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Field Theory in Noncommutative Minkowski Superspace Vahagn Nazaryan∗ and Carl E. Carlson† Particle Theory Group, Department of Physics, College of William and Mary, Williamsburg, VA 23187-8795 (Dated: October2004) There is much discussion of scenarios where the space-time coordinates xµ are noncommutative. The discussion has been extended to include nontrivial anticommutation relations among spinor coordinates in superspace. A numberof authors have studied field theoretical consequences of the deformation of N = 1 superspace arising from nonanticommutativity of coordinates θ, while leaving θ¯’s anticommuting. This is possible in Euclidean superspace only. In this note we present a way to extend the discussion by making both θ and θ¯ coordinates non-anticommuting in Minkowski superspace. We present a consistent algebra for the supercoordinates, find a star- product,andgivetheWess-ZuminoLagrangian LWZ withinourmodel. Ithastwoextratermsdue to non(anti)commutativity. The Lagrangian in Minkowski superspace is always manifestly Hermi- 5 0 tian and for LWZ it preserves Lorentz invariance. 0 2 I. AN OVERVIEW AND INTRODUCTION rectly related to the background field Bµν [5], is just n an antisymmetric array of c-numbers. There has been a J By now, there is a long history of theoretical studies a fair amount of theoretical study learning how to work related to nontrivial, possibly richer structures of space- with fields that are functions of noncommuting coordi- 7 2 time. Under this heading one may include supersymme- nates, and phenomenological studies of possible physi- try and extra-dimensional theories, but we concentrate cal consequences of spacetime noncommutativity. How- 3 here on theories with a noncommutative spacetime alge- ever, theories based on (1.1) with a c-number Θµν suffer v bra. The earliest motivation for such theories was the from Lorentz-violating effects. Such effects are severely 6 hope that divergences in field theory would be be ame- constrained [6]-[16] by a variety of low energy experi- 5 liorated if there were coordinate uncertainty, and coor- ments [17]. 0 dinate uncertainty would follow if coordinate operators Returning to one of ourprevious remarks,in the DFR 0 1 did not commute [1]. The idea did not bear direct fruit, noncommutative algebra [3] xˆµ satisfies [xˆµ,xˆν] = Θˆµν, 4 and Snyder’s paper [1] remained almost alone for many but where here Θˆµν = Θˆνµ transforms as a Lorentz 0 decades. tensor and is in the sa−me algebra with xˆµ. Thus h/ Recently, the idea of noncommutative coordinates has the algebra formulated by DFR is Lorentz-invariant. t blossomed, at least as theoretical speculation, with mo- Carone,Zobin,andoneofthepresentauthors(CEC)[18] p- tivation from several sources. For example, Connes et formulated and studied some phenomenological conse- e al. [2] attempted to make gauge theories of electroweak quences of a Lorentz-conserving noncommutative QED h unification mathematically more natural by using ideas (NCQED) based on a contracted Snyder [1] algebra, v: from noncommutative geometry. Also, Dopplicher, Fre- which has the same Lie algebra as DFR. In [18] light- i denhagen, and Roberts [3] saw general relativity as giv- by-light scattering was studied, and it was found that X ing a naturallimit to the precisionof locating a particle, contributions from noncommutativity can be significant r whichtothemsuggestedanuncertaintyrelationandnon- withrespecttothestandardmodelbackground. Further a commutativity among coordinate operators. They sug- studies of NCQED as formulated in [18] may be found gested a particular algebra of the coordinates now often in [19, 20, 21]. In particular, bounds were obtained on referred to as the “DFR” algebra. However, probably the scale of noncommutativity [20] in the Lorentz con- thegreatestmodernspurtostudyingspacetimenoncom- servingcasefromannumberofQEDprocessesforwhich mutativity was the observation that string theories in a thereexistexperimentsattheCERNLargeElectronand background field can be solved exactly and give coordi- Positroncollider (LEP). nate operators which do not commute [4, 5]. There have also been studies extending noncommuta- Intheorieswithanunderlyingnoncommutativespace- tivity to the full set of supersymmetric coordinates, not time algebra, the position four vector xµ is promoted to just limiting noncommutativity to ordinary spacetime. an operator xˆµ that satisfies the commutation relation In this paper, we wish to continue the study of non- commutativecoordinatesin supersymmetrictheories,by [xˆµ,xˆν]=Θµν. (1.1) givingandstudying consequencesofanalgebraofsuper- spacecoordinatesthatverydefinitelyallowsustoremain The Θµν that comes out of string theory, which is di- in Minkowski space. Recent work (e.g., [22, 23, 24, 25, 26]) has stimulated interest in supersymmetric noncommutativity by show- ∗[email protected] ing, in Euclidean space, how noncommutative superco- †[email protected] ordinates could arise from string theory. Further, some 2 of the recent work [24] defined a star-product from the anticommuting Grassmann variables that we shall pro- commutationrelations. Operatorsmultipliedinnoncom- mote to nonanticommuting operators θˆα and θˆ¯α˙ in some mutative space could then be replaced by their symbols algebra. in commutative space with multiplication replaced by The anticommutation for the θˆ’s will be the star-product. This was then used to study noncommutative modifications to Wess-Zumino and gauge La- θˆα,θˆβ =Cαβ, (2.1) grangians,albeitstillinEuclideanspace. Proofsofrenor- { } malizability of the deformed Wess-Zumino Lagrangian where Cαβ is a symmetric array of c-numbers. We were offered [27], but it was noted that the deformed shall also suppose there is a mapping between the op- Euclidean space Lagrangians, as well as the vector su- erator θˆα and a Grassmann variable θα in ordinary perfield, were not Hermitian [27, 28]. (anti)commutative space. We will soon, as usual, ob- Working in Euclidean space allows coordinates θ with tainusingcommutativevariablesthemultiplicationrules nontrivial anticommutators to be paired with θ¯’s that of the noncommutative algebra by using a star-product anticommute in the normal way; the phrase N = 1/2 rather than the ordinary product for variables and func- supersymmetry describedthis. There is no directanalog tions in commutative space. in Minkowski space, where the θ’s and θ¯’s are tightly In Minkowski space, we relate θˆ¯α˙ to θˆα by connected. Useful formal developments include, using the star- θˆ¯α˙ =(θˆα)† , (2.2) product to define the theory, a display of a number of differentwaystointroducenoncommutativityintosuper- space [29, 30, 31]. Also [32] showed that in Minkowski so that the θˆ¯α˙ are noncommutative also, space nontrivial anticommutation relations for the θ’s and θ¯’s were not compatible with having an associative θˆ¯α˙,θˆ¯β˙ =C¯α˙β˙, (2.3) { } algebra. Hence we have some freedom in the choice of a star-product, but must be open to using a star-product where C¯α˙β˙ =(Cβα)∗. that is non-associative. The commutators of θˆ and θˆ¯ are still unconstrained, In the next section, Sec. II, we present a consistent and we make the simple choice set of (anti)commutation relations among the supercoordinates in Minkowski space. Following that, Sec. III θˆ¯α˙,θˆα =0. (2.4) defines our theory by presenting a star product that { } yields the deformed supercoordinate algebra developed Next we fix the commutation relations among θ’s and in section II. We record the deformed algebra of super- spacetimecoordinates. We define the commutatorofthe symmetry generators, and of the covariant superderivatives. The commutators of the supergenerators and chiral coordinate yˆµ xˆµ+iθˆσµθˆ¯with θˆ, and the com- superderivatives break supersymmetry. In Sec. IV we mutator of the antic≡hiral coordinate yˆ¯µ xˆµ iθˆσµθˆ¯ write down the chiral and antichiral superfields, and with θˆ¯, in such a way that enables us to w≡rite p−roducts show that products of (anti)chiral superfields are them- of chiral fields, and products of antichiral fields, in their selves (anti)chiral superfields. This is a feature retained canonical form. We choose fromcommutativesupersymmetry;someofthechoicesin Sec.IIwereinfactmadeinthehopethatthiswouldhap- [yˆµ,θˆα]=0, (2.5) pen. We construct the Wess-Zumino Lagrangian , andshowhowtoavoidambiguityinourconstructioLnWdZe- [yˆ¯µ,θˆ¯α˙]=0. (2.6) spite the nonassociativity of the products. We end with The nonzero commutators some discussion in section V. [yˆ¯µ,θˆα]= 2[iθˆσµθˆ¯,θˆα]=2iCαβσµ θˆ¯β˙, (2.7) ˙ − ββ II. THE NON(ANTI)COMMUTATIVE SUSY and ALGEBRA [yˆµ,θˆ¯α˙]=2[iθˆσµθˆ¯,θˆ¯α˙]=2iC¯α˙β˙θβσµ , (2.8) ˙ Noncommutativity has usually been studied as the ββ noncommutativity of ordinary spacetime. Here we are considering noncommutativity in superspace1, and for are fixed by the choices already made. The choices and results in (2.1)-(2.6) also constrain Minkowski rather than Euclidean space. The supercoordinate is (xµ,θα,θ¯α˙) where θα and θ¯α˙ are normally thecommutationrelationsofyândofyˆ¯withthemselves. The following condition must be satisfied: [yˆµ,yˆν]−[yˆ¯µ,yˆ¯ν]=4(C¯α˙β˙θˆαθˆβ −Cαβθˆ¯α˙θˆ¯β˙)σαµα˙σβνβ˙ . 1 Wefollowconventions ofWessandBagger [33]. (2.9) 3 Thus, the Hermitian part of [yˆµ,yˆν] is fixed by choices We operationally define our theory by finding a suit- already made. Let us rewrite the previous equation in able star-product. A properly defined star product has the following way, to reproduce the underlying deformed algebra of the supercoordinates in its entirety. We will require that the [yˆµ,yˆν]−[yˆ¯µ,yˆ¯ν]=(4C¯α˙β˙θˆαθˆβ −2CαβC¯α˙β˙)σαµα˙σβνβ˙ star product satisfy the reality condition, that is, the star-product will maintain the usual rules for products +(4Cαβθˆ¯β˙θˆ¯α˙ −2CαβC¯α˙β˙)σαµα˙σβνβ˙ , of involutions, (2.10) (f1 f2)† =f2† f1† . (3.1) ∗ ∗ where eachterm on the right-hand-sideis the Hermitian conjugate of the other. Then we make the choices, We find it convenient to use the supersymmetry generators in defining the star product, and will limit the [yˆµ,yˆν]=(4C¯α˙β˙θˆαθˆβ −2CαβC¯α˙β˙)σαµα˙σβνβ˙ , (2.11) sptaarra-mpreotdeurcCt.toTbheisinigs aatlsomtohste qmuiandimrautmic itnhadtefwoirlml aaltlioown and reproducing the deformed algebra for the supercoordinates. [yˆ¯µ,yˆ¯ν]=(4Cαβθˆ¯α˙θˆ¯β˙ −2CαβC¯α˙β˙)σαµα˙σβνβ˙ , (2.12) Before we define the star product, we find it useful to have before us the well known canonical expressions for which are natural and consistent with already defined covariant derivatives and supercharges. Acting on the commutators. Finally,notethatyândyˆ¯donotcommute right, in this non(anti)commutative algebra, → → [yˆµ,yˆ¯ν]=2CαβC¯α˙β˙σαµα˙σβνβ˙ , (2.13) D→α = ∂∂θα(cid:12)x+iσαµα˙θ¯α˙ ∂∂xµ, →(cid:12) → → ∂(cid:12) ∂ altChoomugmhutthaetiironcormelmatuiotantsogrivisenabcy-n(u2m.1b)-e(r2..8),(2.11)and D¯α˙ = −∂θ¯(cid:12)α˙(cid:12)x−iθασαµα˙∂xµ, (3.2) (cid:12) (2.12) are compete, consistent with each other, and rep- and (cid:12) resent the deformed supersymmetry algebra in terms of (cid:12) chiral and spinor variables. One can summarize this al- → → gebra in terms of (xˆ,θˆ,θˆ¯) as, Q→α = ∂∂θα −iσαµα˙θ¯α˙∂∂xµ, (cid:12)x →(cid:12) → {θˆα,θˆβ}=Cαβ, [xˆµ,θˆα]=iCαβσβµβ˙θˆ¯β˙, (2.14) →Q¯α˙ = −∂∂θ¯(cid:12)(cid:12)α˙ +iθασαµα˙ ∂∂xµ. (3.3) θˆ¯α˙,θˆ¯β˙ =C¯α˙β˙, [xˆµ,θˆ¯α˙]=iC¯α˙β˙θˆβσµ˙, (2.15) (cid:12)(cid:12)x { } ββ In (3.2) and(3.3) derivativ(cid:12)(cid:12)es with respect to θ and θ¯are θˆ¯α˙,θˆα =0, taken at fixed x, and derivatives with respect to x are { } [xˆµ,xˆν]=(Cαβθˆ¯α˙θˆ¯β˙ −C¯α˙β˙θˆβθˆα)σαµα˙σβνβ˙. (2.16) takLeent’astafilsxoedwrθitaenddoθ¯w.n the corresponding equation for two sets of coordinates (y,θα,θ¯α˙) and (y¯,θα,θ¯α˙), where Hence, the space-timecoordinatesxµ do notcommute with each other either, or with the spinor coordinates θ yµ =xµ+iθσµθ¯, y¯µ =xµ iθσµθ¯. (3.4) and θ¯. − Then one can check that → → → III. THE STAR PRODUCT D→ = ∂ +2iσµ θ¯α˙ ∂ , D→ = ∂ , (3.5) α ∂θα αα˙ ∂yµ α ∂θα (cid:12)y (cid:12)y¯ We shall assume that there exists a mapping that re- →(cid:12) → (cid:12) → tlahteeisrctohuentoerrdpianratrsy(xˆv,aθˆr,iaθˆ¯b)liensn(oxn,cθo,mθ¯)minutcaotmivmesuptaactiev,eantdo D→¯α˙ =−∂∂θ¯(cid:12)(cid:12)α˙(cid:12)y , D→¯α˙ =−∂∂θ¯α˙(cid:12)y¯−2iθασαµα(cid:12)(cid:12)˙ ∂∂y¯µ , that relates functions f(x,θ,θ¯) in commutative space (cid:12)(cid:12) (cid:12)(cid:12) (3.6) to operators fˆ(xˆ,θˆ,θˆ¯) in the noncommutative algebra. →Q = →∂ (cid:12), →Q = →∂ (cid:12)2iσµ θ¯α˙ →∂ , (3.7) Products of functions in commutative space will be de- α ∂θα α ∂θα − αα˙ ∂y¯µ fined by a star-product. In noncommutative theories a (cid:12)y (cid:12)y¯ →(cid:12) →(cid:12) → star product is used so that the result of products such → ∂(cid:12) ∂(cid:12) → ∂ as fˆ(xˆ,θˆ,θˆ¯)gˆ(xˆ,θˆ,θˆ¯)hˆ(xˆ,θˆ,θˆ¯) in noncommutative space Q¯α˙ =−∂θ¯(cid:12)α˙(cid:12)y+2iθασαµα˙ ∂y(cid:12)µ , Q¯α˙ =−∂θ¯α˙ (cid:12)y¯ . (3.8) correspondstotheresultoff(x,θ,θ¯) g(x,θ,θ¯) h(x,θ,θ¯) (cid:12) (cid:12) incommutativespace(providedfˆ(xˆ,∗θˆ,θˆ¯)corre∗spondsto Expressions(cid:12)(cid:12) for D←α,D←¯α˙,←Qα, andQ←¯α˙ are (cid:12)(cid:12)obtained f(x,θ,θ¯), etc.). from above by simply changing to , with the fol- → ← 4 lowing definitions, and → ← ∂ 1 ∂ ∂ 1 ∂ ∂ ∂∂θαθβ ≡δαβ , θβ∂∂θα ≡−δαβ , (3.9) ∂θ¯θ¯ ≡ 4∂θ¯α˙ ∂θ¯α˙ =−4ǫγ˙η˙∂θ¯γ˙ ∂θ¯η˙ . (3.16) → ← ∂ yν δν , yν ∂ δν . (3.10) Thefollowingequationsareusefulforderivingcommu- ∂yµ ≡ µ ∂yµ ≡ µ tation relations among various coordinates of deformed superspace, Similar definitions apply derivatives with respect to θ¯α˙ and y¯µ. Now we can write down the star product that we use θα θβ = 1ǫαβθθ+ 1Cαβ, (3.17) for mapping a product of functions fˆgˆ in noncommu- ∗ −2 2 tative space to a product of functions in commutative θ¯α˙ θ¯β˙ = +1ǫα˙β˙θ¯θ¯+ 1C¯α˙β˙. (3.18) space. ∗ 2 2 fˆgˆ⇛f g =f(1+ )g. (3.11) Also, ∗ S Here f and g can be functions of any of the three sets of θα ∗ θθ =Cαβθβ, θ¯α˙ ∗ θ¯θ¯=−C¯α˙β˙θ¯β˙, (3.19) variables mentioned above, and the extra operator is S Cαβ← → C¯α˙β˙← → S =− 2 QαQβ − 2 Q¯α˙Q¯β˙ θθ ∗ θα =−Cαβθβ, θ¯θ¯ ∗ θ¯α˙ =C¯α˙β˙θ¯β˙, (3.20) CαβCγδ← ← → → C¯α˙β˙C¯γ˙δ˙← ← → → + 8 QαQγQδQβ + 8 Q¯α˙Q¯γ˙Q¯δ˙Q¯β˙ + Cαβ4C¯α˙β˙ Q←¯α˙Q←α→Q¯β˙Q→β −Q←αQ←¯α˙→QβQ→¯β˙ . θθ ∗ θθ =−21ǫαα′ǫββ′CαβCα′β′ (cid:16) (cid:17) (3.12) = detC, − (3.21) Itis easyto verifythatthe starproductpresentedabove θ¯θ¯ ∗ θ¯θ¯=−21ǫα˙α˙′ǫβ˙β˙′C¯α˙β˙C¯α˙′β˙′ indeed reproduces the entire noncommutative algebra of = detC¯. supersymmetry parameters, and that it satisfies the re- − ality condition (3.1). and Iff andg arefunctionsonlyofθ oronlyofθ¯,thenthe star product takes the following simple forms, recogniz- 1 1 1 θσµθ¯ θσνθ¯= θθθ¯θ¯ηµν θθC¯µν θ¯θ¯Cµν able from [24], ∗ −2 − 2 − 2 Cαβ ←∂ →∂ − 41CαβC¯α˙β˙σαµα˙σβνβ˙, f(θ) g(θ)=f(θ) 1 ∗ − 2 ∂θα∂θβ (3.22) (cid:18) ← → ∂ ∂ where Cµν and C¯µν are defined as detC g(θ) (3.13) − ∂θθ∂θθ (cid:19) 1 =f(θ)exp Cαβ ←∂ →∂ g(θ), Cµν ≡ 4Cαβǫβγ(σµσ¯ν −σνσ¯µ)αγ =Cαβǫβγ(σµν)αγ, − 2 ∂θα∂θβ! (3.23) and C¯α˙β˙ ←∂ →∂ C¯µν ≡ 41C¯α˙β˙ǫβ˙γ˙(σ¯µσν −σ¯νσµ)γ˙α˙ =C¯α˙β˙ǫβ˙γ˙(σ¯µν)γ˙α˙ . f(θ¯) g(θ¯)=f(θ¯) 1 (3.24) ∗ − 2 ∂θ¯α˙ ∂θ¯β˙ (cid:18) One can now verify, ← → ∂ ∂ −detC¯∂θ¯θ¯∂θ¯θ¯ g(θ¯) (3.14) θα,θβ ∗ =Cαβ, [xµ,θα]∗ =iCαβσµ˙θ¯β˙, (3.25) (cid:19) { } ββ =f(θ¯)exp −C¯2α˙β˙ ∂←∂θ¯α˙ ∂→∂θ¯β˙!g(θ¯), {θθ¯¯αα˙˙,,θθ¯αβ˙}∗ ==C0¯,α˙β˙, [[xxµµ,,xθ¯αν˙]]∗ ==iθ¯Cθ¯¯Cα˙βµ˙θνβ+σβµθβθ˙C,¯µν. ((33..2276)) ∗ ∗ { } where as they should be according to (2.14)-(2.16). Subscript ∂ 1 ∂ ∂ 1 ∂ ∂ “ ” means use star multiplication when evaluating the = ǫγη , (3.15) ∗ ∂θθ ≡ 4∂θ ∂θα 4 ∂θγ ∂θη (anti)commutators. α 5 From (3.7), and (3.8) one may check that in noncom- theory, using the following defining equations for chiral mutative space and antichiral superfields as before, {Qα,Qβ}=−4C¯α˙β˙σαµα˙σβνβ˙∂y¯∂µ∂2y¯ν , (3.28) DD¯α˙ΦΦ¯((yy¯,,θθ¯)) == 00,. ((33..3356)) α 2 ∂ {Q¯α˙,Q¯β˙}=−4Cαβσαµα˙σβνβ˙∂yµ∂yν , (3.29) {Q→α,Q→¯α˙}=2iσαµα˙∂∂yµ . (3.30) IV. THE WESS-ZUMINO LAGRANGIAN A. Chiral and Antichiral Superfields Thus, we see that the first two of the above three anticommutators of supercharges are deformed from their canonical forms. From (3.5), and (3.6) for the covariant ChiralΦ(yˆ,θˆ) andantichiralΦ¯(yˆ¯,θˆ¯) superfields satisfy derivatives we find, (3.35) and (3.36) respectively. We may expand Φ(yˆ,θˆ) and Φ¯(yˆ¯,θˆ¯) as a power series in θˆ and θˆ¯. Just as in D ,D =0, (3.31) { α β} commutativetheory,notermintheserieswillhavemore {D¯α˙,D¯β˙}=0, (3.32) than two powers of θând θˆ¯. In noncommutative theory, {D→α,D→¯α˙}=−2iσαµα˙∂∂yµ . (3.33) tohfiθsîcsantrubeebreedcauucseedptroosduumctssowfittehrmthsreweitohr tmwooreorfafcetwoerrs θˆ, and similarly for θˆ¯. Hence, So, the anticommutators of covariantderivatives are not deformed in this noncommutative superspace. The an- Φ(yˆ,θˆ) = A(yˆ)+√2θˆψ(θˆ)+θˆθˆF(yˆ), (4.1) ticommutators of supercharges and covariantderivatives with each other are not deformed either, Φ¯(yˆ¯,θˆ¯) = A(yˆ¯)+√2θˆ¯ψ¯(yˆ¯)+θˆ¯θˆ¯F¯(yˆ¯). (4.2) {Dα,Qβ}={D¯α˙,Qβ}={Dα,Q¯β˙}={D¯α˙,Q¯β˙}=0. The combination θˆθîs already Weyl ordered, and maps (3.34) simply into θθ in commutative space. Hence, we can still define supersymmetry covariant con- From(3.11),theproductoftwochiralandtheproduct straintsonsuperfieldsasincommutativesupersymmetric of two antichiral fields is, Φ1(y,θ) Φ2(y,θ)=Φ1(y,θ)Φ2(y,θ) Cαβψ1αψ2β detCF1F2 ∗ − − +√2θγCαβ ǫβγ(ψ1αF2−ψ2αF1)+C¯α˙β˙σαµα˙σγνβ˙(∂µA1∂νψ2β −∂µA2∂νψ1β) (4.3) +θθ 2C¯µν∂(cid:2)µA1∂νA2+CαβC¯α˙β˙σαµα˙σβνβ˙(∂µA1∂νF2−∂µA2∂νF1) , (cid:3) (cid:2) (cid:3) and Φ¯1(y¯,θ¯) ∗ Φ¯2(y¯,θ¯)=Φ¯1(y¯,θ¯)Φ¯2(y¯,θ¯)−C¯α˙β˙ψ¯1α˙ψ¯2β˙ −detC¯F¯1F¯2 +√2θ¯γ˙C¯α˙β˙ ǫβ˙γ˙(ψ¯1α˙F¯2−ψ¯2α˙F¯1)+Cαβσαµα˙σβνγ˙(∂µA¯1∂νψ¯2β˙ −∂µA¯2∂νψ¯1β˙) (4.4) +θ¯θ¯ 2Cµν∂(cid:2)µA¯1∂νA¯2+CαβC¯α˙β˙σαµα˙σβνβ˙(∂µF¯1∂νA¯2−∂µF¯2∂νA¯1) . (cid:3) (cid:2) (cid:3) In (4.3) ∂µ ∂/∂yµ, while in (4.4) ∂µ ∂/∂y¯µ. B. Non-associativity and Weyl ordering ≡ ≡ Thus the star product ofchiralfields is chiral,andthe As usual, star product of antichiral fields is antichiral. One may again note that the reality condition is satisfied, Φ1 Φ2 =Φ2 Φ1 (4.6) ∗ 6 ∗ Φ¯1 Φ¯2 =Φ¯2 Φ¯1 (4.7) ∗ 6 ∗ but here the difference persists even if one isolates (say) the θθ terms and integrates over space. (Φ1 Φ2)=Φ¯2 Φ¯1 . (4.5) WhenconstructingaLagrangianthiswouldleadtodif- ∗ ∗ 6 ferent theories, depending on the ordering of the super- It should be clear that for the star product of just two fields. Following [24], for example, the Lagrangian can superfields, the second Weyl ordering leaves the result be specified by requiring products of superfields to be unchanged. We use the double Weyl ordering just de- Weyl ordered. Then a Lagrangianwill get no extra con- scribed to unambiguously define any Lagrangian in the tributions from noncommutativity from terms quadratic noncommutative space given by (2.14)-(2.16). As an ex- inchiralorinantichiralfields,becausethetermspropor- ample, we will write down the Wess-Zumino Lagrangian tional to θθ or θ¯θ¯that involveC or C¯ are antisymmetric in noncommutative Minkowski superspace. under interchange of the two superfields. The situation is more complicated for three or more fields, because the star product (3.11) is not associative, Φ1 (Φ2 Φ3)=(Φ1 Φ2) Φ3 . (4.8) ∗ ∗ 6 ∗ ∗ C. The Lagrangian This is a consequence of having both Q and Q¯ in the star product (3.11), with Q,Q¯ = 0. For discussion of { } 6 It is useful to record some steps in the calculation of associativity of star products see for example [32]. the product of three chiral fields. Since the star product We deal with this by defining for a non-associative of two chiral fields is chiral, from (4.3) we can obtain product a natural Weyl ordering given by the A12, ψ12γ, and F12 components of the chiral field 1 Φ12 =Φ1 Φ2 as W(f1(f2f3)) f1(f2f3)+f2(f1f3)+f2(f3f1) ∗ ≡ 6 +(cid:2)f1(f3f2)+f3(f1f2)+f3(f2f1) = 61 f1(f2f3+f3f2)+f2(f1f3+f3(cid:3)f1) ψA121γ2 ==A(A11Aψ22−γ +CAαβ2ψψ11αγψ)2+βC−αdβetǫCβγF(1ψF12αF2 ψ2αF1) − +(cid:2)f3(f1f2+f2f1) . (4.9) +C¯α˙β˙σαµα˙σγνβ˙(∂µA1∂νψ2(cid:2)β −∂µA2∂νψ1β) (cid:3) F12 =(F1A2+A1F2 ψ1ψ2)+2C¯µν∂µA1∂νA2(cid:3) − WanedylsiomrdilearrilnygftohreWres(u(flt1fin2)tfh3e).noOrmnealcwanayfoalnlodwfinthdisthbayt +CαβC¯α˙β˙σαµα˙σβνβ˙(∂µA1∂νF2−∂µA2∂νF1) (4.11) W[W(f1(f2f3))]=W[W((f1f2)f3)] w(f1f2f3). ≡ (4.10) Then, the star product of three chiral fields is (Φ1(y,θ) Φ2(y,θ)) Φ3(y,θ)=A12A3 Cαβψ12αψ3β detCF12F3+√2θγ A12ψ3γ +A3ψ12γ ∗ ∗ − − +Cαβ ǫβγ(ψ12αF3−ψ3αF12)+C¯α˙β˙σαµα˙σγνβ˙(cid:16)(∂µA12∂νψ3β −∂µA3∂νψ12β) +θθ F(cid:2)12A3+A12F3 ψ12ψ3+2C¯µν∂µA12∂νA3 (cid:3)(cid:17) − +Cα(cid:2)βC¯α˙β˙σαµα˙σβνβ˙(∂µA12∂νF3−∂µA3∂νF12) , (4.12) (cid:3) From(4.12),the onlyC-dependenttermthatwillcon- tribute to the Wess-Zumino Lagrangianfrom the double We find the following simple result for the Wess- Weyl ordered product w(Φ1(y,θ) ∗ Φ2(y,θ) ∗ Φ3(y,θ)) Zumino Lagrangianwith one chiralΦ and one antichiral comes from the A12F3 term. The contribution from this term is proportional to detCF1F2F3, which is Lorentz − invariant. For the star product of three antichiral fields, one finds a contribution proportional to detC¯F¯1F¯2F¯3. − ThereisnoextracontributiontotheWess-ZuminoLa- grangian coming from the kinetic energy term. From Φ¯ Φ there is a termSµν∂ F¯∂ F fromthe starproduct, µ ν wh∗ere Sµν CαβC¯α˙β˙σµ σν is symmetric. However, it ≡ αα˙ ββ˙ is precisely cancelled when one adds Φ Φ¯ in doing the ∗ Weyl ordering. 7 field Φ¯, The star-product in this work is not associative, in keeping with a general theorem of Klemm, Penati, and 1 1 Tomassia [32]. However, this interesting feature causes =w d2θθd2θ¯θ¯Φ¯ Φ+ d2θ mΦ Φ+ gΦ L " ∗ 2 ∗ 3 little trouble after making a natural modification of the Z Z (cid:18) Weyl ordering procedure. Also, the basic commutation 1 1 relation between the components of θ violates Lorentz Φ Φ + d2θ¯ mΦ¯ Φ¯ + gΦ¯ Φ¯ Φ¯ ∗ ∗ (cid:19) Z (cid:18)2 ∗ 3 ∗ ∗ (cid:19)# isnuvpaerri-annocnec.omTmhuetaetxiavmeWpleessL-aZgurmaningoiamnowdeel,stguadinieedd,otnhlye = (C =0) Lorentz invariant modifications, but this cannot be ex- L 1gdetCF3 1gdetC¯F¯3+total derivatives. pected to occur in general. − 3 − 3 Thereareanumberofpotentiallyinterestingdirections (4.13) to pursue in future work. One clearly wants to extend the present supercoordinate algebra to gauge theories, This Lagrangianis Hermitian and Lorentz invariant. andtoexplorepotentialphenomenologicalconsequences. V. SUMMARY One would also like to study connections to string the- oryandattemptaderivationofthepresentcommutation Our goal has been to find a theory that works in relationsfromastringmodel. Onemayalsodefineanex- Minkowskispacethatexploresnon-anticommutativityof plicit connection between operators in noncommutative the supercoordinates θ and θ¯. We have shown a consis- space and their commutative space symbols, and derive tent set of commutation and anticommutation relations the star-product from it. The current star product may for the full setofcoordinatesx, θ, andθ¯(or equivalently be justthe expansionto secondorderindeformationpa- y or y¯, θ, and θ¯). We have found a star product that re- rameter C of one found this way. We should note that produces all the coordinate commutation relations, and if this proves to be the case, the results of the present usethisstarproducttodefinemultiplicationofarbitrary paper will still hold. To this order the star-product we functions. have is unique in satisfying the requirements of giving Thestarproductisreal,meaningitmaintainsthestan- the supercoordinate commutation relations and of being dardrelationsobeyedby involutionsofproductsoffunc- real. tions. This in turn means products that are Hermitian with no star-multiplicationarealsoHermitianwith star- multiplication, after Weyl ordering. Any Lagrangianex- Acknowledgments tendedtononcommutativespaceusingstar-productsand Weyl ordering will necessarily remain Hermitian. Fur- ther, the star-product maintains the chirality of prod- We thank Chris Carone and Marc Sher for useful dis- ucts of chiral fields, and the antichirality of products of cussions. WealsothanktheNSFforsupportunderGrant antichiral fields. PHY-0245056. [1] H. S. Snyder, “Quantized Space-Time,” Phys. Rev. 71, tion of open strings and noncommutativity in branes,” 38 (1947). Nucl. Phys. B 576, 578 (2000) [arXiv:hep-th/9906161]. [2] A. Connes and J. Lott, “Particle Models And Noncom- [5] N. Seiberg and E. Witten, “String theory and noncommutative Geometry (Expanded Version),” Nucl. Phys. mutative geometry,” JHEP 9909, 032 (1999). Proc. Suppl. 18B, 29 (1991). A. Connes, “Noncommu- [6] I.Mocioiu,M.PospelovandR.Roiban,“Low-energylim- tativeGeometryAndReality,”J.Math.Phys.36(1995) its on the antisymmetric tensor field background on the 6194; A. Connes, “Gravity coupled with matter and braneandonthenon-commutativescale,” Phys.Lett.B thefoundationofnon-commutativegeometry,”Commun. 489, 390 (2000). Math. Phys. 182, 155 (1996) [arXiv:hep-th/9603053]; [7] M. Chaichian, M. M. Sheikh-Jabbari and A. Tureanu, A.Connes,M.R.DouglasandA.Schwarz,“Noncommu- “Hydrogen atom spectrum and the Lamb shift in non- tativegeometryandmatrixtheory: Compactification on commutative QED,” Phys. Rev.Lett. 86, 2716 (2001). tori,” JHEP 9802, 003 (1998) [arXiv:hep-th/9711162]; [8] I. Hinchliffe and N. Kersting, “CP violation from non- [3] S. Doplicher, K. Fredenhagen and J. E. Roberts, “The commutative geometry,” Phys. Rev. D 64, 116007 QuantumStructureOfSpace-TimeAtThePlanckScale (2001). And Quantum Fields,” Commun. Math. Phys. 172, 187 [9] S.M.Carroll,J.A.Harvey,V.A.Kostelecky,C.D.Lane (1995); “Space-time quantization induced by classical and T. Okamoto, “Noncommutative field theory and gravity,”Phys. Lett. B 331, 39 (1994). Lorentz violation,” Phys.Rev.Lett. 87, 141601 (2001). [4] F. Ardalan, H. Arfaei and M. M. Sheikh-Jabbari, “Non- [10] C.E.Carlson,C.D.CaroneandR.F.Lebed,“Bounding commutative geometry from strings and branes,” JHEP noncommutative QCD,” Phys. Lett.B 518, 201 (2001). 9902, 016 (1999) [arXiv:hep-th/9810072]. F. Ardalan, [11] A.Anisimov,T.Banks,M.DineandM.Graesser,“Com- H. Arfaei and M. M. Sheikh-Jabbari, “Dirac quantiza- mentsonnon-commutativephenomenology,”Phys.Rev. 8 D 65, 085032 (2002) [arXiv:hep-ph/0106356]. non-commutative QED,” Prog. Theor. Phys. 110, 1003 [12] I. Hinchliffe, N. Kersting and Y. L. Ma, “Review of the (2003) [arXiv:hep-th/0309159]; K. Morita, “A new phenomenology of noncommutative geometry,” Int. J. gauge-invariant regularization schemebased on Lorentz- Mod. Phys. A 19, 179 (2004) [arXiv:hep-ph/0205040]. invariant noncommutative quantum field theory,” Prog. [13] C.E.Carlson,C.D.CaroneandR.F.Lebed,“Supersym- Theor. Phys. 111, 881 (2004) [arXiv:hep-th/0312080]; metric noncommutative QED and Lorentz violation,” K. Morita, “Noncommutative regularization in gauge Phys.Lett. B 549, 337 (2002) [arXiv:hep-ph/0209077]. theories,” arXiv:hep-th/0404038. [14] V. Nazaryan, “Evaluating matrix elements relevant to [20] J. M. Conroy, H. J. Kwee and V. Nazaryan, some Lorentz violating operators,” Phys. Rev. D 67, “Phenomenology of Lorentz-conserving noncommu- 017704 (2003) [arXiv:hep-ph/0210300]. tative QED,” Phys. Rev. D 68, 054004 (2003) [15] O. Bertolami and L. Guisado, “Noncommutative field [arXiv:hep-ph/0305225]. theory and violation of translation invariance,” JHEP [21] M. Haghighat and M. M. Ettefaghi, “Parton model 0312, 013 (2003) [arXiv:hep-th/0306176]. in Lorentz invariant non-commutative space,” [16] C.D.Carone, “Phenomenology ofnoncommutativefield arXiv:hep-ph/0405270. theories,” arXiv:hep-ph/0409348. [22] H. Ooguri and C. Vafa, “The C-deformation of gluino [17] M.E. Rose and H.A. Bethe, “Nuclear Spins and Mag- and non-planar diagrams,” Adv. Theor. Math. Phys. 7, neticMomentsintheHartreeModel”Phys.Rev.51,205 53 (2003) [arXiv:hep-th/0302109]. (1937); J.D. Prestage, J. J. Bollinger, Wayne M. Itano, [23] J.deBoer,P.A.GrassiandP.vanNieuwenhuizen,“Non- and D. J. Wineland, “Limits for Spatial Anisotropy by commutativesuperspacefromstringtheory,”Phys.Lett. Use of Nuclear-Spin-Polarized 9Be+ Ions”, Phys. Rev. B 574, 98 (2003) [arXiv:hep-th/0302078]. Lett. 54, 2387 (1985); S. K. Lamoreaux, J. P. Jacobs, [24] N. Seiberg, “Noncommutative superspace, N = 1/2 su- B. R. Heckel, F. J. Raab and E. N. Fortson, “New Lim- persymmetry, field theory and string theory,” JHEP its On Spatial Anisotropy From Optically Pumped He- 0306, 010 (2003) [arXiv:hep-th/0305248]. 201 And Hg-199,” Phys. Rev. Lett. 57, 3125 (1986); [25] N. Berkovits and N. Seiberg, “Superstrings in gravipho- “Optical pumping technique for measuring small nu- ton background and N = 1/2 + 3/2 supersymmetry,” clear quadrupole shifts in 1S0 atoms and testing spatial JHEP 0307, 010 (2003) [arXiv:hep-th/0306226]. isotropy”; Phys. Rev. A 39, 1082 (1989); T. E. Chupp, [26] A. Imaanpur and S. Parvizi, “N = 1/2 super R.J.Hoare,R.A.Loveman,E.R.Oteiza,J.M.Richard- Yang-Mills theory on Euclidean AdS(2) x S**2,” son, M. E. Wagshul, and A. K. Thompson, “Results of arXiv:hep-th/0403174. a New Test of Local Lorentz Invariance: A Search for [27] R. Britto and B. Feng, “N = 1/2 Wess-Zumino model Mass Anisotropy in 21Ne”, Phys. Rev. Lett. 63, 1541 is renormalizable,” Phys. Rev. Lett. 91, 201601 (2003) (1989); C.J. Berglund, L. R. Hunter, D. Krause, Jr., [arXiv:hep-th/0307165]; D. Berenstein and S. J. Rey, E. O. Prigge, M. S. Ronfeldt, and S. K. Lamoreaux, “Wilsonian proof for renormalizability of N = 1/2 su- “New Limits on Local Lorentz Invariance from Hg and persymmetric field theories,” Phys. Rev. D 68, 121701 Cs Magnetometers,“ Phys. Rev. Lett. 75, 1879 (1995); (2003) [arXiv:hep-th/0308049]; M. T. Grisaru, S. Pe- R.L.WalsworthandD.F.Phillips,“Newclockcompar- nati and A. Romagnoni, “Two-loop renormalization ison testsofLorentzsymmetry,”arXiv:physics/0007062. for nonanticommutative N = 1/2 supersymmetric WZ D.Bear,R.E.Stoner,R.L.Walsworth,V.A.Kostelecky model,”JHEP0308,003(2003)[arXiv:hep-th/0307099]. andC.D.Lane,“LimitonLorentzandCPTviolationof [28] T.Araki,K.ItoandA.Ohtsuka,“Supersymmetricgauge theneutron using a two-species noble-gas maser,” Phys. theories on noncommutative superspace,” Phys. Lett. B Rev. Lett. 85, 5038 (2000) [Erratum-ibid. 89, 209902 573, 209 (2003) [arXiv:hep-th/0307076]. (2002)] [arXiv:physics/0007049]. [29] S. Ferrara and M. A. Lledo, “Some aspects of de- [18] C. E. Carlson, C. D. Carone and N. Zobin, “Noncom- formations of supersymmetric field theories,” JHEP mutativegaugetheorywithoutLorentzviolation,”Phys. 0005, 008 (2000) [arXiv:hep-th/0002084]; S. Ferrara, Rev.D 66, 075001 (2002) [arXiv:hep-th/0206035]. M. A. Lledo and O. Macia, “Supersymmetry in non- [19] K. Morita, “Lorentz-invariant non-commutative commutative superspaces,” JHEP 0309, 068 (2003) QED,” Prog. Theor. Phys. 108, 1099 (2003) [arXiv:hep-th/0307039]. [arXiv:hep-th/0209234]; H. Kase, K. Morita, Y. Oku- [30] J. H. Park, “Superfield theories and dual su- mura and E. Umezawa, “Lorentz-invariant non- permatrix models,” JHEP 0309, 046 (2003) commutative space-time based on DFR algebra,” Prog. [arXiv:hep-th/0307060]. Theor. Phys. 109, 663 (2003) [arXiv:hep-th/0212176]; [31] M. Chaichian and A.Kobakhidze,“Deformed N = 1 su- K.Imai, K.Morita andY.Okumura,“Lorentzinvariant persymmetry,” arXiv:hep-th/0307243. nonabelian gauge theory on noncommutativespace-time [32] D. Klemm, S. Penati and L. Tamassia, and arXiv:hep-th/0309060; K. Morita, Y. Okumura “Non(anti)commutative superspace,” Class. Quant. and E. Umezawa, “Lorentz invariance and unitar- Grav. 20, 2905 (2003) [arXiv:hep-th/0104190]. ity problem in non-commutative field theory,” Prog. [33] J. Wess and J. Bagger, “Supersymmetry and supergrav- Theor. Phys. 110, 989 (2003) [arXiv:hep-th/0309155]; ity,” University Press, Princeton 1992. K. Morita, “Discrete symmetries in Lorentz-invariant