noncommuting electric fields and algebraic consistency in noncommutative gauge theories

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PHYSICAL REVIEW D 67, 105002 ~2003!

Noncommuting electric fields and algebraic consistency in noncommutative gauge theories

Rabin Banerjee*Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organisation (KEK), Tsukuba 305-0801, Japa

~Received 30 January 2003; revised manuscript received 28 February 2003; published 12 May 2003!

We show that noncommuting electric fields occur naturally inu-expanded noncommutative gauge theories.Using this noncommutativity, which is field dependent, and a Hamiltonian generalization of the Seiberg-Wittenmap, the algebraic consistency in the Lagrangian and Hamiltonian formulations of these theories is established.A comparison of results in different descriptions shows that this generalized map acts as a canonical transfor-mation in the physical subspace only. Finally, we apply the Hamiltonian formulation to derive the gaugesymmetries of the action.

DOI: 10.1103/PhysRevD.67.105002 PACS number~s!: 11.10.Nx, 11.10.Ef

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I. INTRODUCTION

Snyder’s@1# old idea that spatial coordinates do not comute has undergone a recent revival due to its appearanstring theory@2#. Inspired by this fact, several papers haappeared discussing different aspects of quantum mechaand field theory on noncommutative space@3#.

There are, however, some issues which have receivedattention than others. For instance, only a few papers@4–7#discuss the Hamiltonian treatment of noncommutative gatheories. Indeed one possible approach to study these tries is to exploit the Seiberg-Witten map@2#, which yields acommutative equivalent to the original noncommutattheory. However, this transition is done at the level of tLagrangian. Also, non-Abelian theories@6,8# have not beenthat widely studied as their Abelian counterparts. The isof algebraic consistency among the various approaches isopen. Likewise, the implications of the Seiberg-Witten min constructing effective theories from noncommutative thries are not completely clear. Indeed it is known thatenergy-momentum tensor obtained directly from noncommtative theory and then using the map is different fromexpression obtained by considering the action of the effectheory @9#.

In this paper we show that noncommuting electric fieoccur in noncommutative gauge theories, mapped to tcommutative equivalents by the Seiberg-Witten transformtion. This algebra is used to obtain the equations of motfrom the Hamiltonian. Its equivalence with the Lagrangiequations of motion is established. By a suitable redefinitwe show that both the Lagrangian and Hamiltonian canput in the usual form as the difference or the sum ofsquares of the electric and magnetic fields. The entire efof noncommutativity is shifted to the nontrivial algebamong these fields. A Hamiltonian generalization of tSeiberg-Witten map is obtained. This is used to show thatcommutative equivalents obtained either from the noncomutative Lagrangian or Hamiltonian are compatible. In coformity with @9# we find that the computation of the Hami

*On leave from S.N.Bose National Center for Basic SciencCalcutta, India. Email address: [email protected]; [email protected]

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tonian density does not commute with the Seiberg-Witexpansion and the star products. The result is differentpending upon whether it is obtained directly from the effetive theory or whether it is obtained in the noncommutatversion, after which the map is exploited. However, we fithat, after implementing the Gauss constraint, this differeis a total boundary, so that the expressions for the Hamtonian agree. The stability of the Poisson algebra amongelectric and magnetic fields, under the generalized SeibWitten map, is examined. This is used to clarify certainsues regarding the possible interpretation of this map acanonical transformation. Our analysis is for Yang-Mitheory with U(N) gauge group, including, in particularN51 ~i.e., Maxwell’s theory!.

An application of the Hamiltonian analysis has been dcussed in detail where the gauge symmetries of the actionsystematically derived, following the Dirac algorithm. Fthe U(1) case it implies that the equations of motion calways be put in a ‘‘Maxwell’’-like form.

After setting up the notation, we carry out the Hamtonian analysis, in both the noncommutative and effecttheories, in Sec. II. Section III contains our analysis of tSeiberg-Witten map as a canonical transformation. THamiltonian formulation is used to derive, in Sec. IV, thgauge symmetries of the action in the noncommutative vables, while the concluding remarks are given in Sec. V.

The ordinary Yang-Mills action is given by

S521

4E d4x Tr~FmnFmn! ~1!

where the non-Abelian field strength is defined as usual,

Fmn5]mAn2]nAm2 i @Am ,An# ~2!

in terms of the gluon field

Am~x!5Ama ~x!Ta. ~3!

HereTa are the generators of aU(N) gauge group satisfying

@Ta,Tb#5 i f abcTc, $Ta,Tb%5dabcTc, Tr~TaTb!5dab.~4!

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RABIN BANERJEE PHYSICAL REVIEW D67, 105002 ~2003!

The noncommutative generalization of this theoryvolves the star product of the noncommutative field strenFmn , expressed in terms of the fieldAm ,

Fmn5]mAn2]nAm2 i ~Am* An2An* Am! ~5!

so that

S521

4E d4x Tr~ Fmn* Fmn!521

4E d4x Tr~ FmnFmn!

~6!

where the second equality follows on using the definitionthe star product,

~A* B!~x!5eiuab]a]b8 /2A~x!B~x8!ux85x , ~7!

and dropping boundary terms. Hereumn is a real and anti-symmetric constant matrix.

II. HAMILTONIAN ANALYSIS

We now carry out a Hamiltonian analysis of this theory.order to avoid higher order time derivatives, henceforthuab

will be chosen to have only spatial components so thatu0a

50 andu i j 5e i jkuk. Also, the ensuing analysis will be confined to the leading order inu only. First, the effective theoryobtained by the Seiberg-Witten map@2# is considered.

A. Effective theory

To first order inu it is possible to relate the variablesthe noncommutative spacetime with those in the usualby the map@2#

Am5Am21

4uab$Aa ,]bAm1Fbm%1O~u2!, ~8!

Fmn5Fmn11

4uab~2$Fma ,Fnb%2$Aa ,DbFmn1]bFmn%!

1O~u2! ~9!

where the covariant derivative is defined as

Dml5]ml1 i @l,Am#. ~10!

On applying this map, the action~6! is written in terms ofan effective theory comprising the usual variables@10#,

S→Se f f521

4E d4x TrS FmnFmn11

2uab~2$Fma ,Fnb%

2$Aa ,DbFmn1]bFmn%)Fmn D1O~u2!.

~11!

The above form is further simplified by dropping a boundaterm that does not affect the equations of motion, to yieldfollowing Lagrangian, expressed solely in terms of the fiestrength:

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4TrS FmnFmn1

1

2uabS 2$Fma ,Fnb%

11

2$Fba ,Fmn% DFmnD1O~u2!. ~12!

In this form the gauge invariance of the theory underusual gauge transformations (dAm5Dma) becomes mani-fest.

Since the Lagrangian is written in terms of the fiestrengths, it is possible to work with the electric and manetic fields, exactly as happened in the case of Abetheory @11#,

Eia52F0i

a , Bia5

1

2e i jkF jk

a . ~13!

In terms of these variables, the effective Lagrangian~12!becomes

Le f f51

2Tr@~E22B2!~12u•B!1u•E~B•E1E•B!#.

~14!

The canonical momentapma , conjugate toAma, are found

to be

p0a50, ~15!

which is the primary constraint of the theory, while the trmomenta are

p ia5Ei

a11

2Tr@~$Bi ,u jEj%2$Ei ,u jBj%1u i$Bj ,Ej%!Ta#.

~16!

Since the canonical variables are (Ai ,p i), it is useful forlater convenience to invert the above relation and solveelectric field in terms of the momenta,

Eia5p i

a21

2Tr@~$Bi ,u jp j%2$p i ,u jBj%1u i$Bj ,p j%!Ta#.

~17!

The Hamiltonian is now obtained in the usual way byLegendre transform,

He f f5E d3x~2p iaAi

a2Le f f!

5E d3x@p iaEi

a1A0a~Dip i !

a2Le f f#. ~18!

Time conservation of the primary constraint yields a sondary constraint. For that it is essential to expressHamiltonian in terms of the canonical variables. In this cait is obvious from Eqs.~14! and~17! that the dependence othe Hamiltonian onA0 has been isolated completely. Thsecondary constraint is therefore the usual Gauss constr

~Dip i !a50. ~19!

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NONCOMMUTING ELECTRIC FIELDS AND ALGEBRAIC . . . PHYSICAL REVIEW D 67, 105002 ~2003!

It is also possible to verify that no further constraints agenerated by this iterative prescription. As expected, thare only first class constraints in the theory, which annihilthe physical states of the theory. The physical Hamiltoniani.e., the Hamiltonian acting on the physical states—is thgiven by dropping the second term of the last expressioEq. ~18!. Using Eqs.~14! and ~16! in Eq. ~18!, we get

He f f51

2E d3x Tr@~E21B2!~12u•B!1u•E~B•E1E•B!#

~20!

which has a very similar structure as the effective Lagraian ~14!. Note also that setting the noncommutative paraeter to zero reproduces the standard expression for YMills theory. For Abelian theory, the above form waobtained and discussed in@12#.

The positive definiteness of the above Hamiltonian ismanifest. It is, however, possible to redefine the electricmagnetic fields,

Eia5Ei

a21

4u jdabc~Bj

bEic22Ej

bBic!, ~21!

Bia5Bi

a21

4u jdabcBj

bBic ~22!

so that the Hamiltonian, in these variables, becomes

He f f51

2E d3x Tr~E21B2!. ~23!

It is now structurally identical to the Hamiltonian for ordnary Yang-Mills theory. The entire effect of noncommutatiity will be shifted to the nontrivial algebra among the electand magnetic fields, a point which we shall consider in sodetails later. Also note that in these variables, the Lagrang~14! has the form of the usual Yang-Mills Lagrangian,

Le f f51

2Tr~E22B2!. ~24!

B. Analysis in the noncommutative variables

We shall now consider the Lagrangian following from taction~6! in the original variables with a caret and obtain tHamiltonian. This will be expressed in terms of the usuvariables by using the map~9!, so that a comparison can bmade with Eq.~20!. We find that although the Hamiltoniadensities are different, the Hamiltonians on the physical sspace become identical. A Hamiltonian analysis of Eq.~6!has also been done in@6#.

The definition of the canonical momenta leads to a pmary constraint:

p0a50 ~25!

and

p ia52F0i

a 5Eia . ~26!

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The Hamiltonian follows from the Legendre transform athe use of certain symmetry operations,

H5E d3x TrS 1

2p ip i1

1

4F i j F i j 1~D ip i !A0D ~27!

where the~careted! covariant derivative is defined as

~D ip i !a5] ip i

a1 i ~p i* Ai2Ai* p i !a

5] ip ia1

1

2f abc$Ai

b ,p ic%*

2i

2dabc@Ai

b ,p ic#* . ~28!

Here both the commutator and anticommutator involvestar multiplication. The above equation defines the Gaoperator, whose vanishing yields the secondary constraAs done earlier we pass to the physical sector by imposthis constraint. In terms of the~careted! electric and magneticfields, the Hamiltonian reduces to

H51

2E d3x Tr~E21B2!. ~29!

It should be pointed out this is an exact result valid toorders in the expansion parameteru. To compare with Eq.~20!, we use the map~9!. Then it would be useful to recasthis map in terms of the electric and magnetic fields,

Eia5Ei

a11

2u lmdabcS e impBp

cElb2Al

b]mEic1

1

2f cdeAl

bAme Ei

dD ,

~30!

Bia5Bi

a21

4u lmdabc~e lmpBp

bBic12Al

b]mBic2 f cdeAl

bAme Bi

d!

~31!

which leads to

Tr E25Tr@E2~12u•B!1u•E~B•E1E•B!2u•“3~E2A!#,~32!

Tr B25Tr@B2~12u•B!2u•“3~B2A!#. ~33!

Using Eqs.~32! and~33! in Eq. ~29!, we see that the Hamil-tonian density is different from that given in Eq.~20!. How-ever, the difference is a boundary term so that the Haminians in the two cases agree, as announced earlier. Alsshould be mentioned that the agreement holds only forphysical Hamiltonians, obtained from the canonical Hamtonians by dropping the terms proportional to the Gauss cstraint in either description.

III. SEIBERG-WITTEN MAP AS A CANONICALTRANSFORMATION

The relations~17! and ~26! connecting the electric fieldwith the momenta, together with Eq.~30!, provide a map

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between the momenta in the noncommutative space andordinary one,

p i5p i1u lm

4~$Flm ,p i%2$Al ,Dmp i1]mp i%!

2u im

2$Flm ,p l% ~34!

which may be regarded as the Hamiltonian generalizationthe usual Seiberg-Witten map~8!. This result is true for anydimensions. If we specialize to 311 dimensions, it simpli-fies to

p i5p i1e i jk

4$F jk ,u lp l%2

e lmnun

4$Al ,Dmp i1]mp i%

~35!

while in 211 dimensions, withu lm5e lmu, it is even sim-pler,

p i5p i2e lmu

4$Al ,Dmp i1]mp i%. ~36!

We shall next discuss the possible interpretation of tmap as a canonical transformation. Instead of working wthe usual canonical coordinates and momenta, we mighwell formulate the discussion in terms of the electric amagnetic fields that have been considered so far. Also, sall the essential features are contained in the Abelian veritself, henceforth we confine ourselves to this case. It asimplifies the algebra and permits a quick check with soexisting results@11#.

As a first step it is necessary to derive the algebra amthe electric and magnetic fields. Using the definition~17! ofthe electric field in terms of the canonical variables, itpossible to get the complete algebra,

$Bi~x!,Bj~y!%50, ~37!

$Ei~x!,Bj~y!%52@11ukBk~x!#D i j ~x2y!

1u$ iBk%~x!Dk j~x2y!, ~38!

$Ei~x!,Ej~y!%5u [ jEk]~y!D ik~x2y!1u [ iEk]~x!D jk~x2y!

1uk„Ek~y!2Ek~x!…D i j ~x2y! ~39!

where

A$ iBk%5AiBk1AkBi , A[ iBk]5AiBk2AkBi ~40!

and the derivative operator has been absorbed in

D i j ~x2y!5e i jk]kd~x2y!. ~41!

Apart from Eq.~37!, the other two brackets involve corection terms which vanish with the vanishing of the expasion parameter. In particular, the electric fields become ncommuting. Moreover, since this noncommutativity is fiedependent, it is essential to verify the Jacobi identity forcomplete algebra. Those involving threeB terms are trivially

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zero. The Jacobi involving threeE terms is also trivial, sincethe terms are ofO(u2). The one involving twoB terms andoneE term is also zero, as may be quickly verified by simpinspection. The only nontrivial check involves twoE termsand oneB term. Some amount of algebra is now requireWe find

$Ei~x!,$Ej~y!,Bk~z!%%

5unD in~x2y!D jk~y2z!2@u jD in~x2y!

1unD i j ~x2y!#Dnk~y2z!. ~42!

The other double bracket is obtained by interchangingi withj andx with y, so that a total of six terms emerges from thalgebra. There is an exact one-to-one cancellation of thterms with the six terms obtained from the finalB-E-Edouble bracket, so that the Jacobi identity is satisfied.

As an illustration of the use of this involved algebra, wshow that it correctly reproduces the Lagrangian equationmotion, in the Hamiltonian formulation. The equationsmotion obtained from the Lagrangian~12! can be expressedin terms of a displacement fieldD and a magnetic fieldH as@11#

¹•D50, ~43!

]

]tD2¹3H50 ~44!

where

D5~12u•B!E1~u•E!B1~B•E!u, ~45!

H5~12u•B!B2~u•E!E11

2~E22B2!u. ~46!

We now reproduce these equations in the Hamiltonformulation. The first of these equations is just the Gaconstraint~19!. This is easily verified by looking at the definition of the canonical momenta~16! and identifying it withthe displacement fieldD. Effectively Eq.~44! is the genuineequation of motion since it involves the accelerations. Hing obtained the complete algebra, the equations of mofor the magnetic and electric fields are obtained by bracking with the Hamiltonian~20!. It yields

]

]tB52¹3E, ~47!

]

]tE5¹3H1M ~48!

where

M5~E•¹3E2B•¹3B!u1u•B~¹3B!1u•E~¹3E!

2u•~¹3E!E2u•~¹3B!B ~49!

andH has been defined in Eq.~46!.The first of these equations is the standard Maxwell eq

tion, which is a consequence of the definition of the elec

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and magnetic fields in terms of the field tensor. The secequation gives a complicated time evolution for the elecfield, where the correction terms to the usual Maxwell eqtion have been isolated. Note that in the limit of vanishingu,the ordinary Maxwell equations are reproduced, as expecIt is now possible to express the additionalM term in Eq.~48! as a total time derivative. To do this the curl of theEfield is replaced by the time derivative of theB field by usingthe identity ~47!. The curl of theB is likewise replaced bythe time derivative of theE field since any correction to thiMaxwell equation must involve terms ofO(u) leading toterms ofO(u2) in M , which can be ignored. Thus we find

M5]

]t@~u•B!E2~u•E!B2~E•B!u#. ~50!

It is now trivial to reproduce Eq.~44!. This completes ourdemonstration of the equivalence of the equations of moobtained in the Lagrangian and Hamiltonian formulations

It may be recalled that we had introduced redefined etric E, Eq. ~21!, and magneticB, Eq. ~22!, fields in terms ofwhich both the Lagrangian~24! and Hamiltonian~23! as-sumed the same structures as in the usual theory. The etions of motion in these variables, as well as their algebcan be easily obtained from our results.

Finally, in order to discuss the role of the Seiberg-Wittmap as a canonical transformation, we consider the issuthe stability of the Poisson algebra among the electricmagnetic fields under the transformation~30! and ~31! andthe algebra~37!, ~38!, ~39!. In the careted variables, the eletric field is the momenta~26! conjugate to the potentialAi ,so that

$Bi~x!,Bj~y!%5$Ei~x!,Ej~y!%50, ~51!

$Ei~x!,Bj~y!%52e i jk]kd~x2y!

1e i jke lmnun] l Ak]md~x2y!. ~52!

Now these brackets are computed in a different way.ing the map~30! and ~31!, the careted variables are epressed in terms of the ordinary variables. The algeamong these variables, given in Eqs.~37!–~39!, is then used.A slightly lengthy algebra leads to the following brackets

$Bi~x!,Bj~y!%50,

$Ei~x!,Bj~y!%52e i jk]kd~x2y!

1e i jke lmnun] lAk]md~x2y!,

$Ei~x!,Ej~y!%5e i jkuk~] lEl !d~x2y!. ~53!

The algebra among the magnetic fields is trivially idencal to Eq.~51!. If we make further use of the Seiberg-Wittemap, we see thatA in Eq. ~52! can be identified with theusualA in Eqs. ~53!, since the corrections will at least involve terms up to (u2). Thus the electric-magnetic fielbrackets in the two cases agree. The last bracket among

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electric fields has to be interpreted with some care. The pis that, on account of Eq.~17!, up to the order we are dealinwith, it is possible to rewrite it as

$Ei~x!,Ej~y!%5e i jkuk~] lEl !d~x2y!

5e i jkuk~] lp l !d~x2y!. ~54!

The bracket is thus proportional to the Gauss constr~in the usual variables!, so that on the physical sector, thsecond equality in Eq.~51! is obtained. This shows that, wita suitable interpretation, the Seiberg-Witten map may begarded as a canonical transformation.

IV. APPLICATION TO GAUGE SYMMETRIES

It is possible to provide an application of the Hamiltoniaformulation to derive the gauge symmetry of the noncommtative theory governed by the action~6!. Moreover, theanalysis is completely general and not confined to any scific order in the expansion parameteru. It is known byinspection that the action~6! is invariant under star gaugtransformations, whose infinitesimal version is given by

dAma 5~Dme!a ~55!

wheree is the gauge parameter. The~careted! covariant de-rivative is defined in Eq.~28!. We now derive this result.

We adopt the same techniques developed earlier for tring conventional gauge theories@13,14#. However, there aresome subtle issues related to the fact that ordinary multication gets replaced by star multiplication. Any gauge thein the Hamiltonian formulation is characterized by the folowing involutive ~first class! algebra involving the con-straintsF and the canonical HamiltonianH:

@H,Fa~x!#5E dyVab~x,y!Fb~y!,

@Fa~x!,Fb~y!#5E dzCabc ~x,y,z!Fc~z! ~56!

whereV,C denote the structure functions which, in the geeral case, may depend on the phase space variables.symbolsa, b, etc., contain the symmetry indices, as wellthe number of the constraints. The brackets denote the uPoisson brackets. Furthermore, in the noncommutative cthe multiplication in the right side is replaced by a star mtiplication. However, since the expressions are within antegral, the star multiplication can be replaced by the ordinone and hence Eq.~56!, as it stands, is correct. Following thDirac algorithm, the gauge generatorG is defined as a lineacombination of the first class constraints,

G5E dxea~x!Fa~x! ~57!

wherea enumerates the constraints. Now all the gaugerametersea are not independent. The number of independ

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parameters is identical to the number of primary first clconstraints. The other ones are fixed by the following contions @13,14#:

deb2~x!

dt5E dyea~y!Va

b2~y,x!

1E dydzea~y!va1~z!Ca1ab2 ~z,y,x!. ~58!

Here the labels 1 and 2 denote the primary and seconsectors, respectively, so that the full set of constraintsnoted by the labela would be divided into two parts:a1denoting the primary constraints~i.e., those obtained fromthe basic definition of the canonical momenta! anda2 denot-ing the secondary constraints~i.e., those found by the consistency requirement of time conservation of the primconstraints!. The Lagrange multipliers entering in the defintion of the total HamiltonianHT are given byva1,

HT5H1E va1Fa1~59!

whereFa1denotes the primary first class constraints. In

noncommutative theory, the products in the various ingrands are replaced by the star products. Then the seintegral in Eq.~58! requires care since it would involve thstar product of three objects. However, in the present prlem, this term actually vanishes, as shown below.

We have one primary constraint~25! and one secondar~Gauss! constraint~28!. Following our conventions, we labethese constraints as

F15p0 ,

F25D ip i . ~60!

The Poisson algebra involving the primary constrainttrivial,

@F1 ,F1#5@F1 ,F2#50. ~61!

This implies that theC function in Eq.~58! vanishes. A non-vanishing piece arises only if the algebra of the primary cstraint is nontrivial. Thus we obtain

deb2~x!

dt5E dyea~y!Va

b2~y,x!. ~62!

The variations of the fields are now defined by bracketwith the generator~57! in the manner

dAm~x!5E dyea~y!* @Am~x!,Fa~y!#, a51,2. ~63!

The variation of the time component of the field is easobtained, with the contribution coming from the primary setor,

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dA0~x!5E dye1~y!* d~x2y!5E dye1~y!d~x2y!

5e1~x!. ~64!

The variation of the space component gets a nonvanishcontribution from the Gauss constraint. This constraintfirst written in terms of the structure constants (f abc,dabc) ofthe symmetry group@see Eq.~28!#, after which the star prod-ucts are expanded in full. Computing the algebra and drping boundary terms, one finally obtains

dAi~x!5D ie2~x!. ~65!

However, e1 and e2 are not independent. It is possibledeterminee1 in terms ofe2 by using Eq.~62!. The first stepis to obtain theV functions. The algebra of the constrainwith the canonical Hamiltonian~27! is given by

@H,p0a#5~D ip i !

a,

@H,~D ip i !a#5

1

2f abc$A0b,~D ip i !

c%*

2i

2dabc@A0b,~D ip i !

c#* ~66!

where the closure of the Gauss constraint involves both~star!commutator and anticommutator. While the closure ofprimary constraint is trivial, that of the Gauss constraintquires some algebra, but the details are given in@6#. The Vfunctions appearing in Eqs.~56! are now obtained from theabove algebra. The only nonvanishing ones are given by

~V12!ab~x,y!5dabd~x2y!,

~V22!ab~x,y!52

1

2f abc$d~x2y!,A0c~y!%*

2i

2dabc@d~x2y!,A0c~y!#* . ~67!

In deriving these relations use has been made of the cyclof the star product within an integral,

E ~A* B* C!5E ~B* C* A!5E ~C* A* B!, ~68!

and the operator identity

A~x!* d~x2y!5d~x2y!* A~y!. ~69!

There is only one equation for Eq.~62! that has to be solvedWriting it out in an expanded form, we find

de2a~x!

dt5E dye1b~y!~V1

2!ba~y,x!

1E dye2b~y!~V22!ba~y,x!. ~70!

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Using the first equation in Eqs.~67!, the first integral in theabove relation is trivially done to yield

E dye1b~y!~V12!ba~y,x!5e1a~x!. ~71!

The second involves the star product of three objects.1 Weevaluate it in some detail:

E dye2b~y!~V22!ba~y,x!

51

2f abcE dye2b~y!* @d~x2y!* A0c~x!

1A0c~x!* d~x2y!#

2i

2dabcE dye2b~y!* @d~x2y!* A0c~x!

2A0c~x!* d~x2y!#. ~72!

The star operation is meaningful when the variables arefined at the same point. Thus the argument of the potenhas to be converted fromx to y. This is done by using theidentity ~69!. Finally, using Eq.~68!, we get

E dye2b~y!~V22!ba~y,x!

51

2f abcE dyd~x2y!* $e2b~y!,A0c~y!%*

2i

2dabcE dyd~x2y!* @e2b~y!,A0c~y!#* . ~73!

Replacing the first star product by an ordinary product,integrals are evaluated by using the delta function to yie

E dye2b~y!~V22!ba~y,x!5

1

2f abc$e2b~x!,A0c~x!%*

2i

2dabc@e2b~x!,A0c~x!#* .

~74!

Using Eqs.~70!, ~71!, and~74!, we obtain

1Note that, although not written explicitly, star products areways implied in such products. It is not written explicitly since tstar product of two objects within an integral can be replaced byordinary product.

10500

e-al

e

e1a5 e2a21

2f abc$e2b~x!,A0c~x!%*

1i

2dabc@e2b~x!,A0c~x!#*

5~D0e2!a. ~75!

Combining Eqs.~64!, ~65! and~75!, we obtain the covariantransformation law

dAma 5~Dme2!a ~76!

which reproduces Eq.~55! oncee2 is identified withe. Thiscompletes our derivation of the infinitesimal gauge transfmations. Note that it has the correct form in the limitu→0, when the noncommutative gauge transformationsduce to the standard non-Abelian gauge transformations

As a further application of the Hamiltonian approach, itpossible to prove that the ‘‘Maxwell’’-type equations~43!and~44! satisfied by noncommutative electrodynamics in tusual variables are actually valid for any order inu and notjust up to first order, which has been done explicitly. The fiof these equations, as discussed there, is the Gaussstraint, with the displacement fieldD identified with the ca-nonical momenta conjugate toA. This constraint is the generator of time-independent gauge transformations. Sincenature of the gauge transformation (dAm5]me) generatingthe Abelian gauge symmetry remains the same, indepenof the specific order ofu, such an equation characterizing thGauss constraint will always occur. Thus we have

¹•D50 ~77!

where

Di5Ei1••• ~78!

where the ellipsis represents additional terms in distinctders of the expansion parameteru. HereDi is the momentaconjugate toAi in the full u-expanded theory. The leadinterm is given by the usual electric field since it is the congate momenta in the conventional Maxwell theory. The ficorrection term inu has been explicitly given in Eq.~45!.

Having obtained the constraint, the genuine equationmotion can now be derived. This is found by bracketing tD with the Hamiltonian. Since the electromagnetic field tesor is the local gauge invariant object, the gauge invariLagrangian will always be constructed in terms of this tesor, as for instance shown in Eq.~12!. Likewise the physicalHamiltonian~i.e., the canonical Hamiltonian after the imposition of the Gauss constraint! will be constructed in terms othe gauge invariant variables, which are the momenta (Di)and the magnetic fieldBi . The Hamiltonian~20! has thisform once the electric fields are replaced in favor of tmomenta by using Eq.~17!. The contribution to the brackeof Di with the Hamiltonian will come only from the magnetic field terms. Since the magnetic field is expressed as

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curl of the vector potential, the result will involve the curl osomething, so that one obtains

]

]tD2¹3H50 ~79!

where

Hi5Bi1•••. ~80!

As before, the ellipsis denotes additional terms in powersu. The leading term is given by the usual magnetic fieldcorrectly reproduce the standard Maxwell equation, in anogy to the Gauss law, in the limitu→0. The first correctionterm is given in Eq.~46!.

Equations~77! and ~79! are the analogues of Eqs.~43!and~44!. The fact that the equations can always be put in tform justifies the terminology ‘‘noncommutative electrodnamics.’’ This feature may be used to extend theO(u) re-sults, as for instance obtained in@11,15#. More recently, ex-plicit computations up toO(u2) in @16#, also confirm thegeneral structure given here.

V. DISCUSSION

We have analyzed noncommutative Yang-Mills theoboth in the original variables defined in the noncommutatspace and also in an effective version by exploitingSeiberg-Witten map to recast the theory in conventional vables. Noncommuting electric fields are a natural outcomthe effective theory. They are essential to correctly reprodthe equations of motion in the Hamiltonian formulation. Thnoncommutativity may be interpreted as a sort of distortin noncommutating electrodynamics, since the electric fieare now a nonlinear function of the canonical variablSomething similar happens when the plane waves ofcommuting Maxwell theory get distorted in noncommutatielectrodynamics@11,15#, leading to a violation of the superposition principle @16#. It was possible, after a suitablchange of variables, to express both the LagrangianHamiltonian in the form of usual gauge theories, with tcomplete effect of noncommutativity relegated to the notrivial algebra. Although noncommuting electric fields haappeared earlier—such as for instance in topologically msive gauge theories@17# or in anomalous gauge theorie@18#—here these are field dependent, in contrast to theamples cited.

Our analysis has clarified certain issues concerningreduction of the noncommutative theory to an effecttheory by the application of the Seiberg-Witten map. Twas possible either through the Lagrangian or throughHamiltonian. As shown in@9#, the results need not agreegeneral. For example, the energy-momentum tensor cputed from the original action and then expanded bySeiberg-Witten map is not the same as obtained by firstpanding the action itself by this map, after which the tenis computed. In our analysis, however, we have shownthe Hamiltonian, which is a component of the energmomentum tensor, agrees by either approach, providepassage to the physical subspace is done by a suitable i

10500

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sition of the appropriate Gauss constraint in the differdescriptions. The mismatch in@9# comes about presumablbecause these constraints~19! and~28! do not get mapped bythe Seiberg-Witten transformation.2 Thus it is essential tofirst pass on to the physical subspace before applyingmap. In that case there is no mismatch. Indeed, the imption of the Gauss law was also crucial for showing thatHamiltonian version of the Seiberg-Witten map discusshere, was a canonical transformation.

The Hamiltonian formulation was applied to explicitly derive the gauge symmetries, which are otherwise postulaon inspection, of the action in the noncommutative variabThe result was valid for any order inu. The nontrivial alge-bra of the first class constraints and the Hamiltonian, as wcertain properties of field variables and distributions operaunder the star multiplication, were essential for getting tresult. Furthermore, based on the structure of the gaugeerators it was possible to show that the equations of mofor the u-expanded noncommutative electrodynamics coalways be cast in the form of ‘‘Maxwell’’ equations. Thisuggests that results, which have been computed up tofirst order inu, could possibly be extended for higher orde

The implications of our analysis at the quantum level mbe discussed both in the effective (u)-expanded theory andin the original noncommutating variables. In the former cathe Poisson brackets among the gauge invariant variaobtained here in Eqs.~37!, ~38!, and~39! can be expressed acommutators since the expressions involve only linear vables so that there is no ordering ambiguity. Expressionsvolving products of field variables should be defined by Weordering. This is the natural prescription once it is realizthat the star operation, through the composition law, direcleads to this ordering. In the noncommutating variables,basic brackets are just the usual Poisson brackets. Comptions arise because of the nontrivial algebra of the constraand the Hamiltonian. However, we have shown here howhandle such complications, precisely obtaining the gasymmetries. The construction of the Becchi-Rouet-StoTyutin ~BRST! charge, which acts as the generator of tBRST transformations, should be possible along these liwith suitable inclusion of ghosts, in analogy to usual YanMills theory. Another aspect is the well-known fact that thprocess of obtaining a quantum theory from a classicalby constraining and quantizing is not commutative; i.e., ocan get different results by first solving the constraints athen quantizing or by reversing the process. Here we hfound that the Hamiltonian in theu-expanded theory and thone in the original variables get mapped only if the costraints are first eliminated. Otherwise we get two distinformulations. A more detailed description of these and otissues related to the quantization procedure are beyondscope of the present paper.

2The fact that the Gauss constraints are not mapped may beeasily in theU(1) case. In the usual picture, this constraint is gauinvariant, while in the noncommutative picture, it is gauge covaant. Since the map connects gauge equivalent classes only, theconstraints do not get identified.

2-8

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thisan

for


We conclude by mentioning that noncommutative fietheory being an emergent topic, it is desirable to explicshow that the Hamiltonian formulation is consistent with tLagrangian one, even at the classical level. Indeed cerdiscrepancies were already mentioned@9#. We have shownhow to obtain a consistent formulation, apart from providicertain applications of the Hamiltonian approach that cobe used for a quantum analysis. Finally, since the SeibWitten map connects gauge equivalent classes, it is expe

od

ys

rov.

a,on

10500

in

dg-ted

to be related to canonical transformations. We provedexplicitly in the physical subspace, using the Hamiltonianalysis.

ACKNOWLEDGMENTS

I thank the Japan Society for Promotion of Science~JSPS!for support and the members of the theory group, KEK,their gracious hospitality.

.’’

ys.

-

@1# H. Snyder, Phys. Rev.71, 38 ~1947!.@2# N. Seiberg and E. Witten, J. High Energy Phys.09, 32 ~1999!.@3# For reviews, see M.R. Douglas and N. Nekrasov, Rev. M

Phys. 73, 977 ~2001!; R. Szabo, ‘‘Quantum field theory onnoncommutative spaces,’’ Phys. Rep.~to be published!,hep-th/0109162.

@4# O. Dayi, Phys. Lett. B481, 408 ~2000!; also see O. Dayi andB. Yapiskan, J. High Energy Phys.10, 022 ~2002!.

@5# J. Gomis, K. Kamimura, and T. Mateos, J. High Energy Ph30, 010 ~2001!.

@6# R. Amorim and F.A. Farias, Phys. Rev. D65, 065009~2002!.@7# For an alternative Hamiltonian approach using embedding p

cedures, see R. Amorim and J. Barcelos-Neto, Phys. Re64, 065013~2001!.

@8# B. Jurco and P. Schupp, Eur. Phys. J. C14, 367 ~2000!.@9# J.M. Grimstrup, B. Kloibock, L. Popp, V. Putz, M. Schwed

and M. Wickenhauser, ‘‘The energy-momentum tensor in ncommutative gauge field models,’’ hep-th/0210288.

.

.

-D

-

@10# A.A. Bichl, J.M. Grimstrup, L. Popp, M. Schweda, and RWulkenhaar, ‘‘Deformed QED via Seiberg-Witten Map,hep-th/0102103.

@11# Z. Guralnik, R. Jackiw, S.Y. Pi, and A.P. Polychronakos, PhLett. B 517, 450 ~2001!.

@12# S.I. Kruglov, Ann. Fond. Louis Broglie27, 343 ~2002!.@13# M. Henneaux, C. Teitelboim, and J. Zanelli, Nucl. Phys.B332,

169 ~1990!.@14# R. Banerjee, H.J. Rothe, and K.D. Rothe, Phys. Lett. B463,

248 ~1999!; 479, 429 ~2000!.@15# R.G. Cai, Phys. Lett. B517, 457 ~2001!.@16# G. Berrino, S.L. Cacciatori, A. Celi, L. Martucci, and A. Vi

cini, Phys. Rev. D~to be published!, hep-th/0210171.@17# S. Deser, R. Jackiw, and S. Templeton, Ann. Phys.~N.Y.! 140,

372 ~1982!.@18# R. Banerjee, Phys. Rev. Lett.61, 514 ~1988!; 63, 337 ~1989!,

and references therein.

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noncommuting electric fields and algebraic consistency in noncommutative gauge theories

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