REVIEWSOFMODERNPHYSICS,VOLUME73,OCTOBER2001 Noncommutative field theory Michael R. Douglas* DepartmentofPhysicsandAstronomy,RutgersUniversity,Piscataway, NewJersey08855 andInstitutdesHautesEtudesScientifiques,35routedesChartres, 91440Bures-sur-Yvette,France NikitaA. Nekrasov InstitutdesHautesEtudesScientifiques,35routedesChartres, 91440Bures-sur-Yvette,France andInstituteforTheoreticalandExperimentalPhysics,117259Moscow,Russia (Published 29November2001) Thisarticlereviewsthegeneralizationoffieldtheorytospace-timewithnoncommutingcoordinates, startingwiththebasicsandcoveringmostoftheactivedirectionsofresearch.Suchtheoriesarenow knowntoemergefromlimitsofMtheoryandstringtheoryandtodescribequantumHallstates.In thelastfewyearstheyhavebeenstudiedintensively,andmanyqualitativelynewphenomenahave beendiscovered,onboththeclassicalandthequantumlevel. CONTENTS G. Otherresults 1004 V. ApplicationstotheQuantumHallEffect 1005 A. ThelowestLandaulevel 1005 I. Introduction 977 B. ThefractionalquantumHalleffect 1005 II. Kinematics 979 VI. MathematicalAspects 1006 A. Formalconsiderations 979 A. Operatoralgebraicaspects 1006 1. Thealgebra 979 B. Othernoncommutativespaces 1008 2. Thederivativeandintegral 980 C. Groupalgebrasandnoncommutativequotients 1009 B. Noncommutativeflatspace-time 980 D. Gaugetheoryandtopology 1009 1. SymmetriesofRud 981 E. Thenoncommutativetorus 1010 2. Plane-wavebasisanddipolepicture 981 F. Moritaequivalence 1012 3. Deformation,operators,andsymbols 981 G. Deformationquantization 1012 4. Thenoncommutativetorus 982 VII. RelationstoStringandMTheory 1013 C. Fieldtheoryactionsandsymmetries 982 A. LightningoverviewofMtheory 1014 D. Gaugetheory 983 B. NoncommutativityinM(atrix)theory 1015 1. Theemergenceofspace-time 984 C. Noncommutativityinstringtheory 1017 2. Observables 985 1. Deformationquantizationfromtheworld 3. Stress-energytensor 985 sheet 1018 4. Fundamentalmatter 986 2. Deformationquantizationfromtopological 5. TheSeiberg-Wittenmap 986 openstringtheory 1018 E. Basesandphysicalpictures 987 3. Thedecouplinglimit 1019 1. Gaussiansandposition-spaceuncertainty 987 4. GaugeinvarianceandtheSeiberg-Witten 2. Fockspaceformalism 988 map 1020 3. Translationsbetweenbases 988 D. Stringyexplanationsofthesolitonicsolutions 1021 4. ScalarGreen’sfunctions 989 E. Quantumeffectsandclosedstrings 1022 5. Themembrane/hydrodynamiclimit 990 F. AdSdualsofnoncommutativetheories 1022 6. Matrixrepresentations 991 G. Timelikeuandexotictheories 1023 III. SolitonsandInstantons 992 VIII. Conclusions 1024 A. Largeusolitonsinscalartheories 992 Acknowledgments 1025 B. Vortexsolutionsingaugetheories 993 References 1025 C. Instantons 995 D. Monopolesandmonopolestrings 996 IV. NoncommutativeQuantumFieldTheory 997 I. INTRODUCTION A. Feynmanrulesandplanarity 997 B. Calculationofnonplanardiagrams 998 Noncommutativity is an age-old theme in mathemat- C. PhysicsofUV/IRmixing 1000 ics and physics. The noncommutativity of spatial rota- D. Gaugetheories 1002 tions in three and more dimensions is deeply ingrained E. Finitetemperature 1003 in us. Noncommutativity is the central mathematical F. Canonicalformulation 1004 concept expressing uncertainty in quantum mechanics, where it applies to any pair of conjugate variables, such as position and momentum. In the presence of a mag- *Electronicaddress:[email protected] netic field, even momenta fail to mutually commute. 0034-6861/2001/73(4)/977(53)/$30.60 977 ©2001TheAmericanPhysicalSociety 978 Douglasetal.: Noncommutativefieldtheory One can just as easily imagine that position measure- theory is not local in any sense we now understand, and ments might fail to commute and describe this using indeed has more than one parameter characterizing this noncommutativityofthecoordinates.Thesimplestnon- nonlocality: in general, it is controlled by the larger of commutativity one can postulate is the commutation re- the Planck length and the string length, the average size lation ofastring.ItwasdiscoveredbyConnesetal.(1998)and Douglas and Hull (1998) that simple limits of M theory @xi,xj#5iuij, (1) andstringtheoryleaddirectlytononcommutativegauge withaparameteruwhichisanantisymmetric(constant) theories, which appear far simpler than the original tensor of dimension (length)2. string theory yet keep some of this nonlocality. As has been realized independently many times, at One might also study noncommutative theories as in- least as early as 1947 (Snyder, 1947), there is a simple terestinganalogsoftheoriesofmoredirectinterest,such modificationtoquantumfieldtheoryobtainedbytaking as Yang-Mills theory. An important point in this regard the position coordinates to be noncommuting variables. isthatmanytheoriesofinterestinparticlephysicsareso StartingwithaconventionalfieldtheoryLagrangianand highly constrained that they are difficult to study. For interpreting the fields as depending on coordinates sat- example, pure Yang-Mills theory with a definite simple isfyingEq.(1),onecanfollowtheusualdevelopmentof gauge group has no dimensionless parameters with perturbative quantum field theory with surprisingly few which to make a perturbative expansion or otherwise changes,todefinealargeclassof‘‘noncommutativefield simplify the analysis. From this point of view it is quite theories.’’ interestingtofindanysensibleandnontrivialvariantsof Itisonthisclassoftheoriesthatourreviewwillfocus. these theories. Until recently, such theories had not been studied very Now, physicists have constructed many variations of seriously. Perhaps the main reason for this is that postu- Yang-Millstheoryinthesearchforregulated(UVfinite) latinganuncertaintyrelationbetweenpositionmeasure- versions as well as more tractable analogs of the theory. ments will a priori lead to a nonlocal theory, with all of Aparticularlyinterestingexampleinthepresentcontext the attendant difficulties. A secondary reason is that is the twisted Eguchi-Kawai model (Eguchi and Na- noncommutativity of the space-time coordinates gener- kayama, 1983; Gonzalez-Arroyo and Okawa, 1983), ally conflicts with Lorentz invariance, as is apparent in which in some of its forms, especially that of Gonzalez- Eq. (1). Although it is not implausible that a theory de- Arroyo and Korthals Altes (1983), is a noncommutative finedusingsuchcoordinatescouldbeeffectivelylocalon gauge theory. This model was developed in the study of lengthscaleslongerthanthatofu,itishardertobelieve the large-N limit of Yang-Mills theory (’t Hooft, 1974) that the breaking of Lorentz invariance would be unob- and we shall see that noncommutative gauge theories servable at these scales. show many analogies to this limit (Filk, 1996; Minwalla Nevertheless, one might postulate noncommutativity etal., 2000), suggesting that they should play an impor- for a number of reasons. Perhaps the simplest is that it tant role in the circle of ideas relating large-N gauge might improve the renormalizability properties of a theoryandstringtheory(Polyakov,1987;Aharonyetal., theory at short distances or even render it finite. With- 2000). out giving away too much of our story, we should say Noncommutativefieldtheoryisalsoknowntoappear thatthisisofcoursenotobviousaprioriandanoncom- naturally in condensed-matter theory. The classic ex- mutativetheorymightturnouttohavethesameoreven ample(thoughnotalwaysdiscussedusingthislanguage) worse short-distance behavior than a conventional isthetheoryofelectronsinamagneticfieldprojectedto theory. thelowestLandaulevel,whichisnaturallythoughtofas Another motivation is the long-held belief that in anoncommutativefieldtheory.Thustheseideasarerel- quantum theories including gravity, space-time must evant to the theory of the quantum Hall effect (Prange change its nature at distances comparable to the Planck and Girvin, 1987), and indeed, noncommutative geom- scale. Quantum gravity has an uncertainty principle etry has been found very useful in this context (Bellis- which prevents one from measuring positions to better sard etal., 1993). Most of this work has treated nonin- accuracies than the Planck length: the momentum and teracting electrons, and it seems likely that introducing energy required to make such a measurement will itself field-theoretic ideas could lead to further progress. modifythegeometryatthesescales(DeWitt,1962).One Itisinterestingtonotethatdespitethemanyphysical might wonder if these effects could be modeled by a motivationsandpartialdiscoverieswejustrecalled,non- commutation relation such as Eq. (1). commutative field theory and gauge theory were first A related motivation is that there are reasons to be- clearly formulated by mathematicians (Connes and lievethatanytheoryofquantumgravitywillnotbelocal Rieffel, 1987). This is rather unusual for a theory of sig- in the conventional sense. Nonlocality brings with it nificant interest to physicists; usually, as with Yang-Mills deep conceptual and practical issues which have not theory, the flow goes in the other direction. been well understood, and one might want to under- An explanation for this course of events might be stand them in the simplest examples first, before pro- found in the deep reluctance of physicists to regard a ceeding to a more realistic theory of quantum gravity. nonlocal theory as having any useful space-time inter- This is one of the main motivations for the intense pretation. Thus, even when these theories arose natu- currentactivityinthisareaamongstringtheorists.String rally in physical considerations, they tended to be re- Rev.Mod.Phys.,Vol.73,No.4,October2001 Douglasetal.: Noncommutativefieldtheory 979 garded only as approximations to more conventional II. KINEMATICS local theories, and not as ends in themselves. Of course suchsociologicalquestionsrarelyhavesuchpatanswers A. Formalconsiderations and we shall not pursue this one further except to re- mark that, in our opinion, the mathematical study of Let us start by defining noncommutative field theory these theories and their connection to noncommutative in a somewhat pedestrian way, by proposing a configu- geometry has played an essential role in convincing ration space and action functional from which we could physicists that these are not arbitrary variations on con- either derive equations of motion or define a functional ventionalfieldtheorybutindeedanewuniversalityclass integral. We shall discuss this material from a more oftheorydeservingstudyinitsownright.Ofcoursethis mathematical point of view in Sec. VI. mathematical work has also been an important aid to Conventions. Throughout the review we use the fol- themoreprosaictaskofsortingoutthepossibilities,and lowing notations: Latin indices i,j,k,... denote space- itisthesourceformanyusefultechniquesandconstruc- time indices, Latin indices from the beginning of the al- tions that we shall discuss in detail. phabet a,b,... denote commutative dimensions, Greek Havingsaidthis,itseemsthatthepresenttrendisthat indices m,n,... enumerate particles, vertex operators, the mathematical aspects appear less and less central to etc., while Greek indices from the beginning of the al- the physical considerations as time goes on. While it is phabet a,b,... denote noncommutative directions. tooearlytojudgetheoutcomeofthistrendanditseems In contexts where we simultaneously discuss a non- certain that the aspects which traditionally have ben- commuting variable or field and its commuting analog, efitedmostfrommathematicalinfluencewillcontinueto weshallusethe‘‘hat’’notation:x isthecommutingana- do so (especially the topology of gauge-field configura- logtoxˆ. However,inothercontexts,weshallnotusethe tions, and techniques for finding exact solutions), we hat. havetosomeextentdeemphasizedtheconnectionswith noncommutative geometry in this review. This is partly 1. Thealgebra tomakethematerialaccessibletoawiderclassofphysi- cists, and partly because many excellent books and re- The primary ingredient in the definition is an associa- viewscoverthematerialfromthispointofview,starting tive but not necessarily commutative algebra, to be de- with that of Connes (1994), and including those of noted A. The product of elements a and b of A will be Nekrasov (2000) focusing on classical solutions of non- denoted ab, a(cid:149)b, or a!b. This last notation (the star commutative gauge theory, Konechny and Schwarz product) has a special connotation, to be discussed (2000b) focusing on duality properties of gauge theory shortly. on a torus, and Gracia-Bondia etal. (2001) and Varilly An element of this algebra will correspond to a con- (1997). We maintain one section which attempts to give figuration of a classical complex scalar field on a space an overview of aspects for which a more mathematical M. Suppose first that A is commutative. The primary point of view is clearly essential. example of a commutative associative algebra is the al- Although many of the topics we discuss were moti- gebra of complex-valued functions on a manifold M, vated by and discovered in the context of string theory, with addition and multiplication defined pointwise: (f we have also taken the rather unconventional approach 1g)(x)5f(x)1g(x) and (f g)(x)5f(x)g(x). In this (cid:149) of separating the discussion of noncommutative field case,ourdefinitionswillreducetothestandardonesfor theory from that of its relation to string theory, to the field theory on M. extent that this was possible. An argument against this Although the mathematical literature is usually quite approachisthattherelationclarifiesmanyaspectsofthe preciseabouttheclassoffunctions(continuous,smooth, theory, as we hope will become abundantly clear upon etc.) to be considered, in this review we follow standard reading Sec. VII. However, it is also true that string physical practice and simply consider all functions that theory is not a logical prerequisite for studying the arise in reasonable physical considerations, referring to theory, and we feel the approach we took better illus- this algebra as A(M) or (for reasons to be explained trates its internal self-consistency (and the points where shortly) as M . If more precision is wanted, for most 0 thisisstilllacking).Furthermore,ifwehopetousenon- purposes one can think of this as C(M), the bounded commutativefieldtheoryasasourceofnewinsightsinto continuous functions on the topological manifold M. stringtheory,weneedtobeabletounderstanditsphys- The most elementary example of a noncommutative ics without relying too heavily on the analogy. We also algebra is Mat , the algebra of complex n3n matrices. n hope this approach will have the virtue of broader ac- Generalizations of this, which are almost as elementary, cessibility and perhaps help in finding interesting appli- are the algebras Mat @C(M)# of n3n matrices whose n cations outside of string theory. Reviews with a more matrix elements are elements of C(M), and with addi- string-theoreticemphasisincludethatofHarvey(2001a) tion and multiplication defined according to the usual which discusses solitonic solutions and their relations to rules for matrices in terms of the addition and multipli- string theory. cationonC(M). ThisalgebracontainsC(M) asitscen- Finally, we must apologize to the many whose work ter (take functions times the identity matrix in Mat ). n wewerenotabletotreatinthedepthitdeservedinthis Clearly elements of Mat @C(M)# correspond to con- n review, a sin we have tried to atone for by including an figurations of a matrix field theory. Just as one can gain extensive bibliography. some intuition about operators in quantum mechanics Rev.Mod.Phys.,Vol.73,No.4,October2001 980 Douglasetal.: Noncommutativefieldtheory by thinking of them as matrices, this example already off at infinity, leading to boundary terms, condition (c) serves to illustrate many of the formal features of non- canbeviolatedforgeneraloperators,leadingtophysical commutative field theory. In the remainder of this sub- consequences in noncommutative theory which we will section we introduce the other ingredients we need to discuss. define noncommutative field theory in this familiar context. To define a real-valued scalar field, it is best to start with Mat @C(M)# and then impose a reality condition n analogoustotherealityoffunctionsinC(M). Themost B. Noncommutativeflatspace-time useful in practice is to take the Hermitian matrices a 5a†, whose eigenvalues will be real (given suitable ad- After Mat @C(M)#, the next simplest example of a n ditionalhypotheses).TodothisforgeneralA,wewould noncommutative space is the one associated with the need an operation a!a† satisfying (a†)†5a and (for c algebra Rud of all complex linear combinations of prod- PC) (ca)†5c*a†, in other words an antiholomorphic ucts of d variables xˆi satisfying involution. ThealgebraMat @C(M)# couldalsobedefinedasthe @xi,xj#5iuij. (2) n tensor product Matn(C)^C(M). This construction gen- The i is present because the commutator of Hermitian eralizes to an arbitrary algebra A to define Matn(C) operators is anti-Hermitian. As in quantum mechanics, ^A, which is just Matn(A) or n3n matrices with ele- this expression is the natural operator analog of the ments in A. This algebra admits the automorphism Poisson bracket determined by the tensor uij, the Pois- groupGL(n,C), actingasa!g21ag (ofcoursethecen- son tensor or noncommutativity parameter. ter acts trivially). Its subgroup U(n) preserves Hermit- By applying a linear transformation to the coordi- ian conjugation and the reality condition a5a†. One nates, one can bring the Poisson tensor to canonical sometimes refers to these as U(n) noncommutative form. This form depends only on its rank, which we de- theories,abitconfusingly.Weshallrefertothemasrank note as 2r. We keep this general as one often discusses n theories. partially noncommutative spaces, with 2r,d. Intherestofthereview,weshallmostlyconsidernon- A simple set of derivatives ] can be defined by the commutative associative algebras which are related to i thealgebrasA(M) bydeformationwithrespecttoapa- relations rameter u, as we shall define shortly. Such a deformed ]xj[dj, (3) algebra will be denoted by Mu, so that M05A(M). @]i ,]#5i 0, (4) i j and the Leibnitz rule. This choice also determines the 2. Thederivativeandintegral integral uniquely (up to overall normalization), by re- A noncommutative field theory will be defined by an quiring that *]f50 for any f such that ]f(cid:222)0. action functional of fields F,f,w,... defined in terms of We shall occaisionally generalize Eq. (4i) to the associative algebra A (it could be elements of A, or @],]#52iF , (5) vectors in some representation thereof). Besides the al- i j ij gebra structure, to write an action we shall need an in- to incorporate an additional background magnetic field. tegral * Tr and derivatives ]i. These are linear opera- Finally, we shall require a metric, which we shall take tions satisfying certain formal properties: to be a constant symmetric tensor g , satisfying ]g 5((]a)iAT)Bhe1Ade(r]iviBat)i.veWiitshalindeearriivtya,titohnis iomnplAie,s t]hi(aAt tBhe) n5o0t.eItnhamtaonnyeecxaanmnoptlebsriwnge btaokthe gthijisantiojdbueij gtoij5cadnioj,niibcuajktl derivative of a constant is zero. form simultaneously, as the symmetry groups preserved (b) The integral of the trace of a total derivative is by the two structures, O(n) and Sp(2r), are different. zero, *Tr]iA50. At best one can bring the metric and the Poisson tensor (c) The integral of the trace of a commutator is zero, to the following form: *Tr@A,B#[*Tr(A B2B A)50. (cid:149) (cid:149) r A candidate derivative ] can be written using an ele- ( ( ment diPA; let ]iA5@di,iA#. Derivations that can be g5a51 dzad¯za1 b dyb2, written in this way are referred to as inner derivations, while those that cannot are outer derivations. 1( Wedenotetheintegralas*Tr,asitturnsoutthat,for u52 a ua]¯za(cid:217) ]za, ua.0. (6) general noncommutative algebras, one cannot separate thenotationsoftraceandintegral.Indeed,onenormally Here za5qa1ipaare convenient complex coordinates. In terms of p,q,y the metric and the commutation rela- useseitherthesinglesymbolTr(asisdoneinmathemat- ics) or * to denote this combination; we do not follow tions Eq. (6) read as this convention here only to aid the uninitiated. @ya,yb#5@yb,qa#5@yb,pa#50, We note that just as condition (b) can be violated in conventional field theory for functions that do not fall @qa,pb#5iuadab, ds25dqa21dpa21dyb2. (7) Rev.Mod.Phys.,Vol.73,No.4,October2001 Douglasetal.: Noncommutativefieldtheory 981 1. SymmetriesofRdu An infinitesimal translation xi!xi1ai on Rud acts on functions as df5ai]f. For the noncommuting coordi- i nates xi, these are formally inner derivations, as ]f5@2i~u21! xj,f#. (8) i ij Oneobtainsglobaltranslationsbyexponentiatingthese, f~xi1«i!5e2iuij«ixjf~x!eiuij«ixj. (9) In commutative field theory, one draws a sharp dis- tinction between translation symmetries (involving the derivatives) and internal symmetries, such as df 5@A,f#. We see that in noncommutative field theory, there is no such clear distinction, and this is why one cannot separately define integral and trace. One often uses only @],f# and if so, Eq. (8) can be simplified further to thie operator substitution ] FIG.1. Theinteractionoftwodipoles. i !2i(u21) xj. This leads to derivatives satisfying Eq. ij (5) with F 52(u21) . More interesting is the interpretation of the multipli- ij ij The Sp(2r) subgroup of the rotational symmetry xi cation law in this basis. This is easy to compute in the !Rjixj which preserves u, Rii8Rjj8ui8j85uij can be ob- plane-wave basis, by operator reordering: tained similarly, as eikx(cid:149)eik8x5e2~i/2!uijkikj8ei(k1k8)(cid:149)x. (14) f~Rjixj!5e2iAijxixjf~x!eiAijxixj, (10) The combination uijkikj8 appearing in the exponent whereR5eiL, Li5A uik, andA 5A . Ofcourseonly comes up very frequently, and a standard and conve- j kj ij ji nient notation for it is the U(r) subgroup of this will preserve the Euclidean metric. k3k8[uijkikj85k3uk8, After considering these symmetries, we might be tempted to go on and conjecture that the latter notation being used to stress the choice of Poisson structure. df5i@f,e# (11) We can also consider for any eis a symmetry of Rud. However, although these eikx f~x! e2ikx5e2uijki]jf~x!5f~xi2uijk !. (15) transformations preserve the algebra structure and the (cid:149) (cid:149) j trace,1 they do not preserve the derivatives. Neverthe- Multiplicationbyaplanewavetranslatesageneralfunc- less, they are important and will be discussed in detail tion by xi!xi2uijk . This exhibits the nonlocality of j below. the theory in a particularly simple way and gives rise to the principle that large momenta will lead to large non- locality. A simple picture can be made of this nonlocality 2. Plane-wavebasisanddipolepicture (Sheikh-Jabbari, 1999; Bigatti and Susskind, 2000) by Onecanintroduceseveralusefulbasesforthealgebra imagining that a plane wave corresponds not to a par- Rud. For discussions of perturbation theory and scatter- ticle (as in commutative quantum field theory) but in- ing, the most useful basis is the plane-wave basis, which stead to a ‘‘dipole,’’ a rigid oriented rod whose extent is consists of eigenfunctions of the derivatives: proportional to its momentum: ]eikx5ik eikx. (12) Dxi5uijp . (16) i i j The solution eikx of this linear differential equation is Ifwepostulatethatdipolesinteractbyjoiningattheir theexponentialoftheoperatorik x intheusualopera- ends (Fig. 1), and grant the usual quantum field theory (cid:149) tor sense. relation p5\k between wave number and momentum, The integral can be defined in this basis as the rule Eq. (15) follows immediately. E Treikx5d , (13) k,0 3. Deformation,operators,andsymbols whereweinterpretthedeltafunctionintheusualphysi- There is a sense in which Rud and the commutative cal way (for example, its value at zero represents the algebra of functions C(Rd) have the same topology and volume of physical space). thesamesize,notionsweshallkeepatanintuitivelevel. In the physical applications, it will turn out that uis typically a controllable parameter, which one can imag- 1Assumingcertainconditionsoneandf;seeSec.VI.A. ineincreasingfromzerotogofromcommutativetonon- Rev.Mod.Phys.,Vol.73,No.4,October2001 982 Douglasetal.: Noncommutativefieldtheory E E commutative(thisdoesnotimplythatthephysicsiscon- tinuous in this parameter, however). These are all Trf!g5 Trfg. (22) reasonstostudytherelationbetweenthesetwoalgebras more systematically. 4. Thenoncommutativetorus There are a number of ways to think about this rela- tion. If uis a physical parameter, it is natural to think of Much of this discussion applies with only minor Rud as a deformation of Rd. A deformation Muof C(M) changestodefineTud, thealgebraoffunctionsonanon- commutative torus. isanalgebrawiththesameelementsandadditionlaw(it To obtain functions on a torus from functions on Rd isthesameconsideredasavectorspace)butadifferent we would need to impose a periodicity condition, say multiplication law, which reduces to that of C(M) as a (multi)parameter ugoes to zero. This notion was intro- f(xi)5f(xi12pni). A nice algebraic way to phrase this istoinsteaddefineTd asthealgebraofallsumsofprod- duced by Bayen etal. (1978) as an approach to quanti- u ucts of arbitrary integer powers of a set of d variables zation, and has been much studied since, as we shall U , satisfying discussinSec.VI.Suchadeformedmultiplicationlawis i often denoted f!g or star product to distinguish it from U U 5e2iuijU U . (23) i j j i the original pointwise multiplication of functions. This notation has a second virtue, which is that it al- The variable U takes the place of eixi in our previous i lowsustoworkwithMuinawaythatissomewhatmore notation, and the derivation of the Weyl algebra from forgiving of ordering questions. Namely, we can choose Eq. (1) is familiar from quantum mechanics. Similarly, a linear map S from Muto C(M), fˆ(cid:176)S@fˆ#, called the we take symbol of the operator. We then represent the original @],U #5idU i j ij j operator multiplication in terms of the star product of and symbols as E fˆgˆ5S21(cid:134)S@fˆ#!S@gˆ#(cid:135). (17) TrU1n1flUdnd5dnW,0. One should recall that the symbol is not ‘‘natural’’ in There is much more to say in this case about the to- the mathematical sense: there could be many valid defi- pological aspects, but we postpone this to Sec. VI. nitionsofS, correspondingtodifferentchoicesofopera- tor ordering prescription for S21. AconvenientandstandardchoiceistheWeylordered C. Fieldtheoryactionsandsymmetries symbol.ThemapS, definedasamaptakingelementsof Rud to A(Rd) (functions on momentum space), and its Field theories of matrix scalar fields are very familiar inverse, are and are treated in most textbooks on quantum field E 1 theory. The matrix generalization is essential in discuss- f~k![S@fˆ#~k!5 Tre2ikxˆfˆ~xˆ!, (18) ing Yang-Mills theory. In a formal sense we shall now ~2p!n/2 makeexplicitanyfieldtheoryLagrangianthatiswritten E 1 intermsofmatrixfields,matrixadditionandmultiplica- fˆ~xˆ!5S21@f#5~2p!n/2 dnkeikxˆf~k!. (19) tion,andthederivativeandintegral,canbeequallywell regarded as a noncommutative field theory Lagrangian, Formally these are inverse Fourier transforms, but the with the same equations of motion and (classical) sym- first expression involves the integral equation (13) on metry properties as the matrix field theory. A(Ru), while the second is an ordinary momentum- Let us consider a generic matrix scalar field theory space integral. with a Hermitian matrix valued field f(x)5f(x)† and One can get the symbol in position space by perform- (Euclidean) action S D ing a second Fourier transform; e.g., E E E A 1 1 S5 ddx g gijTr]f]f1TrV~f! , (24) S@fˆ#~x!5 dnk Treik(x2xˆ)fˆ~xˆ!. (20) 2 i j ~2p!n where V(z) is a polynomial in the variable z, ]5]/]xi i We shall freely assume the usual Fourier relation be- arethepartialderivatives,andgij isthemetric.Thecon- tween position and momentum space for the symbols, straint we require in order to generalize a matrix action while being careful to say (or denote by standard letters to a noncommutative action is that it be written only such as x and k) which we are using. usingthecombination*TrappearinginEq.(24);wedo The star product for these symbols is not allow either the integral * or the trace Tr to appear eikx!eik8x5e2~i/2!uijkikj8ei(k1k8)(cid:149)x. (21) separately.Inparticular,therankofthematrixN cannot appear explicitly, only in the form Tr1 combined with Of course all of the discussion in Sec. II.B.2 above still the integral. applies, as this is only a different notation for the same Under this assumption, it is an easy exercise to check product, Eq. (14). thatifwereplacethealgebraMat @C(M)# byageneral N Another special case that often comes up is associative algebra A with integral and derivative satis- Rev.Mod.Phys.,Vol.73,No.4,October2001 Douglasetal.: Noncommutativefieldtheory 983 fyingtherequirementsabove,thestandarddiscussionof transformation properties (scalar, spinor, vector, and so equationsofmotion,classicalsymmetries,andNoether’s on).However,atthispointweshallonlyconsideragen- theoremallgothroughwithoutchange.Thepointisthat eralizationdirectlyanalogoustothetreatmentofhigher formal manipulations which work for arbitrary matrices spin fields in Euclidean and Minkowski space. We shall offunctionscanalwaysbemadewithoutcommutingthe discuss issues related to curved backgrounds later; at matrices. Another way to think about this result is to present the noncommutative analogs of manifolds with imagine defining the theory in terms of an explicit ma- general metrics are not well understood. trix representation of the algebra A. Althoughonecanbemoregeneral,letusnowassume ThusthenoncommutativetheorywithactionEq.(24) that the derivatives ]i are linearly independent and sat- has the standard equation of motion isfy the usual flat space relations @]i,]j#50. Given Poincare´ symmetry or its subgroup preserving gij]i]jf5V8~f! u,onecanusetheconventionaldefinitionsfortheaction and conservation laws ]Ji50 with the conserved cur- of the rotation group on tensors and spinors, which we i rentJi associatedtoasymmetrydf(e,f) determinedby do not repeat here. In particular, the standard Dirac the usual variational procedures, equation also makes sense over Rud and Tud, so spin-1/2 E particles can be treated without difficulty. dS5 TrJi]e. Thediscussionofsupersymmetryisentirelyparallelto i that for conventional matrix field theory or Yang-Mills For example, let us consider the transformations Eq. theory, with the same formal transformation laws. Con- (11).Inmatrixfieldtheory,thesewouldbetheinfinitesi- straints between the dimension of space-time and the mal form of a U(N) internal symmetry f!U†fU. Al- number of possible supersymmetries enter at the point though in more general noncommutative theories we assume that the derivatives ]i are linearly indepen- @],e#(cid:222)0 and these are not in general symmetries, we dent. With care, one can also use the conventional su- i canstillconsidertheiraction,andbyexponentiationde- perfield formalism, treating the anticommuting coordi- fine an analogous U(N) action. We will refer to this nates as formal variables which commute with elements groupasU(H), thegroupofunitaryoperatorsactingon of A (Ferrara and Lledo, 2000). a Hilbert space H admitting a representation of the al- Finally, as long as time is taken as commutative, the gebra A. In more mathematical terms, discussed in Sec. standard discussion of Hamiltonian mechanics and ca- VI, H will be a module for A. nonical quantization goes through without conceptual Of course, if we do not try to gauge U(H), it could difficulty. On the other hand, noncommutative time im- alsobebrokenbyothertermsintheaction,forexample, pliesnonlocalityintime,andtheHamiltonianformalism source terms *TrJf, position-dependent potentials becomes rather complicated (Gomis etal., 2001); it is *TrfV(f), and so forth. not clear that it has any operator interpretation. Al- Application of the Noether procedure to Eq. (11) though functional integral quantization is formally sen- leads to a conserved current Ti, which for Eq. (24) sible,theresultingperturbationtheoryisproblematicas would be discussed in Sec. IV. It is believed that sensible string theorieswithtimelikenoncommutativityexist,discussed Ti5igij@f,]f#. (25) j in Sec. VII.G. AswediscussedinSec.II.B.1,Eq.(11)includestransla- tions and the rotations which preserve uij, so Ti can be D. Gaugetheory used to define momentum and angular momentum op- erators, for example, Theonlyunitaryquantumfieldtheoriesincludingvec- E torfieldsaregaugetheories,andthestandarddefinitions Pi52i~u21!ij TrxiT0. (26) alsoapplyinthiscontext.However,thereisagreatdeal more to say about the kinematics and observables of Thus we refer to it as the restricted stress-energy tensor. gauge theory. One can also apply the Noether definition to the gen- Agaugeconnectionwillbeaone-formA , eachcom- eral variation xi(cid:176)xi1vi(x), to define a more conven- ponent of which takes values in A andisatisfies A i tional stress-energy tensor Tij, discussed in Gerhold 5A†. (See Sec. VI.D for a more general definition.) i etal. (2000) and Abou-Zeid and Dorn (2001a). In gen- The associated field strength is eral, the action of this stress tensor changes uand the underlying algebra and its interpretation has not been Fij5]iAj2]jAi1i@Ai,Aj#, (27) fully elucidated at present (see Secs. VI.G and VII.C.2 which under the gauge transformation for related issues). Finally, there is a stress-energy tensor for noncommu- dAi5]ie1i@Ai,e# (28) tative gauge theory which naturally appears in the rela- transforms as dF 5i@F ,e#, allowing us to write the ij ij tion to string theory, which we shall discuss in Secs. gauge invariant Yang-Mills action II.D.3 and VII.E. E We could just as well consider theories containing an S52 1 TrF2. (29) arbitrary number of matrix fields with arbitrary Lorentz 4g2 Rev.Mod.Phys.,Vol.73,No.4,October2001 984 Douglasetal.: Noncommutativefieldtheory All this works for the reasons already discussed in Sec. course,thehigherderivativetermsmodifythisresult.In II.C. fact, the full gauge group U(H) is simpler, as we shall Gauge invariant couplings to charged matter fields see in Sec. II.E.2. can be written in the standard way using the covariant Another aspect of this unification of space-time and derivative gaugesymmetryisthatifthederivativeisaninnerderi- D f[]f1i@A ,f#. (30) vation, we can absorb it into the vector potential itself. i i i In other words, we can replace the covariant derivatives Finite gauge transformations act as Di5]i1iAi with connection operators in Rud, ~]i1iAi,F,f!!U†~]i1iAi,F,f!U C [~2iu21! xj1iA (34) i ij i and these definitions gauge the entire U(H) symmetry. such that One can also use Mat (A) to get the noncommutative analogofU(N) gaugeNtheory,though(atthispoint)not Dif!@Ci,f#. (35) the other Lie groups. We also introduce the ‘‘covariant coordinates,’’ As an example, we quote the maximally supersym- Yi5xi1uijA ~x!. (36) metric Yang-Mills (MSYM) Lagrangian in ten dimen- j sions, from which N54 SYM in d54 and many of the If uis invertible, then Yi5iuijC and this is just another j simpler theories can be deduced by dimensional reduc- notation, but the definition makes sense more generally. tion and truncation: In terms of the connection operators, the Yang-Mills E field strength is S5 d10xTr~Fi2j1i¯xID(cid:148) xI!, (31) F 5i@D ,D #!i@C ,C #2~u21! , (37) ij i j i j ij where xis a 16-component adjoint Majorana-Weyl fer- and the Yang-Mills action becomes a simple ‘‘matrix mion. This action satisfies all of our requirements and model’’ action, thus leads to a wide variety of supersymmetric noncom- ( mutative theories. Indeed, noncommutativity is the only S5Tr $i@C ,C #2~u21! %2. (38) i j ij known generalization (apart from adding irrelevant op- i,j erators and taking limits of this) which preserves maxi- Now, although we motivated this from Eq. (29), we mal supersymmetry. couldlookatthistheotherwayaround,startingwiththe Because one cannot separately define integral and action Eq. (38) as a function of matrices C and postu- i trace, the local gauge invariant observables of conven- lating Eq. (34), to derive noncommutative gauge theory tional gauge theory do not carry over straightforwardly: (and Yang-Mills theory in the limit u!0) from a matrix only*TrO isgaugeinvariant.Wenowdiscussthispoint. model. This observation is at the heart of most of the common ways that noncommutative gauge theory arises in particle physics, as the action Eq. (38) and its super- 1. Theemergenceofspace-time symmetrization is simple enough to arise in a wide vari- ety of contexts. For example, it can be obtained as a The first point to realize is that the gauge group in limit of the twisted Eguchi-Kawai model (Eguchi and noncommutative theory contains space-time transla- Nakayama, 1983; Gonzalez-Arroyo and Okawa, 1983), tions. This is already clear from the expression Eq. (8), which was argued to reproduce the physics of large N which allows us to express a translation dA 5vj]A in i j i Yang-Mills theory. The maximally supersymmetric ver- terms of a gauge transformation Eq. (28) with e 5vj(u21) xk. Actually, this produces sion obtained in the same way from Eq. (31), often re- jk ferred to as the ‘‘IKKT model’’ (Ishibashi etal., 1997), dA 5vj]A 1vj~u21! , plays an important role in M theory, to be discussed in i j i ji Sec. VII. but an overall constant shift of the vector potential Havingsoeffectivelyhiddenit,wemightwellwonder drops out of the field strengths and has no physical ef- how d-dimensional space-time is going to emerge again fect in infinite flat space. Takingmoregeneralfunctionsforewillproducemore from Eq. (38). Despite appearances, we do not want to claim that noncommutative gauge theory is the same in general space-time transformations. As position- all dimensions d. Now in the classical theory, to the ex- dependent translations, one might compare these with tentweworkwithexplicitexpressionsforA inEq.(34), coordinate definitions or diffeomorphisms. To do this, i this is generally not a problem. However, in the quan- we consider the products as star products and expand Eq. (21) in u, to obtain tum theory, we need to integrate over field configura- tions C . We shall need to argue that this functional i df5i@f,e#5uij]f]e1O~]2f]2e!. (32) integral can be restricted to configurations which are i j similar to Eq. (34) in some sense. !$f,e%, (33) Thispointisrelatedtowhatatfirstappearsonlytobe ($,% is the Poisson bracket), so at leading order the a technical subtlety involving the u21 terms in Eq. (38). gauge group is the group of canonical transformations They are there to cancel an extra term preserving u(we discuss this further in Sec. II.E.5). Of @(u21x) ,(u21x) #, which would have led to an infinite i j Rev.Mod.Phys.,Vol.73,No.4,October2001 Douglasetal.: Noncommutativefieldtheory 985 E constant shift of the action. The subtlety is that one could have made a mistake at this point by assuming WL~k![ TrW@L1,2#eikxˆ2. that Tr@C ,C #50, as for finite dimensional operators. i j Of course the possibility that Tr@C ,C #(cid:222)0 probably If the distance between the end points of L and the comes as no surprise, but the point wie wjant to make is momentum k satisfy the relation Eq. (16), uijki5(x1 thatTr@Ci,Cj# shouldbeconsideredatopologicalaspect 2x2), this operator will be gauge invariant, as one can of the configuration. We shall argue in Sec. VI.A that it see by using Eq. (15). is invariant under any variation of the fields which (in a Thisprovidesanoperatorwhichcarriesadefinitemo- sense)preservetheasymptoticsatinfinity.Thisinvariant mentum and which can be used to define a version of detects the presence of the derivative operators in C local correlation functions. There is a pleasing corre- i and this is the underlying reason one expects to consis- spondence between its construction and the dipole pic- tently identify sectors with a higher dimensional inter- tureofSec.II.B.2;notonlycanwethinkofaplanewave pretation in what is naively a zero-dimensional theory. as having a dipole extent, we should think of the two ends of the dipole as carrying opposite electric charges whichforgaugeinvariancemustbeattachedtoaWilson 2. Observables line. The straight line has a preferred role in this construc- All this is intriguing, but it comes with conceptual tion,andtheopenWilsonloopassociatedtothestraight problems. The most important of these is that it is diffi- line with length determined by Eq. (16) can be written culttodefinelocalobservables.Thisisbecause,asnoted E E fromthestart,thereisnowaytoseparatethetraceover H(requiredforgaugeinvariance)fromtheintegralover W~k![ Trek3C5 Treik(cid:149)Y, noncommutative space. We can easily enough write gauge invariant observables, such as whereYi arethecovariantcoordinatesofEq.(36).This E construction can be used to covariantize local operators TrF~x!n, as follows: given an operator O(x), transforming in the adjoint, and momentum k, we define E E but they are not local. A step forward is to define the Wilson loop operator. W@O#~k!5 Trek3CO~x!5 TreikmYm(x)O~x!, Given a path L, we write the holonomy operator using E E exactly the same formal expression as in conventional O@y#5 ddk Treik(cid:149)(Y2y)O~x!. (39) gauge theory, S D E W 5Pexp i dsA@x~s!# , 3. Stress-energytensor L L As we discussed above, the simplest analog of the but where the products in the expansion of the path stress-energy tensor in noncommutative field theory is ordered exponential are star products. This undergoes Eq.(25),whichgeneratesthenoncommutativeanalogof the gauge transformation canonical transformations on space-time. However, in W !U†~x !W U~x !, noncommutative gauge theory, this operator is the gen- L 1 L(x ,x ) 2 1 2 eratorofgaugetransformations,soitmustbesettozero wherex1 andx2 arethestartandendpointsofthepath on physical states. This leads to a subtlety analogous to L1,2. one known in general relativity: one cannot define a We can form a Wilson loop by taking for L a closed gaugeinvariantlocalconservedmomentumdensity.This loopwithx15x2, butagainwefacetheproblemthatwe iscompatiblewiththedifficultieswejustencounteredin can only cyclically permute operators, and thus cancel defining local gauge invariant observables. U21(x1) withU(x1), ifwetakethetraceoverH,which One can nevertheless regard Eq. (26) as a nontrivial includes the integral over noncommutative space. global conserved momentum. This is because it corre- We can at least formulate multilocal observables with sponds to a formal gauge transformation with a param- this construction, such as etere;xi whichdoesnotfalloffatinfinity(onthetorus, E it is not even single valued), and as such can be consis- TrO ~x !W@L #O ~x !W@L # O~x !W@L # tently excluded from the gauge group. This type of con- 1 1 1,2 2 2 2,3 fl n n,1 sideration will be made more precise in Sec. VI.A. with arbitrary gauge covariant operators O at arbitrary One can make a different definition of stress-energy i pointsx , joinedbyWilsonloops.Thisallowsustocon- tensor, motivated by the relation Eq. (34) between the i trolthedistancebetweenoperatorswithinasingletrace, connection and the noncommutative space-time coordi- butnottocontrolthedistancebetweenoperatorsindif- nates, as the Noether current associated to the variation ferent traces. Actually one can do better than this, using what are Ci!Ci1ai~k!eik(cid:149)Y called open Wilson loops (Ishibashi etal., 2000). The which for the action Eq. (38) can easily be seen to simplest example is produce Rev.Mod.Phys.,Vol.73,No.4,October2001 986 Douglasetal.: Noncommutativefieldtheory E E ( 1 5. TheSeiberg-Wittenmap Tij~k!5 ds Treisk(cid:149)Y@Ci,Cl#ei(12s)k(cid:149)Y@Cj,Cl#, l 0 Having discovered an apparent generalization of (40) gauge theory, we should ask ourselves to what extent which is conserved in the sense that k umnT (k)50 this theory is truly novel and to what extent we can un- m nl for a solution of the equations of motion. This appears derstanditasaconventionalgaugetheory.Thisquestion tobethenaturaldefinitioninstringtheory,aswediscuss will become particularly crucial once we find noncom- in Sec. VII.E. mutative gauge theory arising from open string theory, as general arguments imply that open string theory can always be thought of as giving rise to a conventional gaugetheory.Isthereaninherentcontradictioninthese 4. Fundamentalmatter claims? Seiberg and Witten (1999) proposed that not only is Another type of gauge invariant observable can be there no contradiction, but that one should be able to obtainedbyintroducingnewfields(bosonsorfermions) write an explicit map from the noncommutative vector which transform in the fundamental of the noncommu- tativegaugegroup.Inotherwords,weconsiderafieldc potential to a conventional Yang-Mills vector potential, explicitly exhibiting the equivalence between the two which is operator valued just as before, but instead of transforming under U(H) as c!UcU†, we impose the classes of theories. One might object that the gauge groups of noncom- transformation law mutative gauge theory and conventional gauge theory c!Uc. are different, as is particularly clear in the rank 1 case. Moregenerally,weneedtodefinemultiplicationa cby However, this is not an obstacle to the proposal, as only (cid:149) anyelementofA,butthiscanbeinferredusinglinearity. the physical configuration space—namely, the set of or- Bilinearssuchasc†c, c†D c, andsoonwillbegauge bits under gauge transformation—must be equivalent in i thetwodescriptions.Itdoesimplythatthemapbetween invariant and can be used in the action and to define the two gauge transformation laws must depend on the newobservables,eitherbyenforcinganequationofmo- tiononcordoingafunctionalintegraloverc(Ambjorn vector potential, not just the parameter. Thus the proposal is that there exists a relation be- etal., 2000; Rajaraman and Rozali, 2000; Gross and tween a conventional vector potential A with the stan- Nekrasov, 2001). i dard Yang-Mills gauge transformation law with param- Although in a strict sense this is also a global observ- eter e, Eq. (28), and a noncommutative vector potential able, an important point (which will be central to Sec. VI.D)isthatonecanalsopostulateanindependentrule Aˆ (A ) and gauge transformation parameter eˆ(A,e) i i for multiplication by A (and the unitaries in A) on the with noncommmutative gauge invariance dˆAˆ 5]eˆ i i right, 1iAˆ *eˆ2ieˆ*Aˆ , such that i i c!ca. Aˆ~A!1dˆeˆAˆ~A!5Aˆ~A1deA!. (41) Indeed,ifwetakecPA, weshallclearlygetanontrivial This equation can be solved to first order in uwithout second action of this type, since left and right multipli- difficulty. Writing u5du, we have cation are different. In this case, we can think of c†cas afunctiononasecond,dualnoncommutativespace.For 1 each fPA one obtains a gauge invariant observable Aˆ i~A!2Ai524dukl$Ak,]lAi1Fli%11O~du2!, Trc†cf, which is local on the dual space in the same (42) sensethatannoncommutativefieldisalocalobservable 1 inTaankuinnggacuPgeAd tihseaorcyh.oice. One could also have taken eˆ~A,e!2e54dukl$]ke,Al%11O~du2!, (43) cPH, which does not lead to such a second multiplica- where $A,B%1[AB1BA. The corresponding first- tionlaw.Thegeneraltheoryofthischoiceisdiscussedin order relation between the field strengths is Sec. VI.D. The two definitions lead to different physics. Let us 1 compare the spectral density. If we take the Dirac op- Fˆij2Fij54dukl~2$Fik,Fjl%12$Ak,DlFij1]lFij%1!. erator giD acting on cPH , this has dr(E) ;dEEr21, ais for a field in r dimernsions. This result even admits a reinterpretation which de- If we take the same Dirac operator with cPA, we finesthemaptoallfiniteordersinu.Considertheprob- wouldgetinfinitespectraldensity.Amoreusefuldefini- lem of mapping a noncommutative gauge field Aˆ (u) de- tion is fined with respect to the star product for u, to a noncommutative gauge field Aˆ (u1du) defined for a Dc5gi~D c2c]!, (cid:148) i i nearby choice of u. To first order in du, it turns out that which fixes this by postulating an ungauged right action the solution to the corresponding relation (41) is again of translations, leading to a spectral density appropriate Eqs. (42) and (43), now with the right-hand side evalu- to 2r dimensions. ated using the star product for u. Thus these equations Rev.Mod.Phys.,Vol.73,No.4,October2001