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TAUP2334-96 ACTION AND PASSION AT A DISTANCEAn Essay in Honor of Professor Abner Shimony∗Sandu PopescuDepartment of Physics,Boston University,Boston,MA02215,U.S.A.Daniel RohrlichSchool of Physics and Astronomy,Tel-Aviv University,Ramat-Aviv69978Tel-Aviv,Israel(May7,1996)AbstractQuantum mechanics permits nonlocality—both nonlocal correlations andnonlocal equations of motion—while respecting relativistic causality.Is quan-tum mechanics the unique theory that reconciles nonlocality and causality?We consider two models,going beyond quantum mechanics,of nonlocality—“superquantum”correlations,and nonlocal“jamming”of correlations—andderive new results for the jamming model.In one space dimension,jammingallows reversal of the sequence of cause and effect;in higher dimensions,how-ever,effect never precedes cause.∗To appear in Quantum Potentiality,Entanglement,and Passion-at-a-Distance:Essays for Ab-ner Shimony,R.S.Cohen,M.A.Horne and J.Stachel,eds.(Dordrecht,Netherlands:Kluwer Academic Publishers),in press.1I.INTRODUCTIONWhy is quantum mechanics what it is?Many a student has asked this question.Some physicists have continued to ask it.Few have done so with the passion of Abner Shimony.“Why is quantum mechanics what it is?”we,too,ask ourselves,and of course we haven’t got an answer.But we are working on an answer,and we are honored to dedicate this work to you,Abner,on your birthday.What is the problem?Quantum mechanics has an axiomatic structure,exposed by von Neumann,Dirac and others.The axioms of quantum mechanics tell us that every state of a system corresponds to a vector in a complex Hilbert space,every physical observable corre-sponds to a linear hermitian operator acting on that Hilbert space,etc.We see the problem in comparison with the special theory of relativity.Special relativity can be deduced in its entirety from two axioms:the equivalence of inertial reference frames,and the constancy of the speed of light.Both axioms have clear physical meaning.By contrast,the numerous axioms of quantum mechanics have no clear physical meaning.Despite many attempts, starting with von Neumann,to derive the Hilbert space structure of quantum mechanics from a“quantum logic”,the new axioms are hardly more natural than the old.Abner Shimony offers hope,and a different approach.His point of departure is a remark-able property of quantum mechanics:nonlocality.Quantum correlations display a subtle nonlocality.On the one hand,as Bell[1]showed,quantum correlations could not arise in any theory in which all variables obey relativistic causality[2].On the other hand,quantum correlations themselves obey relativistic causality—we cannot exploit quantum correlations to transmit signals at superluminal speeds[3](or at any speed).That quantum mechanics combines nonlocality and causality is wondrous.Nonlocality and causality seem prima facie incompatible.Einstein’s causality contradicts Newton’s action at a distance.Yet quan-tum correlations do not permit action at a distance,and Shimony[4]has aptly called the nonlocality manifest in quantum correlations“passion at a distance”.Shimony has raised the question whether nonlocality and causality can peacefully coexist in any other theory2besides quantum mechanics[4,5].Quantum mechanics also implies nonlocal equations of motion,as Yakir Aharonov[6,7] has pointed out.In one version of the Aharonov-Bohm effect[8],a solenoid carrying an isolated magneticflux,inserted between two slits,shifts the interference pattern of electrons passing through the slits.The electrons therefore obey a nonlocal equation of motion:they never pass through theflux yet theflux affects their positions when they reach the screen[9]. Aharonov has shown that the solenoid and the electrons exchange a physical quantity,the modular momentum,nonlocally.In general,modular momentum is measurable and obeys a nonlocal equation of motion.But when theflux is constrained to lie between the slits, its modular momentum is completely uncertain,and this uncertainty is just sufficient to keep us from seeing a violation of causality.Nonlocal equations of motion imply action at a distance,but quantum mechanics manages to respect relativistic causality.Still,nonlocal equations of motion seem so contrary to relativistic causality that Aharonov[7]has asked whether quantum mechanics is the unique theory combining them.The parallel questions raised by Shimony and Aharonov lead us to consider models for theories,going beyond quantum mechanics,that reconcile nonlocality and causality. Is quantum mechanics the only such theory?If so,nonlocality and relativistic causality together imply quantum theory,just as the special theory of relativity can be deduced in its entirety from two axioms[7].In this paper,we will discuss model theories[10–12] manifesting nonlocality while respecting causality.Thefirst model manifests nonlocality in the sense of Shimony:nonlocal correlations.The second model manifests nonlocality in the sense of Aharonov:nonlocal dynamics.Wefind that quantum mechanics is not the only theory that reconciles nonlocality and relativistic causality.These models raise new theoretical and experimental possibilities.They imply that quantum mechanics is only one of a class of theories combining nonlocality and causality;in some sense,it is not even the most nonlocal of such theories.Our models raise a question:What is the minimal set of physical principles—“nonlocality plus no signalling plus something else simple and fundamental”as Shimony put it[13]—from which we may derive quantum mechanics?3II.NONLOCALITY I:NONLOCAL CORRELATIONSThe Clauser,Horne,Shimony,and Holt [14]form of Bell’s inequality holds in any classical theory (that is,any theory of local hidden variables).It states that a certain combination of correlations lies between -2and 2:−2≤E (A,B )+E (A,B )+E (A ,B )−E (A ,B )≤2.(1)Besides 2,two other numbers,2√2and 4,are important bounds on the CHSH sum ofcorrelations.If the four correlations in Eq.(1)were independent,the absolute value of the sum could be as much as 4.For quantum correlations,however,the CHSH sum ofcorrelations is bounded [15]in absolute value by 2√2.Where does this bound come from?Rather than asking why quantum correlations violate the CHSH inequality,we might ask why they do not violate it more .Suppose that quantum nonlocality implies that quantum correlations violate the CHSH inequality at least sometimes.We might then guess that relativistic causality is the reason that quantum correlations do not violate it maximally.Could relativistic causality restrict the violation to 2√2instead of 4?If so,then nonlocalityand causality would together determine the quantum violation of the CHSH inequality,and we would be closer to a proof that they determine all of quantum mechanics.If not,then quantum mechanics cannot be the unique theory combining nonlocality and causality.To answer the question,we ask what restrictions relativistic causality imposes on joint probabilities.Relativistic causality forbids sending messages faster than light.Thus,if one observer measures the observable A ,the probabilities for the outcomes A =1and A =−1must be independent of whether the other observer chooses to measure B or B .However,it can be shown [10,16]that this constraint does not limit the CHSH sum ofquantum correlations to 2√2.For example,imagine a “superquantum”correlation functionE for spin measurements along given axes.Assume E depends only on the relative angle θbetween axes.For any pair of axes,the outcomes |↑↑ and |↓↓ are equally likely,and similarly for |↑↓ and |↓↑ .These four probabilities sum to 1,so the probabilities for |↑↓4and|↓↓ sum to1/2.In any direction,the probability of|↑ or|↓ is1/2irrespective of a measurement on the other particle.Measurements on one particle yield no information about measurements on the other,so relativistic causality holds.The correlation function then satisfies E(π−θ)=−E(θ).Now let E(θ)have the form(i)E(θ)=1for0≤θ≤π/4;(ii)E(θ)decreases monotonically and smoothly from1to-1asθincreases fromπ/4to 3π/4;(iii)E(θ)=−1for3π/4≤θ≤π.Consider four measurements along axes defined by unit vectorsˆa ,ˆb,ˆa,andˆb separated by successive angles ofπ/4and lying in a plane.If we now apply the CHSH inequality Eq.(1)to these directions,wefind that the sum of correlationsE(ˆa,ˆb)+E(ˆa ,ˆb)+E(ˆa,ˆb )−E(ˆa ,ˆb )=3E(π/4)−E(3π/4)=4(2)violates the CHSH inequality with the maximal value4.Thus,a correlation function could satisfy relativistic causality and still violate the CHSH inequality with the maximal value4.III.NONLOCALITY II:NONLOCAL EQUATIONS OF MOTIONAlthough quantum mechanics is not the unique theory combining causality and nonlocal correlations,could it be the unique theory combining causality and nonlocal equations of motion?Perhaps the nonlocality in quantum dynamics has deeper physical signficance.Here we consider a model that in a sense combines the two forms of nonlocality:nonlocal equations of motion where one of the physical variables is a nonlocal correlation.Jamming,discussed by Grunhaus,Popescu and Rohrlich[11]is such a model.The jamming paradigm involves three experimenters.Two experimenters,call them Alice and Bob,make measurements on systems that have locally interacted in the past.Alice’s measurements are spacelike separated from Bob’s.A third experimenter,Jim(the jammer),presses a button on a black box.This event is spacelike separated from Alice’s measurements and from Bob’s.The5black box acts at a distance on the correlations between the two sets of systems.For the sake of definiteness,let us assume that the systems are pairs of spin-1/2particles entangled in a singlet state,and that the measurements of Alice and Bob yield violations of the CHSH inequality,in the absence of jamming;but when there is jamming,their measurements yield classical correlations(no violations of the CHSH inequality).Indeed,Shimony[4]considered such a paradigm in the context of the experiment of Aspect,Dalibard,and Roger[17].To probe the implications of certain hidden-variable the-ories[18],he wrote,“Suppose that in the interval after the commutators of that experiment have been actuated,but before the polarization analysis of the photons has been completed, a strong burst of laser light is propagated transverse to but intersecting the paths of the propagating photons....Because of the nonlinearity of the fundamental material medium which has been postulated[in these models],this burst would be expected to generate exci-tations,which could conceivably interfere with the nonlocal propagation that is responsible for polarization correlations.”Thus,Shimony asked whether certain hidden-variable theories would predict classical correlations after such a burst.(Quantum mechanics,of course,does not.)Here,our concern is not with hidden-variable theories or with a mechanism for jamming; rather,we ask whether such a nonlocal equation of motion(or one,say,allowing the third experimenter nonlocally to create,rather than jam,nonlocal correlations)could respect causality.The jamming model[11]addresses this question.In general,jamming would allow Jim to send superluminal signals.But remarkably,some forms of jamming would not; Jim could tamper with nonlocal correlations without violating causality.Jamming preserves causality if it satisfies two constraints,the unary condition and the binary condition.The unary condition states that Jim cannot use jamming to send a superluminal signal that Alice (or Bob),by examining her(or his)results alone,could read.To satisfy this condition,let us assume that Alice and Bob each measure zero average spin along any axis,with or without jamming.In order to preserve causality,jamming must affect correlations only,not average measured values for one spin component.The binary condition states that Jim cannot use6jamming to send a signal that Alice and Bob together could read by comparing their results, if they could do so in less time than would be required for a light signal to reach the place where they meet and compare results.This condition restricts spacetime configurations for jamming.Let a,b and j denote the three events generated by Alice,Bob,and Jim, respectively:a denotes Alice’s measurements,b denotes Bob’s,and j denotes Jim’s pressing of the button.To satisfy the binary condition,the overlap of the forward light cones of a and b must lie entirely within the forward light cone of j.The reason is that Alice and Bob can compare their results only in the overlap of their forward light cones.If this overlap is entirely contained in the forward light cone of j,then a light signal from j can reach any point in spacetime where Alice and Bob can compare their results.This restriction on jamming configurations also rules out another violation of the unary condition.If Jim could obtain the results of Alice’s measurements prior to deciding whether to press the button,he could send a superluminal signal to Bob by selectively jamming[11].IV.AN EFFECT CAN PRECEDE ITS CAUSE!If jamming satisfies the unary and binary conditions,it preserves causality.These con-ditions restrict but do not preclude jamming.There are configurations with spacelike sep-arated a,b and j that satisfy the unary and binary conditions.We conclude that quantum mechanics is not the only theory combining nonlocal equations of motion with causality.In this section we consider another remarkable aspect of jamming,which concerns the time sequence of the events a,b and j defined above.The unary and binary conditions are man-ifestly Lorentz invariant,but the time sequence of the events a,b and j is not.A time sequence a,j,b in one Lorentz frame may transform into b,j,a in another Lorentz frame. Furthermore,the jamming model presents us with reversals of the sequence of cause and effect:while j may precede both a and b in one Lorentz frame,in another frame both a and b may precede j.To see how jamming can reverse the sequence of cause and effect,we specialize to the7case of one space dimension.Since a and b are spacelike separated,there is a Lorentz frame in which they are simultaneous.Choosing this frame and the pair(x,t)as coordinates for space and time,respectively,we assign a to the point(-1,0)and b to the point(1,0). What are possible points at which j can cause jamming?The answer is given by the binary condition.It is particularly easy to apply the binary condition in1+1dimensions,since in 1+1dimensions the overlap of two light cones is itself a light cone.The overlap of the two forward light cones of a and b is the forward light cone issuing from(0,1),so the jammer, Jim,may act as late as∆t=1after Alice and Bob have completed their measurements and still jam their results.More generally,the binary condition allows us to place j anywhere in the backward light cone of(0,1)that is also in the forward light cone of(0,-1),but not on the boundaries of this region,since we assume that a,b and j are mutually spacelike separated.(In particular,j cannot be at(0,1)itself.)Such reversals may boggle the mind,but they do not lead to any inconsistency as long as they do not generate self-contradictory causal loops[19,20].Consistency and causality are intimately related.We have used the term relativistic causality for the constraint that others call no signalling.What is causal about this constraint?Suppose that an event(a“cause”) could influence another event(an“effect”)at a spacelike separation.In one Lorentz frame the cause precedes the effect,but in some other Lorentz frame the effect precedes the cause; and if an effect can precede its cause,the effect could react back on the cause,at a still earlier time,in such a way as to prevent it.A self-contradictory causal loop could arise.A man could kill his parents before they met.Relativistic causality prevents such causal contradictions[19].Jamming allows an event to precede its cause,but does not allow self-contradictory causal loops.It is not hard to show[11]that if jamming satisfies the unary and binary conditions,it does not lead to self-contradictory causal loops,regardless of the number of jammers.Thus,the reversal of the sequence of cause and effect in jamming is consistent.It is,however,sufficiently remarkable to warrant further comment below,and we also show that the sequence of cause and effect in jamming depends on the space dimension in a surprising way.8The unary and binary conditions restrict the possible jamming configurations;however, they do not require that jamming be allowed for all configurations satisfying the two con-ditions.Nevertheless,we have made the natural assumption that jamming is allowed for all such configurations.This assumption is manifestly Lorentz invariant.It allows a and b to both precede j.In a sense,it means that Jim acts along the backward light cone of j; whenever a and b are outside the backward light cone of j and fulfill the unary and binary conditions,jamming occurs.V.AN EFFECT CAN PRECEDE ITS CAUSE??That Jim may act after Alice and Bob have completed their measurements(in the given Lorentz frame)is what may boggle the mind.How can Jim change his own past?We may also put the question in a different way.Once Alice and Bob have completed their measurements,there can after all be no doubt about whether or not their correlations have been jammed;Alice and Bob cannot compare their results andfind out until after Jim has already acted,but whether or not jamming has taken place is already an immutable fact. This fact apparently contradicts the assumption that Jim is a free agent,i.e.that he can freely choose whether or not to jam.If Alice and Bob have completed their measurements, Jim is not a free agent:he must push the button,or not push it,in accordance with the results of Alice and Bob’s measurements.We may be uncomfortable even if Jim acts before Alice and Bob have both completed their measurements,because the time sequence of the events a,b and j is not Lorentz invariant;a,j,b in one Lorentz frame may transform to b,j,a in another.The reversal in the time sequences does not lead to a contradiction because the effect cannot be isolated to a single spacetime event:there is no observable effect at either a or b,only correlations between a and b are changed.All the same,if we assume that Jim acts on either Alice or Bob—whoever measures later—we conclude he could not have acted on either of them, because both come earlier in some Lorentz frame.9What,then,do we make of cause and effect in the jamming model?We offer two points of view on this question.One point of view is that we don’t have to worry;jamming does not lead to any causal paradoxes,and that is all that matters.Of course,experience teaches that causes precede their effects.Yet experience also teaches that causes and effects are locally related.In jamming,causes and effects are nonlocally related.So we cannot assume that causes must precede their effects;it is contrary to the spirit of special relativity to impose such a demand.Indeed,it is contrary to the spirit of general relativity to assign absolute meaning to any sequence of three mutually spacelike separated events,even when such a sequence has a Lorentz-invariant meaning in special relativity[20].We only demand that no sequence of causes and effects close upon itself,for a closed causal loop—a time-travel paradox—would be self-contradictory.If an effect can precede its cause and both are spacetime events,then a closed causal loop can arise.But in jamming,the cause is a spacetime event and the effect involves two spacelike separated events;no closed causal loop can arise[11].This point of view interprets cause and effect in jamming as Lorentz invariant;observers in all Lorentz frames agree that jamming is the effect and Jim’s action is the cause.A second point of view asks whether the jamming model could have any other interpretation. In a world with jamming,might observers in different Lorentz frames give different accounts of jamming?Could a sequence a,j,b have a covariant interpretation,with two observers coming to different conclusions about which measurements were affected by Jim?(No ex-periment could ever prove one of them wrong and the other right[21].)Likewise,perhaps observers in a Lorentz frame where both a and b precede j would interpret jamming as a form of telesthesia:Jim knows whether the correlations measured by Alice and Bob are nonlocal before he could have received both sets of results.We must assume,however,that observers in such a world would notice that jamming always turns out to benefit Jim;they would not interpret jamming as mere telesthesia,so the jamming model could not have this covariant interpretation.Finally,we note that a question of interpreting cause and effect arises in quantum me-10chanics,as well.Consider the measurements of Alice and Bob in the absence of jamming. Their measured results do not indicate any relation of cause and effect between Alice and Bob;Alice can do nothing to affect Bob’s results,and vice versa.According to the con-ventional interpretation of quantum mechanics,however,thefirst measurement on a pair of particles entangled in a singlet state causes collapse of the state.The question whether Alice or Bob caused the collapse of the singlet state has no Lorentz-invariant answer[11,22].VI.JAMMING IN MORE THAN ONE SPACE DIMENSIONAfter arguing that jamming is consistent even if it allows reversals of the sequence of cause and effect,we open this section with a surprise:such reversals arise only in one space dimension!In higher dimensions,the binary condition itself eliminates such configurations; jamming is not possible if both a and b precede j.To prove this result,wefirst consider the case of2+1dimensions.We choose coordinates(x,y,t)and,as before,place a and b on the x-axis,at(-1,0,0)and(1,0,0),respectively.Let A,B and J denote the forward light cones of a,b and j,respectively.The surfaces of A and B intersect in a hyperbola in the yt-plane.To satisfy the binary condition,the intersection of A and B must lie entirely within J.Suppose that this condition is fulfilled,and now we move j so that the intersection of A and B ceases to lie within J.The intersection of A and B ceases to lie within J when its surface touches the surface of J.Either a point on the hyperbola,or a point on the surface of either A of B alone,may touch the surface of J.However,the surfaces of A and J can touch only along a null line(and likewise for B and J);that is,only if j is not spacelike separated from either a or b,contrary to our assumption.Therefore the only new constraint on j is that the hyperbola formed by the intersection of the surfaces of A and B not touch the surface of J.If we place j on the t-axis,at(0,0,t),the latest time t for which this condition is fulfilled is when the asymptotes of the hyperbola lie along the surface of J.They lie along the surface of J when j is the point(0,0,0).If j is the point(0,0,0),moving j in either the x-or y-direction will cause the hyperbola to intersect the surface of J.We conclude that11there is no point j,consistent with the binary condition,with t-coordinate greater than0. Thus,j cannot succeed both a and b in any Lorentz frame(although it could succeed one of them).For n>2space dimensions,the proof is similar.The only constraint on j arises from the intersection of the surfaces of A and B.At a given time t,the surfaces of A and B are (n−1)-spheres of radius t centered,respectively,at x=−1and x=1on the x-axis;these (n−1)-spheres intersect in an(n−2)-sphere of radius(t2−1)1/2centered at the origin. This(n−2)sphere lies entirely within an(n−1)-sphere of radius t centered at the origin, and approaches it asymptotically for t→∞.The(n−1)-spheres centered at the origin are sections of the forward light cone of the origin.Thus,j cannot occur later than a and b.Wefind this result both amusing and odd.We argued above that allowing j to succeed both a and b does not entail any inconsistency and that it is contrary to the spirit of the general theory of relativity to exclude such configurations for jamming.Nonetheless,wefind that they are automatically excluded for n≥2.VII.CONCLUSIONSTwo related questions of Shimony[4,5]and Aharonov[7]inspire this essay.Nonlocality and relativistic causality seem almost irreconcilable.The emphasis is on almost,because quantum mechanics does reconcile them,and does so in two different ways.But is quantum mechanics the unique theory that does so?Our answer is that it is not:model theories going beyond quantum mechanics,but respecting causality,allow nonlocality both ways.We qualify our answer by noting that nonlocality is not completely defined.Relativistic causality is well defined,but nonlocality in quantum mechanics includes both nonlocal correlations and nonlocal equations of motion,and we do not know exactly what kind of nonlocality we are seeking.Alternatively,we may ask what additional physical principles can we impose that will single out quantum mechanics as the unique theory.Our“superquantum”and “jamming”models open new experimental and theoretical possibilities.The superquantum12model predicts violations of the CHSH inequality exceeding quantum violations,consistent with causality.The jamming model predicts new effects on quantum correlations from some mechanism such as the burst of laser light suggested by Shimony[4].Most interesting are the theoretical possibilities.They offer hope that we may rediscover quantum mechanics as the unique theory satisfying a small number of fundamental principles:causality plus nonlocality“plus something else simple and fundamental”[13].ACKNOWLEDGMENTSD.R.acknowledges support from the State of Israel,Ministry of Immigrant Absorption, Center for Absorption in Science.13REFERENCES[1]J.S.Bell,Physics1,195(1964).[2]The term relativistic causality denotes the constraint that information cannot be trans-ferred at speeds exceeding the speed of light.This constraint is also called no signalling.[3]G.C.Ghirardi,A.Rimini and T.Weber,Lett.Nuovo Cim.27(1980)263.[4]A.Shimony,in Foundations of Quantum Mechanics in Light of the New Technology,S.Kamefuchi et al.,eds.(Tokyo,Japan Physical Society,1983),p.225.[5]A.Shimony,in Quantum Concepts in Space and Time,R.Penrose and C.Isham,eds.(Oxford,Claredon Press,1986),p.182.[6]Y.Aharonov,H.Pendleton,and A.Petersen,Int.J.Theo.Phys.2(1969)213;3(1970)443;Y.Aharonov,in Proc.Int.Symp.Foundations of Quantum Mechanics,Tokyo, 1983,p.10.[7]Y.Aharonov,unpublished lecture notes.[8]Y.Aharonov and D.Bohm,Phys.Rev.115(1959)485,reprinted in F.Wilczek(ed.)Fractional Statistics and Anyon Superconductivity,Singapore:World-Scientific,1990;[9]It is true that the electron interacts locally with a vector potential.However,the vectorpotential is not a physical quantity;all physical quantities are gauge invariant.[10]S.Popescu and D.Rohrlich,Found.Phys.24,379(1994).[11]J.Grunhaus,S.Popescu and D.Rohrlich,Tel Aviv University preprint TAUP-2263-95(1995),to appear in Phys.Rev.A.[12]D.Rohrlich and S.Popescu,to appear in the Proceedings of60Years of E.P.R.(Work-shop on the Foundations of Quantum Mechanics,in honor of Nathan Rosen),Technion, Israel,1995.14[13]A.Shimony,private communication.[14]J.F.Clauser,M.A.Horne,A.Shimony and R.A.Holt,Phys.Rev.Lett.23,880(1969).[15]B.S.Tsirelson(Cirel’son),Lett.Math.Phys.4(1980)93;ndau,Phys.Lett.A120(1987)52.[16]For the maximal violation of the CHSH inequality consistent with relativity see also L.Khalfin and B.Tsirelson,in Symposium on the Foundations of Modern Physics’85,P.Lahti et al.,eds.(World-Scientific,Singapore,1985),p.441;P.Rastall,Found.Phys.15,963(1985);S.Summers and R.Werner,J.Math.Phys.28,2440(1987);G.Krenn and K.Svozil,preprint(1994)quant-ph/9503010.[17]A.Aspect,J.Dalibard and G.Roger,Phys.Rev.Lett.49,1804(1982).[18]D.Bohm,Wholeness and the Implicate Order(Routledge and Kegan Paul,London,1980);D.Bohm and B.Hiley,Found.Phys.5,93(1975);J.-P.Vigier,Astr.Nachr.303,55(1982);N.Cufaro-Petroni and J.-P.Vigier,Phys.Lett.A81,12(1981);P.Droz-Vincent,Phys.Rev.D19,702(1979);A.Garuccio,V.A.Rapisarda and J.-P.Vigier,Lett.Nuovo Cim.32,451(1981).[19]See e.g.D.Bohm,The Special Theory of Relativity,W.A.Benjamin Inc.,New York(1965)156-158.[20]We thank Y.Aharonov for a discussion on this point.[21]They need not be incompatible.An event in one Lorentz frame often is another eventin another frame.For example,absorption of a virtual photon in one Lorentz frame corresponds to emission of a virtual photon in another.In jamming,Jim might not only send instructions but also receive information,in both cases unconsciously.(Jim is conscious only of whether or not he jams.)Suppose that the time reverse of“sending instructions”corresponds to“receiving information”.Then each observer interprets the sequence of events correctly for his Lorentz frame.15。
维的准确数学定义及其意义（最新）(含“杨－米尔斯(Yang-Mills)存在性和质量缺口”的数学理论依据）南京市秦淮区瓮家营汤家坝14号马双焕msh2003nj@摘要：该文对数学中维的准确定义和意义做了系统介绍。
关键词：维、多维数、素数、论证。
MR（2000）主题分类：00A05一、维的准确数学定义看到维，人们的第一个联想是几维空间。
上网检索，可以搜到：维是因素，是参数，是衡量比较的尺度。
对于维的进一步的认识是很重要的。
首先不难知道，维的学问就是关于如何衡量比较的学问。
从上面的因素、参数、尺度的概念可以看出这并不是维的根本数学定义，为什么呢？因为因素、参数、尺度有各种各样，有很多种，不同种因素、参数、尺度之间如何比较呢？仅以前述的介绍是不行的，所以维的根本数学定义是很重要的。
本人在探讨哥德巴赫猜想、庞加莱猜想的论证过程中得出了维的根本数学定义：在某前提条件下可无限任意，该前提条件就是某一维。
试想一下，无限还是有限，任意还是确定是可以衡量比较的。
什么意识呢，在某维上，如果有其他限制因素，也就是如果不是无限或是确定的，必定在该维上留下印记或标志，这就可以衡量比较了。
所以在某前提条件下可无限任意，该前提条件就是某一维。
这是维的数学定义。
而且这里的无限还是有限，任意还是确定可以是各种各样的，是跨越行业之别的。
二、维的合并与分解从前面的介绍可以知道，只要在某前提条件下可无限任意，该前提条件就是某一维。
这样维的种类数量很多很多，哪么各维之间的关系是什么样的呢？首先有些维之间是可以分解合并的。
例如学生这一维与大学生这一维就是这样的关系。
学生这一维，只要是学生，都在这一维内，无论这个学生是中学生，是大学生，是男，是女，是哪里的学生，．．．，都在这一维上。
大学生这一维仅限于大学生，显然学生这一维包含了大学生维、中学生维、男学生维、女学生维。
这就是维之间的分解合并关系。
三、不同类别维的关系我们知道，维是衡量比较的尺度，如果不同类别的维之间，不同行业单位之间不能通过维的概念进行衡量比较，那么这个维的概念是不到位的。
Region-based non-local means algorithm for noise removalW.L.Zeng and X.B.LuThe non-local means (NLM)provides a useful tool for image denoising and many variations of the NLM method have been proposed.However,few works have tried to tackle the task of adaptively choos-ing the patch size according to region characteristics.Presented is a region-based NLM method for noise removal.The proposed method first analyses and classifies the image into several region types.According to the region type,a local window is adaptively adjusted to match the local property of a region.Experimental results show the effectiveness of the proposed method and demonstrate its superior-ity to the state-of-the-art methods.Introduction:The use of the non-local means (NLM)filter for noise removal has been extensively studied in the past few years.The NLM filter was first addressed in [1].The discrete version of the NLM is as follows:u (k ,l )=(i ,j )[N (k ,l )w (k ,l ,i ,j )v (i ,j )(1)where u is the restored value at pixel (k,l )and N (k,l )stands for theneighbourhood of the pixel (k,l ).The weight function w (k,l,i,j )is defined asw (k ,l ,i ,j )=1exp −||T k ,l v −T i ,j v ||22,a(2)where T k,l and T i,j denote two operators that extract two patches of sizeq ×q centred at pixel (k,l )and (i,j ),respectively;h is the decay para-meter of the weights; . 2,a is the weighted Euclidean norm using a Gaussian kernel with standard deviation a ,and Z (k,l )is the normalised constantZ (k ,l )= (i ,j )exp −||T k ,l v −T i ,j v ||22,ah 2(3)The core idea of the NLM filter exploits spatial correlation in the entireimage for noise removal and can produce promising results.This method is time consuming and not able to suppress any noise for non-repetitive neighbourhoods.Numerous methods were proposed to accel-erate the NLM method [2–4].Also,variations of the NLM method have been proposed to improve the denoising performance [5–7].In smooth areas,a large matching window size could be used to reduce the influ-ence of misinterpreting noise as local structure.Conversely,a small matching window size could be used for the edge /texture region,which means not only the local structure existing within a neighbour-hood can be effectively used but can also speed up the matching process.To the best of our knowledge,few works have tried to tackle the task of adaptively choosing the patch size according to region characteristics.To overcome the disadvantage of the NLM method and its variances,in this Letter we present an adaptive NLM (ANLM)method for noise removal.The proposed method first analyses and classifies the image into several region types based on local structure information of a pixel.According to the region type,a local window is adaptively adjusted to match the local property of a region.Experimental results show the effectiveness of the proposed method.Proposed NLM algorithm:The adaptive patches based non-local means algorithm is conducted according to the region classification results,owing to the fact that the structure tensor can obtain more local structure information [8].Therefore,we use it to classify the region.For each pixel (i,j )of the region,the structure tensor matrix is defined asT s =t 11t 12t 12t 22 =G s ∗(g x (i ,j ))2G s ∗g x (i ,j )g y (i ,j )G s ∗g y (i ,j )g x (i ,j )G s ∗(g y (i ,j ))2where g x and g y stand for gradient information in the x and y directions,G s denotes a Gaussian kernel with a standard deviation s .Theeigenvalues l 1and l 2of T s are given byl 1=12t 11+t 22+ (t 11−t 22)2+4t 212 and l 2=1t 11+t 22− (t 11−t 22)2+4t 212 For a pixel in the smooth region,there is a small eigenvalue difference;for a pixel in an edge /texture region,there is a large eigenvalue differ-ence.Therefore,region classification can be achieved by examining the eigenvalue difference of each pixel.Let l (i ,j )=|l 1(i ,j )−l 2(i ,j )|.We propose the following classifi-cation scheme to partition the whole image region into n classes {c 1,···,c n }:(i ,j )[c 1,if l (i ,j )≤l min +(l max −l min )n c 2,if l (i ,j )≤l min +2(l max −l min )n ...c n ,if l (i ,j )≤l min +n (l max −l min )n ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩where l min and l max are the minimum and maximum of {l (i ,j ):(i ,j )[V },respectively.To exploit the local structure information and reduce noise in different regions,we adaptively choose the matching window based on the region classification result.The scheme for selecting the matching window is asfollows:if (k ,l )[c r ,T k ,l :=T r k ,l ,where T rk ,l denotes an operator of the r-type region that extracts one patch of size q r ×q r .To reduce the influ-ence of misinterpreting noise as local structure,a larger patch size is adopted for a smooth region.In contrast,a small patch size is employed for the edge /texture region.Intuitively,the number of the class n should be as big as possible.In practice,the gain is insignificant for n greater than 4.Therefore,we choose n ¼4in our experiments.Table 1:PSNR performance comparison of ‘Lena’,‘Barbara’,‘Peppers’imagesFig.1Comparison of results with additive Gaussian noise of s ¼35a Original image b Noisy image c NLM d WUNLM e ANLMExperimental results:In this Section,we compare our proposed ANLM method with the NLM method [2]and the weight update NLM (WUNLM)method [3].We test the proposed method on ‘Lena’,‘Barbara’,and ‘Peppers’,which were taken from the USC-SIPI Image Database (/database/base).The performance of the method was evaluated by measuring the peak signal-to-noise ratio (PSNR).In general h corresponds to the noise level and is usuallyELECTRONICS LETTERS 29th September 2011Vol.47No.20,1125-1127fixed to the standard deviation of the noise.The size of the search window is21×21.Table1shows results obtained with three methods across four noise levels.Figs.1a and b,show the‘Barbara’image and the corresponding noisy image generated by adding Gaussian white noise with variance s¼35,respectively.Figs.1c–e show denoised images by using the NLM,WUNLM,and ANLM methods,respectively.From the standpoint of perceptual view and PSNR values,the proposed ANLM method produced the best quality. Conclusions:An adaptive NLM(ANLM)method for noise removal is presented.In the method,an image isfirst analysed and classified into several region types.According to the region type,a local window is adaptively adjusted to match the local property of a region. Experimental results show the effectiveness of the proposed method and demonstrate its superiority to the state-of-the-art methods. Acknowledgments:This work was supported by the National Natural Science Foundation of China under grant60972001,the National Key Technologies R&D Program of China under grant2009BAG13A06 and the Scientific Innovation Research of College Graduate in Jiangsu Province under grant CXZZ_0163.#The Institution of Engineering and Technology20115August2011doi:10.1049/el.2011.2456W.L.Zeng(School of Transportation,Southeast University,Nanjing 210096,People’s Republic of China)X.B.Lu(School of Automation,Southeast University,Nanjing210096, People’s Republic of China)E-mail:xblu2008@References1Budades,A.,Coll,B.,and Morel,J.M.:‘A review of image denoising algorithms,with a new one’,Multiscale Model Simul.,2005,4,(2), pp.490–5302Mahmoudi,M.,and Sapiro,G.:‘Fast image and video denoising via nonlocal means of similar neighborhoods’,IEEE Signal Process.Lett., 2005,12,(12),pp.839–8423Vignesh,R.,Oh,B.T.,and Kuo,C.-C.J.:‘Fast non-local means(NLM) computation with probabilistic early termination’,IEEE Signal Process.Lett.,2010,17,(3),pp.277–2804Brox,T.,Kleinschmidt,O.,and Cremers,D.:‘Efficient nonlocal means for denoising of textural patterns’,IEEE Trans.Image Process.,2008, 17,(7),pp.1083–10925Kervrann,C.,and Boulanger,J.:‘Optimal spatial adaptation for patch-based image denoising’,IEEE Trans.Image Process.,2006,15,(10), pp.2866–28786Ville,D.V.D.,and Kocher,M.:‘SURE-based non-local means’,IEEE Signal Process.Lett.,2009,16,(11),pp.973–9767Park,S.W.,and Kang,M.G.:‘NLM algorithm with weight update’, Electron.Lett.,2010,16,(15),pp.1061–10638Brox,T.,Weickert,J.,Burgeth,B.,and Mrazek,P.:‘Nonlinear structure tensors’,Image put.,2006,24,pp.41–55ELECTRONICS LETTERS29th September2011Vol.47No.20。
Accelerating non-local means algorithm with random projectioni and Y.T.YangThe non-local means (NLM)algorithm suppresses noise via replacing the noisy pixel by the weighted average of all the pixels with similar neighbourhood vectors.However,the weights calculation is computa-tionally expensive,as a result of which the NLM algorithm is quite slow for practical applications.A random projection approach is intro-duced to reduce the dimension of neighbourhood vectors used in the NLM filtering process,which yields a faster and more accurate denois-ing effect.Experimental results illustrate the effectiveness of the pro-posed method.Introduction:Image denoising,although it has been studied over a long time,is still an unsolved problem that remains a wide-open field of research in image processing.Recently,Buades et al.[1]established a new non-local means (NLM)denoising framework based on the idea that averaging repeated structures contained in an image will reduce the noise.In this framework,each pixel is estimated as the weighted average of all the pixels in the image,and the weights are determined by the similarity between the neighbourhood vectors surrounding the pixels.Although the NLM denoising method suppresses noise and preserves details in an image well,its implementation is computationally expens-ive owing to the tedious weights calculation process in denoising.The above-mentioned problem can be solved in the following ways.First,reducing the number of candidate neighbourhoods for weights calcu-lation,which has been realised in [2]by limiting the weight calculation to a sub-image surrounding the pixel being processed,moreover,[3]and [4]report further efforts to eliminate unrelated neighbourhoods in the search window.Secondly,reducing the complexity of weight calcu-lation,which is addressed in [5]by using the fast Fourier transform (FFT)combined with the summed squares image (SSI)and in [6]by introducing a multi-resolution pyramid architecture.Motivated by the above-mentioned ideas,we introduce a random projection approach to project neighbourhood vectors onto lower-dimensional subspaces,which promotes both the accuracy and the computational performance of the NLM algorithm considerably.Non-local means algorithm:Consider a discrete noisy imagey i ,j =x i ,j +n i ,j ,(i ,j )[I(1)where y i ,j ,x i ,j ,n i ,j denote the observed value,the ‘true’value and the noise perturbation at the pixel (i ,j ),respectively.The NL-means filter is written asˆxi ,j =1Z k ,l(i ,j )[ˆN d ×d (k ,l )w k ,l ,i ,j y i ,j (2)where the normalising termZ k ,l =(i ,j )[ˆNd ×d (k ,l )w k ,l ,i ,j(3)and the weight for each pair of neighbourhood vectors isw k ,l ,i ,j =exp −ˆN d ×d (k ,l )−ˆN d ×d (i ,j ) 22,ah 2(4)where ˆNd ×d (m ,n )represents the square neighbourhood vector of size d ×d centred at location (m ,n ), · 22,a denotes the Gaussian-weighted-semi-norm,and h is a global smoothing parameter used to control the amount of blurring introduced in the denoising process.Proposed method:To accelerate the non-local means filtering,we propose an alternative strategy that employs the random projection approach introduced in [7]to perform dimensional reduction for each neighbourhood vector.In detail,we recalculate the Euclidean distanceˆN d ×d (k ,l )−ˆN d ×d (i ,j ) 22,ain (4)by replacing the vectors ˆN d ×d (i ,j )and ˆNd ×d (k ,l )with lower-dimensional ones f RP [i ,j ]and f RP [k ,l ]deter-mined by random projection,respectively.In random projection,the original d -dimensional data is projected to a k -dimensional subspace via using a random k ×d matrix R ,which can be mathematicallyrepresented asf RP [·]=R k ×d ·ˆNd ×d (·)(5)In order to guarantee the random matrix R k ×d is orthogonal,whichyields a projection with no significant distortion,the elements of R k ×d are often Gaussian distributed.Thereafter,the weight for each pair of vectors can be defined asw RP k ,l ,i ,j =exp − f RP [k ,l ]−f RP[i ,j ] 22,ah 2(6)Finally,the non-local means filtering can be performed by replacingw k ,l ,i ,j in (2)with w RP k ,l ,i ,j .The computational complexity of the search window limited non-local means algorithm [2]is of O(N 2M 2d 2),where N 2,M 2and d 2are the number of pixels in the image,in the search window and in the neighbourhood,respectively.In comparison,the complexity when using a k -dimensional subspace is O(N 2M 2kd ).The additional cost in performing random projection is O(N 2kd 2).Therefore,the total com-plexity for the proposed method is O(N 2kd (M 2+d )).In the case of d ..k ,the saving of computational cost is very significant.Experimental results:We performed two sets of experiments to gauge the advantage of using the proposed method rather than the search window limited NLM algorithm:one set to test execution time,and the other set to test denoising quality.In the experiments,three types of images (Lena,Fingerprint,and Texture)with different sizes were used to implement the performance assessment in MATLAB.Moreover,the execution times were run on an IBM T61p 2.2GHz laptop,and the denoising performance is quantitatively evaluated by the criterions of PSNR and SSIM [8].For the following simulation,the search window size is set to be M ¼21,and each neighbourhood vector is projected to a k -dimensional subspace (k ¼1).In the experiments of execution time test,the neighbourhood size d took on the value of 5,7and 9,respectively.Table 1shows the speed acceleration afforded by the proposed method,from which we can see the acceleration is more obvious with the increase of neighbourhood size.In the following experiments of denoising quality assessment,the neighbourhood size d is fixed to be 7,and the variation value of Gaussian additional noise is 0.03.The results in Table 2show that our proposed method achieves higher PSNR and SSIM,which indicates that the accuracy of the proposed method is superior to the search window limited NLM algorithm.Table 1:Comparison of computational performance under differentneighbourhood sizes (the values listed in the Table are the execution times,units seconds)Table 2:Comparison of denoising quality with quantitativecriterionsConclusions:In this Letter,we recalled the non-local means filtering algorithm for image denoising;we then accelerated the NLM by intro-ducing a random projection based dimensional reduction approach toproject the neighbourhood vectors ˆNd ×d (·)onto a lower-dimensional subspace,which results in considerable savings in the cost of weight calculation.Experimental results validate that the accuracy and compu-tational cost of the NLM image denoising algorithm can be improved by computing weight after a random projection.Acknowledgments:This work is partly supported by the National Natural Science Fund of China (nos 60902080and 60725415)and the Fundamental Research Funds for Central Universities (no.72104909).ELECTRONICS LETTERS 3rd February 2011Vol.47No.3#The Institution of Engineering and Technology201116September2010doi:10.1049/el.2010.2618i and Y.T.Yang(Department of Microelectronics,Xidian University,Xi’an,People’s Republic of China)E-mail:rlai@References1Buades,A.,Coll,B.,and Morel,J.M.:‘A review of image denoising algorithms,with a new one’,Multiscale Model.Simul.SIAM Interdiscip.J.,2005,4,(2),pp.490–5302Buades,A.,Coll,B.,and Morel,J.M.:‘Denoising image sequences does not require motion estimation’.Proc.IEEE2005Int.Conf.Advanced Video and Signal Based Surveillance,Teatro Sociale,Italy,pp.70–74 3Mahmoudi,M.,and Sapiro,G.:‘Fast image and video denoising via nonlocal means of similar neighborhoods’,IEEE Signal Process.Lett., 2005,12,(12),pp.839–8424Orchard,J.,Ebrahimi,M.,and Wong,A.:‘Efficient non-local means denoising using the SVD’.Proc.2008IEEE Int.Conf.Image Processing,San Diego,CA,USA,pp.1732–17355Wang,J.,Guo,Y.W.,Ying,Y.T.,Liu,Y.L.,and Peng,Q.S.:‘Fast non-local algorithm for image denoising’.Proc.2006IEEE Int.Conf.Image Processing,Atlanta,GA USA,pp.1429–14326Karnati,V.,Uliyar,M.,and Dey,S.:‘Fast non-local algorithm for image denoising’.Proc.2008IEEE Int.Conf.Image Processing,San Diego, CA,USA,pp.3873–38767Bingham,E.,and Mannila,H.:‘Random projection in dimensional reduction:applications to image and text data’.Proc.2001ACM Int.Conf.Knowledge Discovery and Data Mining,San Francisco,CA, USA,pp.245–2508Wang,Z.,Bovik,A.C.,Sheikh,H.R.,and Simoncelli,E.P.:‘Image quality assessment:from error visibility to structural similarity’,IEEE Trans.Image Process.,2004,13,(4),pp.600–612ELECTRONICS LETTERS3rd February2011Vol.47No.3。
a rXiv:h ep-ph/53198v12Mar25Renormalizability of the massive Yang-Mills theory rin Institute for Nuclear Research of the Russian Academy of Sciences,60th October Anniversary Prospect 7a,Moscow 117312,Russia Abstract It is shown that the massive Yang-Mills theory is on mass-shell renormalizable.Thus the Standard Model of electroweak interactions can be modified by removing terms with the scalar field from the Lagrangian in the unitary gauge.The resulting electroweak theory without the Higgs particle is on mass-shell renormalizable and unitary.1The massive Yang-Mills theory[1]is considered to be non-renormalizable [2],see also[3,4]and references therein.The known way to get renormaliz-able and unitary theory with massive Yang-Mills bosons is due to the Higgs mechanism of spontaneous symmetry breaking[5].The mechanism is used in the Standard SU(2)×U(1)Model of electroweak interactions[6]which is established to be renormalizable[7],see also[8]and references therein.In this way one introduces in the Model the scalar Higgs particle which one can hope to see in experiments.The purpose of the present paper is to show that the massive Yang-Mills theory is in fact on mass-shell renormalizable.Hence the Standard Model can be modified by discarding from the Lagrangian in the unitary gauge all terms containing the scalarfield.Let us consider the massive Yang-Mills theory of gaugefields W aµ(x)de-fined by the generating functional of Green functions in the path integral formZ(J)=14F aµνF aµν+1(2π)4gµν−kµkν/m2To establish on mass-shell renormalizability of the massive Yang-Mills theory (1)oneshould show that the S-matrix elements can be made finite by means of counterterms which can be absorbed into renormalization constants of the parameters g and m although the Green functions are divergent.We will work within perturbation theory.To regularize ultraviolet divergences we will use for convenience dimen-sional regularization [9]with the space-time dimension d =4−2ǫ,ǫbeing the regularization parameter.Let us consider the known model given by the initial SU (2)-invariant Lagrangian possessing the spontaneously broken symmetryL =−12aW a µ Φis the covariant derivative,τa are the Pauli matrices,λ>0,v 2>0.To get the complete Lagrangian one makes the shift of the scalar fieldΦ(x )=12 iφ1(x )+φ2(x )√NdW dφdχd 4F a µνF a µν+m 22∂µφa ∂µφa +12χ2+g 2χW a µW a µ+g 24mχ(χ2+φa φa )−g 2M 22ξ(∂µW a µ+ξmφa )23+∂µc a c a −g c a c a +g c a c b φc+countertermsThis theory describes three physical massive vector bosons with the mass m =gv/√4F a µνF a µν+m 22∂µχ∂µχ−M 22χW a µW a µ+g 24m χ3−g 2M 2N dW dφdχexp i dx L R +J a µW a µ+Kχ ∆L (W )δ(∂µW µ)(6)where ∆L (W )is the Faddeev-Popov determinant[13]and L R is obtained from L R ξby omitting terms depending on ξand c a (and by corresponding modification of counterterms).The Lagrangian L R is invariant under the following gauge transformationsW a µ→ W ωµ a =W a µ+∂µωa +˜g f abc W b µωc +O (ω2)(7)φa →(φω)a =φa −˜mωa −˜g 2χωa +O (ω2)4χ→χω=χ−˜g z 2g ˜m =z 1NdW dφdχexp i dx L U +J µW ˜ωµ+Kχ˜ω ∆U (χ)δ(φ)(9)where ˜ωis defined from the equation ∂µ W ˜ωµ a =∂µW a µ+∂µ˜ωa +˜g f abc W b µ˜ωc +O (˜ω2)=0(10)The Lagrangian L U is given ineq.(5).The functional ∆U (χ)can be presented on the surface φa =0as ∆U (χ)=det |˜m +˜g 2m χ(x ))3dxIn dimensional regularization this functional is just a constant and can be absorbed in the normalization factor N although this simplification is not essential for the following derivation.One obtainesZ L (J,K )=1N dW dχexp idx (L U +J µW µ+Kχ) (12)5only by source terms.It is known that this difference is not essential for the S-matrix elements,see e.g.[8].Thus the physical equivalence of the L-gauge and the U-gauge is proved.From eq.(11)one sees that the counterterms of L U are given by the coun-terterms of L R atφa(x)=0.To consider renormalization for our purpose it is convenient to use the Bogoliubov-Parasiuk-Hepp subtraction scheme[14].As it is well known in this scheme a counterterm of e.g.a primitively divergent Feynman diagram is the truncated Taylor expansion of the diagram itself at somefixed values of external momenta.Hence counterterms of mass dependent diagrams are also mass dependent.Let us now analyze the dependence of the Green functions on the Higgs mass M.We will use for this purpose the expansion in large M.The algo-rithm for the large mass expansion of Feynman diagrams is given in[15],it can be rigorously derived e.g.with the technique of[16].The representation(11)ensures for the regularized Green functions of the fields W andχthat the large M-expansion of M-dependent contributions contain either terms with integer negative powers of M2or terms with non-integer powers of M2(non-integer powers containǫ).This is because each vertex with the factor M2has three or four attachedχ-lines due to the structure of L U.Corresponding counterterms(i.e.counterterms relevant for L U)have the same property within the large-M expansion.(In contrast, counterterms of e.g.the four-φvertex in L R contain polynomial in M terms because of the M2-factors in the couplings of L R.)Let us further consider Green functions with external W-bosons only.We willfirst shortly formulate the result.The eq.(11)ensures that if one removes from a renormalized Green function M-dependent terms then the remaining part isfinite.On the Lagrangian level it means that one removes from L U all terms containing thefieldχ.Thus one obtains the theory˜Z(J)=1F aµνF aµν+m2W aµW aµ+counterterms4withfinite off-shell Green functions,where W˜ωµis given by(10).Since the difference between W˜ωµand Wµin the source term is not essential for S-matrix elements the massive Yang-Mills theory is renormalizable on mass-shell.6Let us elaborate these arguments in more detail.The representation (11)ensures,seee.g.[8],that the following on mass-shell expressions for the renormalized Green functions (relevant for the S-matrix elements)should coincide1znn i =1(k 2i −m 2)G a 1...a n µ1...νn (k 1...k n )|L −gauge k 2i =m 2=(14) 1z n n i =1(k 2i −m 2)G a 1...a n µ1...νn (k 1...k n )|U −gauge where z is the residue of the propagator poleδab g µν−k µk νk 2 e ikx δ2Z (J )NdW exp i dx L Y M +J a µW a µ (16)L Y M =−1z 2gf abc W b µW c ν)2+z m m 2W a µW a µAfter renormalizability is established one can fix renormalization con-stants z 1,z 2and z m within the theory (16)(without referring to the L-gauge)by proper normalization conditions.It is known that the Higgs theories of vector mesons posses so called tree level unitarity,see e.g.[3]and references therein.Tree level cross sections of such theories grow at high energies slowly enough and do not exceed the7so called unitary limit imposed by the unitarity condition.The reversed statement is also proved:from the condition of tree level unitarity follows that a theory of vector mesons should be a Higgs theory[17].But one can see that tree level unitarity is not the necessary condition for renormalizability. Tree level unitarity is violated in the massive Yang-Mills theory.It indicates that higher order contributions become relevant at high energies.The above derivation of on mass-shell renormalizability is applicable also to other gauge groups.It can be straightforwardly applied to the Standard SU(2)×U(1)Model of electroweak interactions.The presence of the U(1) gauge boson and of fermions does not change the derivation.One can remove from the Lagrangian in the unitary gauge all terms containing the scalarfield. 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a rXiv:076.452v1[he p-th]27J un27FTI/UCM 75-2007Renormalisability of the matter determinants in noncommutative gauge theory in the enveloping-algebra formalism C.P.Mart´ın 1and C.Tamarit 2Departamento de F´ısica Te´o rica I,Facultad de Ciencias F´ısicas Universidad Complutense de Madrid,28040Madrid,Spain We consider noncommutative gauge theory defined by means of Seiberg-Witten maps for an arbitrary semisimple gauge group.We compute the one-loop UV divergent matter contributions to the gauge field effective ac-tion to all orders in the noncommutative parameters θ.We do this for Dirac fermions and complex scalars carrying arbitrary representations of the gauge group.We use path-integral methods in the framework of dimen-sional regularisation and consider arbitrary invertible Seiberg-Witten maps that are linear in the matter fields.Surprisingly,it turns out that the UV divergent parts of the matter contributions are proportional to the noncom-mutative Yang-Mills action where traces are taken over the representation of the matter fields;this result supports the need to include such traces in the classical action of the gauge sector of the noncommutative theory.PACS:11.10.Nx;11.10.Gh;11.15.-qKeywords:Renormalisability,Seiberg-Witten map,noncommutative gauge theories.The issue of renormalisability of noncommutative gauge theories in the enveloping-algebraapproach has been a subject of intense research in the last years[1,2,3,4,5,6,7,8,9].Theoutcome of this research so far shows that NC Yang Mills is one-loop renormalisable up tofirst order inθ[5,9]—in fact,up to order two for the case of NC U(1)Yang-Mills[3]—butrenormalisability is spoiled by the presence of Dirac fermions in the fundamental repesentationor complex scalars in the U(1)case[2,3,4,8].However,in all cases the gauge sector of thetheory remains renormalisable despite the presence of matter;this has also been checked for anoncommutative extension of the Standard Model[6]in which the traces in the gauge sector aretaken over all the different particle representations.This renormalisability of the gauge sectoris quite intriguing and far from trivial since BRS invariance and power-counting do not accountfor it.Indeed,take a simple compact gauge group,then,power-counting and BRS invariancedo not restrict the one-loop UV divergent part of the effective action of the gaugefield in thebackground-field gauge to the noncommutative Yang-Mills action,but to a linear combinationwith arbitrary UV divergent coefficients of the noncommutative Yang-Mills action and termslikeθαβTr d4x Fαβ⋆Fµν⋆Fµν,θαβTr d4x Fµα⋆Fβν⋆Fµν,etc...The confirmation of the renormalisablity we have mentioned at higher orders inθand its understanding–perhaps,asa by-product of an as yet undiscovered symmetry of the theory–,as well as the study of itsdependence on the choice of traces for the noncommutative Yang-Mills action,are still openproblems.As afirst step in this direction,in this paper,we compute to all orders inθthe UV part ofthe one-loop effective action obtained by integrating out the matterfields in noncommutativegauge theory for arbitrary semisimple gauge group.By“matter”we mean Dirac fermions andcomplex scalars in an arbitrary unitary irreducible representation of the gauge group.Whatwe have obtained is that,in both cases and in dimensional regularisation with D=4+2ǫ,the pole part of the effective action of the gaugefield turns out to be proportional to thenoncommutative Yang-Mills action with traces taken over the representation of the gaugegroup acting on the matterfields,namely:=1Γf[A]one-looppole,A-dep.Tr d4x Fµν⋆Fµν,Fµν=∂µAν−∂νAµ−i[Aµ,Aν]⋆.(1)192π2ǫThis result supports the need to consider such types of traces for models aiming to have therenormalisability property;in fact,all models considered so far with a one-loop,order-θrenor-malisable gauge sector have these types of traces.A relevant example is the noncommutativeversion of the Standard Model in ref.[6],whose gauge sector involves a non-trivial sum oftraces over all the particle representations of the model.Our result holds for the class ofSeiberg-Witten applications for which the map between the noncommutative and ordinary matterfields is linear and invertible;the computation to all orders inθis feasible due to the possibility of changing variables in the functional integrals from the ordinaryfields to the noncommutativefields.The result we have obtained is quite surprising since BRS invariance and power-counting of the theory formulated in terms of the ordinaryfields do not enforce it,and,it is relevant in the phenomenological applications of noncommutative gauge theories, since it supports the robustness of the predictions based on the gauge sector of the theory. These phenomenological predictions can certainly be tested at the LHC[10,11,12,13].Using a similar notation as that employed in ref.[14],we consider a noncommutative gauge theory with a semisimple gauge group of the form G1×···×G N with G i simple for i=1...s and abelian for i=s+1,...,N.Then the ordinary gaugefield will be of the formaµ=sk=1g k(a kµ)a(T k)a+Nl=s+1g l a lµT l,where the T′s are generators of unitary irreducible representations of the group factors.The matterfields to consider are Dirac fermionsψand complex scalarsφin an irreducible rep-resentation of G and therefore carrying multi-indices I=i1...i s for the irreducible factors. In multi-index notation we can define generators and a“global”trace Tr as follows(T k)a IJ=δi1j1···(T k)a ik j k···δis j s,k=1,...,s,T l IJ=δi1j1···δis j sY l,l=s+1,...,N,Tr(T k)a(T k′)a′=(T k)a IJ(T k′)a′JI.In order to build noncommutative actions for the matterfields we need the Seiberg-Witten maps for the noncommutative gaugefield Aµand the matterfields,i.e.,noncommutative fermionsΨαI and complex scalarsΦI.We make no assumption on the map for the gauge field,but for the matterfields we consider maps of the formΨαI=(δIJδαβ+M[aµ,∂,γ;θ]αβIJ)ψβJ,ΦI=(δIJ+N[aµ,∂;θ]IJ)φJ,(2)and analogously for the Dirac adjoint fermion¯Ψ=¯Ψ†γ0and the complex conjugateΦ∗of Φ.γµdenotes the gamma matrices satisfying{γµ,γν}=2ηµν.With this notation we define next the actions for the matterfields in noncommutative spacetimeS f= d4x(¯Ψ⋆iγµDµΨ−m¯Ψ⋆Ψ),S sc= d4x((DµΦ)∗⋆DµΦ−m2Φ∗Φ)−V(Φ),(Dµ)IJ=δIJ∂µ−iAµIJ⋆,(3)where⋆denotes the usual Moyal product and V(Φ)is an arbitrary noncommutative gauge-invariant potential that will not contribute to the Aµ-dependent part of the gauge effective action.Our objective is to compute the divergent part of the one-loop effective actions for the gaugefield that are formally defined byΓf/sc[a;θ]=−i ln Z f/sc[a;θ],(4)Z f[a;θ]=N f [d¯ψ][dψ]exp(iS f),Z sc[a;θ]=N sc [dφ∗][dφ]exp(iS sc),N f/sc=Z f/sc[0;θ]−1.The previous expressions relate the gauge effective actions to the determinants of the operators appearing in the actions for the matterfields.We will make sense out of these formal definitions by using dimensional regularisation in D=4+2ǫdimensions;this will make the divergent contributions appear as poles inǫ.Note that,in order to work out the UV divergence ofΓf/sc to all orders inθby integrating out the matterfields in the functional integrals in eq.(4), one would need to know the Seiberg-Witten map to all these orders.We can avoid this by performing a change of variables in the functional integrals from the ordinaryfields to the noncommutative ones.Recalling eq.(2),[d¯ψ][dψ]=det(I I+M)det(I I+¯M)[d¯Ψ][dΨ],[dφ∗][dφ]=det(I I+N)−1det(I I+N∗)−1[dΦ∗][dΦ].The above determinants are defined in dimensional regularisation by a diagrammatic expansion where the propagators are equal to the identity.As a consequence they are given in momentum space by tadpole-like integrals,which are zero.Therefore,Z f[a;θ]=N f [d¯Ψ][dΨ]exp(iS f),Z sc[a;θ]=N sc [dΦ∗][dΦ]exp(iS sc)(5) and,since the matter actions given in eq.(3)depend on the noncommutative gaugefield Aµ, we have that the dependence of the effective actionsΓon aµis through a dependence on Aµ:Γf/sc[a;θ]=Γf/sc[A;θ].In particular,when the eqs.(4)and(5)definingΓf/sc are interpreted diagramatically,it is clear that the potential V(Φ)makes no contribution to the Aµ-dependent part of the gauge effective actions.Thus,eqs.(4)and(5)allow us to obtain the Aµ-dependent parts of the effective actionsΓf/sc as the determinants of the operators that appear in the actions for the matter in terms of the noncommutativefields.Integrating over[d¯Ψ],[dΨ]and[dΦ∗],[dΦ]neglecting V(Φ)we obtain:iΓf[A]A-dep.=ln det[∂/+im−iA/⋆]nTr[(∂/+im)−1iA/⋆]n,iΓsc[A]A-dep.=−ln det[iD2+im2]nTr[(i∂2+i m2)−1((∂·A)⋆+2A·∂⋆−iAµ⋆Aµ⋆))]n.(6)In the previous expressions Tr denotes a trace over discrete indices and integration over the continuous indices of the corresponding operators.The operators(∂/+im)−1and(i∂2+i m2)−1 have matrix elements given by ordinary propagators:y|(∂/+im)−1|x = d D p p2−m2+i0+, y|(i∂2+im2)−1|x = d D p p2−m2+i0+.(7) Since we are interested in the divergent part ofΓf/sc,we need to identify the contributions in eq.(6)that yield the poles at D=4.This can be done by using power-counting arguments as follows.Let usfirst note that the propagators in eq.(7)are diagonal in the colour indices, then,one realises that the trace in eq.(6)forces a trace in the colour indices of the background gaugefields AµIJ.Therefore we can writeiΓf/sc[A]A-dep.≡∞n=1d d x1... d d x n Tr[Aµ1(x1)...Aµn(x n)]Γf/sc(n)µ1...µn[x1,...,x n],(8)where the mass dimension ofΓf/sc(n)µ1...µn[x i]is,for D=4,4+3n.In momentum space Γf/sc(n)µ1...µn[p i]will be of dimension4−n and,due to the translation invariance of the propagators in eq.(7),it is given by a single loop integral.Then,power-counting tell us that Γf/sc(n)with n≥5arefinite.Thus we only need to work out contributions with up to4 background gaugefields Aµ.Let us start with the case of fermions.Going over to momentum space and starting from eq.(6),we can expressΓf(n)[A]in terms of a loop integral as followsΓf(n)µ1...µn [x1,...,x n]=(−1)n+1(2π)D(2π)Dδ i p i e i P n i=1p i x i e−i P n i<j p i◦p j×× dq[(q+p1)−m2][q2−m2][(q−p2)2−m2]···[(q− n−1i=2p i)2−m2].(9) p◦q≡ito the presence of interaction vertices with one and two A ′µs .Instead of a closed formula for Γ(n )as the one just given for fermions in eq.(9),we provide formulae for the potentially divergent contributions Γsc (k ),1≤k ≤4.Γsc (n )µ1...µn [x 1,...,x n ]= n i =1d D p i(2π)D2q µ(2π)D e −p 1◦p 2ηµν2 d D q [(q +p 1)2−m 2]2,˜Γ(3)µνρ[p 1,p 2,p 3]= d D q [q 2−m 2][(q −p 1)2−m 2]−1(2π)D e −P i<j p i ◦p j (2q +p 1)µ(2q −p 2)ν(2q −p 2+p 1)ρ2 d D q [q 2−m 2][(q −p 3−p 4)2−m 2]− d D q [q 2−m 2][(q −p 3)2−m 2][(q −p 3−p 4)2−m 2]+1(2π)De −P i<j p i ◦p j (2q +p 1)µ(2q −p 2)ν(2q −2p 2−p 3)ρ(2q −p 2−p 3+p 1)σReferences[1]A.Bichl,J.Grimstrup,H.Grosse,L.Popp,M.Schweda and R.Wulkenhaar,JHEP0106(2001)013[arXiv:hep-th/0104097].[2]R.Wulkenhaar,JHEP0203(2002)024[arXiv:hep-th/0112248].[3]M.Buric and V.Radovanovic,JHEP0210(2002)074[arXiv:hep-th/0208204].[4]M.Buric and V.Radovanovic,JHEP0402(2004)040[arXiv:hep-th/0401103].[5]M.Buric,tas and V.Radovanovic,JHEP0602(2006)046[arXiv:hep-th/0510133].[6]M.Buric,V.Radovanovic and J.Trampetic,JHEP0703,030(2007)[arXiv:hep-th/0609073].[7]X.Calmet,Eur.Phys.J.C50(2007)113[arXiv:hep-th/0604030].[8]C.P.Martin, D.Sanchez-Ruiz and C.Tamarit,JHEP02(2007)065[arXiv:hep-th/0612188].[9]tas,V.Radovanovic and J.Trampetic,arXiv:hep-th/0703018.[10]M.Buric,tas,V.Radovanovic and J.Trampetic,Phys.Rev.D75(2007)097701.[11]A.Alboteanu,T.Ohl and R.Ruckl,Phys.Rev.D74(2006)096004[arXiv:hep-ph/0608155].[12]M.Mohammadi Najafabadi,Phys.Rev.D74(2006)025021[arXiv:hep-ph/0606017].[13]W.Behr,N.G.Deshpande,G.Duplancic,P.Schupp,J.Trampetic and J.Wess,Eur.Phys.J.C29(2003)441[arXiv:hep-ph/0202121].[14]F.Brandt,C.P.Martin and F.R.Ruiz,JHEP0307(2003)068[arXiv:hep-th/0307292].[15]C.P.Martin and D.Sanchez-Ruiz,Nucl.Phys.B572(2000)387[arXiv:hep-th/9905076].。
a r X i v :h e p -t h /0012024v 3 25 J a n 2001FT/UCM–60–2000The BRS invariance of noncommutative U (N )Yang-Mills theoryat the one-loop levelC.P.Mart´ın*andD.S´a nchez-Ruiz †Departamento de F´ısica Te´o rica I,Universidad Complutense,28040Madrid,SpainWe show that U (N )Yang-Mills theory on noncommutative Minkowski space-timecan be renormalized,in a BRS invariant way,at the one-loop level,by multiplicative dimensional renormalization of its coupling constant,its gauge parameter and its fields.It is shown that the Slavnov-Taylor equation,the gauge-fixing equation and the ghost equation hold,up to order ¯h ,for the MS renormalized noncommutative U (N )Yang-Mills theory.We give the value of the pole part of every 1PI diagram which is UV divergent.1.-IntroductionNoncommutative field theories occur both in the ordinary (commutative space-time)field theory setting and in the realm of string theory.The study of the large N limit of ordinary field theories naturally leads to field theories over noncommutative space [1,2].General relativity and Heisenberg’s uncertainty principles give rise,when strong gravita-tional fields are on,to space-times defined by noncommuting operators [3],whereupon it arises the need to define quantum field theories over noncommutative space-times.Su-per Yang-Mills theories on noncommutative tori occur in compactifications of M(atrix)-theory [4],M(atrix)-theory on noncommutative tori being the subject of a good many papers [5].Theories of strings ending on D-branes in the presence of a NS-NS B-field lead to noncommutative space-times;their infinite tension limit being –if unitarity allows it–certain noncommutative field theories [6].It is therefore no wonder that a sizeable amount of work has been put in understanding,either in the continuum [7,8]or on the lattice [9],whether quantum field theories make sense on noncommutative space-times.Applications to collider physics and Cosmology have just begun to come up [10].Quantum field theories on noncommutative space-time present a characteristic con-nection between UV and IR scales:the virtual high-momenta modes contributing to a given Green function yield,when moving around a planar loop,an UV divergence,butgive rise to an IR divergence-even if the classical Lagrangian has only massivefields-as they propagate along a nonplanar loop.This is the UV/IR mixing unveiled in ref.[11], which has been further investigated in refs.[12].The new–as regards to quantumfield theory on commutative space-time–IR divergences that occur in noncommutativefield the-ories makes it impossible[13],beyond a few loops,that these theories be renormalizable ´a la Bologiubov-Parasiuk-Hepp-Zimmerman[14],if supersymmetry is not called in[15]. Besides,they lack a Wilsonian action[11],which puts in jeopardy the implementation in noncommutativefield theories of Wilson’s renormalization group program[16];and,hence, the existence of a continuum limit for these theories.The existence of a continuum limit for noncommutativefield theories has been studied in ref.[17].It is well known[7]that,at the one-loop level,only if a diagram is planar it can be UV divergent and that the momentum structure of this divergence,should it exist,is the product of a polynomial of the appropriate dimension of the external momenta(the UV degree of divergence of the Feynman loop integral)by suitable Moyal phases.Besides,if the noncommutative diagram has an ordinary counterpart(the diagram when space-time is commutative),the polynomial of the external momenta which carries the UV divergence is the same for both diagrams.This might lead us to think that noncommutativefield theories are always one-loop renormalizable,if their ordinary counterpart is;which would in turn render almost trivial the issue of the one-loop renormalizability of noncommutative field theories.One cannot be more mistaken.There are certain∗-deformations ofλφ4that are shown not to be renormalizable at the one-loop level:see ref.[18].It is common lore that U(N)Yang-Mills theories on noncommutative Minowski are one-loop renormalizable.Indeed,if one assumes that gauge,better,BRS invariance is preserved at the one-loop level,it is difficult to think otherwise.However,statements about the BRS invariance of afield theory are rigorous only if they are based either on explicit computations or on the Quantum Action Principle[19]plus BRS cohomology techniques [20].Since we lack a Quantum Action Principle for noncommutativefield theories,we should better carry out explicit computations,lest our statements will be erected on shaky ground.Even if we had a Quantum Action Principle at our disposal,it would always be advisable to check general results by performing explicit computations up to a few loops.In this paper we shall compute the complete UV divergent contribution to the1PI functional of4-dimensional noncommutative U(N)Yang-Millsfield theory for an arbitrary Lorentz gauge-fixing condition.We shall use dimensional regularization to carry out the computations.We shall thus show by explicit computation that this1PI functional is the sum of two integrated polynomials(with respect to the Moyal product)of thefield and its derivatives.Thefirst term is the noncommutative Yang-Mills action.This term is nontrivial in the cohomology of the noncommutative Slavnov-Taylor operator and gives rise to the renormalization of the coupling constant.The second term is exact in the cohomology of the noncommutative Slavnov-Taylor operator and gives rise to the wave function and gauge-fixing parameter renormalizations.This result constitutes a highly nontrivial check of the one-loop BRS invariance of the the theory;the high nontriviality of the check stemming from the fact that the UV divergence of each planar diagram contributing to the4-point function of the gaugefield has a structure which differs very much from that of the4-point tree-level contribution.The BRS invariance of the MS2(minimal subtraction)UV divergent part of the one-loop1PI functional leads,as we shall see,to a renormalized BRS invariant1PI functional up to order¯h.The layout of this paper is as follows.In Section2,we set the notation and display the Feynman rules for noncommutative U(N)Yang-Millsfield theory in an arbitrary Lorentz gauge.In Section3,we give the UV divergent divergent contribution to the one-loop1PI Green functions in the MS scheme of dimensional regularization and,from these data we obtain the MS UV divergent part of1PI functional written in an explicitly BRS invariant form.Section4is devoted to comments and conclusions.In the Appendices we give for the record the UV divergent contribution to each one-loop1PI diagram in the MS scheme.2.-Notation and Feynman rulesThe classical U(N)field theory on noncommutative Minkowski space-time is given by the Yang-Mills functionalS Y M=−1(2π)4d4p2θµνqµpνf(q)g(p).Here f(q)and q(p)are,respectively,the Fourier transforms of f(x)and g(x),the latter being two rapidly decreasing functions at infinity.θµνwill be taken either magnetic or light-like,thus unitarity holds[21].The reader is referred to ref.[22]for introductions to Noncommutative geometry.We shall denote1where S Y M has been given in eq.(1)andS gf=12B+∂µAµ)](x),S ext= d4x Tr Jµ⋆sAµ+H⋆sc (x),whereλis the gauge-fixing parameter.Taking into account thatN2−1a=0(T a)j1i1(T a)j2i2=T Rδj1i2δj2i1,one readily shows that the gaugefield propagator readsd4p p2T R g2δj2i1δj1i2 gµ1µ2+(λ′−1)pµ1pµ2δJµδΓδHδΓδ¯c =0.(3)For the one-loop,Γ1,contribution toΓ[Aµ,B,c,¯c;Jµ,H],the Slavnov-Taylor equation boils down toBΓ1=0,(4) whereB= d4x Tr δS clδAµ+δS clδJµ+δS clδc+δS clδH+Bδµ11j i 1µ22j i2G (0)µ1µ2AAj 1j 2i 1i 2(k )=T R g 2δj 2i 1δj 1i 2(−i )k 2µ11j i µ22i 233j i 3i S µ1µ2µ3AAAj 1j 2j 3i 1i 2i 3(k 1,k 2,k 3=−k 1−k 2)=ig 2T Rδj 4i 1δj 1i 2δj 2i 3δj3i 4e i [ω(k 1,k 2)+ω(k 3,k 4)](2g µ1µ3g µ2µ4−g µ1µ4g µ2µ3−g µ1µ2g µ3µ4)+(1243)+(1324)+(1342)+(1423)+(1432)1j i 12j i 2G (0)c ¯c j 1j 2i 1i 2(k )=T R δj 2i 1δj 1i2i T Re −iω(k 1,k 2)δj 2i 1δj i 2δj 1i −e iω(k 1,k 2)δj i 1δj 1i 2δj2iFigure 1:Feynman rules for noncommutative U (N )Yang-Mills theory:Propagators andvertices with no external fields.(i j k l )denotes a permutation of (1234).To regularize the Feynman integrals of our theory will shall use dimensional regu-larization.To define θµνin dimensional regularization we shall follow the philosophy in ref.[24]and say that θµνis an algebraic object which satisfies the following equationsθµν=−θνµ,ˆηµρθρν=0,p µθµρηρσθσνp ν≥0,∀p µ.Here ηρσand ˆηµρare,respectively,the “D-dimensional”and “D-4-dimensional”metrics asdefined in ref.[24].It is not difficult to convince oneself that,with the previous definition of θµν,the one-loop Feynman integrals do have a mathematically well-defined expressions and that the techniques used in ref.[24]to prove the Quantum Action Principle for dimen-sionally regularized ordinary field theories can also be employed here to conclude that at5µ22j i 21j i 11S (0)c ;sA j 1j 2i 1;i 2µ2(k 1)=i k 1µ2δj 2i 1δj1i 23µ3jµ22j i 21j i 1S (0)cA ;sA j 1j 2i 1i 2µ2;j3i 3µ3(k 1,k 2,k 3)=−iδj 2i 1δj 3i 2δj 1i 3e −iω(k 1,k 2)−δj 3i 1δj 1i 2δj2i 3e iω(k 1,k 2)g µ2µ31j i12j i 2j 3S (0)cc ;sc j 1j 2j3i 1i 2;i 3(k 1,k 2,k 3)=iδj 2i 1δj 3i 2δj 1i 3e −iω(k 1,k 2)−δj 3i 1δj 1i 2δj2i 3e iω(k 1,k 2)Figure 2:Feynman rules for noncommutative U (N )Yang-Mills theory:Vertices withinsertions of BRS variations.the one-loop level our dimensionally regularized noncommutative U (N )is BRS invariant.The so regularized Feynman diagrams are meromorphic functions of D ,with simple poles at D =4,if they are planar,and no poles,if they are nonplanar and P µθµρηρσθσνP ν>0for any linear combination,P ,of the external momenta.3.-The MS UV divergent part of the 1PI functionalThe fact that at the one-loop level only planar diagrams can be UV divergent and the fact that planar diagrams have the same UV degree as their ordinary field theory counterparts readily leads to the conclusion that the one-loop UV divergent 1PI functions are following:ΓAA ,ΓAAA ,ΓAAAA ,Γ¯c c ,Γ¯c Ac ,ΓJc ,ΓJAc and ΓHcc ,with obvious notation.We have computed the one-loop UV divergent contribution to all the divergent 1PI Green functions.Taking into account the results presented in the Appendices,one obtains the following values for these UV divergent part in the MS scheme:Γ(AA ),(pole )µ1µ2j 1j 2i 1i 2(p )=13−(λ′−1)N δj 2i 1δj 1i 2 p 2g µ1µ2−p µ1p µ2Γ(AAA ),(pole )µ1µ2µ3j 1j 2j 3i 1i 2i 3(p 1,p 2,p 3)=13−3Γ(AAAA ),(pole )µ1µ2µ3µ4j 1j 2j 3j 4i 1i 2i 3i 4(p 1,p 2,p 3,p 4)=13+2(λ′−1)N δj 4i 1δj 1i 2δj 2i 3δj 3i 4e i [ω(p 1,p 2)+ω(p 3,p 4)](2g µ1µ3g µ2µ4−g µ1µ4g µ2µ3−g µ1µ2g µ3µ4)+(1243)+(1324)+(1342)+(1423)+(1432)Γ(c ¯c ),(pole )j 2j 1i 2i 1(p 1)=− 12N δj 2i 1δj 1i 2p 21Γ(c ¯c A ),(pole )j 2j 1i 2i 1j 3i 3µ(p 2,p 1,p 3)=1(4π)2ε g 2T R 1−λ′−1(4π)2εg 2T R1+(λ′−1)Ne −iω(p 1,p 2)δj 2i 1δj 3i 2δj 1i 3−eiω(p 1,p 2)δj 3i 1δj 1i 2δj 2i 3 g µ2µ3Γ(ccH ),(pole )j 1j 2j 3i 1i 2i 3(p 1,p 2,p 3)=i1δΦi 11δΦi 22...δΦi n n,Γ(φ1φ2...φnK φ)i 1i 2...i ni =δsφi ·Γso that−2Z(1)g+2Z(1)A=13−1(4π)2ε2N 24(λ′−1) g2T R,−2Z(1)g+4Z(1)A=13−(λ′−1) g2T R.Hence,Z g=1−16g2T R,Z A=1−14(λ′−1)]g2T R.(7)Analogously,Z¯c Z c=1+12g2T R N,Z¯c Z A Z c=1−1(4π)2ελ′g2T R N,Z B=Z−1A,Zλ=Z2A and Z J=Z¯c.(8)That the Z s in eqs.(7)and(8)render UVfinite the one-loop1PI functional is a consequence of BRS invariance.Indeed,in view of eq.(6),it is not difficult to show that the singular contribution,Γ(pole),to the dimensionally regularized1PI functional can be cast into the formΓ(pole)=a(4π)2ε22(4π)2ε3+λ′(4π)2ελ′N T R g2,and B D is the linearized Slavnov-Taylor operator acting upon the space of formal∗-polynomials constructed with“D-dimensional”monomials of thefields and their deriva-tives.B D is defined as followsB D= d D x Tr δS clδAµ+δS clδJµ+δS clδc+δS clδH+Bδthe sum of two terms:thefirst term,the Yang-Mills term,is B D-closed and the second term is B D-exact(recall that B2D=0).The analogy with ordinary SU(N)Yang-Mills theory is apparent.And,indeed,as as in ordinary four-dimensional Yang-Mills theory we haveZ g=1−a∂µ+β∂∂λ− φγφ d4xφ(x)δdµ=−13N T R g4.9The other renormalization group coefficients read at the one-loop level:γA=+12 N T R g2,γc=+1δB=0,δΓ(pole)δJµ=0.Hence,up to order¯h,the MS renormalized1PI functional does satisfy both the gauge-fixing equation and the ghost equation:T R δΓMSrenδ¯c+∂µδΓMSrenThat the one-loop4-point function of the gaugefield has a MS UV divergent part which does not spoil the Slavnov-Taylor equation is,though,highly nontrivial.It demands that very delicate cancellations occur upon adding all the UV divergent4-point diagrams:unlike 2-and3-point diagrams,the sum of4-point diagrams of the same type is not proportional to tree-level4-point gauge vertex.That these delicate cancellations do happen is,of course, a consequence of the fact that BRS invariance holds indeed.In this regard,we invite the reader to go to Appendix C and eq.(6)and get acquainted with the momentum and colour structure of the UV divergent contributions reported there.Note that every4-point UV divergent contribution has an overall factor equal to N,for they come only from planar diagrams.As regards the actual value of the beta function,the anomalous dimensions of thefields and gauge-fixing parameter,we have found that they are those of SU(N)’s.This result is, of course,almost trivial[7,11],once it is shown that BRS holds.Indeed,all1PI planar diagrams but the4-point diagrams can be grouped in classes of planar diagrams of the same type,the sum of the diagrams in each class being proportional to the corresponding tree-level contributions.However,taking into account what it is at stake,computations which are both explicit and thorough are much welcomed.Finally,the computations presented in this paper werefinished more than a year ago. In the meantime two papers which overlap with it have appeared[25].Ourfindings are in agreement with theirs,but ours are more general.Appendix A.Gaugefield2-point functionThe diagrams which are UV divergent in dimensional regularization are the planar diagrams in Figure3.Note that the planar tadpole diagram is not singular at D=4in dimensional regularization.µ11ji1µ22ji2µ11ji1µ22ji2(i)(ii)Figure3:1PI UV divergent2-point Feynman diagrams for the gaugefield. The minimal UV divergent part of each diagram reads (i)=2i 112−(λ′−1)6−(λ′−1)(4π)2ε Nδj2i1δj1i216pµ1pµ2 .Appendix B.Gauge field 3-point functionThe UV divergent part of the 3-point function of the gauge field is obtained from the planar diagrams in Figure 4.µ11j µ22i 23j i 333p 2µ2i 2p 33µ3j i 3µ11j p 2µ2i 2p 33µ3j i 3µ11j (i )123(ii )123(iii )123Figure 4:1PI UV divergent 3-point Feynman diagrams for the gauge field.The UV divergent part of these diagrams read (i )123=i12+3(λ′−1)(4π)2ε134Ne −iω(p 1,p 2)δj 2i1δj 3i2δj 1i 3−e iω(p 1,p 2)δj 3i1δj 1i2δj 2i 3×(p 2−p 1)µ3g µ1µ2−(p 1+2p 2)µ1g µ2µ3+(p 2+2p 1)µ2g µ1µ3 ,(iii )123=i 112Ne −iω(p 1,p 2)δj 2i 1δj 3i 2δj 1i 3−e iω(p 1,p 2)δj 3i 1δj 1i 2δj 2i 3 ×(−p 1−2p 2)µ3g µ1µ2+(2p 1+p 2)µ1g µ2µ3+(p 2−p 1)µ2g µ1µ3 .Summing over permutations one obtains (I )=(i )123+(i )231+(i )312=i12+3(λ′−1)(4π)2ε134Ne −iω(p 1,p 2)δj 2i 1δj 3i 2δj 1i 3−e iω(p 1,p 2)δj 3i 1δj 1i 2δj 2i 3×(p 2−p 1)µ3g µ1µ2−(p 1+2p 2)µ1g µ2µ3+(p 2+2p 1)µ2g µ1µ3 ,(III )=(iii )123+(iii )213=i 112Ne −iω(p 1,p 2)δj 2i 1δj 3i 2δj 1i 3−eiω(p 1,p 2)δj 3i 1δj 1i 2δj 2i 3 × (p 1−p 2)µ3g µ1µ2+(p 1+2p 2)µ1g µ2µ3+(−2p 1−p 2)µ2g µ1µ3 .The sum of (I ),(II )and (III )yields the UV divergent contribution to the 1PI 3-point function of the gauge field.Note that (I),(II)and (III)are proportional to the 3-point vertex of the gauge field.Appendix C.Gauge field 4-point functionThe UV divergent contribution to the 4-point 1PI function of the gauge field is com-puted by summing over the UV divergent parts of the diagrams,and the appropriate permutations of these diagrams,in Figure 5.µ11j 444j i 4µ22i 23j i 3331p µ11j p 444j i 4p 2µ22i 2p 33µ3j i 31p µ11jp 444j i 4p 2µ2i 2p 33µ3j i 31p µ11jp 444j i 4p µ2i 2p 33µ3j i 3(i )1234(ii )1234(iii )1234(iv )1234Figure 5:1PI UV divergent 4-point Feynman diagrams for the gauge field.The UV divergent part of the diagrams in Figure 5read (i )1234=i124g µ1µ4g µ2µ3+5+5(λ′−1)24g µ1µ3g µ2µ4+−4−2(λ′−1)+(λ′−1)224 gµ1µ4gµ2µ3+ −4−2(λ′−1)+(λ′−1)22+(λ′−1)2(4π)2ε N×e i[ω(p1,p2)−ω(p1,p3)−ω(p2,p3)]δj3i1δj1i2δj4i3δj2i4+e i[−ω(p1,p2)+ω(p1,p3)+ω(p2,p3)]δj2i1δj4i2δj1i3δj3i4 2+17(λ′−1)24 gµ1µ4gµ2µ3+−58−(λ′−1)24−13(λ′−1)24 gµ1µ2gµ3µ4+e i[−ω(p1,p2)−ω(p1,p3)−ω(p2,p3)]δj2i1δj3i2δj4i3δj1i4+e i[ω(p1,p2)+ω(p1,p3)+ω(p2,p3)]δj4i1δj1i2δj2i3δj3i4 −58−(λ′−1)28−(λ′−1)24−13(λ′−1)24 gµ1µ2gµ3µ4,(iii)1234=i 112+5(λ′−1)12 gµ1µ4gµ2µ3+172+(λ′−1)2472+7(λ′−1)224 1(4π)2ε N×e i[−ω(p1,p2)−ω(p1,p3)+ω(p2,p3)]δj3i1δj4i2δj2i3δj1i4+e i[ω(p1,p2)+ω(p1,p3)−ω(p2,p3)]δj4i1δj3i2δj1i3δj2i4 7+7(λ′−1)12 gµ1µ4gµ2µ3+7+7(λ′−1)12 gµ1µ3gµ2µ4+−8−4(λ′−1)+(λ′−1)22+7(λ′−1)212 gµ1µ3gµ2µ4+7+7(λ′−1)12 gµ1µ2gµ3µ4 +e i[ω(p1,p2)−ω(p1,p3)−ω(p2,p3)]δj3i1δj1i2δj4i3δj2i4+e i[−ω(p1,p2)+ω(p1,p3)+ω(p2,p3)]δj2i1δj4i2δj1i3δj3i4 −8−4(λ′−1)+(λ′−1)22+7(λ′−1)22+7(λ′−1)2(4π)2ε N×e i[ω(p1,p2)−ω(p1,p3)−ω(p2,p3)]δj3i1δj1i2δj4i3δj2i4+e i[−ω(p1,p2)+ω(p1,p3)+ω(p2,p3)]δj2i1δj4i2δj1i3δj3i48+17(λ′−1)6 gµ1µ4gµ2µ3+−236 gµ1µ3gµ2µ4+−236 gµ1µ2gµ3µ4 +e i[−ω(p1,p2)−ω(p1,p3)−ω(p2,p3)]δj2i1δj3i2δj4i3δj1i4+e i[ω(p1,p2)+ω(p1,p3)+ω(p2,p3)]δj4i1δj1i2δj2i3δj3i4−236 gµ1µ4gµ2µ3+8+17(λ′−1)6 gµ1µ3gµ2µ4+−236 gµ1µ2gµ3µ4 +e i[−ω(p1,p2)−ω(p1,p3)+ω(p2,p3)]δj3i1δj4i2δj2i3δj1i4+e i[ω(p1,p2)+ω(p1,p3)−ω(p2,p3)]δj4i1δj3i2δj1i3δj2i4−236 gµ1µ4gµ2µ3+−236 gµ1µ3gµ2µ4+8+17(λ′−1)6 gµ1µ2gµ3µ4 ,(III)=(iii)1234+(iii)1324+(iii)1243==i 112+5(λ′−1)12 gµ1µ4gµ2µ3+172+(λ′−1)212+5(λ′−1)12 gµ1µ2gµ3µ4+e i[−ω(p1,p2)−ω(p1,p3)+ω(p2,p3)]δj3i1δj4i2δj2i3δj1i4+e i[ω(p1,p2)+ω(p1,p3)−ω(p2,p3)]δj4i1δj3i2δj1i3δj2i4472+7(λ′−1)212+5(λ′−1)12 gµ1µ3gµ2µ4+172+(λ′−1)212−(λ′−1)12 gµ1µ4gµ2µ3+472+7(λ′−1)212+5(λ′−1)12 gµ1µ2gµ3µ4,(IV)=(iv)1234+(iv)1324+(iv)1243+(iv)1432+(iv)1342+(iv)1423==−i(4π)2ε N×e i[−ω(p1,p2)−ω(p1,p3)−ω(p2,p3)]δj2i1δj3i2δj4i3δj1i4+e i[ω(p1,p2)+ω(p1,p3)+ω(p2,p3)]δj4i1δj1i2δj2i3δj3i4+e i[−ω(p1,p2)−ω(p1,p3)+ω(p2,p3)]δj3i1δj4i2δj2i3δj1i4+e i[ω(p1,p2)+ω(p1,p3)−ω(p2,p3)]δj4i1δj3i2δj1i3δj2i4+e i[ω(p1,p2)−ω(p1,p3)−ω(p2,p3)]δj3i1δj1i2δj4i3δj2i4+e i[−ω(p1,p2)+ω(p1,p3)+ω(p2,p3)]δj2i1δj4i2δj1i3δj3i4(gµ1µ4gµ2µ3+gµ1µ3gµ2µ4+gµ1µ2gµ3µ4).The sum(I)+(II)+(III)+(IV)is the UV divergent part of the1PI4-point gauge function in the MS scheme.Note that (I),(II),(III)and(IV)are not proportional to the tree-level4-point vertex.1j i 1122j i 21j i 1122j i 23µj i 331j i 1122j i 23µj i 33(i )(ii )(iii )Figure 6:1PI UV divergent Feynman digrams for the ghost functions.Appendix D.Ghost functionsThe diagrams needed to compute the ghost self-energy and the ghost-boson vertex function are shown in Figure 6.From them,we have obtained the following results(i )=−i12N δj 2i1δj 1i 2p 21,(ii )=3i(4π)2εg21+(λ′−1)Ne −iω(p 1,p 2)δj 2i 1δj 3i 2δj 1i 3−e iω(p 1,p 2),δj 3i 1δj 1i 2δj 2i 3 p 1µ3,(10)(iii )=i (4π)2εg 2 1+(λ′−1)Ne −iω(p 1,p 2)δj 2i 1δj 3i 2δj 1i 3−eiω(p 1,p 2)δj 3i 1δj 1i 2δj 2i 3 p 1µ3.Appendix E.Functions with external fieldsThe diagrams which contribute at the one-loop level are shown in Figure 7.The results are:(i )=−i12N δj 2i 1δj 1i 2p 1µ2,(ii 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第五讲⾃对偶的Yang-Mills ⽅程及Polyakov 和tHooft 解以下我们考虑的是R 4或者S 4上的Yang-Mills 泛函，它们是共形不变的。
⼀.⾃对偶和反⾃对偶我们寻找R 4或S 4上的⼀个重要问题：Yang-Mills 泛函在何时取得最⼩值？于是我们考虑R 4上微分形式∗:∧2R 4→∧2R 4,我们有∗∗=1。
那么定义F +A =12(F A +∗F A )以及F −A =12(F A −∗F A )。
我们不难验证∗F +A =F +A ，以及∗F −A =−F −A 。
以及有⟨F +A ,F −A ⟩=0，我们相当于给出了∗的特征值所对应的分解。
也就是说YM (D A )=12∫M ⟨F A ,F A ⟩∗1=12∫R 4‖F +A ‖2+‖F −A ‖2dV .⽽注意到p 1(E )[M ]=18π2∫M ‖F +A ‖2+‖F −A ‖2dV为流形的第⼀Pontrjagin 类。
这是⼀个拓扑不变量，不依赖于联络的选取。
所以YM (D A )=∫R 4‖F +A ‖2dV −4π2p 1当p 1≤0∫R 4‖F −A ‖2dV +4π2p 1当p 1≥0可见当p 1≤0时，在F +A =0时取得最⼩值，这也是Yang-Mills 泛函的临界点。
对于p 1≥0同理，我们就有如下的定义：定义 F +A =0⇔F A =∗F A 称为⾃对偶的Yang-Mills 联络，F −A =0⇔F A =−∗F A 称为反对偶的Yang-Mills 联络也就是说，我们解出了⾃对偶或反对偶的Yang-Mills 联络（这是⼀个⼀阶线性⽅程），就相当于解出了原来的⼆阶线性⽅程D ∗A F A =0。
这与调和⽅程的解和柯西-黎曼⽅程的解两者关系类似。
注1：四维球⾯在去除⼀点的情况下共形于R 4。
⽽由于Yang-Mills 泛函在共形下不变，所以在R 4上⾃对偶解也可以放到S 4上变成⼀个⾃对偶解（对于反⾃对偶也类似）。
Unit OneTask 1⑩④⑧③⑥⑦②⑤①⑨Task 2① be consistent with他说，未来的改革必须符合自由贸易和开放投资的原则。
② specialize in启动成本较低，因为每个企业都可以只专门从事一个很窄的领域。
③ d erive from以上这些能力都源自一种叫机器学习的东西，它在许多现代人工智能应用中都处于核心地位。
④ A range of创业公司和成熟品牌推出的一系列穿戴式产品让人们欢欣鼓舞,跃跃欲试。
⑤ date back to置身硅谷的我们时常淹没在各种"新新"方式之中，我们常常忘记了，我们只是在重新发现一些可追溯至涉及商业根本的朴素教训。
Task 3T F F T FTask 4The most common viewThe principle task of engineering: To take into account the customers ‘ needs and to find the appropriate technical means to accommodate these needs.Commonly accepted claims:Technology tries to find appropriate means for given ends or desires;Technology is applied science;Technology is the aggregate of all technological artifacts;Technology is the total of all actions and institutions required to create artefacts or products and the total of all actions which make use of these artefacts or products.The author’s opinion: it is a viewpoint with flaws.Arguments: It must of course be taken for granted that the given simplified view of engineers with regard to technology has taken a turn within the last few decades. Observable changes: In many technical universities, the inter‐disciplinary courses arealready inherent parts of the curriculum.Task 5① 工程师对于自己的职业行为最常见的观点是：他们是通过应用科学结论来计划、开发、设计和推出技术产品的。
a rXiv:mat h-ph/04738v12J ul24Yang-Mills Theory for Noncommutative Flows Addendum Hiroshi TAKAI Department of Mathematics,Tokyo Metropolitan University MinamiOhsawa,Hachiohji,Tokyo,JAPAN.Abstract This supplementary manuscript is to describe an im-portant nontrivial example,which appears in the matrix model of type IIB in the super string theory in order to apply a new duality for the moduli spaces of Yang-Mills connections on noncommutative vector bundles.Actually,the moduli space of the instanton bundle over noncommutative Euclidean 4-spaces with respect to the canonical action of space translations is computed pre-cisely without using the ADHM-construction.July,20041§1.Introduction In the manuscript[6],we have found a new duality for the moduli spaces of Yang-Mills connections on noncommutative vector bundles with re-spect to noncommutativeflows.As we have also an-nounced in[6]with a short proof that such a duality was also affirmatively shown for noncommutative multiflows. In this addendum,we apply it to compute the moduli space in the case of the instanton bundles on the noncom-mutative Euclidean4-space with respect to the canonical space translations without using the ADHM construction (cf:[1],[2]).§2.Preliminaries Let(A,R n,α),(n≥1)be a F∗-system,andΞafinitely generated projectiveα-invariant right A-module.AsΞ=P(A m)for a projection P∈M m(A)over A(m≥1),and let Ξ= P( A m)where A=A⋊αR n and P=P×I∈M m(M( A))where M( A) is the F∗-algebra consisting of all A-valued bounded C∞-functions on R n withα-twisted*-convolution.Then Ξis afinitely generated projective right A-module.In[6], we have shown the following theorem:Theorem1([6]).Let(A,R n,α)be a F∗-dynamical system with a faithfulα-invariant continuous traceτ, andΞafinitely generated projective right A-module. Then there exist a dual F∗-dynamical system( A,R n,α-invariant trace τand afinitely gener-ated projective right A-module Ξwith the property that the moduli space M(A,R n,α,τ)(Ξ)of the Yang-Mills con-nections ofΞfor(A,R n,α,τ)is homeomorphic to the dual moduli space M( A,R n,α, τ).2Theorem2([6]).Let( A,R n,β)be a F∗-dynamical system with a faithfulβ-invariant continuous traceτ,and Ξafinitely generated projective right A-module.Ifβcommutes withalgebra R4θdepends essentially on one positive real num-ber denoted by the same symbolθ,which satisfy the following relation:(2)[z∗i,z i]=θ,[z i,z j]=[z∗i,z j]=0(i,j=0,1,i=j)where z0=x1+√−1x4and z∗iare theconjugate operators of z i.Let us consider the canonical actionαof R4on R4θdefined by(3)αti(x i)=x i+t i(t i∈R,i=1,···,4).Then it is easily seen that the triplet(R4θ,R4,α)is a F∗-dynamical system,and we eas-ily see that(4)αwi(z i)=z i+w i(w i∈C,i=0,1).By(2),R4θis nothing but the F∗-tensor product A0⊗A1where A i are the F∗-algebras generated by z i(i=0,1).We now check the algebraic structure of A i.By(2),it follows from[3](cf.[4])that there exist two Fock spaces H i such thatz i(ξi n)= nθξi n−1,where{ξi n}are complete orthonormal systems of H i with respect to the following inner product:<f|g>= (n+1)θf(n)for i=0,1.We may assume that the A i act on H i irreducibly.Then it also follows from[4]that the F∗-algebras A i are isomorphic to the F∗-algebras K∞(H i) defined byK∞(H i)={T∈K(H i)|{λk}∈S(N)} where{λk}are all eigen values of T and S(N)are the set of all sequences{c n}of C with sup n≥1(1+|n|)k|c n|<∞for all k≥0.Therefore,the F∗-algebra R4θis isomorphic to K∞(H0⊗H1).We then have the following proposition: Proposition3(cf:[5]).Ifθ=0,then R4θis isomor-phic to K∞(L2(C2))as a F∗-algebra.By the above Proposition,K∞(L2(C2))is the F∗-crossed product S(C2)⋊τC2of S(C2)by the shift actionτof C2.We then consider the actionαdefined before.By (4),it follows from[R]thatαplays a role of the dual ac-tion ofτ.Then the F∗-crossed product R4θof R4θby the actionαof R4is isomorphic to the F∗-crossed product K∞(L2(C2))⋊ τC2,where τis the dual action ofτ.Then it is isomorphic to S(C2)⊗K∞(L2(C2))as a F∗-algebra. We now consider afinitely generated projective right R4θ-moduleΞ.Then there exist an integer n≥1and a pro-jection P∈M n(M(R4θ))such thatΞ=P((R4θ)n).where M(R4θ)is the F∗-algebra consisting of all bounded linear operators T on L2(C2)whose kernel functions T(·,·)are C-valued bounded C∞-functions of C2×C2.Let us take the canonical faithful trace T r on R4θbecause of Propo-sition1.Then we consider the moduli space:M(K∞(L2(C2)),C2,α,T r)(Ξ)ofΞfor(K∞(L2(C2)),C2,α,T r).5We want to describe P cited above as a precise fashion. Actually,we know thatK∞(L2(C2))∼=S(C2)⋊λC2where∼=means isomorphism as a F∗-algebra.λis the shift action of C2on S(C2).Then it follows thatM n(K∞(L2(C2)))∼=M n(S(C2))⋊λn C2whereλn w(f)(w′)=f(w′−w)for f∈M n(S(C2)),w,w′∈C2.Letλw= λw◦ λw,(w∈C2) where λis the dual action ofλandλw(x)(w′)=λw x(w′)6for all x∈S(C2)⋊λC2and w,w′∈C2.Hence λcom-mutes withSumming up the argument discussed above,we deduce thatM(R4θ,R4,α,T r)(Ξ)≈M(C,C2,ι,1)(R(C m)).By the definition of the moduli space,we deduce thatM(C,C2,ι,1)(R(C m))≈End C(R(C m))sk/U(End C(R(C m)), whereEnd C(R(C m))sk(resp.U(End C(R(C m)))is the set of all skew adjoint(resp.unitary)elements in End C(R(C m)).Since End C(R(C m))=M k(C)for some natural number k(m≥k),it follows by using diagonal-ization thatEnd C(R(C m))sk/U(End C(R(C m))≈R k,which implies the following theorem:Theorem4.Let R4θbe the deformation quantization of R4with respect to a skew symmetric matrixθand take the F∗-dynamical system(R4θ,R4,α)with a canon-ical faithfulα-invariant trace T r of R4θ,whereαis thetranslation action of R4on R4θ.SupposeΞis afinitely generated projective right R4θ-module,then there exists a natural number k such thatM(R4θ,R4,α,T r)(Ξ)≈R k.Remark.The above theorem only states the topo-logical data of the moduli spaces of Yang-Mills connec-tions.We would study their both differential and holo-morphic structures in a forthcoming paper(cf:[2]).8References[1]K.Furuuchi:Instantons on Noncommutative R4and Projection Operators.arXiv:hep-th/9912047.[2]H.Nakajima:Resolutions of Moduli Spaces of Ideal Instantons on R4.World Scientific.129-136(1994).[3]N.Nekrasov and A.Schwarz:Instantons on Noncom-mutative R4,and(2,0)Superconformal Six Dimen-sional mun.Math.Phy.198,689-703(1998),[4]C.R.Putnam:Commutation Properties of Hilbert Space Operators and Related Topics.Springer-Verlag(1967).[5]M.A.Rieffel:Deformation Quantization for Actions of R d.Memoires AMS.506(1993).[6]H.Takai:Yang-Mills Theory for Noncommutative Flows. arXiv:math-ph/040326.9。
a rXiv:h ep-th/014115v28M a y21Duality Equivalence Between Self-Dual And Topologically Massive Non-Abelian Models A.Ilha and C.Wotzasek Instituto de F´ısica,Universidade Federal do Rio de Janeiro,Caixa Postal 68528,21945Rio de Janeiro,RJ,Brazil.(February 7,2008)The non-abelian version of the self-dual model proposed by Townsend,Pilch and van Nieuwenhuizen presents some well known difficulties not found in the abelian case,such as well defined duality operation leading to self-duality and dual equivalence with the Yang-Mills-Chern-Simons theory,for the full range of the coupling constant.These questions are tackled in this work using a distinct gauge lifting technique that is alternative to the master action approach first proposed by Deser and Jackiw.The master action,which has proved useful in exhibiting the dual equivalence between theories in diverse dimensions,runs into trouble when dealing with the non-abelian case apart from the weak coupling regime.This new dualization technique on the other hand,is insensitive of the non-abelian character of the theory and generalize straightforwardly from the abelian case.It also leads,in a simple manner,to the dual equivalence for the case of couplings with dynamical fermionic matter fields.As an application,we discuss the consequences of this dual equivalence in the context of 3D non-abelian bosonization.I.INTRODUCTIONThe bosonization technique that expresses a theory of interacting fermions in terms of free bosons provides a powerful non-perturbative tool for investigations in different areas of theoretical physics with practical applications [1].In two-dimensions these ideas have been extended in an interpolating representation of bosons and fermions which clearly reveals the dual equivalence character of these representations[2].In spite of some difficulties,the bosonization program has been extended to higher dimensions[3,4].In particular the2+1dimensional massive Thirring model(MTM)has been bosonized to a free vectorial theory in the leading order of the inverse mass ing the well known equivalence between the self-dual[5] and the topologically massive models[6]proved by Deser and Jackiw[7]through the master action approach[8],a correspondence has been established between the partition functions for the MTM and the Maxwell-Chern-Simons (MCS)theories[9].The situation for the case of fermions carrying non-abelian charges,however,is less understood due to a lack of equivalence between these vectorial models,which has only been established for the weak coupling regime [10].As critically observed in[11]and[12],the use of master actions in this situation is ineffective for establishing dual equivalences.In this paper we intend tofill up this gap.We propose a new technique to perform duality mappings for vectorial models in any dimensions that is alternative to the master action approach.It is based on the traditional idea of a local lifting of a global symmetry and may be realized by an iterative embedding of Noether counter terms.This technique was originally explored in the context of the soldering formalism[13,14]and is exploited here since it seems to be the most appropriate technique for non-abelian generalization of the dual mapping concept.Using the gauge embedding idea,we clearly show the dual equivalence between the non-abelian self-dual and the Yang-Mills-Chern-Simons models,extending the proof proposed by Deser and Jackiw in the abelian domain.These results have consequences for the bosonization identities from the massive Thirring model into the topologically massive model,which are considered here,and also allows for the extension of the fusion of the self-dual massive modes[14]for the non-abelian case[15].We also discuss the case where charged dynamical fermions are coupled to the vector bosons.For fermions carrying a global U(1)charge we reproduce the result of[16]but the result for the non-abelian generalization is new.The technique of local gauge lifting is developed in section II through specific examples.In section III we show how the gauge embedding idea solves the problem of non-abelian dual equivalence.For completeness wefirst discuss the abelian case showing how the well known results are easily reproduced.The case of dual equivalence between the SD and the MCS when dynamical fermionfields are coupled to the gaugefields is also considered,both in the abelian and in the non-abelian cases.The remaining sections are dedicated to explore this result in the non-abelian bosonization program and to present our conclusions.II.NOETHER GAUGE EMBEDDING METHODRecently there has been a number of papers examining the existence of gauge invariances in systems with second class constraints[17].Basically this involves disclosing,using the language of constraints,hidden gauge symmetries in such systems.This situation may be of usefulness since one can consider the non-invariant model as the gauge fixed version of a gauge theory.By doing so it has sometimes been possible to obtain a deeper and more illuminating interpretation of these systems.Such hidden symmetries may be revealed by a direct construction of a gauge invariant theory out of a non-invariant one[18].The former reverts to the latter under certain gaugefixing conditions.The associate gauge theory is therefore to be considered as the embedded one.The advantage in having a gauge theory lies in the fact that the underlying gauge invariant theory allows us to establish a chain of equivalence among different models by choosing different gaugefixing conditions.In this section we shall review a different technique to achieve this goal:the iterative Noether gauging procedure. For pedagogical reasons,we develop simple illustrations making use of scalarfield theories living in a Minkowski space-time of dimension two.This will allow us to discuss some subtle technical details of this method,regarding the connection between the implementing symmetries and the Noether currents,which are necessary for its application in the2+1dimensional self-dual model.The important point to stress in this review of the iterative Noether procedure is its ability to implement specific symmetries leading to distinct models.To avoid unnecessary complications let us consider the case of a free two dimensional scalarfield theory,S(0)=1and choose to gauge either the axial shift,(i)ϕ→ϕ+ǫ,(2) or the conformal symmetry,(ii)ϕ→ϕ+ǫ∂−ϕ,(3) (∂0±∂1)andǫis a global parameter.In thefirst case it is simple to identify the Noether current as, where∂±=12Jµ=∂µϕ,(4) which comes from the variation of the scalarfield actionδS(0)= d2x Jµ∂µǫ,(5) which is non-vanishing ifǫis lifted to its local version.To compensate for this non-vanishing result we introduce a counter-term together with an ancillary gaugefield Bµ(also called as compensatoryfield)as,S(1)=S(0)− d2x JµBµ,(6) such that its variation reads,δS(1)=− d2x BµδJµ,(7) which is achieved if Bµtransforms as a vectorfield simultaneously with(2).Introducing an extra counter-term as,S(2)=S(1)+1III.DUAL EQUIV ALENCE OF SD AND MCS THEORIESIn this section we discuss the application of the gauge invariant embedding to show the dual equivalence between the SD model with the MCS theory both abelian and non-abelian including the coupling with charged dynamical matter fields.As mentioned in the introduction this has the advantage of possessing a straightforward extension to the non-abelian case for all values of the coupling constant.Let us recall that the essential properties manifested by the three dimensional self-dual theory such as parity breaking and anomalous spin,are basically connected to the presence of the topological and gauge invariant Chern-Simons term.The abelianself-dual model for vector fields was first introduced by Townsend,Pilch and van Nieuwenhuizen[5]through the following action,S χ[f ]= d 3x χ2f µf µ ,(15)where the signature of the topological terms is dictated by χ=±1and the mass parameter m is inserted for dimensional reasons.Here the Lorentz indices are represented by greek letters taking their usual values as µ,ν,λ=0,1,2.The gauge invariant combination of a Chern-Simons term with a Maxwell actionS (MCS )= d 3x 12mǫµνλf µ∂νf λ ,(16)is the topologically massive theory,which is known to be equivalent [7]to the self-dual model (15).f µνis the usual Maxwell field strength,f µν≡∂µf ν−∂νf µ.(17)The non-abelian version of the vector self-dual model (15),which is our main concern in this work,is given byS χ= d 3x tr −14mǫµνλ F µνF λ−24m 2F µνF µν+χ3F µF νF λ ,(20)only in the weak coupling limit g →0so that the Yang-Mills term effectively vanishes 1.To study the dual equivalence of (18)and (20)for all coupling regimes and the consequences over the bosonization program is main contribution of this work.Next we analyze the dualization procedure in the massive spin one self-dual theories using the Noether gauging procedure.To begin with,it is useful to clarify the meaning of the self duality inherent in the action (15).The equation of motion in the absence of sources is given by,f µ=χ1Here we are using the bosonization nomenclature that relates the Thirring model coupling constant g 2with the inverse mass of the vector model;see discussion after Eq.(62)∂µfµ=0,2+m2 fµ=0.(22) From(21)and(22)we see that there is only one massive excitation whose value is m.Afield dual to fµis defined as,⋆fµ=1mǫµνλ∂ν⋆fλ=fµ,(24)obtained by exploiting(22),thereby validating the definition of the dualfibining these results with(21),we conclude that,fµ=χ⋆fµ.(25) Hence,depending on the signature ofχ,the theory will correspond to a self-dual or an anti self-dual model.To prove the exact equivalence between the self-dual model and the Maxwell Chern-Simons theory,we start with the zeroth-iterated action(15)which is non-invariant by gauge transformations of the basic vectorfield fµ.To construct from it an abelian gauge model,we have to consider the gauging of the following symmetry,δfµ=∂µξ,(26) whereξis an infinitesimal local parameter.Under such transformations,the action(15)change as,δSχ= d3x Jµ(f)∂µξ,(27) where the Noether currents are defined by,Jµ(f)≡−fµ+χ2BµBµ ,(32)which is invariant under the combined gauge transformations(26)and(29).The gauging of the U(1)symmetry is complete.To return to a description in terms of the original variables,the auxiliary vectorfield is eliminated from (32)by using the equation of motion,Bµ=−Jµ.(33)Note that taking variations on both sides of this equation and using the gauge invariance of the Chern-Simons form we obtain consistency with the conditionδfµ=δBµ.It is now crucial to note that,by using the explicit structures for the currents,the above action(32)forms a gauge invariant combination expressed by the action(16)which is the Maxwell Chern-Simons theory.Our goal has been achieved.The iterative Noether dualization procedure has precisely incorporated the abelian gauge symmetries in the self-dual model to yield the gauge invariant Maxwell Chern-Simons theory.The free case considered above can also be extended to couplings with external,field-independent sources.To illustrate this point,we consider a coupling between dynamical U(1)charged fermions and self-dual vector bosons [16].In the abelian case,this model is written asS[f,ψ]=Sχ[f]+ d3x −e fµJµ+¯ψ(i∂/−M)ψ ,(34)where Jµ=¯ψγµψand M is the fermionic mass.Sχis the self-dual action(15).The Noether current associated with this action isχKµ=−fµ+BµBµ .(36)2After the elimination of Bµthrough its equations of motion,we get ourfinal theory,S=S(MCS)+ d3x e2JµJµ+eχfµGµ+L D ,(37) where S(MCS)is the Maxwell Chern-Simons action(16)and L D is the free Dirac Lagrangian.Here Gµ=1ǫµνλ(∂µ+Fµ) Fν,(39)mwhere the operator inside the square brackets in the right-hand side acts on the basicfield Fνdefining⋆Fλas the dual of Fν.Repeating this operation,and using the equations of motion obtained by varying(18)with respect to FλχFλ=with the Noether currents being defined as,Jµ=−Fµ+χ2δ(JµJµ)−δ(JµBµ)−JµδBµ ,(45) where we have used the following transformation rule for the gaugingfield,δFµ=−δBµ−δJµ.(46) This prompt us to define the following second iterated action,S(2)=Sχ+ d3x tr 14mǫµνλ FµνFλ−22BµBµ (47) which is gauge invariant after noticing that the transformation rule(46)fixes the Bµfield asBµ=−χ2m ǫµνλFµνJλ+e2Bosonization was developed in the context of the two-dimensional scalarfield theory and has been one of the main tools available to investigate the non-perturbative behavior of some interactivefield theories[1].For some time this concept was thought to be an exclusive property of two-dimensional space-times where spin is absent and one cannot distinguish between bosons and fermions.It was only recently that this powerful technique were extended to higher dimensional space-times[24,25][9,26].The bosonization mapping in D=3,first discussed by Polyakov[27],shows that this is a relevant issue in the context of transmutation of spin and statistics in three dimensions.The equivalence of the three dimensional effective electromagnetic action of the CP1model with a charged massive fermion to lowest order in inverse(fermion)mass has been proposed by Deser and Redlich[28].Using their results bosonization was extended to three dimensions in the1/m expansion[9].These endeavor has led to promising results in diverse areas such as,for instance,the understanding of the universal behavior of the Hall conductance in interactive fermion systems[29].For higher dimensions,due to the absence of an operator mapping a la Mandelstan,the situation is more complicate and even the bosonization identities extracted from these procedures relating the fermionic current with the bosonic topological current is a consequence of a non-trivial current algebra.Moreover,contrary to the two dimensional case, in dimensions higher than two there are no exact results with the exception of the current mapping[22,23].Besides, while the two-dimensional fermionic determinant can be exactly computed,here it is neither exact nor complete, having a non-local structure.However,for the large mass limit in the one-loop of perturbative evaluation,a local expression materializes.This procedure,is in a sense,opposite to what is done in1+1dimensions where bosonization is a set of operator identities valid at length scales short compared with the Compton wavelength of the fermions while in D=3only the long distance regime is considered.In this section we review how the low energy sector of a theory of massive self-interacting,G-charged fermions,the massive Thirring Model in2+1dimensions,can be bosonized into a gauge theory,the Yang-Mills-Chern-Simons gauge theory thanks to the results of the preceeding section.A.The Non-Abelian MappingIn the sequence we investigate the problem of identifying a bosonic equivalent of a three dimensional theory of self-interacting fermions with symmetry group G and show how it is possible to bosonize the low-energy regime of the theory.We follow the same strategy as in reference[9]but follow the notation of[10]that is slightly different than [9].We seek a bosonic theory which reproduces correctly the low-energy regime of the massive fermionic theory.To begin with we define the G-current,j aµ=¯ψi t a ijγµψj,(51) whereψi are N two-component Dirac spinors in the fundamental representation of G,i,j=1,...,N and a= 1,...,dim G.Here t a and f abc are the generators and the structure constants of the symmetry group G,respectively and j aµis a G-current.The(Euclidean)fermionic partition function for the three-dimensional massive Thirring model is,Z T h= D¯ψDψe− ¯ψi(∂/+m)ψi−g22 d3x j aµj aµ= D aµe− d3x tr(12g2 d3x tr(aµaµ).(54)The determinant of the Dirac operator is an unbounded operator and requires regularization.For D=2this deter-minant can be computed exactly,both for abelian and non-abelian symmetries.Based on general grounds only,one may say that this determinant consists of a Chern-Simons action standing as the leading term plus an infinite series of terms depending on the dual of the vectorfield,˜Fµ∼ǫµνλ∂νAλ,including those terms that are non-local andnon-quadratic in˜Fµ.For the D=3the actual computation of this determinant will give parity breaking and parity conserving terms that are computed in powers of the inverse mass,χln det(i∂/+m+a/)=aµaνaλ),(56)3is the non-abelian Chern-Simons action and the parity conserving contributions,infirst-order,is the Yang-Mills action1I P C[a]=−16πS CS[a](59) Using this result we can write Z T h in the formZ T h= D aµexp(−S SD[a]),(60) where S SD is the non-abelian version of the self-dual action introduced in[5],S SD[a]=116πS CS[a](61)Therefore,to leading order in1/m we have established the identification Z T h≈Z SD.Now,recalling that the model with dynamics defined by the non-abelian self-dual action is equivalent to the Yang-Mills-Chern-Simons theory,we use this connection to establish the equivalence of the non-abelian massive Thirring model and the YMCS theory asZ T h≈Z Y MCS.(62) It is interesting to observe that the Thirring coupling constant g2/N in the fermionic model is mapped into the inverse mass spin1massive excitation,m=π/g2.Now comes an important observation.Unlike the master approach,our result is valid for all values of the coupling constant.The proof,based on the use of an“interpolating Action”S I,is seen to run into trouble in the non-abelian case.That the non-abelian extension of this kind of equivalences is more involved was already recognized in[7]and [10],and shown that the non-abelian self-dual action is not equivalent to a Yang-Mills-Chern-Simons theory(the natural extension of the abelian MCS theory)but to a model where the Yang-Mills term vanishes in the limit g2→0 [10].B.Current IdentitiesTo infer the bosonization identities for the currents which derive from the equivalence found in the last section,we add a source for the Thirring current leading to the following functional generator,Z T h[b]= D¯ψDψD aµe− d3x ¯ψ(i∂/+m+a/+b/)ψ+1tr d3x bµbµ· D aµe−S SD[a]+12g2after shifting aµ→aµ−bµ.In order to connect this with the Yang-Mills-Chern-Simons system we repeat the steps of the last section to obtain,Z T h[bµ]≈Z MCS[bµ](64) We have therefore established,to order1/m,the connection between the Thirring and self-dual models in the non-abelian context,now in the presence of sources.This is,in its most general form,the result we were after.It provides a complete low-energy bosonization prescription,valid for any g2,of the matrix elements of the fermionic current.From(64)we see,from simple differentiation w.r.t.the source,that the bosonization rule for the fermion current,to leading order in1/m,readsj aµ→iχπ)ǫµν∂νφwhile in this case it should be considered as the analog of the Wess-Zumino-Witten currents.Notice that as in the abelian case the bosonized expression for the fermion current is topologically conserved.We thus see that the non-abelian bosonization of free,G-charged massive fermions in2+1dimensions leads to the non-abelian Chern-Simons theory,with the fermionic current being mapped to the dual of the gaugefield strength. This result holds only for length scales large compared with the Compton wavelength of the fermion,since our results were obtained for large fermion mass.It is important to notice that the limit g2→0used in earlier approaches corresponding to free fermions(but not to an abelian gauge theory)was not taken here at any stage.This is important since Yang-Mills coupling is proportional to g2,which is why we are left with a Yang-Mills-Chern-Simons action and not the pure Chern-Simons theory of[10].V.CONCLUSIONSThe rationale of different phenomena in planar physics have greatly benefitted from the use of2+1dimensionalfield theories including the parity breaking Chern-Simons term.In this scenario it is important to establish connections among different models so that a unifying picture emerges.In this context we have shown,in earlier work,that the soldering formalism has established a direct link between self-dual models of opposite helicities with the Proca model [14].Other instances includes the recent extension of the functional bosonization program interpolating from fermions to bosons in a coherent picture[9,10].In the context of thefirst it has been argued that the soldering formalism is equivalent to canonical transformation albeit in the Lagrangian side while for the later the mentioned mapping between SD and MCS models has been used to establish a formal equivalence between the partition functions of the abelian version of MTM and a theory of interacting bosons.The non-abelian extension of this analysis,for the full range of the coupling constant,has been the main concern of the present work since only partial results were reported in the literature.Other directions have also been investigated,with new results,that includes the proof of the self-duality property of(18)and the coupling with G-charged dynamical matterfields.Our analysis has shown how the gauge lifting approach sheds light on the question of dual equivalence between SD and topologically massive theories with new results for the non-abelian case.This discussion becomes the central issue when deriving bosonization rules in D=3,for fermions carrying non-abelian charges since,up to date,only prescriptions based on the Master action of Ref.[7]were used,apart from[12]with conclusions consistent with[10]. These derivations of the bosonization mappings suffered from well known difficulties related to the dual equivalence, restricting the results to be limited to weak coupling constant only.Therefore,regarding the non-abelian bosonization in D=3dimensions,we believe that the method developed here,which is simpler and better suited to deal with non-abelian symmetries,completes the program initiated in[10]and confirms the exact identities found in[22,23].This new approach has also been used with dynamical fermionicfields leading naturally to the necessity of a Thirring like term to establish the equivalence of the fermionic sectors in both sides.Such equivalence may be extended to the scalar case[21].The bosonization for D≥4poses no difficulties as long as the fermionic determinant can be evaluated in some approximation and is expected to yield a gauge invariant piece.This is of importance since the description of chargedfermionicfields in terms of gaugefields has brought new perspectives and a deeper insight on the non perturbative dynamics of planar physics[30]that might be extended to higher dimensions.。
鲁班造锯英语作文六年级Carpenter Lu Ban and His Invention of the SawIn the annals of Chinese history, the name Lu Ban stands out as a legendary figure whose ingenuity and craftsmanship have left an indelible mark on the world of carpentry and woodworking. Born in the state of Lu during the Spring and Autumn period, Lu Ban was a skilled carpenter and inventor whose contributions to the field of woodworking have endured for centuries.One of Lu Ban's most significant achievements was his invention of the saw, a tool that revolutionized the way in which wood was worked and shaped. Prior to Lu Ban's innovation, carpenters and woodworkers relied on a variety of tools, such as axes and adzes, to cut and shape wood. However, these tools were often unwieldy and inefficient, requiring a great deal of physical exertion and skill to use effectively.Lu Ban recognized the limitations of these traditional tools and set out to create a more efficient and effective means of cutting wood.Drawing upon his extensive knowledge of mechanics and engineering, he designed a tool that could slice through wood with ease, reducing the amount of physical labor required and increasing the speed and precision of the cutting process.The saw that Lu Ban invented was a remarkable feat of engineering, incorporating a number of innovative features that set it apart from the tools that had come before. At the heart of the saw was a serrated blade, which was constructed from high-quality steel and designed to cut through wood with minimal effort. The teeth of the blade were carefully shaped and angled to provide maximum cutting power, while the overall design of the tool was optimized for balance and control.In addition to the blade, Lu Ban's saw also featured a sturdy wooden handle that allowed the user to grip the tool securely and apply the necessary force to the cutting surface. The handle was ergonomically designed to minimize fatigue and strain on the user's hands and arms, making it possible to work for extended periods without experiencing significant discomfort.One of the most remarkable aspects of Lu Ban's saw was its versatility. Unlike the specialized tools that had been used in the past, Lu Ban's saw could be used for a wide range of woodworking tasks, from rough-cutting large pieces of timber to fine-tuning the detailsof a finished project. This versatility made the saw an indispensable tool for carpenters and woodworkers, who quickly adopted it as an essential part of their toolkit.As word of Lu Ban's invention spread throughout China, the saw quickly became a ubiquitous tool in the world of carpentry and woodworking. Carpenters and builders incorporated the saw into their work, using it to construct a wide range of structures, from simple furniture to elaborate buildings and temples.The impact of Lu Ban's invention was not limited to the realm of carpentry, however. The saw also had a profound influence on the development of other technologies and industries. For example, the saw's ability to cut through wood with precision and efficiency made it an invaluable tool in the construction of ships and other watercraft, which were essential for trade and transportation in ancient China.Moreover, the saw's versatility and effectiveness also led to its adoption in other fields, such as the production of agricultural tools and the construction of infrastructure like bridges and roads. As a result, Lu Ban's invention played a crucial role in the economic and social development of ancient China, helping to drive the growth of various industries and supporting the expansion of trade and commerce.Despite the immense impact of his invention, Lu Ban's life and work were not without their challenges. As a skilled craftsman and inventor, he faced the skepticism and resistance of those who were skeptical of his innovations and unwilling to embrace change. However, Lu Ban's unwavering commitment to his craft and his ability to overcome these obstacles ultimately led to the widespread adoption of the saw and the lasting legacy of his work.Today, the saw remains an essential tool in the world of woodworking and construction, with Lu Ban's original design serving as the foundation for a wide range of modern saw technologies. From the handheld power saws used in home workshops to the massive industrial saws used in lumber mills and construction sites, the legacy of Lu Ban's invention continues to shape the way we work with wood and build the structures that define our world.In many ways, the story of Lu Ban and his invention of the saw serves as a testament to the power of human ingenuity and the transformative potential of technological innovation. Through his dedication, creativity, and problem-solving skills, Lu Ban was able to revolutionize an entire industry and leave an indelible mark on the course of human history. His legacy continues to inspire and guide the work of carpenters, woodworkers, and innovators around the world, reminding us of the transformative power of human creativity and the enduring importance of the tools we use to shape our world.。
a rXiv:h ep-th/04769v19J u l24The Non-Local Massive Yang-Mills Action as a Gauged Sigma Model Mboyo Esole ∗African Institute for Mathematical Sciences,6Melrose Road,Muizenberg,7945Cape Town,RSA and Instituut-Lorentz,Universiteit Leiden,P.O.Box 9506,2300RA Leiden,The Netherlands February 1,2008Abstract We show that the massive Yang–Mills action having as a mass term the non-local operator introduced by Gubarov,Stodolsky,and Zakharov is classically equivalent to a principal gauged sigma model.The non-local mass corresponds to the topological term of the sigma model.The latter is obtained once the degrees of freedom implicitly generated in the non-local action are explicitly implemented as group elements.The non-local action is recovered by integrating out these group elements.In contrast to the usual gauge-fixed treatment,the sigma model point of view provides a safe framework in which calculation are tractable while keeping a full control of gauge-invariance.It shows that the non-local massive Yang–Mills action is naturally associated with the low-energy description of QCD in the Chiral Perturbation Theory approach.Moreover,the sigma model admits solutions called center vortices familiar in different (de)-confinementand chiral symmetry breaking scenarios.This suggests that the non-local operator introduced by Gubarov,Stodolsky,and Zakharov might be sensitive to center vortices configurations.1IntroductionNon-locality appears in a wide range of areas from astrophysics and cosmology to string theoriesand non-commutative theories[1,2,3,4,5,6,7].However,it can frighten the basic principles ofa relativistic theory.The assumption of locality pervades most methods of quantumfield theoryand is related to unitarity,causality and renormalizability[8,9,12,10].It is then far from obviousthat a given non-local theory can make sense physically.In this letter we are interested in the non-local mass generation scheme developed recently forYang–Mills theories in the Landau gauge[11].The interest in non-local mass generation scenariosis mainly due to the absence of local gauge-invariant mass term in pure Yang–Mills theories in fourdimensions1.Consequently,a gauge-invariant massive Yang-Mills action requires an extension ofthe number offields,like in the Higgs mechanism,or the introduction of non-local quantities.TheHiggs mechanism is not appropriate when there is no room for new particles in the spectrum of thetheory.Therefore,although non-locality is a tricky concept in quantumfield theory,it naturallyappears in mass generation without Higgsfields.A non-local massive Yang–Mills model has emerged recently2following the introduction byGubarov,Stodolsky,and Zakharov[13]of a new gauge-invariant operator having the dimensionof a mass term.That operator is defined as the minimum along the gauge orbit of the squarepotential A2and is usually denoted A 2m in(‘A2minimum’).Gauge-invariance is ensured in principle since the minimum is the same for any two points on the same gauge orbit.But theprice to pay is the loss of locality.A non-local massive action is obtained by adding A 2m in tothe Yang–Mills action. A 2m in is a Morse functional which contains topological information onthe gauge-orbit[14].These type of functionals obtained from minimizing a given local quantitiesalong the gauge orbit reveals interesting properties of the structure of the configuration space ofYang–Mills theories.The analysis of A 2m in for example,requires a careful look at issues related toGribov copies[14,15].It also emphasises the role of the the center of the Lie group to understandthe structure of the configuration space,the distinction between reducible and irreducible gaugepotentials and the relevance of the stratification of the configuration space[14].It is not verysurprising that a mass term is closely related to the topology.For example in three dimensions,the Chern-Simons term provides important topological information. A 2m in is supposed to giveas well information on the topological structures of gauge theories.This has been illustrated bylattice simulations which seems to indicate that A 2m in is sensitive to phase transitions in compactQED[13].It was also argued on the lattice that the expectation value of the square potential isassociated to instantons in the Landau gauge[16].All these lattice results are not understood inthe continuous formulation of Yang–Mills theory.It is convenient to think of the non-local massive Yang–Mills action as a(low-energy)effec-tive action.The possible dangerous effects associated with non-locality[1,2,3,4,5]are then notnecessarily the result of fundamental violations offirst principles.They can follow from the approx-imative nature of the model[3].In particular,unitarity and causality might be preserved under a certain cut-offand renormalizability is not an issue.Since Yang–Mills theories have a mass gap, an effective massive theory provides an accurate description of the physics at low energy.Another point of view is to treat the massive action as an effective action used for example to renormalize the operator A 2m in.The functional A 2m in is difficult to analyze in a gauge-invariant way because of its unconven-tional definition.Therefore,the associated non-local massive action is often defined by the trivial massive action F2+m2A2considered as a gauge-fixed action in the Landau gauge.Indeed,the square potential( A2)reaches its local minima when it is transverse and this corresponds to the Landau gauge condition(∂·A=0)[15].Atfirst sight,the gauge-fixed action in the Landau gauge seems to be easier to handle because of its locality.However,this type of‘locality’is a pure gauge-fixed mirage that can lead to erroneous conclusions when the usual techniques of(local)quantum field theory are used without checking their relevance in a non-local context3.The topology of the configuration space is very important to understand low-energy physics of Yang–Mills theories. Therefore gauge-invariance should be kept manifest as far as possible since gauge-fixing conditions can constrain the value of boundary terms and therefore interfere with the topology.A better understanding of the non-local massive model based on the functional A 2m in calls for a tractable gauge-invariant formulation which takes into account the consequences of the non-local nature of the gauge-invariant theory.Providing such a description is the aim of this letter.We show that the non-local massive Yang–Mills action associated with the operator introduced by Gubarev,Stodolsky,and Zakharov is classically equivalent to a Yang–Mills principal gauged sigma model also known as the Stueckelberg model[17,18].In this reformulation,the operator A 2m in corresponds to the so-called topological term of the sigma model.The non-local action contains implicitly new degrees of freedom.Locality emerges once these degrees of freedom are explicitly implemented in the action.They corresponds to the group elements in the sigma model action.Conversely,the non-local action is retrieved from the sigma model once the group elements are integrated out.A similar scheme is familiar in the Schwinger model in1+1-dimensions:a non-local mass term appears once the fermionicfield is integrated out.The sigma model formulation provides a local theory that encodes many effects coming from the non-locality of the original model in a gauge-invariant way.The local and non-local formulation are equivalent only at the classical level.We shall see that the local theory fulfilled the requirement of tree unitarity and is‘renormalizable in the modern sense’.These two properties are lost in the non-local model. The sigma model formulation is valid under a certain ultra-violet cut-offwhich is determined so that the loop corrections are irrelevant in comparison to the tree level amplitudes.The classical reformulation helps to understand many features of the non-local theories in a gauge-invariant way.Moreover,it provides a safe framework in which calculation are tractable.We shall see that the non-local massive action although quite odd atfirst look,happens to be naturally related to two more conventional approaches of low-energy QCD.The sigma model is relevant in the chiral effective theory of low-energy QCD[22]which describes gluons and the lightest quarks at low-energy QCD.Interestingly enough,the Stueckelberg action admits a type of solutions which correspond in the usual Yang-Mills theory to configurations called center vortices[23].The latter are familiarin different (de)-confinement and chiral symmetry breaking scenarios [25,26,27,28].We shallseethat non-localityalso constrains the topology.To illustrate our method,we shall first treat the Abelian theory.We then generalize to the non-Abelian case.We consider the simple form of the action in the Landau gauge and we obtain a gauge-invariant formulation by computing its gauge-invariant extension [20].That is,we obtain a gauge-invariant action that reduces to F 2+m 2A 2in the Landau gauge.Gauge-invariant extension is a general technique that has been recently applied to non-local Yang–Mills theories[20]and effective field theories for massive gravitons [29].2Sigma model reformulation in the Abelian caseThe action of QED:S [A µ]=−12F µνF µνand F µν=∂µA ν−∂νA µ)is invariantunder an irreducible Abelian gauge symmetry δεA µ=∂µεwhich is not compatible with the trivial mass term A 2.The gauge-invariant extension of the trivial mass term A 2from the Landau gauge is the non-local functional A 2+∂·A ∂·A2e 2 F 2−m 2 A 2+∂·A ∂·A2e 2 F 2−m 2(A 2−2A µ∂µϕ+∂µϕ∂µϕ)=−12e 2 F 2−m 2D µφ(D µφ)†,(3)where D µX =∂µX −i A µX and φ=e iϕ.The shift symmetry becomes a shift of phase :δεφ=iεφ.An elegant way to understand the connection between the Proca action and the sigma model is to use the theory of constrained system.It is easy to see that the Proca action is a gauge fixed version of the Stuekelberg action.As usual,fixing the gauge replaces first-class constraints by second-class ones.Similarly,an Hamiltonian analysis of the Proca action revels that it possesses second-class constraints.The Stueckelberg action is obtained by trading these second-class constraints for first class ones.This implies the addition of a compensating field.The gauge transformation of φis generated by the first class constraints.Therefore it is often stated that “the Proca action possesses a hidden symmetry”.The replacement of a second class constrained system by an equivalent first class one is a general method nowadays known under the name of conversion .It is used for the quantification of theories with second class constraints See [30]and reference therein.3Generalization to Yang–Mills theoryThe free Yang-Mills action is S=12δab,and T a†=−T a.In the Abelian case,the Stueckelberg action S[Aµ,ϕ]can be obtained from the Proca action 1g2 tr F2−m2A2 we getS[Aµ,U]=1g2 tr F2+m2DµU(DµU)† .(5)The non-local operator A 2m in corresponds here to the so-called topological term of the sigma modelD U 2≡− tr D U(D U)† =− tr(Aµ−U−1∂µU)2.(6) The equations of motion of the sigma model areDνFνµ+m2[Aµ−U−1∂µU]=0,(7)m2Dµ[Aµ−U−1∂µU]=0,(8) where DµX=∂µX−[Aµ,X]is the(adjoint)-covariant derivative.These two equations are not independent as the second can be derived from thefirst one by acting with the adjoint-covariant derivative.If the matrix U is expanded in terms of thefieldϕ,one can easily solve the equation of motion ofϕ(8)in term of the Yang-Millsfield Aµorder by order inΦ≡{Aµ,ϕ}usingU(ϕ)−1∂µU(ϕ)=∞n=0(−)n−12[∂·A,∂·A] +O(A3)(10)Using the equations(6)and(9)we get the non-local mass term as a funtional of Aµtr A2+∂·A∂·A,∂·A4Discussion and ConclusionWe shall present two point of view which comes naturally from the reformulation of the effective action as a sigma model.The sigma modelfits naturally in the description of gluons and the lightest quarks for low energy QCD in the spirit of the chiral effective theory.It can also be viewed as an effective action in the center vortices description of the features of low-energy QCD.We shall also discuss the shortcoming of our methods.This is mainly related to the existence of Gribov copies. We shall see that non-locality constrains the topology.4.1The Perturbative Chiral Effective Action Approach.The sigma model approach shows that the massive Yang–Mills action associated with the non-local operator A 2m in is essentially the same as the model introduced as an alternative to the Higgs model by Delbourgo and Thompson[32].However,it has been found not to be satisfactory as a fundamental theory because of problems with unitarity[33].This explains why we considered the non-local massive action as an effective action for low-energy QCD.This point of view makes sense as unitarity problem can be disregarded once an appropriate cut-offis considered.In the spirit of effective action[31]it is interested to look at all the possible local terms that are consistent with the symmetries of the action.Indeed,they might be used as corrections to the action.For the principal gauged sigma model this question has been carefully analyzed by Henneaux and Wilch [18]:in spite of the usual curvature terms coming from pure Yang-Mills theory there are also ‘winding number terms’that depend on the group elements and cannot be eliminated when the topology is non-trivial.The Stueckelbergfieldsϕa,play the role of unphysical Goldstone-bosonfields since they decouple in the so-called unitary gauge(ϕ=0⇐⇒U=I).The unitary gauge is equivalent to the Landau gauge thanks to equation(8).If we are not in the Landau gauge(which corresponds here to the unitary gaugeϕ=0),equation(9)shows clearly that they are an infinite number of non-renormalizable vertices involving(adϕ)n∂µϕ.However,if we were working only in the Landau gauge,we would have the impression that the action is power counting renormalizable.The gauge invariant analysis shows that renormalizability in the Landau gauge is a gauge artifact.However, although the sigma model is not power counting renormalizable in four spacetime dimensions it is renormalizable in the modern sense of Gomis and Weinberg[31]as shown in[18].That is,all the infinities can be regulated by counterterms respecting the symmetries of the original action.But we might need an infinite number of them.Renormalizability in the modern sense is specially well appropriate in the analysis of effective actions[31].It is interesting to write equation(8)atfirst order in thefieldsΦ∈{Aµ,ϕ}∂·A a∂·A= ϕ+O(Φ2)=⇒∂µϕa=∂µparameter.The effective theory breaks down at the scale Λsuch that the loop correction becomes relevant.The naive dimensional analysis (NDA)gives Λ∼4πf where f =m√∂λd Φexp [S (Φ)+λO (Φ)] λ=0(13)where the functional S (Φ)+λO (Φ)is called the effective action for the operator O (Φ).This is a very practical concept.For example,the correlation functions φ(y 1)···φ(y r )O (z ) are just the first order λ-coefficients in the usual Schwinger functions for the action S (Φ)+λO (Φ).The effectiveaction for the operator A 2m in is the non-local massive Yang-Mills action F 2+λ A 2m in ,which canbe reformulated as a principal gauged sigma model of equations(4)and (5).In view of the relationbetween the sigma model and center vortices[23],it is natural to ask if A 2m in is sensitive to center vortices configurations.It would be possible to study this possiblity with a lattice simulation. 4.3Local vs non-local formulation.The non-local massive Yang–Mills theory having as a mass term the minimum of the square po-tential along the gauge potential is equivalent to a gauged-principal sigma model in the following sense.The non-local theory is obtained from the sigma model by integration out the group ele-ments present in the sigma model action.In the other direction,the sigma model emerges once the degrees of freedom implicitly generated by non-locality are explicitly included in the action.Although the local and non-local formulation are equivalent in the sense explain above,they do have important differences.For example,the Stueckelberg model satisfies the tree unitarity conditions[35]4and is renormalizable in the modern sense of Gomis and Weinberg[31]5as shown in[18].This two properties disappear in the non-local theories[33,20].This can be understood from the fact that the transformation from one model to the other is highly non-local whereas tree unitarity and renormalizability in the modern sense are mostly valid for(perturbatively)local theories.Quantum correction can be seen as part of the measure of the path-integral.The latter is known to be invariant underfield redefinitions.However,when non-locality is introduced,the physics can be altered.This is well illustrated by the re-investigation of the Equivalence Theorem by Tuytin[36].It is shown there that twofield theories related by afield redefinition can have different physical content even when thefield redefinitions are local or perturbatively local.4.4The gauged-principal sigma model as a collective description of the localminima of the square-potential.The sigma model that we present in this letter is the gauge-invariant extension of the trivial massive YM theory from the Landau gauge.Therefore its equivalence to the gauge-invariant definition of the A 2m in-massive YM action(which requires the computation of the absolute minimum along the gauge orbit)relies on the ability of the Landau gauge to detect the absolute minimum.However, the Landau gauge condition does not single out a unique value of the potential Aµas it pocesses Gribov copies.That is,there exists different gaugefields related by a gauge transformation but satisfying all the Landau gauge condition.The Landau condition characterizes local minima of the square potential along the gauge orbit.The absolute minimum among all possible Gribov copies is never selected.Since the sigma model is obtained from the Landau gauge description,it inherits the Landau gauge ambiguities:the sigma model encodes only information associated with local minima of the square potential and does not identify the global minimum.Still it is the gauge-invariant formulation of the‘Landau gauge dynamical mass generation’as discuss in the literature. Indeed,the latter is only defines by the Landau gauge-fixed action and therefore suffer as well of the Gribov ambiguities.The difficulties of identifying the absolute minima of the square potential is a common feature of all the methods used to evaluate A 2m in.The problem of Gribov copies in the definition of A 2m in is carefully analyzed in[15]to which we refer for more details.The ambiguities in the definition of A 2m in are not only a technical problem.They are related to miscellaneous properties of the configuration space of Yang–Mills theory.They depend on particular featuresof the gauge potential that we start with.In particular,they are unavoidable when we deal with reducible gauge potential[14].We would like as well to point that topological consideration are crucial to make sense of the non-local mass term(1),(11).Taking the inverse of a differential operator makes sense only if the latter has no non-trivial zero modes.The possible boundary conditions are therefore constrained. For consistency the set of gauge transformations has to be limited to those that preserve these boundary conditions.In the Hamiltonian analysis of the operator A 2m in,restrictions are imposed in the Landau gauge to avoid Gribov copies[15].We do not think that a model that determines the exact absolute minimum can be defined in a tractable way.When the connectionfield Aµis reducible,the spectrum of absolute minima is degenerated[14].In this respect,the sigma model reformulation is a fair alternative.It is local,manifestly gauge-invariant,renormalizable in the modern sense,and fulfills the tree unitarity condition.It provides a gauge-invariant description in which calculations are possible while the effects of non-locality are taken into account through the additionalfields.The gauged principal sigma model is also interesting in its own without any link with the A 2m in operator.At low energy the gauged principal sigma model can be used as an effective action for gluons and lightest hadrons in the spirit of perturbative chiral effective theory.In the spirit of the center vortices pictures of low energy QCD an effective action relevant to understand confinement and chiral symmetry breaking. In view of these features of the gauged principal sigma model,it would be interesting to study the sensitivity of the expectation value of A 2m in to center vortices configurations. 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