS PHYSICAL REVIEW D 77, 048702 (2008)

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the particle model. In these kinds of situations the currentsM�� will fail to satisfy the continuity equation in thesefield theoretical systems as certain components of j� or��� can be readily related to identifiable appropriate ex-ternal ‘‘torques’’ ~�. For example, one can get the time-derivative of Ji � 1

2 "ijkR

d3xM0jk from Eq. (20) as,

~� �d ~Jdt� ~R� ~j; (22)

where ~R �R

d3x ~A�x�. This has indeed the same structuralform as that of the particle model (2), with ~j indeed playingthe role of force ~F. The same holds for the space compo-nents of the constant background field P� introduced inEq. (7) of the preceding Comment [1]. Likewise one canalso relate appropriate components of � to ~� as well for themodel (19). We will not require the explicit form here.

All these analyses clearly show that the angular momen-tum tensor will not satisfy the continuity equation inpresence of such transforming additional constant parame-ters in the theory which act as constant background fieldsand give rise to torque on the system. In [3], the variationsof the parameters j� or ��� were not considered. Also, itwas demanded that [1,3]

�fj� � fj�;Qg � 0; �f��� � f���;Qg � 0 (23)

following from their requirement of dynamicalconsistency.

But as we have pointed out earlier the Poisson bracket ofany nondynamical variable like Fi, j�, or ��� can possiblybe defined only in an extended space in a more fundamen-tal theory that goes beyond the usual phase space discussedhere. For example, one can envisage a situation where theadditional variables describing the extended space corre-spond to an appropriate external system which is ‘‘robust’’enough, i.e. which is very weakly influenced by the dy-namics of the original variables. One can then possiblyconstruct the ‘‘total angular momentum,’’ taking into con-sideration the contribution of these additional variables,which can generate transformations in Fi, j�, or ���. Inother words, the ‘‘algebraic’’ transformations of j� or ���

given by

�j� � !��j�; ���� � !�

�� �!�

�� (24)

cannot be generated by the angular momentum M�� �

Rd3xM0��, obtained solely from the variables occurring

within the theory, through a Poisson bracket like (23), justas the transformation �Fi (5) could not be generated by anaive Poisson bracket of Fi with the angular momentumoperator.

The SUSY model considered in the Comment cannot becompared to any of the above-mentioned models, as theequation of motion for the D-field can be used to eliminateit from the Wess-Zumino model (11) in the precedingComment [1] to yield a meaningful on-shell version (17)in [1]. In contrast, neither our j�, nor author’s P� nor ���

can be eliminated in this manner, as the entire theorycollapses—as has been noted in the preceding Commentas well. Consequently, they cannot be regarded as variablesin the configuration space and Euler-Lagrange equation ofmotion of these fields does not make any sense. On thecontrary, the D-field has some similarity with Lagrangemultipliers which are counted in the configuration spacevariables and enforce meaningful constraints on the theory,like A0 in the Maxwell theory. Thus we see that thisprocedure has to be implemented case by case and onlyin those models where it can be done consistently.

We conclude here by summarizing that all we wanted todemonstrate in our paper [2] was that to preserve Lorentzinvariance of these models, where j� or ��� can be re-garded as constant background fields, it is necessary toconsider a transforming j� or ���. This is manifest fromthe structure of the actions (18) and (19) themselves. Wehave also seen that despite the Lorentz invariance, themodels do not admit angular momentum tensor M��

satisfying @�M�� � 0, as these background fields actlike forces, thereby generating external torque on the sys-tem, as can be seen by considering spatial componentsM0ij. Also one cannot impose the requirement of dynami-cal consistency on any background fields.

Finally, we would like to mention that there are reasonsfrom black hole physics to expect that the length scaledetermined by j���j has a lower bound [4]. It thus becomesnecessary to demand ��� to be constant from other physi-cal considerations. But as we have shown this cannot bereconciled with the usual Poincare invariance.Nevertheless, it has been pointed out recently that a twistedPoincare symmetry can be reconciled with constant ���

[5]. However these issues were not discussed by us in [2].

[1] A. Iorio, preceding Comment, Phys. Rev D 77, 048701(2008).

[2] R. Banerjee, B. Chakraborty, and K. Kumar, Phys. Rev. D70, 125004 (2004).

[3] A. Iorio and T. Sykora, Int. J. Mod. Phys. A 17, 2369

(2002).[4] S. Doplicher, K. Fredenhagen, and J. E. Roberts,

Commun. Math. Phys. 172, 187 (1995).[5] M. Chaichian, P. P. Kulish, K. Nishijima, and A. Tureanu,

Phys. Lett. B 604, 98 (2004).

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