Nonlocal Yang—Millshome.iitk.ac.in/~sdj/HAQUE/Kleppe_Woodard.pdf · G. Kleppe, R.P. Woodard / Nonlocal Yang—Mills 83 Although the method is no more difficult to implement than

N U CLEARNuclearPhysicsB388 (1992)81—112 PH VS I C S BNorth-Holland ________________

NonlocalYang—Mills

GaryKieppeDepartmentof Physics, Virginia PolytechnicInstitute, Blacksburg, VA 24061, USA

R.P. WoodardDepartmentofPhysics,Universityof Florida, Gainesville,FL 32611, USA

Received17 September1991(Revised9 December1991)

Acceptedfor publication24 July 1992

We presenta very simple and explicit procedurefor nonlocalizingthe action of any theorywhich can be formulated perturbatively.When the resulting nonlocal field theory is quantizedusingthe functional formalism — with unit measurefactor — its Greenfunctionsarefinite to allorders. The construction also ensuresperturbative unitarity to all orders for scalars withnonderivativeinteractions, however, decoupling is lost at one loop when vector and tensorquantaarepresent.Decouplingcanberestored(again,to all orders)if a suitablemeasurefactorexists.We computethe requiredmeasurefactor for pureYang—Mills at orderg

2 andthenuse itto evaluatethevacuumpolarizationatoneloop. A peculiarfeatureof our regularizationschemeis that the on-shell treeamplitudes are completelyunaffected.This implies that the nonlocalfield theory can be viewed as a highly noncanonicalquantizationof the original, local fieldequations.

1. Introduction

At any point in the evolution of sciencethereis always a “simplest way” toperceivetrue results,yet insightsof greatpowersometimesemergefrom adoptingnovel perspectives.The putative finiteness of superstring loop amplitudes iscurrently best understoodthrough the Polyakov representation[1], a formalismwhich makes frequent and essentialuse of the fact that the first quantizeddynamicalvariable is an extendedobject.Few would care to view finitenessfromthe perspectiveof stringfield theory; indeedit is not evensimple to computetreeamplitudesin this way [2]. Howeverwhenthe attempt is madea verycuriousfactemerges:thefinitenessofstringfield theorywouldfollow trivially from the nonlocal-ity of its interactions[31were it notfor the infinite numberof componentfields .

Sincethis last featureis a direct consequenceof the infinite numberof coordinates

* We have suppresseda necessaryqualifier in the interest of rhetorical trenchancy.There are, of

course,many Sorts of nonlocalvertices and not all of them serveto quenchloop divergences.Theultravioletfinitenessof stringfield theorycan beascribedto thefact that its verticesincludenonlocalconvergencefactorsof the form exp(— a’p2).

0550-3213/92/$05.OO© 1992 — Elsevier SciencePublishersB.V. All rights reserved

82 G. Kleppe,R.P. Woodard/ NonlocalYang—Mills

neededto describean extendedobject we seethat the much toutedpropertyof

“stringyness”is not only of no usein quenchingdivergences,it is actually the onlything which can threatenthe otherwisemanifestfinitenessof a suitably nonlocalfield theory.

To this observationmustbe addedthe fact that mostof the extrafields of stringtheoryserveno otherpurposethan to distort its phenomenologyin oftenuntestableand sometimesunacceptableways. So why do we not instead considernonlocaltheoriesbasedon a finite numberof fields? Until very recentlythe answerwould

have been that strings provide the only perturbatively consistentand nonlocaltheorieswhichcontain interactingvectorandtensorquanta.

In fact it was long believed impossible to constructperturbatively consistent,purely scalar theorieswith nonlocalinteractions,despiteyearsof heroiceffort [4].The problem in this casewas how to constructan operatorformalism throughcanonicalquantization.By exploiting the functional formalism directly Polchinskiexhibited a satisfactory model in 1984 [51,and a procedurefor deriving theassociatedoperatorformalism wasfinally given (in a different context)by Jaénetal. in 1986 [61.Thoughhighly significant,thesedevelopmentsstill do not suffice forthe consistentincorporationof vectorandtensorquantabecausenonlocalinterac-

tions disrupt the local gauge symmetrieswhich are used to reconcile Lorentzinvarianceandperturbativeunitarity.

A two-stageprocedurefor avoiding this problem was discoveredin early 1990[7]. In the first stagea gaugeinvariant action is nonlocalizedand then fortifiedwith an infinite seriesof carefully chosenhigherinteractions.Thesenew interac-tions endow the theory with a strangesort of nonlinear and nonlocal gaugeinvariance which does the crucial job of reconciling Lorentz invariance and

perturbativeunitarity at treeorder.The secondstageconsistsof finding a measurefactor to make the functional formalism invariant under the nonlocal gaugesymmetrywithout compromisingperturbativeunitarity “. If such a measurefactorcan be found then the resulting functional formalism defines a set of Greenfunctions which are ultraviolet finite and Poincarécovariant to all orders, andwhich reduceto give perturbativelyunitary scatteringamplitudes.For QED thefirst stagewasworked out explicitly and a proof was given that the secondstagecouldbe carriedoutaswell. For Yang—Mills andgravity only an indirect argumentwaspresentedthat the first stagecould be completedandno resultswerereportedregardingthe secondstage.

The purposeof this paper is to give a very simple procedurefor constructingnonlocalversionsof Yang—Mills, gravity or any otherlocal field theorywhichhas a

perturbativeformulation.This procedure,thework of sect.2, hastwo applications.First, it provides a regularization techniquewhich can be used in any theory.

* We thankA. Polychronakosfor showingus that a measurefactorcan alwaysbe foundwhich restores

invariance.The nontrivial thing is to findonewhoseinteractionsareentire functionsof the derivativeoperator.Otherwiseperturbativeunitary is threatened.

G. Kleppe,R.P. Woodard/ NonlocalYang—Mills 83

Althoughthe method is no more difficult to implementthandimensionalregular-ization it is alsono easier;its selling point is that if a suitablemeasurefactor existsthenthe nonlocalizedtheorypreservesphysicallyequivalentversionsof any andall

continuoussymmetriesof its parent.This fact is provenin appendixB. The secondapplicationfor nonlocalizationis the generationof new fundamentaltheories.If asuitablemeasurefactor existsthen the nonlocalizedtheoryappearsto meetall therequirementsfor a fundamentalmodel andthereis really no reasonto take the

local limit.In sect. 3 we implementthe secondstagefor Yang—Mills to the first nontrivial

order. The resultingFeynmanrules areused in sect. 4 to computethe one loopcorrectionto the gluon propagator.Sect.5 discussesthe possibility that nonlocal-ization producesconsistentfundamentaltheories.To do so the resultingmodelsmust avoid the nonperturbativeproblemswhich occur in mostnonlocal theories,including stringfield theory [8]. Theseproblemsderivelargely from the existenceof higher derivative solutions to the classical field equations;we will show inappendixA that nonlocalizationdoesnot engendersuch solutions. This promptsus to advancea radical proposalfor extricatingphysics from the apparentimpassebetweengravitationand quantummechanics.Our conclusionscomprisesect.6.

Before proceedingwe shouldcommenton the distinctionbetweennonlocaliza-

tion and two other invariantschemeswith which it is sometimesconfused.In thehigher derivativemethodoneattemptsto regulatethe theoryby addingtermswithhigher covariant derivatives and/or higher powers of the field strength. Thismakes the propagatorfall off faster for large euclideanmomenta but it alsoinducesinteractionswith morefactors of the momenta.The two effectsbalanceatone loop so that divergencesremain; it is only at higher loops that this methodregulates[9]. Higher derivativesmust thereforebe used in conjunctionwith someother techniqueto providecompleteregularization.One exampleof such a hybridschemehasbeengiven by Slavnov[10].

Nonlocalizationresults in propagatorswhich fall off exponentiallyin euclideanmomentumspace,as opposedto the power law fall off of the higher derivativemethod.A direct consequenceis that the higher derivativepropagatorcontainsnew poles, some of which necessarilyrepresentghostparticles. Nonlocalizationintroducesno new singularities.This is why nonlocalizationcan preserveperturba-tive unitarity before the unregulated limit is taken while the higher derivativemethod cannot. Although both methods induce new interactions those fromnonlocalizationareno worsebehavedthan the interactionsof the original theory;where they havemore derivativesthereare also compensatinginverse powersofthe regularizationscale. (This is why nonlocalizedloop amplitudesare consistentwith power counting in the local theory.)Another important differencebetweenthe two methodsis that higherderivativespreservelocal gaugeinvariancewhereasnonlocalizationpreservesonly its physical import, namely decoupling.The actualtransformationlaw is generallydistorted into a nonlocalandnonlinearrule.

84 G. Kleppe,R.P. Woodard/ NonlocalYang— Mills

The secondschemewith which weshouldcontrastnonlocalizationis continuumregularization [11]. In this methodspacetimeis endowedwith an extra dimensionand the fields are determinedclassically in terms of stochasticfields, of whichthereis onefor eachof the old fields. Onetakesexpectationvaluesof productsoffields on the higherdimensionalspacetimeby simply expandingthe fields in terms

of the stochasticvariablesandthensubstitutingthe appropriatestochasticcorrela-tion functions. The regularizationis imposedby nonlocalizing the correlationfunctions. RegulatedGreen functions in the lower dimensional theory are ob-tainedby taking the extracoordinateto infinity.

Although the role of the stochasticcorrelationfunctionsis somewhatsimilar tothe smearingoperatorof nonlocalizationthe two methodshavemany differences.Nonlocalization leavesthe tree amplitudesof any theory unchangedwhereascontinuumregularizationdoesnot. Nonlocalizationdistorts gaugeinvarianceintoa nonlocal andnonlinearsymmetrywhich only closeson-shellwhereascontinuumregularizationpreservesthe original transformationrule. Nonlocalizationproducesan action in the original dimension whereascontinuum regularization is notderivable from an action on this spacetime.Finally, if a suitable measurefactorexists thennonlocalizationgives a theorywhich is perturbativelyunitary beforetheunregulatedlimit is takenwhereasthis is not the casefor continuumregularization.We wish to emphasizethat our point in making this comparisonis that the twomethodsdiffer, not that one is superiorto the other. In fact it seemsto us thatwhichmethodis preferabledependsvery much on what onedesiresto do; andit isin any caseuseful to havedifferentperspectivesfrom which thesameproblemcanbe viewed.

2. General first stageconstruction

Wewill describeour methodin termsof a genericactionS which is a functionalof generalizedfields 4~.Thesefields may be commuting or anticommuting,ortheremay be someof eachtype.The index whichwe write as i may representanysortof label, includingLorentz tensoror spinor indices.The only restrictionis thatthe theorymust makesenseperturbatively,that is, the actioncanbe decomposedinto free and interactingparts:

S[~]=F[4]+I[4], (2.la)

F[~] = dDx ~ (2.lb)

where I[4] is analyticin 4 aroundthe point 4 = 0.

G. Kleppe,R.P. Woodard / NonlocalYang— Mills 85

Our constructionfor nonlocalizing S[4.] dependsupon the specificationof asmearing operator ~. For simplicity we shall take this to have the simpleexponentialform:

~~~exp[,~/2A2j, (2.2)

but the constructionwould work as well for any entire function of .~A2which is

unity for .9~= 0 andwhich falls fasterthanany powerof ST. We settledupon(2.2)only becauseit makes loop integralssimple. The regularizationparameteris A;taking A —s gives the unregulatedlimit.

Two useful quantitieswhich follow from the smearingoperatorare

(2.3)

~_ (~2 — 1)ST’ = fIT

2 (2.4)

Note that ~‘ is alwayswell definedand invertible,even for ST = 0. It is especiallysignificant that ~ still makessensewhen ST is the kinetic operator of a gaugetheory.

For each field 4, we now introducean auxiliary fields I~/ of the sametype.Toavoid confusionwith other auxiliaries which may be presentwe shall call the ui’s,“shadowfields”. The auxiliary actionwhich couples t~and ~ is

ST[~,~]~F[ç~]-A[~]+I[çb+~], (2.5a)

A[~] ~fdDx ~ (2.5b)

We provein appendixA that the classicalshadowfield equations:

&9’[& ~ií1=0 (2.6a)

~5~ (x)

uniquely determinethe shadowfields as functions,~ of the original fields 4~—up to termswhich vanishwith the ~, field equations.Multiplying by the invertibleoperator5~:

31[4+çti](2.6b)

gives an expressionfrom which we candevelopa seriesexpansionfor ~ In factthe quantity(~+ ~‘) is nothing morethanthe classicalfield for the action —A + Iin the presenceof the background~.


The nonlocalizedaction is obtainedby substitutingthis classicalsolution intothe auxiliary action:

~s~[~]~5”[& ~[~]]. (2.7)

Expanding5~in powersof 4’ gives the kinetic term F[4] plus an infinite seriesofinteraction terms, the first of which is just I[q5]. There is a simple graphicalrepresentationfor the higher interactionvertices: they are the connectedtrees

which follow from using the local interactionvertexbut replacingthe propagatorby — i~.Note that since ~5’ is an entire function of ST so too are thesehigherinteractions.This point is crucial in seeingthat the methodcan preserveperturba-tive unitarity.

We show in appendixB that if the infinitesimal transformation

= T1[4] (2.8a)

generatesa symmetry of S[~] then the following transformationgeneratesasymmetryof 5~[4]:

(2.8b)

Thereforenonlocalizationpreservesgenerallydistorted versions of any and allcontinuoussymmetries.Another importantresult derivedin appendixB is that the

shadowfield transformsas follows:

STk 55~[4~]= (1 —~‘

2),~T.[cb+~1/[~]] +K~,[t~+~/i[4]] -~--—[4 + ~/‘[4]]~’,~~uttjj ‘t~1

(2.9a)

~ 2j [4~]K’[4]~~.T’— . (2.9b)

Ii Ii

A direct consequenceof the secondterm in (2.9a) is that closure of the localsymmetry— by which we mean[si, ~2] = — only implies on-shell closure:

[t~i, 62j4i = t534)~+ + ~ , (2.lOa)

~T.’[cb] 6T1

2[4] ~T2[4] _____

K11~[4] — K.k[4] ~4k (2.lOb)

G. Kleppe,R.P. Woodard/ NonlocalYang— Mills 87

It follows that enforcing off-shell closure generally requires the inclusion ofgeneratorswhich vanishwith the field equations.In this sensethe nonlocalizedsymmetryalgebrais not only distortedbut also somewhatenlarged.

When a field of the local theory does not appear in the transformationfunctional T

1[4] there is no reason it need be endowedwith a shadow. Thenonlocalizationof thesefields canbeaccomplishedmerelyby smearingthe kineticterm.An exampleis providedby the vectorpotentialof QED. The greatadvantageof this shortcut in QED is that it allows one to solve explicitly for the shadowelectronfield. This is why a closedform for the actionof nonlocal QED could begiven [7]. Onedisadvantageof the shortcutis that the associatednonlocalizedfieldequationspossessnonperturbative,higherderivativesolutionsof the samesort thatwreck string theory [81.

Quantizationis accomplishedthroughthe definition

(T *(O[~])>~~f[D~5],i[41I(gaugefixing) °[~Iexp(i5~[4]). (2.11)

The object on the left is the nonlocally regulatedvacuumexpectationvalueof theT *~orderedproductof an arbitraryoperatorO[~]. It hasa subscript ~ to signifythe dependenceof nonlocalizationupon the choice of smearingoperator.The twoundefinedquantitieson the right are the measurefactor, ~ and the gauge

fixing corrections.Both of theseare neededto maintain perturbativeunitarity ingauge theories; for scalarswith nonderivative interactions there is no gaugesymmetryto fix andthe measurefactor can be set to unity ~.

The issueof perturbativeunitarity wasdiscussedextensivelyin the treatmentofQED [71.The basic idea is that the Cutkosky rules apply to nonlocal theorieswhosepropagatorshavepolesonly whereST = 0 andwhoseinteractionverticesareentire functions of ST. Since S[4d meets both requirements this guaranteesunitarity on a largespaceof stateswhich includesunphysicalpolarizations.If thequantumtheory is gaugeinvariant then theseunphysical polarizationsdecoupleand it is also unitaryon the physical spaceof states.We will explainin sect.3 thatit is simplest to gaugefix the local actionand then nonlocalizethe resultingBRS

theory. Since our proceduremakesno distinction betweeninvariant actionsandBRSgauge-fixedonesit follows that a nonlocalizedversionof BRS invarianceisinduced.

If the local action is invariantundera continuoussymmetry ~ we haveseenthatthe nonlocalizedaction will be invariant under the symmetry ~ which is given ineq. (2.8b).The quantumtheorywill possessthis symmetryif andonly if a suitable

measurefactor can be found such that 6([Dq~]~4~])= 0. This condition can be

* Experiencewith the nonlinearsigmamodelleadsus to expectthat scalarswith derivativeinteractions

requirea nontrivial measurefactor in order to maintain perturbativeunitarity.

88 G. Kleppe,R.P. Woodard/ Nonlocal Yang—Mills

re-expressedas follows usingsimplematrix manipulationsandrelation(B.12b):

4~.= _Tr{~-~} (2.12a)

= _Tr{~~[~ + ~[~]]Kk,[~ + ~[~]}~1}. (2.12b)

In additionthe interactionvertices of themeasurefactor mustbe entirefunctionsof the operator.Y~,andof coursethey must not spoil the regularization.Sucha

choice may not exist. In this case one of the local symmetries is potentiallyanomalous;one must renormalizeand take the unregulatedlimit to be sure. Anexplicit exampleof the appearanceof such an anomaly in nonlocal regularizationhasbeendescribedby Hand in the chiralSchwingermodel [12].

A curiosity of definition (2.11) is that the field functionson its right-handside

arehattedwhile thefield operatorson the left arenot.The reasonfor this canbeseenby taking O[4] to be a simpleproduct:

(2.13)

andthenacting .S~on bothsidesof (2.11). The zerothorder result on the left isjust the various commutatorterms; the right-handside gives a sumoverpairingsofpropagators,plus interactionterms.Correspondenceat zerothorder impliesthat afactor of ~‘ resideson eachfield in the functional integrand.

We emphasizethat it is importantto eliminatethe shadowfields at the classicallevel. Functionallyintegratingoverthem would engenderdivergencesfrom shadowloops; nor doesperturbativeunitarity require us to quantizethesefields becausetheir propagatorscontain no poles.Nevertheless,the Feynmanrules of S[4] arevery cumbersomebecauseof all the inducedinteractionswhile the Feynmanrulesof ST[4, i/i] are essentially as simple as those of the local theory. We havethereforechosento usethe simpleFeynmanrulesof .ST[4, ~i] andto enforcetheconditionthat i~i~obey its classicalfield equation(~/‘~= tfr~[4]) by merely omittingclosedloopscomposedsolely of shadowlines.

The resulting nonlocalizedFeynmanrules are a trivial extensionof the localones.Exceptfor the measurefactor thevertices areunchanged,but everyleg cannow connecteitherto a smearedpropagator*:

ST+IE = I exp(~~) (2.14a)

* A digressionis necessaryto commenton thevarious factorsof i which appearin what thereaderwas

assuredwould be a euclideanformulation of perturbationtheory.From theearliestwork on nonlocalregularization[71it hasbeenconvenientto setperturbationtheoryup in Minkowski spaceanddefineloop integrationsby formal Wick rotation.Thisconventionwill be followed here as well. We stressthat it is only a trick to facilitate the analytic continuation.If rigor is desiredthen thewhole thingcan be done in euclideanspacewhere~ is a nonpositiveoperator.


aa “\J\j’\/\/\f\J\/\, b)3 ~~‘I ~ exp (~P2)

aa~/\f\J~\f\J\f\~ b)3 ~11 {1_exp(_~1~~)}

(a)Fig. Ia. Thesmearedand shadowgluon propagatorsof nonlocalYang—Mills.

or to a shadowpropagator:

i(1 _,~~2) i dT ST

ST = —~J—~ exP(r~). (2.14b)

The two sortsof propagatorsare representedgraphically for Yang—Mills theory infigs. la andlb. Weplacea baracrossthe shadowline becauseits propagatorlacks

a pole and so carries no quanta. For this reason all external lines must beunbarred.(This morethancompensatesthefactorsof ~ which arisebecauseweuseO[~] ratherthan O[4] on the right-handside of eq.(2.11).)Verticesfrom themeasurefactor are representedgraphicallywith a circled “M”, as in figure 2.Measureverticescanbe connectedonly to unbarredlines.The symmetryfactorofany diagram is computedwithout distinguishingbetweenbarred and unbarredlines.

A number of points deservemention. First, tree order Greenfunctions are

unchangedexceptfor external line factors which are unity on shell. This followsbecauseevery internal line of a tree graphcan be either barred or unbarred.Henceit is only the sumof (2.14a)and(2.14b)which enters,andthis sumjust givesthe local propagator.Second,the fact that on-shelltreeamplitudesareunchangedby nonlocalizationmeansthat S[4] preservesany andall symmetriesof S[4d onshell. A direct proofof this fact is given for continuoussymmetriesin appendixB.Third, all loops contain at least one of the smearedpropagators(2.14a)and are

a ~--~———--~-—— b H~ exp(~-~)

a -~--.-—-- b ‘~ {i_exp(-~-)}

(b)Fig. lb. The smearedandshadowghostpropagatorsof nonlocalYang—Mills.


(a)~ ig2~~(p)

Fig. 2a.The2-pointmeasurevertexof nonlocalYang—Mills.

aap1~~~ igI~ (p~,p~,p~)

/ C~D3

fi

aap1~ iglabc (Pi,P2,P3)

(b) bp~Fig. 2b. The3-point verticesof local Yang—Mills.

thereforeconvergentin the ultraviolet. Finally,whenapplying the cutting rulesoneallows only unbarred lines to be cut becausethey alone contain poles. Thisautomatically implies unitarity on the extendedspaceof stateswhich includesunphysicalpolarizations.

Whenthe measurefactor is unity nonlocal regularizationis no moredifficult tousethan dimensionalregularization.We haveshownthis by carrying out the fullone andtwo loop renormalizationprogramfor scalar4,4 theory in four spacetimedimensionsand for 4,3 theory in six dimensions [131.We refer the interested

aap1-~~ ~j-~JdSp4

1g21 ~ (P

1,P2,P3’~)

b)3p2 cyp3(c)

Fig. 2c. The4-point vertexof local Yang—Mills.


readerto this work for a very detailed study of the method in operation. Inparticularit shouldbe notedthat renormalizationproceedsin nonlocal regulariza-tion as in any other method: the bare parametersare determinedas functionsofthe renormalizedonesand the scaleA such that the limit A —* ~ is finite for allnoncoincidentGreenfunctions.EventhoughS[4,] possessesnonlocalinteractionsits bareparametersare still thoseof a local theory. Oncea smearingoperatorhasbeenchosenthe nonlocalinteractionsare completelydeterminedby the local ones.We remark also that the method correctly handlesoverlappingdivergencesandthat powercountingworks asit would if the scaleA hadbeena momentumcutoff.

3. The secondstagefor Yang—Mills

When vectorand/or tensorquantaare allowedto interactsome form of gaugeinvariancemustbe presentin orderto reconcilePoincaréinvarianceandperturba-tive unitarity. For this invarianceto continue operatingin loops the functionalformalism must respect it. This generally requires that the functional measurefactor should dependnontrivially upon the fields. The measurefactor must alsomeetthe requirementsfor being an acceptableinteractionin theregulatedtheory.This meansthat it musthavemanifestPoincaréinvariance,that it caninvolve onlyentirefunctionsof thekinetic operator,andthat diagramsinvolving it mustbewelldampedin euclideanmomentumspace.If no such measurefactor exists then theregulatedtheory has an anomaly “. For the appearanceof such an anomaly innonlocalregularizationwe referthe readerto Hand’sstudyof thechiral Schwingermodel [12].

We shall take pure Yang—Mills for the unregulatedtheory. Its lagrangianandfield strengthtensorare

ST= ~ (3.la)

Fap,,~Aap~~ (3.lb)

where g is the coupling constantand the fab,. are the structureconstants.It isinvariantunderthe familiar transformation:

~5üAap~=~ (3.2)

Note that the local symmetryalgebraclosesfor all field configurations:

[c~o,, 5O2]AQ~.= OaIL ~ (3.3a)

~a ~gf~b~°lh°2~. (3.3b)

* The anomalymay, however,vanishin the unregulatedlimit.

92 G. Kleppe,R.P.Woodard/ NonlocalYang— Mills

To nonlocalizewe identify the kinetic operator:

— a’~aP) (3.4a)

andthe interaction:

I[A] f dDx(gf~~~AQAb~AC~— ~ (3.4b)

Comparisonwith (2.5) and(2.6) gives thefollowing expansionfor the shadowfield:

~I[A +B]Baa[A1=t9.~~j

3

~ +ACYA d~’~— 2A~~A/’YI+ O(g2A3). (3.5)

The resultingnonlocalizedaction follows immediatelyfrom (2.5) and (2.7):

~[A] = ~fdDX{AaaSTa?A*b~_Baa[A16~5~_I~Bb$[A1}+I[A +B[A]]. (3.6)

We seefrom (2.8) that the nonlocalizedgaugesymmetry is

6~0A~T~i~(0a”~+gf~~d(A~~+B~O[A])Od}

= ~a” + ~J~gfb~d(A~4+BC4[A])6d. (3.7)

As was the case for QED [7], nonlocalizing Yang—Mills results in a gaugetransformationwhich is neither linear nor local. Another point in common withQED is the failure of the symmetrygeneratorsto closeundercommutation,exceptfor field configurationswhich obey the equationsof motion. Comparisonwithrelations(2.9) and(2.10) shows

* * * 5S~[A][6~, ~ Aa” = 8(~)Aa+ ~‘ ab~Mbcf3y[A + B[A]]~’

2~ ~Ad8 (3.8a)

~ — O2dK~f~[A]OIf}, (3.8b)

—t 521[A]

(K~) [A]=r~— (38c)aa bj3

This meansthat the spaceof generators(3.7) is only a subsetof a true algebrawhich includesgeneratorsthat vanishwith the field equations.


The mostobvious applicationof the procedurelaid out in sect. 2 would be tofunctionally quantizeas follows:

(T*(O[A]))~=f[DA]~s[A] (gaugefixing) O[A] exp(i5~[A]). (3.9)

The trivial problem with this approachis that the measurefactor definedby eq.(2.12)would be singularon accountof the unsuppressed,longitudinal componentsof transformationlaw (3.7). This could be circumventedby the simpleexpedientofredefiningwhatwe calledthe “gaugeparameter”so as to absorbthe unsuppressedtermsinto the zerothorder piece. *

The harderproblemis how to fix the gauge.It is here that the nonclosureof Shasits only significant impact. We might parallel the usualderivation of Faddeevand Popovby introducingunity in the form:

So~A°b1 = f [D19]S[3 A~ fa] det , (3.10)

where by A°we mean the transformedvector potential. Onewould then changevariables A —~A’=A° and exploit the gaugeinvarianceof S[A] and [DA]i..r[A],

and the presumedinvariance of the determinant, to factor out the functionalintegrationover 0. The final stepwould be to functionally averageoverfa with thedesiredweighting factor, usually a simple gaussian.Unfortunately the Faddeev—Popovdeterminantis not invariantowing to thefailure of two 5-transformationstogive anotherS-transformation.Hencethe integrationover 0 doesnot quite factorout and the naivederivation fails.

This problemis a familiar onefrom supergravityandthe answeris well known

[14]. The physically essentialsymmetryof a gauge-fixedtheory is BRS invariance.When the algebraof gaugetransformationsclosesfor all field configurationstheghoststructureof the BRSaction comesfrom exponentiatingthe Faddeev—Popovdeterminant, and the BRS transformationof the gauge field is just a gaugetransformationwith the parameterequalto a constantGrassmannnumbertimestheghostfield. Foran algebrawhich fails to closeoff-shell oneneedsto introducehigher ghost terms into both the action and the BRS transformation.There willalso generallyneedto be a correctionto the measurefactor. One could probablyderive the necessarychangesby formally integratingthe Faddeev—Popovdetermi-nant with respectto 0, howevera much simpler approachis to nonlocalizetheBRS lagrangian directly.

In Feynmangaugethe local BRS lagrangianis

~BRS — ~a *Aa”~ — ~a’~7a,~* +

+ ~ — ~ (3.11)


It is invariant underthe following global symmetrytransformation:

SAaa = (‘Tla,a ~ (3.12a)

= — ~ (3.12b)

= ~Aa~*S~, (3.12c)

where ~ is a constantanticommutingC-number.

To nonlocalizethis actionwe identify the gluon andghostkinetic operators:

= ~at~71’~~, (3.13a)

‘~ab5ab3. (3.13b)

Since the indices and derivativesseparatewe havefound it convenientto use asmearingoperatorwith no indices:

(3.14a)

- ~2_1

(3.14b)

The BRS interactionis

I[A, ~, ~1=gfabcf ~

— ~ (3.15)

Comparisonwith (2.5) and (2.6) gives the following expansionsfor the shadowfields:

_S1 —

B0~[A,~, ~1 ~SAaa[A +B, ~+4,, ~

= ~ +AbOAC1~’a— 2AbOA C — ~1b’~1c] + 0(g2),

(3.16a)

— _SI —

4,a[A, ~, ~]= _~~[A +B, ~+4,; ~ +4,]~~7a

= ~ + 0(g2), (3.16b)


_SI —

4,~[A,~, 71]=9~[A +B, ~+4,, ~ +4,]~

= ~ + 0(g2). (3.16c)

To economizeon spacewe will henceforthsuppressthe functional argumentliston theshadowfields. From expressions(2.5) and(2.7) we seethat the nonlocalizedBRS action is

~[A, ~, ~]= f dDx{ — ~4aa,s~~a”~ — ~Baa(f~_IBaa— ~a’~a,a

(3.17)

The nolocalizedBRS symmetry which follows from applying (2.8) to (3.12) iseasyto give in termsof the shadowfields:

~Aaa ~~~2((71a,a + 4’a,a) gfa~e(A~~+Bca)(77c + 4,~)}S~, (3.18a)

~~1a= — ~gfahc~’2(71b + ~b)(71~ + 4,~)S~, (3.18b)

~tia ~ (3.18c)

Becausethe shadowfields — Baa[A, ~1’711’ ‘4’aH, ~, ~1]and 4ia[A, ~, 77] — dependupon the ghostsnot every occurrenceof 71~in (3.18a) derivesfrom the replace-ment O~—~ ~17aS~Tin (3.7). These “extra” terms appearas well in the BRS

transformationsof supergravitymodelswhosealgebrasonly closeon-shell [141.The transformationlaw (3.18) is inconvenientbecausethe variation of the

antighostno longer producesjust a longitudinal gluon. This property plays aspecial role in using BRS invarianceto prove decoupling.We thereforefold in adynamicallyirrelevantsymmetryof the classdescribedin eqs. (B.16) and(B.17):

S2OAaa= (1 ~ (3.19a)

~Ol7a=0, (3.19b)

= —(1 ~ (3.l9c)

Combining the two gives a physically equivalentbut moreconvenienttransforma-tion:

~1 Aaa={?7a,a_gfabcT2(Aba+Bc~)(77c+4,c))S~, (3.20a)

71a —~gfabc~’2(71b+4i~)(71ç+4ic)S~, (3.20b)

~ ~a = ~Aa°~p~S~. (3.20c)


That sucha form canbe reachedalsofollows from the generalargumentof de Witandvan Holten [14].

Since the formalism of sect. 2 applies to gaugefixed theoriesas well as toinvariant oneswe can read off the rule for functional quantizationdirectly fromexpression(2.11):

KT*(O[A, ~, 77]))~c

~,71]O[A, ~, ~ exp(i~[A, ~, n]). (3.21)

Note, however, that is shouldbe the transformation51, not ~, which goes intoexpression(2.12) to partially determinethe measurefactor. Of courseone canfreely modify 1s[A, ~, 77] by termsthat are invariantunder~l, butwith a naturalsort of minimality conditionthe answeris

ln(~[A, 77, 77]) = _~g2fCdfbCdf dDXAa~J(A~+ 0(g3), (3.22a)

1 ~ A’~2 TI dT exp———

2~)1r~~2fo (r+1)~2 T+l A2

(2x~— — (D— 1) +2(D— 1)— ~. (3.22b)

~r+l T+l)

Thereis no proof that the requisitehigherordertermsexist, althoughthis is whatwe expectbecausethe local theory is free of anomalies.

The presumedexistenceof a suitable measurefactor implies Slavnov—Taylor

identities in the usual way. Consider, for example,the following choice for theoperatorO[A, ~, 771 of (3.21):

O[A, ~, ~ ~a(~).~4b”,,(y). (3.23)

Of course the functional integral vanishesby ghostnumbercounting,but a BRStransformation gives a nontrivial relationbetweenGreenfunctions:

* Ss= —A (x)A~(y)S~+~ij~(x)~ ~. ~S~. (3.24)

17bk y~

Upon functionally integratingby parts we infer the transversalityof the vacuumpolarizationat oneloop ~. This resultwill be explicitly verified in sect.4.

* Although we expecttransversalityto persistto all orderswe haveno proof of this owing to lack of

knowledgeaboutthehigherordertermsin themeasurefactor. In particularwe havenot beenabletodecidewhetheror not themeasuredependsupon theghosts.


4. The gluon self-energyat one loop

The Feynmanrules canbe read off from the functional formalism (3.21) usingrelations (3.17) and (3.22). The barred and unbarredgluon propagatorsarerepresentedgraphically in fig. la andhavethe following forms:

af3 2 dT 2

~ ex~(_~) = ~ ~ ex~(_r~)~ (4.la)

~ d~p2jE {l_exP(_~)} = ~5ah71~f ~ ex~(_r~). (4.lb)

The barred and unbarredghost propagatorsare shown in fig. lb and can beexpressedas follows:

2 d 2

~ exp(_~) = ~_Sabf~~~ exp(_r~)~ (4.2a)

1dT p2

P2~E{P(A2)} = lSabf ~ ex~(_r~). (4.2b)

The obviousadvantageof this gaugeis that the tensorandgaugeindicesdecouplefrom the nonlocalfactors.The 2-pointmeasurevertexis representedin fig. 2a and

hasthe form:

f f af3

Mao! \ = — J acdi bcd71ab~P1

A’~2 T p2 D—2>KfdrD/

2exP(_~~)[(D_l)_2+1]. (4.3)

The local3-pointvertices areshownin fig. 2b andcanbe expressedas follows:

I~(p1, P2, p3) _ifa6c{71a0(p2 Pi) + 710Y(p p2) + 77Ya(p1—p3)°}

(4.4a)

‘~h~(PI,P2 ~3) =*lfabCP3. (4.4b)


The local 4-pointvertex is depictedin fig. 2c and hasthe form:

I~0~(p1,P2, P3, p4) — {fabefcde(71~’7i

0~— 7770715a)

+faCefdbe(71a~77YO — 778777a0)

+fadefbCe(77a0?7S7 —

711Th777a)} (4.5)

The only thing missing is the highermeasurevertices.Sincethe N-point measurevertexcomesin at order g N it is clear that only for N = 2 can it contribute to the(orderg

2) one loop vacuumpolarization.The one loop vacuumpolarization can be naturally divided into four parts.

Thesecorrespondto figs. 3—6 andwe shall term them A, B, C and D respectively:

~ (4.6)

The threebarringvariationsdepictedin fig. 3 comefrom the 3-gluonvertex.Theycanbe summedup to give the following:

d’3k k2

= 2J~ d2rf (2~)~~’ —k, _~){_iSce71YP ex~(_ri~)}xigI,~7(—p,k, ~){_iSdf715U exP(_T

2~)} (4.7a)

ig2f~Cdfb~d d2T A’~4 T

1T2 p2 ~ A2

=

2~D/2 (Tl+r2)0/2 exp — T1+T2A

2

— T1T2 [p271a0 + (2D — 3)pap0] + [~p271a0 + ~(D — 6)PaPO]}

+

(4.7b)

~+ ~+ ~

Fig. 3. ThecomponentA~(p)of theone loopvacuumpolarization.

G. Kieppe,R.P. Woodard/ NonlocalYang— Mills 99

r\J\/\ + J\J\J r\x~J~’+ ~/V~J

Fig. 4. ThecomponentB~(p)of theoneloop vacuumpolarization.

The symbol R~in this expressiondenotesthe standard2-parameterregion ofintegrationat one loop, namelythe first quadrantminus the square0 � T

1, ~ < ~•

A very similar contribution comes from the three ghost loops of fig. 4:

dDk k2

iB~(p)=f d2Tf ~igI~d(p, —k, _~){_iSceexP(_TI~)}

XigI~f(—p,k, ~){_iSdf exP(_T2~)} (4.8a)

,~2 AD4— ig Jacdibcd ç T

— 2D~D/2 JR~(TI+T2)0/2

T1T2 p

2 A2 T

1T2Xexp — —~. _.~ 77af3~ papO . (4.8b)

+ r2 A T1 + T2 T1 + ~2

The diagramof fig. 5 comes from the 4-gluon vertex:

ic~(p)=fd~ig2I~(p, -p, -k, k)(~~) ex~(_~) (4.9a)

= i~f~fbcdo(D_i)D2 (4.9b)

This contribution would vanish in dimensional regularization; here it plays anessential role in canceling the quadraticdivergence.The samecommentspertain

Fig. 5. ThecomponentC~(p) of theone loop vacuumpolarization.

100 G. Kleppe, R.P. Woodard/ NonlocalYang— Mills

Fig. 6. ThecomponentD,~%3(p) of theoneloop vacuumpolarization.

as well to the measurefactor contributionwhich comprisesfig. 6:

i~2facdfbcdf1 drAD_2iD~(p)= —

2D~D/2 ~ (r+ 1)D/2

I TP2){ D—2\

Xexpl—--————-— (D—l)—2 ~‘. (4.10)T+lJ

To checkthat the vacuumpolarization is transversefirst add the longitudinalparts of A and B. Now reduce the parameter integral to which the sum isproportional:

d2r / T

1T2 p

2 \fR~(~+ T

2)D/2 exp~ T

1 + T2

1 T1T2 p2 1(D1)1’\

x{(~D2) -(2D-3)A2J

d2T= _(D_1)f

D/2Rf (T

1 + T2)

T1T2 P2 \ f — T

1T2 P2 1 p2 ‘~

xexP(_ —I, +T

1 + T2 A2 ) ~ T~+ r

2 (r1 + T2) A2 — 2 A2J

d2T- (D -2) f

D/2R~(T

1 + T2)

T1T2 p

2\ (1 (D—2) T1T2

xexP(_ —Ii + (4.lla)Ti+T2A

2J~2T1+T2 (r1+T2)

2A~J

l~ 0 0 —D/2 / T

1T2 p

2 \ ~=_(D_1)f d2T—~——+——~(T’+T2) expi— H

R~ 2 Or1 0r21 ~ + r2 A

2) J

+Jfldxfdy+fdxf dy\ D—2 3 1 (_ ~ p2\~D/2+1 ~~y1_D/2 exp0 1 1 1/x J (x+1) Oy ~ x+l

(4.llb)


D—1 1 ‘r p2 D—2

=2(D2) +f dT(T+1)D/2 exP(_~~){(D_ 1) _2( ~+ 1 )}‘

(4.llc)

These final terms serve to cancel the longitudinal contributionsfrom C and D,

proving transversality.The integral identitywe havejust establishedcanbeusedas well to simplify the

transverse part. The result is

~ P) = 2DD/2facdfhcd(p271a0 _papP )H(p2), (4.12a)

H(p2) fR~d2r +r2)~

2

T1T2 p

2 ( T

1T2

Xexp — —i ~2(D—2) 2—~(D—6)T1+T2A (r1+r2)

= 2f1/2 dx 1(2 —D/2, xp2/A2)[x(l _x)p2]~22

>< {2(D — 2)x(l —x) — ~(D — 6)}, (4.12b)

where the incomplete gamma function is

F(n, z) f dt t’~ et. (4.13)

The dimensionallyregulatedresult is obtainedby the replacement

F(2—D/2, xp2/A2)—*F(2—D/2). (4.14)

This is an important checkon accuracy.The same relation occurred for the oneloop vacuumpolarizationof nonlocal QED [7]; andthe generalrule relating thetwo methodshas beeninferred for any diagramat any order in nonlocal scalarfield theory [13]. We conjecturethe samerule appliesfor nonlocalYang—Mills aswell. If true, this defines the general measure factor and the formalism iscomplete.

Note that evenif our conjectureis true nonlocalregularizationwould still differfrom the dimensionalscheme.The latter regulatesby deformingthe argumentsofdivergent gamma functions while the former works by trading these gammafunctions in for incomplete gammafunctions evaluated at predictable functions of


the Feynmanparameters.Oneobvious distinction is that the replacement(4.14)discards gauge invariant divergences for D> 4. Dimensional regularization issensitive only to logarithmic divergences whereas nonlocal regularization seeseverything.

5. Acceptable nonlocalization as noncanonicalquantization

The first stage procedure of sect. 2 can be used to regulate any theory. The

result is basically a systematized version of proper time regularization.At this levelnonlocalization functions like any other regulator: the original theory is changed tomake it finite, but at the cost of introducing a physically unacceptable propertywhich persists as long as the regulator is on. In order to obtain a physicallyacceptable theory one must take the unregulated limit after the appropriaterenormalizations have been performed.

For the case of nonlocal regularizationtheproblemis that perturbativeunitarityis violated by loops which contain vector and tensor quanta. The first stagecontrives to enforce tree order decoupling through the agency of a peculiarnonlinear gauge invariance. Since the naive measure does not preserve thissymmetry decoupling must fail in loop amplitudes. If we are only interested indeveloping a noninvariant regulator this is as irrelevant as the ghost states of thePauli—Villars method; all will be set right in the unregulated limit. However thesituation changes radically if the second stage can be carried out — andwe havebeen some evidence that it can be for Yang—Mills. In this case the regulatedtheory is not only finite but also Poincaré invariant andperturbativelyunitary. Ithas all the attributes of a fundamental theory; why then must one take the

unregulatedlimit?One answer is that at fixed loop order our nonlocal amplitudesviolate the

bounds implied by partial wave unitarity — though of course the trees are those of

the local theory. This is perhaps not as damning as it seems because a similarbreakdown occurs even at tree order in string theory [151.The problemtherehasnot beenmuch studiedbut the feeling is that it can be resolved by an ingeniousresummationof perturbationtheory [16]. Pendingan argumentagainstthe exis-tenceof the samemechanismin nonlocalfield theory it is obviouslyprematuretofret overmuch about the violation of nonperturbative bounds by finite orderperturbationtheoryat extremelyhighenergies.

The sameconsiderationsapply to theperturbative,off-shellviolationsof causal-ity which appearas quantumeffects in nonlocal field theory. 0ff-shell acausalityoccurs even at tree order in string field theory [8]. The effect hasreceivedalmostno attentionbecauseit doesnot give acausalpoles in the perturbative S-matrix,becauseit is extremelysmall (in nonlocal field theory the strengthof a signalpropagated an invariant interval 12> 0 outside the lightcone would be suppressed


by factors of exp[—12A2]), and becauseessentiallymystical considerationshaveencouragedthe belief that very small length scalescannot be probed in stringtheory. In the rarified atmosphereof critical thinking which now prevails onemight advance any of these points with equal smugness as a reasonfor ignoringtheacausality of nonlocal field theory. Of greater import to us is the fact that it is onlythrough quantum effects that the acausality of nonlocal field theory manifestsitself. If this acausality could be viewed as the consequence of a noncanonicalquantization procedure — andwe will shortly argue that it can be — then it mightbe both very small and logically consistent. We note, in passing, the observation ofref. [8] that Planck scale acausalitymight actually be desirable in providing anexplanationof the horizonproblemof cosmology[171.

A more serious problem would seemto be the nonpertubativeinstabilitieswhich tend to occur in any fundamentallynonlocal theory. The origin of theseinstabilities is explainedat greatlength in ref. [8]. Basically they are due to theextra canonicaldegreesof freedom associatedwith higher time derivatives.lt issimple to show that if the lagrangiancontainsup to N time derivatives then theassociated hamiltonian is linear in N — 1 of the correspondingcanonicalvariables.A fully nonlocal lagrangianwhoseinteractionsare entire functionsof the deriva-tive operatorcanbe viewed as an Nth derivativelagrangianin the limit that N

becomesinfinite. Such a nonlocal theorywill then inherit the instabilities of itshigher derivative limit sequencesprovided the time dependenceof the extrasolutions does not becomearbitrarily choppy in the limit. The condition for theextrasolutionsto havesmoothlimits is that no invertible field redefinition shouldexistwhich takesthe nonlocal field equationsinto local ones.That string theoryobeysthis condition canbe inferredfrom its treeorderS-matrix; that the nonlocal

actions produced by the construction of section 2 do not obey this conditionfollows from the fact that their tree amplitudes agree with those of their localancestors.

In fact we show in appendixA that the solutionsof the nonlocalfield equations

are in one-to-onecorrespondencewith thoseof theoriginal theory. The relation isvery simple:

4,nonloc ~24,I.OC (5.1)

Comparisonof the actionsshows that for solutionswe also have 5[4~]=

Westressthat theseare nonperturbative,albeit classical, results. There are no extraclassical solutions. With no extra solutions there is no necessaryproblem witheither stability — although the theory might still be unstable— or with a finiteinitial valueformalism.

Sincethe on-shellamplitudesof our nonlocalizedactiondiffer from thoseof itslocal parent only at loop order it must be that the nonlocal field theorycan beviewed as a highly noncanonical quantization of fields which obey the local

104 G. Kieppe,R.P. Woodard/ NonlocalYang—Mills

equationsof motion.The logical possibility of alternatequantizationschemeshaslong beenrealizedbut they haveattractedlittle attentionbecausethey tendnot toresult in unitary time evolution. Westressthat if thesecondstagegoes through thenour constructiondoesmaintainunitarity, at leastperturbatively.

We close with a discussionof another reason for the lack of interest innoncanonicalquantization,namelythe spectacularphenomenologicalsuccessob-tained by applying the canonicalmethod to familar gaugetheories.The leftoverinteractionin all this is gravity. Indeed,one might regardthe last thirty yearsofeffort in quantum gravity as a sort of proof by exhaustionof a fundamentalincompatibility between general relativity and quantum mechanics. It is usuallyassumed that the former must be changed to accommodate the latter; we wish hereto proposenonlocalizationas a specific mechanismfor achievingthe converse.

This is far too radical a proposal to be fully realized. For one thing it isunaestheticallyarbitrary; onecannonlocalizewith any smearingoperatorprovidedit is bothanalyticandconvergent.Another problemis thebackgrounddependencewhich seemsto infiltrate the constructionof sect. 2 from the distinction it makes

betweenthe free andinteractingpartsof the action. Finally, thereis the issue ofunification. Even if quantum gravity should owe its ultraviolet finiteness to nonlo-cality we still expect it to form part of a larger structure that includes the otherforces in some way more fundamental than as optional matter couplings.

Theseare all valid criticismsof the proposal— and we have thought it good toraise them in advance of anyone else. Nevertheless we have been unable to dismissthe notion that certain forms of nonlocalizationmight, when applied to specialtheories, yield a nonperturbatively acceptable theory. Nor have we been able to ridourselvesof the feeling that noncanonicalquantization,as outlined here,mightplay a decisive role in the quantum theory of gravitation.

6. Conclusions

The major result of this paper is the procedureof sect. 2 for constructingaregulated,nonlocal action S[4,] from any local action S[4,] which canbe formu-lated perturbatively. The nonlocalized theory can either be thought of as aregularizationof the local theoryor as a candidatefundamentalmodel in its ownright. Though S[4,] contains an infinite tower of induced interactions these can becompletely subsumed into an auxiliary field line which couples using only the localinteraction. Loop calculations in this formulation of the theory are no moredifficult than with dimensionalregularizationbut the greatthing aboutnonlocal-ization is its potential for preservingsymmetries.We prove in appendixB thatS[4,] preservessuitablynonlocalizedversionsof any andall continuoussymmetriesof S[4,].

Whetheror not thesesymmetriesare preservedin path integral quantization


dependsupon the existenceof an otherwisesuitablemeasurefactor i44,] havingthe property that [D4,]~[4,] is invariant. One mustnot expectthat all symmetriescan be preservedin all theories;for example,gaugeinvarianceis broken in thechiral Schwingermodel [12]. However, somesymmetriesarecertainlypreservedinsome theories;for example,the requiredmeasurefactor has beenshownto existfor nonlocalQED [7].No conclusionhasbeenreachedyet for nonlocalYang—Mills,although we did computeboth the invariant measureand the BRS measureatorder g2.

Quantizationalso requiresthat the gaugebe fixed. A peculiarityof the nonlo-

calizedgaugesymmetryis that its generatorstypically fail to closeexcepton-shell.This means that the Faddeev—Popovdeterminantis not gaugeinvariant, thuscomplicatingtheusualfunctional gaugefixing procedure.The simplestsolution tothis problem is to nonlocalize the BRS theory directly and then functionallyquantizeusinga BRS invariantmeasure.Whenthis is done onefinds bothhigherghostinteractionsand higherghostcontaminationin the nonlocalizedBRStrans-

formation.Similar effectshavebeennotedwith formulationsof local supersymme-try which fail to close off-shell [141.

Our partial determinationof the measurefactor was sufficient to evaluatethevacuumpolarization at one loop, which we did in sect. 4. As was the casewithnonlocal QED [7], the result bears a striking resemblanceto that obtainedfromdimensionalregularization.One merelyreplacesthe divergentgammafunction ofthe latter method with an incomplete gammafunction whoseargumentdependsupon the Schwingerparametersand ratios of momentuminvariantsto the regular-ization parameterA2. For the vacuumpolarization the necessaryreplacementisgiven in eq. (4.14). In another work we have presented the general relation for any

amplitudesat any order in scalar field theory [13]. If this samerule appliesas wellto nonlocalYang—Mills then the measurefactor is determined.

The constructionof sect.2 canbe appliedto supersymmetricgaugetheoriesand

to supergravity.Since it preservesthe on-shell tree amplitudesit will necessarilypreserveglobal supersymmetry(untouchedsince it is linearly realized)and somesort of nonlocal and nonlinear generalizationof gauge invariance and localsupersymmetry.If a supersymmetricmeasurefactorcan be found aswell thenwewill haveproducedthe long-soughtgaugeandsupersymmetricinvariant regulator.

Perhapsthe moststimulating featureof our methodis that it leavesunchanged

the treeamplitudesof the local theory. A consequenceis thatoneoughtto be ableto view nonlocalregularizationasa noncanonicalquantizationof fields obeyingthelocal equationsof motion. The connectioncan be made explicit and we havesketchedit in sect.5. Thoughthe idea is obviously incompleteat this point, we feelit mayplay a role in reconcilinggravitationand quantummechanics.

We have benefittedfrom conversationswith M. Chu, A. Polychronakos,P.Ramond and D. Zoller. This work was partially supportedby the Institute for

106 G. Kleppe, R.P. Woodard / NonlocalYang— Mills

FundamentalTheory and by DOE contractsDE-FGO5-86-ER40272and ASO5-8OER-10713.

Appendix A: Classicalsolutions

Here we prove a series of theorems concerning classical solutions to theEuler—Lagrangeequationsassociatedwith the localaction,S[~] — for the form ofwhich see(2.1) — the auxiliary action, 5°[4,, 4,] — see (2.5) — and the nonlocalizedaction,S[4,] — see (2.7).

TheoremA.1. The shadowfields 4,~[çb] can be expressedas follows:

,w2_1= - ( ~2 + 54, (A.1)

To seethis considerthe variationsof ST[4,, 4,]:

______ SI[çb + 4’]54, =~JSTJ~4,~+54, (A.2a)

______ SI[4, + i/i]= _~‘TI4J + . (A.2b)

S’IJi j

Subtracting(A.2b) from (A.2a) and multiplying by the invertible operator~ gives:

~2_1 SST[4,,4,] SST[4,,4i]

= — ~2 4,~+ 64, — 34, (A.3)

We can neglect the rightmost term because4,~[4,] is defined to enforce thevanishingof (A.2b). The resultfollows from the fact that at 4,, = i/i

1[4,] the variationof S[4,] is the sameas (A.2a):

SS[4,] SST54, =~-[4,,4,[4,]]+~-[4,,4,[4,]] ~

=~-[4,,4,[4,]] (A.4)

The significanceof this theoremis that 4,~[4,] is unique up to termswhich vanish

with the 4,, field equations.Sinceany suchextratermscanbeabsorbedinto a field


redefinition — which only changesthe still undeterminedmeasurefactor — itfollows that the first stageof nonlocalizationis unique.

TheoremA.2. If the fields 4,, and 4’, obey the Euler—Lagrange equationsof5°[4,, 4’] then thefield x, = 4,, + 4’, obeysthe Euler—LagrangeequationsofS[xJ.

Multiplying (A.2a)by ~2 addingit to (1 — ~2) times(A.2b), andsettingthesum tozerogives the equation:

SI[cb+4i]=0. (A.5)

Substituting x1 = 4,, + 4’, proves the theorem.TheoremA.3. If x, obeysthe Euler—Lagrangeequationof S[xJ then thefollow-

ingfields:

4,, = ~x1, (A.6a)

4’, = (1 —T2),

1x~, (A.6b)

obeythe Euler—Lagrangeequationsof S [4,, 4’].By adding(A.6a) and(A.6b) we seethat 4,, + 4’, = x,. Substitutioninto (A.2a) gives

~-[~‘2x~ (1 ~2)x] =~

1x1~~ . (A.7a)‘1~~j xi

Doing the samefor (A.2b) gives

85” [~2x(1~2)x] =5~+ ~ . (A.7b)xi

The result follows from imposingthe x, field equations.TheoremA.4. If 4,, obeysthe Euler—Lagrangeequation of S[4,] then x, = 4,, +

4i~[4,] obeysthe Euler—Lagrangeequationof S[x].First notethat 4,, and4’, = 4’~[4,] enforcethevanishingof (A.2a) — by relation(A.4)— andof (A.2b) — from the definition of 4’~[4,]. The result follows from theoremA.2.

Theorem.4.5. If x, obeystheEuler—Lagrangeequation ofS[~] then thefield 4,,,definedby (A.6a), obeysthe Euler—Lagrangeequation of S[4,].

This result follows from (A.4) and theoremA.3. TheoremsA.4 andA.5 togetherprovetheveryimportantfactthat the solutionsof the nonlocalizedEuler—Lagrangeequationsare in one-to-onecorrespondencewith those of the local action. Itfollows that the nonlocalized theory is free of the sorts of higher derivativesolutions that wreck string field theory. Note also that the smearingoperatorin(A.6a) generally smoothsout small scale phenomena.Therefore it is entirely

108 G. Kleppe,R.P. Woodard / NonlocalYang — Mills

possiblefor a singular solution of thelocal theoryto give a nonsingularsolution inthe nonlocalizedtheory.

Appendix B: Classicalsymmetries

Herewe provea seriesof theoremsconcerningclassicalsymmetriesof the localaction,S[~]— for the form of which see(2.1) — the auxiliaryaction, 5”[4,, 4’] — see(2.5) — andthe nonlocalizedaction, S[4,] — see(2.7).

TheoremB.1. If S[4,] is invariant underthe infinitesimal transformation:

(B.1)

then thefollowing transformationis a symmetryof 5”[ 4,, 4’]:

(B.2a)

(B.2b)

First note that by adding (B.2a) and (B.2b) we find

(B.3)

Now from the definition of 5”[4,, 4’] — expression(2.5) — it follows that

(B.4)

Thereforewe seethat ~5”[4,, i/i] = 0 as a consequenceof the assumedrelation= 0. A significant corollary is that [8~, ~2] = 83 implies [~, ~2] = ~13.

TheoremB.2. If ST[4,, 4,] is invariant under

~4,~=r~[4,, 4,], (B.5a)

44~~=o[4,,4,], (B.5b)

then thefollowingtransformationis a symmetryof S[ 4,]:

~4,~=r~[4,, 4,[4,]]. (B.6)

From definition (2.7) andthe fact that 4,,[4,] is constructedto enforceeq. (A.2b)we seethat

~ S4’J[~]~4,

= (zlS”)[4,, 444,]]. (B.7)

G. Kleppe, R.P. Woodard / NonlocalYang — Mills 109

Note that onegenerallyobtainsa different result from applying (B.6) to 4i,[4,] thanby first transforming 4, accordingto (B.5b) and thensetting 4,, =

* 8444,](B.8a)

‘�‘o~,[4,, 4’[q5]]. (B.8b)

The equalityof (B.8a) and(B.8b)wasunnecessaryin proving theoremB.2 because4’~[4,] is defined to make the variation of 5”[4,, 41] with respect to 4,, vanish. Thesame is not the case for closure: a simple exercise shows that [z.i~,Li2] = Li3 implies

[&, ~2] = 33 if andonly if S4,~[4,]= o-,[4,, 4i[q5]].

TheoremB.3. If S[4,] is invariant under (B.l) then S[4,] is invariant under

(B.9)

The sametransformationtakesthe shadowfield to

ST. SS[4,]= (1 ~2),~7.[4, +4,[4,]] —K~1[4,+4,[4,]J~—~-[4,+4,[4,]]~, ~

‘I’]

(B.lOa)

wherethe operator K,1[4,] is

52J [4,]K~[4,] ~!~J’ — 54, S4~ (B.lOb)

Of courserelation(B.9) is a trivial consequenceof the two precedingtheorems.Toprove (B.10) first apply the transformation to expression (A.1):

* 1_~~2 884, 85~[4,]= ( ~ ) 84,~ Scbk (B.lla)

= (1 ~2)~

1T[4,+4,[4,]]

_~~{~kj+ 5%4,] ~ (B.llb)

110 G. Kleppe,R.P. Woodard/ NonlocalYang—Mills

Now functionally differentiate the 4,, equation of motion:

84,[4,] ~2j

54, = 54,84,[4’+4’[~]]{8k+ 5J4, }. (B.12a)

This can be rearrangedto obtain an expressionfor 84’i/S4,k involving only 4, and

4’:

~2j —1

____ = {~~i_ 54, 54,[~+4’[~]]} 54, 54,[~+4’[~]]. (B.12b

Substitution into (B.llb) and a few simplifications gives (B.10). A significantcorollary is that if the local symmetryobeys[Si, ~2] = 33 then the nonlocalized oneobeys

[~,~ (B.13a)

ST1[4,] 5T1

2[4,] ST.2[4,] 8T1

1[4,]

KJ~[4,] S4,k — 84, KJ~[çb] 54,~ (B.l3b)

Thereforea local symmetrygroupnonlocalizesto a symmetrywhich must includegeneratorsproportionalto the field equationsin order to give a closedcommuta-tion algebra.As with so much elsethesegeneratorsaremore easily studiedat thelevel of the auxiliary action.

TheoremB.4. Thefollowing transformationgeneratesa symmetryof 5”[ 4,, i/i]:

SST[4,,4’] SST[4,,4i]Li4,~~A~

1[4,,4’]{ S4,~ — ~ }, (B.14a)

&7[4,,4i] 35”[4,,4i]—AJ4,, i/i] 84,~ — 34’, (B.14b)

providedA,,[4,, 4’]= —A,,[4,, 4,].

Of course = —Li4,~.By simply transformingthe auxiliary action:

zlS”[4,, 4,] = —

85” 35” 65” 55”= ~ A..[4, 4’] ~ (B.15)541i SI/I~ ‘~ 34,~

G. Kleppe, R.P. Woodard / Nonlocal Yang — Mills 111

we seethat the theoremfollows trivially from the antisymmettyof As,. Since 4i~[4,]is defined to make 85”/StJi, = 0 the image of this transformationunder theoremB.2 gives themostgeneraloff-shell symmetryof S[4,]. Suchsymmetriesoccurin allfield theories— even conventional,local ones.Of coursegeneratorswhich havenoeffect upon solutionsof the field equationsarewithout dynamicalsignificance.Weconsiderthem here only becausesome subsetof thesesymmetriesmustbe addedto the dynamically significantgeneratorsin order to obtain a closedcommutationalgebra.

Comparisonwith relation (A.2) reveals that transformation(B.14a) can berecastin the following simpleform

Li4,~=A,~[4,, 4’]~2~91[(1 — ~2)~4, 4’m]. (B.16)

An importantspecialcaseis given by the choice

A~1[4,,4,] =M~J~,.k%’~, (B.17)

where M~1is an antisymmetricoperatorwhich commuteswith ST11. When S is theBRS symmetrythe nonlocalizedtransformationrule inducedby theoremsB.1 andB.2 turns out not to agreewith the local rule evenat lowest order. Insteadthe

lowest order termsdiffer by a factor of ~2 Wehavefound it convenientto absorbthis factor by usingthe imageof oneof the dynamicallytrivial symmetriesof type(B.17).

References

[1] A.M. Polyakov, Phys.Lett. B103 (1981)207;D. Friedan, in Recentadvancesin field theoryand statisticalmechanics,ed. J.-B. Zuber andR.Stora(North-Holland,Amsterdam,1984);0. Alvarez, NucI. Phys.B216 (1983) 125

[2] S.B. Giddings, Nucl. Phys. B278 (1986) 242

[31D.J. GrossandA. Jevicki, Nucl. Phys.B283 (1987) 1;E. Cremmer,A SchwimmerandC.B. Thorn, Phys. Lett. B179 (1986)468

[4] C. Hayashi,Prog.Theor. Phys.10 (1953)533; 11(1954)226;R. Marnelius,Phys.Rev. D8 (1973) 2472; D10 (1974)3411

[5] J. Polchinski,NucI. Phys.B231 (1984)269[61 X. Jaén,J. Llosaand A. Molina, Phys.Rev. D34 (1986)2302[7] D. Evens,J.W. Moffat, G. Kleppeand R.P. Woodard,Phys.Rev. D43 (1991)499[8] D.A. Eliezer and R.P.Woodard,NucI. Phys.B325 (1989) 389[9] B.W. Lee andJ. Zinn-Justin,Phys.Rev. D5 (1972)3121

[10] L.D. Faddeevand A.A. Slavnov, Gaugefields: introduction to quantumfield theory(Benjamin/Cummings,Reading,1980)

[11] Z. Bern, M.B. Halpern,L. Sadunand C. Taubes,Phys.Lett. B165 (1985) 151; Nucl. Phys.B284(1987) 1,35

[12] B. Hand, Phys.Lett. B275 (1992)419

112 G. Kieppe,R.P. Woodard / NonlocalYang— Mills

[13] G. Kleppe and R.P.Woodard,Ann. Phys.(N.Y.) 221 (1993) 106[14] B. de Wit andJ.W. van Holten, Phys.Lett. B79 (1978)389[15] M. Soldate,Phys.Lett. B186 (1987)321[16] I.J. Muzinich and M. Soldate,Phys. Rev. D37(1988) 359[17] A.H. Guth, Phys.Rev. D23 (1981) 347

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Nonlocal Yang—Millshome.iitk.ac.in/~sdj/HAQUE/Kleppe_Woodard.pdf · G. Kleppe, R.P. Woodard / Nonlocal Yang—Mills 83 Although the method is no more difficult to implement than