Dilatonic Supergravity in Two Dimensions and the Disappearance of Quantum Black Hole
逻辑斯蒂增长英语
逻辑斯蒂增长英语
逻辑斯蒂增长（Logistic Growth）的英文是：Logistic Growth。
逻辑斯蒂增长模型，又称自我限制增长模型，是一种描述种群增长速率先增加后减小，呈“S”型曲线的数学模型。
它是生物学、生态学和数学等学科中常用的一种模型。
这种模型在生态学和流行病学等领域中尤为重要，因为它能够描述资源有限的情况下种群或疾病的增长情况。
在逻辑斯蒂增长模型中，种群的增长率与种群大小成反比，当种群大小接近环境容纳量时，增长率逐渐减小，最终趋于零。
这个模型可以用微分方程来描述，也可以通过离散时间递推公式来模拟。
Massive IIA Supergravity as a Non-linear Realisation
a r X i v :h e p -t h /0204207v 1 24 A p r 2002KCL-MTH-02-08hep-th//0204207Massive IIA Supergravity as a Non-linear RealisationIgor Schnakenburg and Peter WestDepartment of Mathematics King’s College,London,UK Abstract A description of the bosonic sector of massive IIA supergravity as a non-linear realisation is given.An essential feature of this construction is that the momentum generators have non-trivial commutation relations with the generators associated with the gauge fields.Igor Schnakenburg is financially supported by DAAD (D/00/09914).email:schnake,pwest@0.IntroductionLong ago Nahm[1]pointed out that supergravity theories could only exist in eleven and less space-time dimensions and that the maximum number of supersymmetries they could possess were contained in spinors that had in total no more than32real components. The supergravity theories with32components are called maximal supergravity theories. There is a unique such theory in eleven dimensions which was constructed[2]as an ap-plication of the Noether method that was used to construct thefirst supergravity theory, the N=1,D=4supergravity theory.However,in ten dimensions there are two such maximal supergravity theories called IIA and IIB.These two theories possess different su-persymmetries which when decomposed in terms of Majorana-Weyl spinors are of opposite and the same chirality respectively.While the construction of the IIA theory[3]was found by dimensional reduction of the eleven dimensional supergravity theory,the construction of the IIB theory[4,5,6]required new techniques.There is a modification of the IIA theory that preserves the number of supersymmetries,but introduces a dimensionfull parameter [7].This theory is called massive IIA supergravity and it possesses a cosmological constant.One of the most remarkable features of supergravity theories is that the scalars in the supergravity multiplets always occur in a coset structure[8].While this can be viewed as a consequence of supersymmetry,the groups that occur in these cosets are rather mysterious [9,10,11].It has been conjectured[12]that the symmetries found in these cosets are symmetries of the associated non-perturbative string theory.The coset construction for the description of the scalars was extended to include the gaugefields for the maximal supergravity theories[13].In this construction all gaugefields and scalars were introduced along with their duals.The advantage of this approach is that the equations of motion can be reduced tofirst order equations in form of a generalised self-duality condition.This method was subsequently applied to massive IIA[14].However,in these papers the vector indices on the gaugefields did not arise from the underlying group, but were introduced by hand.As such,it is difficult to see how this construction could be extended to include the other degrees of freedom of the theory,namely the graviton and the fermions.Recently,it was shown that the entire bosonic sectors,including gravity,of the eleven dimensional supergravity theory and the ten dimensional IIA[15]and IIB supergravity theories[16],could be formulated as a non-linear realisation.It was also conjectured, that when suitably formulated eleven dimensional supergravity would be invariant under a Kac-Moody algebra[17].Although,this conjecture was not proved in[17]some evidence was given and the Kac-Moody algebra was identified.It had rank eleven and was denoted E11[17].A similar analysis found that E11was also the Kac-Moody algebra that would underlie the ten dimensional IIA and IIB theories[16,17].As has been pointed out by one of the authors(PCW)in a number of seminars it could be that E11is part of an evenlarger underlying algebra that is a Borcherds algebra.This is consistent with the general belief that all these theories are part of a larger theory which has been called M theory[18].It was conjectured[17]that E11over an appropriatefield was a symmetry of M theory.However,as seen from this perspective M theory does not necessarily live in eleven dimensions,but rather has its dimension undetermined.A particular theory results from a choice of”vacuum”in M theory and the dimension of the resulting theory is a consequence of the Lorentz group that is contained in the subgroup of E11that is preserved by the”vacuum”under consideration.In this paper we extend some of these considerations to the massive IIA theory and as afirst step show that its entire bosonic sector can be described as a non-linear realisation.1.Massive IIA SupergravityThe bosonic sector of massive IIA supergravity was originally[7]described in terms of the samefields as in the ten-dimensional chiral theory without the cosmological constant.Thesefields were the graviton,h a b,the scalar A(dilaton)and the p-forms A a1···a pforp=1,2,3.However,the massive theory also contains a constant which is related to the cosmological constant.The bosonic part of the original Lagrangian given by Romans in [7]wase−1L=R−112e−1/2A FµνρσFµνρσ−12m2e−5/2A+e−15m2BµνBρσBµ1µ2Bµ3µ4Bµ5µ6),(1.1)whereGµνρ=3∂[µBνρ],and Fµνρσ=4(∂[µA′νρσ]+6mB[µνBρσ]).(1.2) By redefining thefields according toBµν=Aµν+2mA[µ∂µAρ],(1.3)as explained in[19],it can be rewritten in the forme−1L=R−112e1/2A FµνρσFµνρσ−12m2e5/2A+e−19+mAµν),and Gµνρ=3∂[µAνρ],(1.5)23Fµνρσ=4(∂[µAνρσ]+6A[µ∂νAρσ]+The generators K a b satisfy the commutation relations of GL(10,R)and the non-zero commutation relations between all generators mentioned above are given by[K a b,K c d]=δc b K a d−δa d K c b,[K a b,P c]=−δa c P b,[K a b,R c1···c p]=δc1bR ac2···c p+···, [R,R c1···c p]=c p R c1···c p,[R c1···c p,R c1···c q]=c p,q R c1···c p+q,(1.9) where+···means the appropriate anti-symmetrisations.We also include the momentum generator P a in the symmetry algebra and its non-trivial relations with the other generators are given by[P a,R c1···c p]=−mb p(δc1a R c2···c p+···),[P a,R]=−mb0P a.(1.10) In what follows it will often be useful to denote c−1=mb0,since in this way one can view the last commutator of equation(1.10)as an extension of those of equation(1.9).If the coefficients are taken to bec3=−c5=−12,c1=−c7=−34,c1,2=−c2,3=−c3,3=c1,5=c2,5=2,c3,5=1,c2,6=2,c1,7=3,c2,7=−4,b2=−12,b9=5whereg A=e(1/9!)A a1...a9R a1...a9e(1/8!)A a1...a8R a1...a8e(1/7!)A a1...a7R a1 (7)e(1/6!)A a1···a6R a1···a6e(1/5!)A a1···a5R a1···a5e(1/3!)A a1a2a3R a1a2a3e(1/2!)A a1a2R a1a2e A a1R a1e AR.(1.14) Of course,we could have chosen any other representation,but the calculations turn out to be simplest in this particular exponential representation.We now construct a non-linear realisation of the G mIIA algebra taking the local sub-algebra to be the Lorentz group.As such,we demand that the theory is invariant underg→g0gh−1,(1.15) where g0is a rigid element from the whole group G mIIA and h is a local Lorentz transfor-mation.We calculate the Maurer-Cartan formV=g−1dg−ω(1.16) in the presence of the Lorentz connectionω=1p!e−c p−1A˜DµA a1···a pR a1···a p),(1.20)whereΩab c≡(ˆe−1)aµ (e−1∂µe)b c−ωµb c ,(1.21) andˆeµa=e−5/4A(e h)µa,eµa=(e h)µa.(1.22) We see that the object in front of the momentum generator gets altered with respect to IIA (where it was eµa)due to the non-vanishing commutator in equation(1.10).We also see that the vielbeins in theΩµb c equation(1.21)are the unhatted vielbeins.The additional factor of e−5/4A just multiplies the usual expression ofΩab c from the massless case[16].As we will see in chapter2,the physical vielbein therefore remains unchanged.The objects˜D µA a1···a pdefined in(1.20)will be explicitly stated below.Massive IIA supergravity is the non-linear realisation of the group that is the closure of the G mIIA algebra given above and the conformal group in ten dimensions.We therefore take only those combinations of the Cartan forms of G mIIA that can be rewritten as Cartan forms of the conformal group(see section2).Lorentz covariant objects which are also covariant under the full non-linear realisation of the closure of the conformal and the G mIIA algebra are then for example the completely anti-symmetrised derivatives˜Fa1···a p =pe−5/4A e−c p−1A˜D[a1A a2···a p],(1.23)where we have to use˜D a=ˆe aµ˜Dµto convert the curved index to aflat index group-covariantly.As the physical vielbein is the unhatted one,we gain an additional factor e−5/4A in front of everyfield strength in comparison with the IIA case.A discussion of the closure with the conformal group is postponed to the second section.We now give the explicit form of thefield strengths.They are given for the scalar by:˜Fa=e−5/4A D a A,(1.24) for the1-form:˜Fa1a2=2e−5/4A e(3/4)A˜D[a1A a2]=2e−5/4A e(3/4)A(D[a1A a2]+12mA[a1a2A a3a4])(1.27)for the5-form:˜Fa1···a6=6e−5/4A e−(1/4)A D[a1A a2···a6]+20(D[a1A a2···a4+17mA a1a2···a7−20A[a1a2a3D a4A a5a6a7]++12A[a1(D[a2A a3 (7)+20A a2a3(D[a4A a5a6a7]+12mA a3a4A a5a6A a7a8]) (1.30)for the 8-form:˜F a 1···a 9=9e −5/4A D [a 1A a 2···a 9]−24A [a 1D a 2A a 3···a 9]+56A [a 1a 2D a 3A a 4···a 9]−−56A [a 1a 2a 3D a 4A a 5···a 9]+1008A [a 1A a 2a 3D a 4A a 5···a 9]+8mA [a 1a 2A a 3···a 9]++7!4A [a 1A a 2a 3A a 4a 5D a 6A a 7a 8a 9]−−1120A [a 1a 2A a 3a 4a 5˜D a 6A a 7a 8a 9]−518A [a 1a 2A a 3a 4A a 5a 6D a 7A a 8a 9a 10]++3024mA [a 1a 2A a 3a 4A a 5a 6A a 7a 8A a 9a 10] ,(1.32)where D a is the covariant derivative D a A a 1...a p =e a µ(∂µA a 1...a p +(e −1∂µe )a 1c A ca 2...a p +...)(1.33)and ...indicates the terms where (e −1∂µe )acts on the other indices of the gauge field.Also,we have written the exponential e −5/4A separately in front of every field strengths to indicate that it is common to all of them.Using the Cartan forms which transform only under the local Lorentz group in a manner that their indices suggest we must write down a set of invariant equations.If we ask that they be first order in derivatives they can only be given by˜F a 1···a p =110!ǫa 1···a 10˜F a 1···a 10.(1.35)All the above equations of motion and the Einstein equation are equivalent to those one can derive from the Lagrangian formulation given at the beginning of this chapter.We can see that the simple field strengths of equations (1.5),(1.6)indeed match with those given in our group approach of equations (1.25)-(1.27)and one can indeed verify that the aboveequations of motion for the gauge sector(1.34)and(1.35)are the same as those one can derive from the Lagrangian(1.4)once we take m to be a dynamicalfield.To recover the massless case,we simply switch offthe commutators with the momen-tum generator by setting m=0.However,because c−1=mb0we also have c−1=0.Then using the Jacobi relationc−1=−c9(1.36) we deduce c9=0,and then asc2,7c9=c2,7(c2+c7),(1.37) we also need c2,7=0.Thereby the nine form is made redundant(˜F(10)=dA(9))and we are indeed left with the massless case.2.Closure with the Conformal GroupThe massive IIA supergravity theory is the non-linear realisation of the group that is the closure of the G mIIA group given above with the ten-dimensional conformal group. The closure of these two groups is an infinite dimensional group,but rather than working with this group we can perform a simultaneous realisation of the G mIIA and the conformal group.What this actually means is that we construct the equations of motion only from combinations of the Cartan forms of the G mIIA group,given above,that can be rewritten in terms of the Cartan or other covariant forms of the conformal group.In doing this one gains invariance under both conformal group and G mIIA,and so necessarily wefind invariance under the group which is the closure of G mIIA and the conformal group.This is discussed at length in reference[15],but here we briefly discuss the novel features that arise in this procedure when applied to the massive IIA theory.The two groups only have one Goldstone boson in common namely the trace of h a b which is related to the conformalfieldσ.In fact we have to identify these twofields via eµa≡(e h)µa=(e¯h+δσ)µa(as in[15]).All the otherfields that occur as Goldstone bosons in the G mIIA algebra are viewed as matterfields from the conformal group viewpoint.The conformal covariant derivative of afield B transforming under a representationΣof the Lorentz group is:∆µB=e−σ(∂µ+∂νσΣµν)B.(2.1) In contrast,the G mIIA covariant derivative of a matterfield is given by˜D a B=(e−1)aµ(∂µ+1(2.1)for∂µB and plugging the result into(2.2),we get˜D a B=(e−1)aµ(eσ∆µB−∂νσΣµνB+12mAµa2) ,where the vielbein with the overbar stands for the traceless part eµa=e¯hµa+δaµσ=(¯e·eσ)µa. We realise that only if we take the combination˜D[a A b]does theσdependence only appear through the conformal covariant derivative alone,as then the3σ-dependent terms in the second bracket vanish since they are symmetric inµand a2.If we want to use expressions covariant under both the conformal group and G mIIA,then we have to demand that all σdependence be implicitly through the conformal derivative alone.As such we conclude that only the totally antisymmetrised object2e3/4Aˆe[a|µ˜DµA|b]≡˜F ab is covariant under both groups.We know that the closure of the G mIIA group and the conformal group generates gauge transformations and general coordinate transformations and so the above object should be covariant under these transformations.We observe that˜D[aA b]=2 ∂[a A b]+(e−1∂[a e)b]c A c+mb2A ab =2e aµe bν(∂[µAν]+mb2Aµν),(2.6) making it clear that it is covariant under gauge and general coordinate transformations. The G mIIA covariant derivatives of IIA supergravity only differ from those of massive IIA supergravity by terms containing m,and the nine form potential.However,as these new terms do not contain derivatives the closure with the conformal group is not spoilt by the presence of these terms.We now turn to the gravity sector of the theory.Clearly,the constraintΩa,[bc]−Ωb,(ac)+Ωc,(ab)=0.(2.7) is G mIIA covariant,but one can show in much the same way as for the IIA and the eleven dimensional supergravity cases[15]that it is also conformally covariant.Examining the definition ofΩa,bc in equation(1.21),we see that it involves an undifferentiated factor of ˆe aµ=e−5/4A e aµ.The factor of e−5/4A can then be removed and wefind that it is exactly the same constraint as in the other cases(eleven dimensional supergravity,IIA and IIB supergravity)and therefore results in the usual expression for the spin connection in terms of the vielbein,namelyωµbc=12(e bρ∂ρeµc−e cρ∂ρeµb)−116m2e5/2Aηab+112ηab F(3)cd f F(3)cd f)+2me3/2A(F(2)a c F(2)bc−13e1/2A(F(4)acd f F(4)bcd f−3massive IIA theory we also have included,following[19],a nine form gauge which is associated with the introduction of the cosmological constant.The correct theory requires that the momentum generator has non-trivial commutation relations with the generators associated with the gaugefields as given in equation(1.10).This is natural as the nine form is associated with the cosmological constant and so with gravity.This is in contrast to the work of reference[14]which takes a different approach and does not include gravity,but does introduce a dual form for the nine form gaugefield which was called a“minus one form”.The properties of this minus one form are not very explicitly spelt out.In effect wefind in this paper that the momentum generator plays the role of the generator associated with the“minus one form”of reference[14].It would be interesting to examine if the non-linear realisation could be extended,in ways explained in reference[17],to be invariant under a Kac-Moody,or Borcherds algebra, and to conjecture what this algebra is.In the previous non-linear realisation of the maximal supergravities the momentum generator has not played a central part in the Kac-Moody algebra that has been identified.However,the non-trivial relations of equations(1.10) imply that this generator must occur in a non-trivial way in the corresponding algebra. Progress in this direction may also shed light on the place that the massive IIA theory has in M theory.Acknowledgements:IS would like to thank Andr´e Miemiec,who has given support when calculating the equations of motion for various SUGRAs.IS is also supported by DAAD(D/00/09914).References[1]W.Nahm,”Supersymmetries and their representations”,Nucl.Phys.B135(1978),p.149[2]E.Cremmer,B.Julia,and J.Scherk,“Supergravity theory in11dimensions”,Phys.Lett.B76(1978)409–412.[3]I.C.G.Campbell and P.C.West,“N=2d=10nonchiral supergravity and its spon-taneous compactification”,Nucl.Phys.B243(1984)112.;M.Huq and M.Namazie,“Kaluza–Klein supergravity in ten dimensions”,Class.Q.Grav.2(1985).;F.Giani and M.Pernici,“N=2supergravity in ten dimensions”,Phys.Rev.D30(1984)325.[4]J.H.Schwarz and P.C.West,“Symmetries and transformations of chiral N=2,D=10supergravity,”Phys.Lett.B126(1983)301.[5]J.H.Schwarz,“Covariantfield equations of chiral N=2D=10supergravity,”Nucl.Phys.B226(1983)269.[6]P.S.Howe and P.C.West,“The complete N=2,d=10supergravity,”Nucl.Phys.B238(1984)181.[7]L.J.Romans,“Massive N=2a Supergravity in Ten Dimensions,”Phys.Lett B169(1986)374.[8]S.Ferrara,J.Scherk and B.Zumino,“Algebraic Properties of Extended Supersym-metry”,Nucl.Phys.B121(1977)393;E.Cremmer,J.Scherk and S.Ferrara,“SU(4) Invariant Supergravity Theory”,Phys.Lett.B74(1978)61.[9]E.Cremmer and B.Julia,“The N=8supergravity theory.I.The Lagrangian”,Phys.Lett.B80(1978)48[10]B.Julia,“Group Disintegrations”,in Superspace&Supergravity,p.331,eds.S.W.Hawking and M.Roˇc ek,Cambridge University Press(1981).[11]H.Nicolai,Phys.Lett.B187(1987)316.[12]C.M.Hull,P.K.Townsend,“Unity of Superstring Dualities,”Nucl.Phys.B438(1995),109,hep-th/9410167.[13]E.Cremmer,B.Julia,H.L¨u,and C.N.Pope,“Dualisation of dualities.I&II:Twisted self-duality of doubledfields and superdualities,”Nucl.Phys.B535(1998) 242,hep-th/9806106[14]vrinenko,H.L¨u,C.N.Pope,K.Stelle,“Superdualities,Brane Tensions andMassive IIA/IIB Duality,”Nucl.Phys.B555(1999)201,hep-th/9903057[15]P.C.West,“Hidden superconformal symmetry in M theory”,JHEP08(2000)007,hep-th/0005270[16]I.Schnakenburg,P.West,“Kac-Moody Symmetries of IIB Supergravity,”,Phys.Lett.B517(2001)421,hep-th/0107181[17]P.West,“E(11)and M theory,Classical and Quantum Gravity,18(2001)4443,hep-th/0104081[18]E.Witten,String theory dynamics in various dimensions,Nucl.Phys.B443(1995)85,hep-th9503124[19]E.Bergshoeff,M.de Roo,M.B.Green,G.Papadopoulos,P.K.Townsend“Dualityof Type II7-branes and8-branes,”Nucl.Phys.B470(1996)113,9601150。
BLACK HOLES ] _ Physics Papers - Steven Hawking (1993), The Superscattering Matrix For Two Dimension
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However when one goes to quantum gravity, the possibility of forming real or virtual black holes means that the evolution is given by what I call a super scattering operator, $. This
Most of the calculations that have been done on two dimensional black holes, have been carried out in Lorentzian spacetime. They have assumed no horizons or singularities in the past. One can show[2] that this implies that there must be horizons or naked singularities in the future. Thus the calculations are manifestly not CPT invariant. This is reflected in the fact that the outgoing energy flux they predict is always below a certain critical level. However, the ingoing energy flux can have any value. Thus the super scattering operator given by these calculations will not be CPT invariant. To lose quantum coherence is bad enough, but to lose CPT as well, sounds like carelessness.
A Comprehensive Survey of Multiagent Reinforcement Learning
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 38, NO. 2, MARCH 2008
A Comprehensive Survey of Multiagent ReinfoN
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MULTIAGENT system [1] can be defined as a group of autonomous, interacting entities sharing a common environment, which they perceive with sensors and upon which they act with actuators [2]. Multiagent systems are finding applications in a wide variety of domains including robotic teams, distributed control, resource management, collaborative decision support systems, data mining, etc. [3], [4]. They may arise as the most natural way of looking at the system, or may provide an alternative perspective on systems that are originally regarded as centralized. For instance, in robotic teams, the control authority is naturally distributed among the robots [4]. In resource management, while resources can be managed by a central authority, identifying each resource with an agent may provide a helpful, distributed perspective on the system [5].
Quantum dynamics of $N=1$, $D=4$ supergravity compensator
a rXiv:h ep-th/96793v112J ul1996Quantum dynamics of N =1,D =4supergravity compensator I.L.Buchbinder and A.Yu.Petrov Department of Theoretical Physics State Pedagogical Institute Tomsk 634041,Russia TSPI-TH32/96Talk given by I.L.Buchbinder.To be published in the Proceedings of the SUSY96conferenceQuantum dynamics of N=1,D=4supergravity compensatorI.L.Buchbinder and A.Yu.PetrovDepartment of Theoretical PhysicsTomsk State Pedagogical InstituteTomsk634041,RussiaTSPI-TH32/96AbstractA new N=1superfield theory in D=4flat superspace is suggested.It describes dynamics ofsupergravity compensator and can be considered as a low-energy limit for N=1,D=4superfieldsupergravity.The theory is shown to be renormalizable in infrared limit and infrared free.A quantumeffective action is investigated in infrared domain.It is well known that the superfield supergravity has been constructed in refs.[1-2]as a dynamical theory of vector superfield and chiral and antichiral superfield compensators in N=1,D=4curved superspace.A general consideration of quantum aspects of superfield supergravity has been carried out in refs.[3].Purpose of this paper is to develop a simplified theory describing dynamics of only compensator su-perfields inflat superspace.We show that such a theory can be treated as a natural low-energy limit of quantum supergravity with matter and possesses remarkable properties in infrared domain.The model under consideration is based on an idea of gravity induced by conformal anomaly of matter fields in curved space-time[4](see also[5]).A superconformal anomaly of matter superfields in N=1 curved superspace has been found in ref.[6]and a superfield action generating this anomaly has been constructed in ref.[7](see also[8]).We investigate a model an action of which is a sum of the anomaly generating action and the action of N=1,D=4superfield supergravity(see f.e.[8]).Being transformed to conformallyflat superspace the action of the model takes the formS= d8z(−Q2eσ+¯σ)+(Λ d6ze3σ+h.c.)(1) 2Here Dα,¯D˙α,∂α˙αare theflat supercovariant derivatives,σ=lnΦandΦis a chiral compensator of N=1,D=4supergravity.Q2,ξ1,ξ2,m2,Λare the arbitrary parameters of the model playing a role of the couplings.We begin with consideration of renormalization structure of the model(1).One can prove that the superficial degree of divergencesωhas the form[10]ω≤2−3L c−2V3−3V4(2) Here V3is a number of vertices proportional to m2,V4is a number of vertices proportional toΛ,L c is a number of lines corresponding to the propagators<σσ>and<¯σ¯σ>.The condition of divergenceω≥0means that the divergent supergraphs cannot contain the vertices proportional toΛand the above lines.They can include no more than one vertex proportional to m2. As a result we have some sort of non-renormalization theorem according to which the vertex propor-tional toΛis alwaysfinite.Let us consider now an one-loop renormalization.The supergraphs leading to one-loop divergences are given by the Fig.1,Fig.2and Fig.3&%'$ ¯D ˙αd d D α¯D ˙αD β¯D˙βG +−G −+Fig.1&%'$ d d d d ¯D ˙αD β¯D ˙β¯D ˙αD β¯D ˙βG +−G −+Fig.2&%'$∂α˙α∂β˙βD α¯D˙α¯D ˙βG +−G −+Fig.3Here G +−and G −+are the <σ¯σ>-and<¯σσ>propagators respectively.After straightforward calculations of above supergraphs in framework of dimensional reduction regularization we obtainQ 2(0)=µ−ǫZ Q Q 2,ξ(0)1,2=µ−ǫZξ1,2(3)Z Q =1+21432π6ξ21Q 4ǫThe eqs.(3)lead to the following equations for running couplingsdξ1(t )Q 4(t );dξ2(t )Q 4(t )dQ 2(t )Q 4(t )(4)where a =21132π2,b =32214π4.These equation show that the ξ1(t )=ξ2(t )=0is an infrared fixed point and Q 2(t )→Q 2=const in infrared domain t →−∞.Now we consider a renormalization of the coupling m 2in infrared limit where ξ1=ξ2=0.The corresponding divergent supergraphs are given by Fig.4.&%'$ d d ...G +−Fig.4Here the dots mean externalσ,¯σ-lines.A straightforward calculation lead to m20=Z m2m2;Z m2=1+2dt=(2−2α+2α22Q2α2d6ze3ασ+h.c.)corresponds tofinite in infrared domain theory which can be treated as a natural infrared limit of the quantum supergravity with matter.We investigate now a structure of effective action for the model(7).Since the model isfinite we face a problem of effective action infinite quantumfield theory.An effective action in N=1,D=4superfield theory is written in the formΓ= d8zL+( d6z L c+h.c.)(8)Here L is called the general effective lagrangian,L c is called the chiral effective lagrangian.The L depends on thefieldσand¯σand their supercovariant derivatives.In particular there can be a term independent of the derivatives.This term is called a kahlerian effective potential K(see the discussion in[8,9]).The L c depends only onσand its space-time derivatives of any order.We represent L(σ,¯σ)and L c(σ)in the form of power expansion in derivatives ofσ,¯σand willfind the lower contributions in this expansion. The action(7)is invariant under transformationsσ→σ+γ,¯σ→¯σ+β,Λ→e−3γΛ,¯Λ→e−3βΛ(9) One can expect that the effective action should be also invariant under these transformations(see dis-cussion in ref.[10]).Taking into account the dimensions ofσ,Λ,¯Λ,L and L c we obtain the lower contributions to L and L c in the formK=c ΛQ2/3{(c1e−σ+c2e2σ+(10) +c3e−σ+c4e−4σ)∂mσ∂mσ++2σ(c3e−σ+c4e−4σ)}where¯Λ=Λ;c,c1,c2,c3,c4are arbitrary dimensionless constants.Explicit form of these constants can be found by straightforward calculations in one-loop approximation(see ref.[10],where a special technique for calculation of one-loop effective action for the model(7)has been developed).As a result the effective action with lower contributions has a form of classical action(7)and quantum corrections defined by K and L c.One can prove that for the superfieldsσ,¯σslowly varying in space-time this effective action can be rewritten in the formΓ= d8zφ¯φ+ λ d6zφ3+h.c.)(11) whereλ=Q2c−3/2,φ=(ΛThe work was supported by Russian Foundation for Basic Research under the project No.96-02-16017.References[1]V.I.Ogievetsky,E.S.Sokatchev.Phys.Lett.B79,222(1978).Yad.Fiz.(Sov.J.Nucl.Phys)31,264(1980);32,862(1980);32,870(1980);32,1142(1980).[2]W.Siegel,S.J.Gates.Nucl.Phys.B147,77(1979).[3]M.T.Grisaru,W.Siegel.Nucl.Phys.B187,149(1981);B201,292(1982).[4]A.M.Polyakov.Phys.Lett.B103,207(1981).R.J.Reigert.Phys.Lett.B134,56(1984)E.S.Fradkin,A.A.Tseytlin.Phys.Lett.B134,187(1984).I.Antoniadis,E.Mottola.Phys.Rev.D45,2013(1992).[5]I.L.Buchbinder,S.D.Odintsov,I.L.Shapiro.Effective Action in Quantum Gravity,IOP PublishingLtd,Bristol and Philadelphia,1992.[6]I.L.Buchbinder and S.M.Kuzenko.Nucl.Phys.B274,653(1986)[7]I.L.Buchbinder and S.M.Kuzenko.Phys.Lett.B202,233(1988).[8]I.L.Buchbinder,S.M.Kuzenko.Ideas and Methods of Supersymmetry and Supergravity,IOPPublishing Ltd,Bristol and Philadelphia,1995.[9]I.L.Buchbinder,S.M.Kuzenko,J.V.Yarevskaya.Nucl.Phys.B411,665(1994).I.L.Buchbinder,S.M.Kuzenko,A.Yu.Petrov.Phys.Lett.B321,372(1994).[10]I.L.Buchbinder,A.Yu.Petrov.Quantum dynamics of N=1,D=4supergravity chiral compensator.hep-th9501125(Class.Quant.Grav.,to be published);I.L.Buchbinder,A.Yu.Petrov.On structure of effective action in four-dimensional quantum dilaton supergravity.hep-th9604154.。
Glueballs and Their Kaluza-Klein Cousins
a r X i v :h e p -t h /9806171v 1 19 J u n 1998UCB-PTH-98/35,LBNL-41948,NSF-ITP-98-070hep-th/9806171Glueballs and Their Kaluza-Klein CousinsHirosi Ooguri,Harlan Robins and Jonathan Tannenhauser Department of Physics,University of California at Berkeley,Berkeley,CA 94720Theoretical Physics Group,Mail Stop 50A-5101,Lawrence Berkeley National Laboratory,Berkeley,CA 94720Institute for Theoretical Physics,University of California,Santa Barbara,CA 93106Abstract Spectra of glueball masses in non-supersymmetric Yang-Mills theory in three and four dimensions have recently been computed using the conjectured duality between superstring theory and large N gauge theory.The Kaluza-Klein states of supergravity do not correspond to any states in the Yang-Mills theory and therefore should decouple in the continuum limit.On the otherhand,in the supergravity limit g 2Y M N →∞,we find that the masses of theKaluza-Klein states are comparable to those of the glueballs.We also showthat the leading (g 2Y M N )−1corrections do not make these states heavier than the glueballs.Therefore,the decoupling of the Kaluza-Klein states is not evident to this order.1IntroductionSpectra of glueball masses in non-supersymmetric Yang-Mills theory in three and four dimensions have recently been calculated[1]using the conjectured duality between string theory and large N gauge theory[2–5].The results are apparently in good numerical agreement with available lattice gauge theory data,although a direct comparison may be somewhat subtle,since the supergravity computation is expected to be valid for large ultraviolet couplingλ=g2Y M N,whereas we expect that QCD in the continuum limit is realized forλ→0[5,6].As explained in[6,1],the supergravity computation atλ≫1 gives the glueball masses in units of thefixed ultraviolet cutoffΛUV.Forfiniteλ,the glueball mass M is expected to be a function of the formM2=F(λ)Λ2UV.(1.1)In the continuum limitΛUV→∞,M should remainfinite and of orderΛQCD.This would require F(λ)→0asλ→0.In[1],the leading string theory corrections to the masses were computed and shown to be negative and of orderλ−3/2,in accordance with expectation.Witten has proposed[5]that three-dimensional pure QCD is dual to type IIB string theory on the product of an AdS5black hole and S5.This proposal requires that certain states in string theory decouple in the continuum limitλ→0.One class of such states are Kaluza-Klein excitations on S5.The supergravityfields on the AdS5black hole×S5 can be classified by decomposing them into spherical harmonics(the Kaluza-Klein modes) on S5[7,8].They fall into irreducible representations of the isometry group SO(6)of S5, which is the R-symmetry of the four-dimensional N=4supersymmetric gauge theory from which QCD3is obtained by compactification on a circle.Consequently,only SO(6) singlet states should correspond to physical states in QCD3in the continuum limit.These are the glueball states studied in[1].However,wefind that,in the supergravity limit, masses of the SO(6)non-singlet states are of the same order as the SO(6)singlet states. Since these states should decouple in the limitλ→0,it was speculated in[1]that the string theory corrections should make the non-singlet states heavier than the singlet states.The purpose of this paper is to test this idea.We compute the masses of the SO(6)non-singlet states coming from the Kaluza-Klein excitations of the dilaton in ten dimensions. Wefind the masses in the supergravity limit to be of the same order as those of the SO(6) singlet states.We then calculate the leading string theory corrections to the masses.We find that the leading corrections do not make the Kaluza-Klein states heavier than theglueballs.Therefore,the decoupling of the Kaluza-Klein states is not evident to this order.This suggests that the quantitative agreement between the glueball masses from supergravity and the lattice gauge theory data should be taken with a grain of salt.2The Supergravity LimitWe calculate the masses of the Kaluza-Klein states following the analysis of[1].Ac-cording to[5],QCD3is dual to type IIB superstring theory on the AdS5black hole×S5 geometry given bydx24πg2Y M N =dρ2ρ2 +ρ2−b4dρ (ρ4−1)ρd f0this shooting method can be used to compute k 20and the wavefunction f 0(ρ)to arbitrarilyhigh precision.The results of the numerical work are listed in Table 1.As expected,the masses are all of the order of the ultraviolet cutoffΛUV =b .l135711.5929.2654.9388.6034.5363.60100.6145.668.98109.5157.9214.3l 2sρ2−b 4ρ2 dτ2+ρ23 i =1dx 2i +d Ω25,(3.1)where δ1=+15γ 5b 4ρ8−19b 12ρ4+5b 8ρ12 ,(3.2)and γ=18γ b 42ρ8+b 1216πG 10 d 10x √2g µν∂µΦ∂νΦ+γe −3periodicity2πR ofτis also modified toR= 1−152b.(3.5) It is the inverse radius R−1that serves as the ultraviolet cutoffof QCD3.To solve the dilaton wave equation in theα′-corrected geometry(3.1),we writeΦ=Φ0+f(ρ)e ikx Y l(Ω5),(3.6) whereΦ0is the dilaton background given by(3.3),and expand f(ρ)and k2inγasf(ρ)=f0(ρ)+γh(ρ),k2=k20+γδk2.(3.7) Here f0(ρ)obeys the lowest order equation(2.3)and is a numerically given function,and k20is likewise determined from(2.3).The second-order differential equation obtained from the action(3.4)in the background metric(3.1)and dilatonfield(3.3)is,in units in which b=1,ρ−1ddρ −(k20+l(l+4)ρ2)h==(75−240ρ−8+165ρ−12)d2f0dρ+(δk2−120(k20+l(l+4)ρ2)ρ−12−405ρ−14)f0(ρ).(3.8) With f0(ρ)and k20given,one may regard this as an inhomogeneous version of the equation (2.3).We solve this equation for h(ρ)andδk2.We are now ready to present our results.Let us denote the lowest mass of the l-th Kaluza-Klein state by M l.In units of the ultraviolet cutoffΛUV=(2R)−1,with R given by(3.5),wefindM20=11.59×(1−2.78ζ(3)α′3+···)Λ2UVM21=19.43×(1−2.66ζ(3)α′3+···)Λ2UVM22=29.26×(1−2.62ζ(3)α′3+···)Λ2UVM23=41.10×(1−2.61ζ(3)α′3+···)Λ2UVM24=54.93×(1−2.63ζ(3)α′3+···)Λ2UVM25=70.76×(1−2.66ζ(3)α′3+···)Λ2UVM26=88.60×(1−2.69ζ(3)α′3+···)Λ2UVM27=108.4×(1−2.72ζ(3)α′3+···)Λ2UV.(3.9)Similar behavior is observed for the excited levels of each Kaluza-Klein state.Thus the corrections do not make the Kaluza-Klein states heavier than the glueballs, and the decoupling of the Kaluza-Klein states is not evident to this order.According to Maldacena’s duality,theλ−1/2expansion of the gauge theory corresponds to theα′-expansion of the two-dimensional sigma model with the AdS5black hole×S5as its target space.It is possible that the decoupling of the Kaluza-Klein states takes place only non-perturbatively in the sigma model.AcknowledgmentsWe thank Csaba Cs´a ki,Aki Hashimoto,Yaron Oz,John Terning,and especially David Gross for useful discussions.We thank the Institute for Theoretical Physics at Santa Barbara for its hospitality.This work was supported in part by the NSF grant PHY-95-14797and the DOE grant DE-AC03-76SF00098,and in part by the NSF grant PHY-94-07194through ITP.H.R. and J.T.gratefully acknowledge the support of the A.Carl Helmholz Fellowship in the Department of Physics at the University of California,Berkeley.Appendix:The Boundary Condition at the HorizonIn this appendix,we show that the boundary condition at the horizonρ=b used in the shooting method[1]is consistent,and that the eigenvalue k2and the wavefunction f(ρ)can be evaluated to an arbitrarily high precision using this method.In the neighborhood ofρ=b,the dilaton wave equation takes the form∂ρ(ρ−b)∂ρf(ρ)+···=0.(3.10) Its general solution is of the formf(ρ)=c1[1+α(ρ−b)+···]+c2[log(ρ−b)+···](3.11)with arbitrary coefficients c1,2(the constantαis determined by the wave equation and is in general non-zero).The regularity of the dilatonfield requires c2=0.In the shooting method,we numerically integrate the differential equation starting from a sufficiently large value ofρdown to the horizon.For generic k2,the function thus obtained,when expanded as in(3.11),would have c2=0.The task is to adjust k2so that c2=0.Since f(ρ)is divergent atρ=b for generic k2,it is numerically difficult to impose the boundary condition directly atρ=b.Instead,in[1]and in this paper,we required f′=0 atρ=b+ǫfor a given smallǫ(for example,ǫ=0.0000001b in this paper).By(3.11), this condition impliesc2=−c1αǫ+···.(3.12)Therefore,c2can be made arbitrarily small by adjustingǫ.This justifies the numerical method used in[1]and in this paper.We thank Aki Hashimoto for discussions on the numerical method.References[1]C.Cs´a ki,H.Ooguri,Y.Oz and J.Terning,“Glueball Mass Spectrum from Supergravity,”hep-th/9806021.[2]J.M.Maldacena,“The Large N Limit of Superconformal Field Theories and Supergravity,”hep-th/9711200.[3]S.S.Gubser,I.R.Klebanov and A.M.Polyakov,“Gauge Theory Correlators from Non-Critical String Theory,”hep-th/9802109.[4]E.Witten,“Anti-de Sitter Space and Holography,”hep-th/9802150.[5]E.Witten,“Anti-de Sitter Space,Thermal Phase Transition,And Confinement in GaugeTheories,”hep-th/9803131.[6]D.J.Gross and H.Ooguri,“Aspects of Large N Gauge Theory Dynamics as Seen by StringTheory,”hep-th/9805129.[7]H.J.Kim,L.J.Romans and P.Van Nieuwenhuizen,“Mass Spectrum of Chiral Ten-Dimensional N=2Supergravity on S5,”Phys.Rev.D32(1985)389.[8]M.G¨u naydin and N.Marcus,“The Spectrum of The S5Compactification of Chiral N=2,D=10Supergravity and The Unitary Supermultiplets of U(2,2/4),”Class.Quant.Grav.2(1985)L11.[9]S.S.Gubser,I.R.Klebanov and A.A.Tseytlin,“Coupling Constant Dependence in theThermodynamics of N=4Supersymmetric Yang-Mills Theory,”hep-th/9805156.[10]M.T.Grisaru,A.E.M.van de Ven and D.Zanon,“Four-Loop Beta Function for theN=1and N=2Supersymmetric Nonlinear Sigma-Model in Two Dimensions,”Phys.Lett.B173(1986)423;M.T.Grisaru and D.Zanon,“Sigma-Model Superstring Corrections to the Einstein-Hilbert Action,”Phys.Lett.B177(1986)347.[11] D.J.Gross and E.Witten,“Superstring Modifications of Einstein Equation,”Nucl.Phys.B277(1986)1.。
凝聚态物理相关诺贝尔奖(可编辑修改word版)
凝聚态物理相关诺贝尔化学奖1970-1985 年赫伯特·豪普特曼（美）杰罗姆·卡尔勒（美）在测定晶体结构的直接方法上的贡献2000 年艾伦·黑格（美）艾伦·麦克迪尔米德（美/新西兰）白川英树（日）对导电聚合物的研究2011 年丹·谢赫特曼（以）准晶的发现[5]凝聚态物理相关诺贝尔物理学奖1970-1972年约翰·巴丁美国“他们联合创立了超导微观理论，即常说的BCS 理论”"for their jointly developed theory ofsuperconductivity, usually called theBCS-theory"[73]利昂·库珀美国约翰·罗伯特·施里弗国美1973年江崎玲于奈日本“发现半导体和超导体的隧道效应”"for their experimental discoveriesregarding tunneling phenomena insemiconductors and superconductors,respectively"[74]伊瓦尔·贾埃弗挪威国英 “他理论上预测出通过隧道势垒的超电流的性质，特别是那些通常被称为约瑟夫森效应 布赖 的现象” 恩·戴"for his theoretical predictions of the 维·约瑟 properties of a supercurrent through a 夫森 tunnel barrier, in particular those phenomena which are generally known as the Josephson effect"[74]1977 年菲利普·沃伦·安德森国美“对磁性和无序体系电子结构的基础性理论研究”"for their fundamental theoretical investigations of the electronicstructure of magnetic and disorderedsystems"[78]内维尔·莫特 国英约翰·凡扶累克 国美年彼得·列昂尼多维奇·卡皮查苏联“低温物理领域的基本发明和发现” "for his basic inventions and discoveries in the area of low- temperature physics"[79]1982 年肯尼斯·威尔逊美国“对与相转变有关的临界现象理论的贡献”"for his theory for critical phenomena in connection with phase transitions"[83]1985 年克劳斯·冯·克利青德国“发现量子霍尔效应”"for the discovery of thequantized Hall effect"[86]特·鲁斯卡德国“电子光学的基础工作和设计了第一台显微镜"for his fundamental work in electron optics,electron microscope"格尔德·宾宁德国“研制"for their design of the scanningtunneling microscope"海因里希·罗雷尔瑞士约翰内斯·贝德诺尔茨德国“在发现"for their important break-through the discovery of superconductivity ceramic materials"卡尔·米勒瑞士皮埃尔吉勒纳法国“发现研究简单系统中有序现象的方法可以被推广到比较复杂的物质形式，特别是推广到"forfor studying order phenomena in simple systems can be generalized to morecomplexliquid crystals and polymers"1996年戴维·李美国“发现了在氦-3 里的超流动性”"for their discovery ofsuperfluidity inhelium-3"[97]道格拉斯·奥谢罗夫美国罗伯特·理查森美国2000年若雷斯·阿尔费罗夫俄罗斯“发展了用于高速电子学和光电子学的半导体异质结构”"for developing semiconductorheterostructures used in high-speed-and optoelectronics"[101]赫伯特·克勒默德国杰克·基尔比美国“在发明集成电路中所做的贡献”"for his part in the invention of theintegrated circuit"[101]2001年埃里克·康奈尔国美“在碱性原子稀薄气体的玻色-爱因斯坦凝聚态方面取得的成就，以及凝聚态物质属性质的早期基础性研究”"for the achievement of Bose-Einsteincondensation in dilute gases of alkaliatoms, and for early fundamental studiesof the properties of the condensates"[102]卡尔·威曼国美沃尔夫冈·克特勒国德2003年阿列克谢·阿布里科索夫美国俄罗斯“对超导体和超流体理论做出的先驱性贡献”"for pioneering contributions tothe theory of superconductors andsuperfluids"[104]维塔利·金俄罗斯兹堡安东尼·莱格特英国美国2007 年艾尔伯·费尔法国“发现巨磁阻效应”"for the discovery of giant magnetoresistance"[108]彼得·格林贝格德国2009 年高锟英国 美国 [110] “在光学通信领域光在纤维中传输方面的突破性成就” "for groundbreaking achievementsconcerning the transmission of light infibers for optical communication"[111]威拉 德·博伊尔 美国“发明半导体成像器件电荷耦合器件” "for the invention of an imaging semiconductor circuit – the CCD sensor"[111]乔治·史密斯美国2010 年安德烈·海姆荷兰俄罗斯“在二维石墨烯材料的开创性实验”"for groundbreaking experimentsregarding the two-dimensionalmaterial graphene"[112]康斯坦丁·诺沃肖洛夫英国俄罗斯。
薛定谔—麦克斯韦尔方程径向解的存在性和多重性(英文)
In 1887, the German physicist Erwin Schrödinger proposed a radial solution to the Maxwell-Schrödinger equation. This equation describes the behavior of an electron in an atom and is used to calculate its energy levels. The radial solution was found to be valid for all values of angular momentum quantum number l, which means that it can describe any type of atomic orbital.The existence and multiplicity of this radial solution has been studied extensively since then. It has been shown that there are infinitely many solutions for each value of l, with each one corresponding to a different energy level. Furthermore, these solutions can be divided into two categories: bound states and scattering states. Bound states have negative energies and correspond to electrons that are trapped within the atom; scattering states have positive energies and correspond to electrons that escape from the atom after being excited by external radiation or collisions with other particles.The existence and multiplicity of these solutions is important because they provide insight into how atoms interact with their environment through electromagnetic radiation or collisions with other particles. They also help us understand why certain elements form molecules when combined together, as well as why some elements remain stable while others decay over time due to radioactive processes such as alpha decay or beta decay.。
Supersymmetric Gauge Theories and Gravitational Instantons
a rXiv:h ep-th/987237v13J ul1998CALT-68-2190SUPERSYMMETRIC GAUGE THEORIES AND GRA VITATIONAL INSTANTONS SERGEY A.CHERKIS California Institute of Technology,Pasadena CA 91125,USA E-mail:cherkis@ Various string theory realizations of three-dimensional gauge theories relate them to gravitational instantons 1,Nahm equations 2and monopoles 3.We use this correspondence to model self-dual gravitational instantons of D k -type as moduli spaces of singular monopoles,find their twistor spaces and metrics.This work provides yet another example of how string theory unites seem-ingly distant physical problems.(See references for detailed results.)The central object considered here is supersymmetric gauge theories in three di-mensions.In particular,we shall be interested in their vacuum structure.The other three problems that turn out to be closely related to these gauge theories are:•Nonabelian monopoles of Prasad and Sommerfield,which are solutions of the Bogomolny equation ∗F =D Φ(where F is the field-strength of a nonabelian connection A =A 1dx 1+A 2dx 2+A 3dx 3and Φis a nonabelian Higgs field).•An integrable system of equations named after Nahm dT i2εijk [T j ,T k ],(1)for T i (s )∈u (n ).These generalize Euler equations for a rotating top.•Solutions of the Euclideanized vacuum Einstein equation called self-dual gravitational instantons ,which are four-dimensional manifolds with self-dual curvature tensorR αβγδ=1The latter provide compactifications of string theory and supergravity that preserve supersymmetry and are of importance in euclidean quantum gravity.The compact examples are delivered by a four-torus and K3.The noncom-pact ones are classified according to their asymptotic behavior and topology.Asymptotically Locally Euclidean (ALE)gravitational instantons asymptoti-cally approach R 4/Γ(Γis a finite subgroup of SU (2)).These were classified by Kronheimer into two infinite (A k and D k )series and three exceptional (E 6,E 7and E 8)cases according to the intersection matrix of their two-cycles.Asymp-totically Locally Flat (ALF)spaces approach the R 3×S 1 /Γmetric.(To be more precise S 1is Hopf fibered over the two-sphere at infinity of R 3.)Sending the radius of the asymptotic S 1to infinity we recover an ALE space of some type,which will determine the type of the initial ALF space.For example,the A k ALF is a (k+1)-centered multi-Taub-NUT space.Here we shall seek to describe the D k ALF space.M theory on an A k ALF space is known to describe (k+1)D6-branes of type IIA string theory.Probing this background with a D2-brane we obtain an N =4U (1)gauge theory with (k+1)electron in the D2-brane worldvolume.As the D2-brane corresponds to an M2-brane in M theory,a vacuum of the above gauge theory corresponds to a position of the M2-brane on the A k ALF space we started with.Thus the moduli space of this gauge theory is the A k ALF.Next,considering M theory on a D k ALF one recovers 4k D6-branes parallel to an orientifold O 6−.On a D2-brane probe this time we find an N =4SU(2)gauge theory with k matter multiplets.Its moduli space is the D k ALF.So far we have related gauge theories and gravitational instantons .D3NS5NS5pp p k D3Figure 1:The brane configuration corresponding to U (2)gauge theory with k matter mul-tiplets on the internal D3-branes.There is another way of realizing these gauge theories.Consider the Chalmers-Hanany-Witten configuration in type IIB string theory (Figure 1).2In the extreme infrared limit the theory in the internal D3-branes will appear to be three-dimensional with N=4supersymmetry.This realizes the gauge theory we are interested in.A vacuum of this theory describes a particular position of the D3-branes.In the U(2)theory on the NS5-branes the internal D3-branes appear as nonabelian monopoles,while every external semiinfinite D3-brane appears as a Dirac monopole in the U(1)of the lower right corner of the U(2).Thus the moduli space of such monopole configurations of non-abelian charge two and with k singularities is also a moduli space of the gauge theory in question.Another way of describing the vacua of the three-dimensional theory on the D3-branes is by considering the reduction of the four-dimensional theory on the interval.For this reduction to respect enough supersymmetry thefields (namely the Higgsfields of the theory on the D3-branes)should depend on the reduced coordinate so that Nahm Equations(1)are satisfied.Thus the Coulomb branch of the three-dimensional gauge theory is described as a moduli space of solutions to Nahm Equations.At this point we have two convenient descriptions of D k ALF space as a moduli space of solutions to Nahm equations and as a moduli space of sin-gular monopoles.It is the latter description that we shall make use of here. Regular monopoles can be described5by considering a scattering problem u·( ∂+ A)−iΦ s=0on every lineγin the three-dimensional space di-rected along u.The space of all lines is a tangent bundle to a sphere T=TP1. Let(ζ,η)be standard coordinates on T,such thatζis a coordinate on the sphere andηon the tangent space.Then the set of lines on which the scat-tering problem has a bound state forms a curve S∈T.S is called a spectral curve and it encodes the monopole data we started with.In case of singular monopoles some of the linesγ∈S will pass through the singular points.These lines define two sets of points Q and P in S,such that Q and P are conjugate to each other with respect to the change of orientation of the lines.Analysis of this situation1shows,that in addition to the spectral curve,we have to consider two sectionsρandξof the line bundles over S with transition functions eµη/ζand e−µη/ζcorrespondingly.Alsoρvanishes at the points of Q andξat those of P.Since we are interested in the case of two monopoles the spectral curve is given byη2+η2(ζ)=0whereη2(ζ)=z+vζ+wζ2−¯vζ3+¯zζ4.z,v and w are the moduli.z and v are complex and w is real.The sectionsρand ξsatisfyρξ= k i=1(η−P i(ζ)),where P i are quadratic inζwith coefficients given by the coordinates of the singularities.The above equations provide the description of the twistor space of the singular monopole moduli space.3Knowing the twistor space one can use the generalized Legendre transform techniques6tofind the auxiliary function F of the moduliF(z,¯z,v,¯v,w)=1ζ3+2 ωr dζ√ζ2−a1ζ2(√−η2−za(ζ)).(2)Imposing the consistency constraint∂F/∂w=0expresses w as a function of z and v.Then the Legendre transform of FK(z,¯z,u,¯u)=F(z,¯z,v,¯v)−uv−¯u¯v,(3) with∂F/∂v=u and∂F/∂¯v=¯u,gives the K¨a hler potential for the D k ALF metric.This agrees with the conjecture of Chalmers7.AcknowledgmentsThe results presented here are obtained in collaboration with Anton Kapustin. This work is partially supported by DOE grant DE-FG03-92-ER40701. References1.S.A.Cherkis and A.Kapustin,“Singular Monopoles and GravitationalInstantons,”hep-th/9711145to appear in Nucl.Phys.B.2.S.A.Cherkis and A.Kapustin,“D k Gravitational Instantons and NahmEquations,”hep-th/9803112.3.S.A.Cherkis and A.Kapustin,“Singular Monopoles and SupersymmetricGauge Theories in Three Dimensions,”hep-th/9711145.4.A.Sen,“A Note on Enhanced Gauge Symmetries in M-and String The-ory,”JHEP09,1(1997)hep-th/9707123.5.M.Atiyah and N.Hitchin,The Geometry and Dynamics of MagneticMonopoles,Princeton Univ.Press,Princeton(1988).6.N.J.Hitchin,A.Karlhede,U.Lindstr¨o m,and M.Roˇc ek,“Hyperk¨a hlerMetrics and Supersymmetry,”Comm.Math.Phys.108535-589(1987), Lindstrom,U.and Roˇc ek,M.“New HyperK¨a hler metrics and New Su-permultiplets,”Comm.Math.Phys115,21(1988).7.G.Chalmers,“The Implicit Metric on a Deformation of the Atiyah-Hitchin Manifold,”hep-th/9709082,“Multi-monopole Moduli Spaces for SU(N)Gauge Group,”hep-th/9605182.4。
狭义相对论 英语
狭义相对论英语
狭义相对论英语：（Special Theory of Relativity）
狭义相对论（Special Theory of Relativity）是阿尔伯特·爱因斯坦在1905年提出的理论，它基于两个基本的假设：物理定律在所有惯性参照系中都是相同的（相对性原理），以及光在真空中的速度在所有惯性参照系中都是相同的（光速不变原理）。
以下是狭义相对论的一些关键概念和术语的英语表达：Principle of Relativity（相对性原理）
Principle of the Constancy of the Velocity of Light（光速不变原理）
Lorentz Transformation（洛伦兹变换）
Time Dilation（时间膨胀）
Length Contraction（尺缩效应）
Mass-Energy Equivalence（质能等价）
Rest Mass（静质量）
Relativistic Mass（相对论质量）
Inertial Frame of Reference（惯性参照系）
这些术语和概念构成了狭义相对论的基础，帮助我们理解时间和空间如何在不同的参照系中相对变化，以及质量和能量之间的关系。
From Data Mining to Knowledge Discovery in Databases
s Data mining and knowledge discovery in databases have been attracting a significant amount of research, industry, and media atten-tion of late. What is all the excitement about?This article provides an overview of this emerging field, clarifying how data mining and knowledge discovery in databases are related both to each other and to related fields, such as machine learning, statistics, and databases. The article mentions particular real-world applications, specific data-mining techniques, challenges in-volved in real-world applications of knowledge discovery, and current and future research direc-tions in the field.A cross a wide variety of fields, data arebeing collected and accumulated at adramatic pace. There is an urgent need for a new generation of computational theo-ries and tools to assist humans in extracting useful information (knowledge) from the rapidly growing volumes of digital data. These theories and tools are the subject of the emerging field of knowledge discovery in databases (KDD).At an abstract level, the KDD field is con-cerned with the development of methods and techniques for making sense of data. The basic problem addressed by the KDD process is one of mapping low-level data (which are typically too voluminous to understand and digest easi-ly) into other forms that might be more com-pact (for example, a short report), more ab-stract (for example, a descriptive approximation or model of the process that generated the data), or more useful (for exam-ple, a predictive model for estimating the val-ue of future cases). At the core of the process is the application of specific data-mining meth-ods for pattern discovery and extraction.1This article begins by discussing the histori-cal context of KDD and data mining and theirintersection with other related fields. A briefsummary of recent KDD real-world applica-tions is provided. Definitions of KDD and da-ta mining are provided, and the general mul-tistep KDD process is outlined. This multistepprocess has the application of data-mining al-gorithms as one particular step in the process.The data-mining step is discussed in more de-tail in the context of specific data-mining al-gorithms and their application. Real-worldpractical application issues are also outlined.Finally, the article enumerates challenges forfuture research and development and in par-ticular discusses potential opportunities for AItechnology in KDD systems.Why Do We Need KDD?The traditional method of turning data intoknowledge relies on manual analysis and in-terpretation. For example, in the health-careindustry, it is common for specialists to peri-odically analyze current trends and changesin health-care data, say, on a quarterly basis.The specialists then provide a report detailingthe analysis to the sponsoring health-care or-ganization; this report becomes the basis forfuture decision making and planning forhealth-care management. In a totally differ-ent type of application, planetary geologistssift through remotely sensed images of plan-ets and asteroids, carefully locating and cata-loging such geologic objects of interest as im-pact craters. Be it science, marketing, finance,health care, retail, or any other field, the clas-sical approach to data analysis relies funda-mentally on one or more analysts becomingArticlesFALL 1996 37From Data Mining to Knowledge Discovery inDatabasesUsama Fayyad, Gregory Piatetsky-Shapiro, and Padhraic Smyth Copyright © 1996, American Association for Artificial Intelligence. All rights reserved. 0738-4602-1996 / $2.00areas is astronomy. Here, a notable success was achieved by SKICAT ,a system used by as-tronomers to perform image analysis,classification, and cataloging of sky objects from sky-survey images (Fayyad, Djorgovski,and Weir 1996). In its first application, the system was used to process the 3 terabytes (1012bytes) of image data resulting from the Second Palomar Observatory Sky Survey,where it is estimated that on the order of 109sky objects are detectable. SKICAT can outper-form humans and traditional computational techniques in classifying faint sky objects. See Fayyad, Haussler, and Stolorz (1996) for a sur-vey of scientific applications.In business, main KDD application areas includes marketing, finance (especially in-vestment), fraud detection, manufacturing,telecommunications, and Internet agents.Marketing:In marketing, the primary ap-plication is database marketing systems,which analyze customer databases to identify different customer groups and forecast their behavior. Business Week (Berry 1994) estimat-ed that over half of all retailers are using or planning to use database marketing, and those who do use it have good results; for ex-ample, American Express reports a 10- to 15-percent increase in credit-card use. Another notable marketing application is market-bas-ket analysis (Agrawal et al. 1996) systems,which find patterns such as, “If customer bought X, he/she is also likely to buy Y and Z.” Such patterns are valuable to retailers.Investment: Numerous companies use da-ta mining for investment, but most do not describe their systems. One exception is LBS Capital Management. Its system uses expert systems, neural nets, and genetic algorithms to manage portfolios totaling $600 million;since its start in 1993, the system has outper-formed the broad stock market (Hall, Mani,and Barr 1996).Fraud detection: HNC Falcon and Nestor PRISM systems are used for monitoring credit-card fraud, watching over millions of ac-counts. The FAIS system (Senator et al. 1995),from the U.S. Treasury Financial Crimes En-forcement Network, is used to identify finan-cial transactions that might indicate money-laundering activity.Manufacturing: The CASSIOPEE trou-bleshooting system, developed as part of a joint venture between General Electric and SNECMA, was applied by three major Euro-pean airlines to diagnose and predict prob-lems for the Boeing 737. To derive families of faults, clustering methods are used. CASSIOPEE received the European first prize for innova-intimately familiar with the data and serving as an interface between the data and the users and products.For these (and many other) applications,this form of manual probing of a data set is slow, expensive, and highly subjective. In fact, as data volumes grow dramatically, this type of manual data analysis is becoming completely impractical in many domains.Databases are increasing in size in two ways:(1) the number N of records or objects in the database and (2) the number d of fields or at-tributes to an object. Databases containing on the order of N = 109objects are becoming in-creasingly common, for example, in the as-tronomical sciences. Similarly, the number of fields d can easily be on the order of 102or even 103, for example, in medical diagnostic applications. Who could be expected to di-gest millions of records, each having tens or hundreds of fields? We believe that this job is certainly not one for humans; hence, analysis work needs to be automated, at least partially.The need to scale up human analysis capa-bilities to handling the large number of bytes that we can collect is both economic and sci-entific. Businesses use data to gain competi-tive advantage, increase efficiency, and pro-vide more valuable services to customers.Data we capture about our environment are the basic evidence we use to build theories and models of the universe we live in. Be-cause computers have enabled humans to gather more data than we can digest, it is on-ly natural to turn to computational tech-niques to help us unearth meaningful pat-terns and structures from the massive volumes of data. Hence, KDD is an attempt to address a problem that the digital informa-tion era made a fact of life for all of us: data overload.Data Mining and Knowledge Discovery in the Real WorldA large degree of the current interest in KDD is the result of the media interest surrounding successful KDD applications, for example, the focus articles within the last two years in Business Week , Newsweek , Byte , PC Week , and other large-circulation periodicals. Unfortu-nately, it is not always easy to separate fact from media hype. Nonetheless, several well-documented examples of successful systems can rightly be referred to as KDD applications and have been deployed in operational use on large-scale real-world problems in science and in business.In science, one of the primary applicationThere is an urgent need for a new generation of computation-al theories and tools toassist humans in extractinguseful information (knowledge)from the rapidly growing volumes ofdigital data.Articles38AI MAGAZINEtive applications (Manago and Auriol 1996).Telecommunications: The telecommuni-cations alarm-sequence analyzer (TASA) wasbuilt in cooperation with a manufacturer oftelecommunications equipment and threetelephone networks (Mannila, Toivonen, andVerkamo 1995). The system uses a novelframework for locating frequently occurringalarm episodes from the alarm stream andpresenting them as rules. Large sets of discov-ered rules can be explored with flexible infor-mation-retrieval tools supporting interactivityand iteration. In this way, TASA offers pruning,grouping, and ordering tools to refine the re-sults of a basic brute-force search for rules.Data cleaning: The MERGE-PURGE systemwas applied to the identification of duplicatewelfare claims (Hernandez and Stolfo 1995).It was used successfully on data from the Wel-fare Department of the State of Washington.In other areas, a well-publicized system isIBM’s ADVANCED SCOUT,a specialized data-min-ing system that helps National Basketball As-sociation (NBA) coaches organize and inter-pret data from NBA games (U.S. News 1995). ADVANCED SCOUT was used by several of the NBA teams in 1996, including the Seattle Su-personics, which reached the NBA finals.Finally, a novel and increasingly importanttype of discovery is one based on the use of in-telligent agents to navigate through an infor-mation-rich environment. Although the ideaof active triggers has long been analyzed in thedatabase field, really successful applications ofthis idea appeared only with the advent of theInternet. These systems ask the user to specifya profile of interest and search for related in-formation among a wide variety of public-do-main and proprietary sources. For example, FIREFLY is a personal music-recommendation agent: It asks a user his/her opinion of several music pieces and then suggests other music that the user might like (<http:// www.ffl/>). CRAYON(/>) allows users to create their own free newspaper (supported by ads); NEWSHOUND(<http://www. /hound/>) from the San Jose Mercury News and FARCAST(</> automatically search information from a wide variety of sources, including newspapers and wire services, and e-mail rele-vant documents directly to the user.These are just a few of the numerous suchsystems that use KDD techniques to automat-ically produce useful information from largemasses of raw data. See Piatetsky-Shapiro etal. (1996) for an overview of issues in devel-oping industrial KDD applications.Data Mining and KDDHistorically, the notion of finding useful pat-terns in data has been given a variety ofnames, including data mining, knowledge ex-traction, information discovery, informationharvesting, data archaeology, and data patternprocessing. The term data mining has mostlybeen used by statisticians, data analysts, andthe management information systems (MIS)communities. It has also gained popularity inthe database field. The phrase knowledge dis-covery in databases was coined at the first KDDworkshop in 1989 (Piatetsky-Shapiro 1991) toemphasize that knowledge is the end productof a data-driven discovery. It has been popular-ized in the AI and machine-learning fields.In our view, KDD refers to the overall pro-cess of discovering useful knowledge from da-ta, and data mining refers to a particular stepin this process. Data mining is the applicationof specific algorithms for extracting patternsfrom data. The distinction between the KDDprocess and the data-mining step (within theprocess) is a central point of this article. Theadditional steps in the KDD process, such asdata preparation, data selection, data cleaning,incorporation of appropriate prior knowledge,and proper interpretation of the results ofmining, are essential to ensure that usefulknowledge is derived from the data. Blind ap-plication of data-mining methods (rightly crit-icized as data dredging in the statistical litera-ture) can be a dangerous activity, easilyleading to the discovery of meaningless andinvalid patterns.The Interdisciplinary Nature of KDDKDD has evolved, and continues to evolve,from the intersection of research fields such asmachine learning, pattern recognition,databases, statistics, AI, knowledge acquisitionfor expert systems, data visualization, andhigh-performance computing. The unifyinggoal is extracting high-level knowledge fromlow-level data in the context of large data sets.The data-mining component of KDD cur-rently relies heavily on known techniquesfrom machine learning, pattern recognition,and statistics to find patterns from data in thedata-mining step of the KDD process. A natu-ral question is, How is KDD different from pat-tern recognition or machine learning (and re-lated fields)? The answer is that these fieldsprovide some of the data-mining methodsthat are used in the data-mining step of theKDD process. KDD focuses on the overall pro-cess of knowledge discovery from data, includ-ing how the data are stored and accessed, howalgorithms can be scaled to massive data setsThe basicproblemaddressed bythe KDDprocess isone ofmappinglow-leveldata intoother formsthat might bemorecompact,moreabstract,or moreuseful.ArticlesFALL 1996 39A driving force behind KDD is the database field (the second D in KDD). Indeed, the problem of effective data manipulation when data cannot fit in the main memory is of fun-damental importance to KDD. Database tech-niques for gaining efficient data access,grouping and ordering operations when ac-cessing data, and optimizing queries consti-tute the basics for scaling algorithms to larger data sets. Most data-mining algorithms from statistics, pattern recognition, and machine learning assume data are in the main memo-ry and pay no attention to how the algorithm breaks down if only limited views of the data are possible.A related field evolving from databases is data warehousing,which refers to the popular business trend of collecting and cleaning transactional data to make them available for online analysis and decision support. Data warehousing helps set the stage for KDD in two important ways: (1) data cleaning and (2)data access.Data cleaning: As organizations are forced to think about a unified logical view of the wide variety of data and databases they pos-sess, they have to address the issues of map-ping data to a single naming convention,uniformly representing and handling missing data, and handling noise and errors when possible.Data access: Uniform and well-defined methods must be created for accessing the da-ta and providing access paths to data that were historically difficult to get to (for exam-ple, stored offline).Once organizations and individuals have solved the problem of how to store and ac-cess their data, the natural next step is the question, What else do we do with all the da-ta? This is where opportunities for KDD natu-rally arise.A popular approach for analysis of data warehouses is called online analytical processing (OLAP), named for a set of principles pro-posed by Codd (1993). OLAP tools focus on providing multidimensional data analysis,which is superior to SQL in computing sum-maries and breakdowns along many dimen-sions. OLAP tools are targeted toward simpli-fying and supporting interactive data analysis,but the goal of KDD tools is to automate as much of the process as possible. Thus, KDD is a step beyond what is currently supported by most standard database systems.Basic DefinitionsKDD is the nontrivial process of identifying valid, novel, potentially useful, and ultimate-and still run efficiently, how results can be in-terpreted and visualized, and how the overall man-machine interaction can usefully be modeled and supported. The KDD process can be viewed as a multidisciplinary activity that encompasses techniques beyond the scope of any one particular discipline such as machine learning. In this context, there are clear opportunities for other fields of AI (be-sides machine learning) to contribute to KDD. KDD places a special emphasis on find-ing understandable patterns that can be inter-preted as useful or interesting knowledge.Thus, for example, neural networks, although a powerful modeling tool, are relatively difficult to understand compared to decision trees. KDD also emphasizes scaling and ro-bustness properties of modeling algorithms for large noisy data sets.Related AI research fields include machine discovery, which targets the discovery of em-pirical laws from observation and experimen-tation (Shrager and Langley 1990) (see Kloes-gen and Zytkow [1996] for a glossary of terms common to KDD and machine discovery),and causal modeling for the inference of causal models from data (Spirtes, Glymour,and Scheines 1993). Statistics in particular has much in common with KDD (see Elder and Pregibon [1996] and Glymour et al.[1996] for a more detailed discussion of this synergy). Knowledge discovery from data is fundamentally a statistical endeavor. Statistics provides a language and framework for quan-tifying the uncertainty that results when one tries to infer general patterns from a particu-lar sample of an overall population. As men-tioned earlier, the term data mining has had negative connotations in statistics since the 1960s when computer-based data analysis techniques were first introduced. The concern arose because if one searches long enough in any data set (even randomly generated data),one can find patterns that appear to be statis-tically significant but, in fact, are not. Clearly,this issue is of fundamental importance to KDD. Substantial progress has been made in recent years in understanding such issues in statistics. Much of this work is of direct rele-vance to KDD. Thus, data mining is a legiti-mate activity as long as one understands how to do it correctly; data mining carried out poorly (without regard to the statistical as-pects of the problem) is to be avoided. KDD can also be viewed as encompassing a broader view of modeling than statistics. KDD aims to provide tools to automate (to the degree pos-sible) the entire process of data analysis and the statistician’s “art” of hypothesis selection.Data mining is a step in the KDD process that consists of ap-plying data analysis and discovery al-gorithms that produce a par-ticular enu-meration ofpatterns (or models)over the data.Articles40AI MAGAZINEly understandable patterns in data (Fayyad, Piatetsky-Shapiro, and Smyth 1996).Here, data are a set of facts (for example, cases in a database), and pattern is an expres-sion in some language describing a subset of the data or a model applicable to the subset. Hence, in our usage here, extracting a pattern also designates fitting a model to data; find-ing structure from data; or, in general, mak-ing any high-level description of a set of data. The term process implies that KDD comprises many steps, which involve data preparation, search for patterns, knowledge evaluation, and refinement, all repeated in multiple itera-tions. By nontrivial, we mean that some search or inference is involved; that is, it is not a straightforward computation of predefined quantities like computing the av-erage value of a set of numbers.The discovered patterns should be valid on new data with some degree of certainty. We also want patterns to be novel (at least to the system and preferably to the user) and poten-tially useful, that is, lead to some benefit to the user or task. Finally, the patterns should be understandable, if not immediately then after some postprocessing.The previous discussion implies that we can define quantitative measures for evaluating extracted patterns. In many cases, it is possi-ble to define measures of certainty (for exam-ple, estimated prediction accuracy on new data) or utility (for example, gain, perhaps indollars saved because of better predictions orspeedup in response time of a system). No-tions such as novelty and understandabilityare much more subjective. In certain contexts,understandability can be estimated by sim-plicity (for example, the number of bits to de-scribe a pattern). An important notion, calledinterestingness(for example, see Silberschatzand Tuzhilin [1995] and Piatetsky-Shapiro andMatheus [1994]), is usually taken as an overallmeasure of pattern value, combining validity,novelty, usefulness, and simplicity. Interest-ingness functions can be defined explicitly orcan be manifested implicitly through an or-dering placed by the KDD system on the dis-covered patterns or models.Given these notions, we can consider apattern to be knowledge if it exceeds some in-terestingness threshold, which is by nomeans an attempt to define knowledge in thephilosophical or even the popular view. As amatter of fact, knowledge in this definition ispurely user oriented and domain specific andis determined by whatever functions andthresholds the user chooses.Data mining is a step in the KDD processthat consists of applying data analysis anddiscovery algorithms that, under acceptablecomputational efficiency limitations, pro-duce a particular enumeration of patterns (ormodels) over the data. Note that the space ofArticlesFALL 1996 41Figure 1. An Overview of the Steps That Compose the KDD Process.methods, the effective number of variables under consideration can be reduced, or in-variant representations for the data can be found.Fifth is matching the goals of the KDD pro-cess (step 1) to a particular data-mining method. For example, summarization, clas-sification, regression, clustering, and so on,are described later as well as in Fayyad, Piatet-sky-Shapiro, and Smyth (1996).Sixth is exploratory analysis and model and hypothesis selection: choosing the data-mining algorithm(s) and selecting method(s)to be used for searching for data patterns.This process includes deciding which models and parameters might be appropriate (for ex-ample, models of categorical data are differ-ent than models of vectors over the reals) and matching a particular data-mining method with the overall criteria of the KDD process (for example, the end user might be more in-terested in understanding the model than its predictive capabilities).Seventh is data mining: searching for pat-terns of interest in a particular representa-tional form or a set of such representations,including classification rules or trees, regres-sion, and clustering. The user can significant-ly aid the data-mining method by correctly performing the preceding steps.Eighth is interpreting mined patterns, pos-sibly returning to any of steps 1 through 7 for further iteration. This step can also involve visualization of the extracted patterns and models or visualization of the data given the extracted models.Ninth is acting on the discovered knowl-edge: using the knowledge directly, incorpo-rating the knowledge into another system for further action, or simply documenting it and reporting it to interested parties. This process also includes checking for and resolving po-tential conflicts with previously believed (or extracted) knowledge.The KDD process can involve significant iteration and can contain loops between any two steps. The basic flow of steps (al-though not the potential multitude of itera-tions and loops) is illustrated in figure 1.Most previous work on KDD has focused on step 7, the data mining. However, the other steps are as important (and probably more so) for the successful application of KDD in practice. Having defined the basic notions and introduced the KDD process, we now focus on the data-mining component,which has, by far, received the most atten-tion in the literature.patterns is often infinite, and the enumera-tion of patterns involves some form of search in this space. Practical computational constraints place severe limits on the sub-space that can be explored by a data-mining algorithm.The KDD process involves using the database along with any required selection,preprocessing, subsampling, and transforma-tions of it; applying data-mining methods (algorithms) to enumerate patterns from it;and evaluating the products of data mining to identify the subset of the enumerated pat-terns deemed knowledge. The data-mining component of the KDD process is concerned with the algorithmic means by which pat-terns are extracted and enumerated from da-ta. The overall KDD process (figure 1) in-cludes the evaluation and possible interpretation of the mined patterns to de-termine which patterns can be considered new knowledge. The KDD process also in-cludes all the additional steps described in the next section.The notion of an overall user-driven pro-cess is not unique to KDD: analogous propos-als have been put forward both in statistics (Hand 1994) and in machine learning (Brod-ley and Smyth 1996).The KDD ProcessThe KDD process is interactive and iterative,involving numerous steps with many deci-sions made by the user. Brachman and Anand (1996) give a practical view of the KDD pro-cess, emphasizing the interactive nature of the process. Here, we broadly outline some of its basic steps:First is developing an understanding of the application domain and the relevant prior knowledge and identifying the goal of the KDD process from the customer’s viewpoint.Second is creating a target data set: select-ing a data set, or focusing on a subset of vari-ables or data samples, on which discovery is to be performed.Third is data cleaning and preprocessing.Basic operations include removing noise if appropriate, collecting the necessary informa-tion to model or account for noise, deciding on strategies for handling missing data fields,and accounting for time-sequence informa-tion and known changes.Fourth is data reduction and projection:finding useful features to represent the data depending on the goal of the task. With di-mensionality reduction or transformationArticles42AI MAGAZINEThe Data-Mining Stepof the KDD ProcessThe data-mining component of the KDD pro-cess often involves repeated iterative applica-tion of particular data-mining methods. This section presents an overview of the primary goals of data mining, a description of the methods used to address these goals, and a brief description of the data-mining algo-rithms that incorporate these methods.The knowledge discovery goals are defined by the intended use of the system. We can distinguish two types of goals: (1) verification and (2) discovery. With verification,the sys-tem is limited to verifying the user’s hypothe-sis. With discovery,the system autonomously finds new patterns. We further subdivide the discovery goal into prediction,where the sys-tem finds patterns for predicting the future behavior of some entities, and description, where the system finds patterns for presenta-tion to a user in a human-understandableform. In this article, we are primarily con-cerned with discovery-oriented data mining.Data mining involves fitting models to, or determining patterns from, observed data. The fitted models play the role of inferred knowledge: Whether the models reflect useful or interesting knowledge is part of the over-all, interactive KDD process where subjective human judgment is typically required. Two primary mathematical formalisms are used in model fitting: (1) statistical and (2) logical. The statistical approach allows for nondeter-ministic effects in the model, whereas a logi-cal model is purely deterministic. We focus primarily on the statistical approach to data mining, which tends to be the most widely used basis for practical data-mining applica-tions given the typical presence of uncertain-ty in real-world data-generating processes.Most data-mining methods are based on tried and tested techniques from machine learning, pattern recognition, and statistics: classification, clustering, regression, and so on. The array of different algorithms under each of these headings can often be bewilder-ing to both the novice and the experienced data analyst. It should be emphasized that of the many data-mining methods advertised in the literature, there are really only a few fun-damental techniques. The actual underlying model representation being used by a particu-lar method typically comes from a composi-tion of a small number of well-known op-tions: polynomials, splines, kernel and basis functions, threshold-Boolean functions, and so on. Thus, algorithms tend to differ primar-ily in the goodness-of-fit criterion used toevaluate model fit or in the search methodused to find a good fit.In our brief overview of data-mining meth-ods, we try in particular to convey the notionthat most (if not all) methods can be viewedas extensions or hybrids of a few basic tech-niques and principles. We first discuss the pri-mary methods of data mining and then showthat the data- mining methods can be viewedas consisting of three primary algorithmiccomponents: (1) model representation, (2)model evaluation, and (3) search. In the dis-cussion of KDD and data-mining methods,we use a simple example to make some of thenotions more concrete. Figure 2 shows a sim-ple two-dimensional artificial data set consist-ing of 23 cases. Each point on the graph rep-resents a person who has been given a loanby a particular bank at some time in the past.The horizontal axis represents the income ofthe person; the vertical axis represents the to-tal personal debt of the person (mortgage, carpayments, and so on). The data have beenclassified into two classes: (1) the x’s repre-sent persons who have defaulted on theirloans and (2) the o’s represent persons whoseloans are in good status with the bank. Thus,this simple artificial data set could represent ahistorical data set that can contain usefulknowledge from the point of view of thebank making the loans. Note that in actualKDD applications, there are typically manymore dimensions (as many as several hun-dreds) and many more data points (manythousands or even millions).ArticlesFALL 1996 43Figure 2. A Simple Data Set with Two Classes Used for Illustrative Purposes.。
金兹堡朗道理论
Ginzburg–Landau theoryFrom Wikipedia, the free encyclopediaIn physics, Ginzburg–Landau theory, named after Vitaly Lazarevich Ginzburg and Lev Landau, is a mathematical physical theory used to describe superconductivity. In its initial form, it was postulated as a phenomenological model which could describe type-I superconductors without examining their microscopic properties. Later, a version of Ginzburg–Landau theory was derived from the Bardeen-Cooper-Schrieffer microscopic theory by Lev Gor'kov, thus showing that it also appears in some limit of microscopic theory and giving microscopic interpretation of all its parameters.Contents•1Introduction•2Simple interpretation•3Coherence length and penetration depth•4Fluctuations in the Ginzburg–Landau model•5Classification of superconductors based on Ginzburg–Landau theory•6Landau–Ginzburg theories in string theory•7See also•8References•8.1PapersIntroduction[edit]Based on Landau's previously-established theory of second-order phase transitions, Ginzburg and Landau argued that the free energy, F, of a superconductor near the superconducting transition can be expressed in terms ofa complex order parameter field, ψ, which is nonzero below a phase transition into a superconducting state and isrelated to the density of the superconducting component, although no direct interpretation of this parameter was given in the original paper. Assuming smallness of |ψ| and smallness of its gradients, the free energy has the form ofa field theory.where F n is the free energy in the normal phase, α and β in the initial argument were treated as phenomenologicalparameters, m is an effective mass, e is the charge of an electron, A is the magnetic vector potential, and is the magnetic field. By minimizing the free energy with respect to variations in the order parameter and the vector potential, one arrives at the Ginzburg–Landau equationswhere j denotes the dissipation-less electric current density and Re the real part. The first equation — which bears some similarities to the time-independent Schrödinger equation, but is principally different due to a nonlinear term —determines the order parameter, ψ. The second equation then provides the superconducting current.Simple interpretation[edit]Consider a homogeneous superconductor where there is no superconducting current and the equation for ψ simplifies to:This equation has a trivial solution: ψ = 0. This corresponds to the normal state of the superconductor, that is for temperatures above the superconducting transition temperature, T>T c.Below the superconducting transition temperature, the above equation is expected to have a non-trivial solution (that is ψ ≠ 0). Under this assumption the equation above can be rearranged into:When the right hand side of this equation is positive, there is a nonzero solution for ψ (remember that the magnitude of a complex number can be positive or zero). This can be achieved by assuming the following temperature dependence of α: α(T) = α0 (T - T c) with α0/ β > 0:•Above the superconducting transition temperature, T > T c, the expression α(T) / β is positive and the right hand side of the equation above is negative. The magnitude of a complex number must be a non-negative number, so only ψ = 0 solves the Ginzburg–Landau equation.•Below the superconducting transition temperature, T < T c, the right hand side of the equation above is positive and there is a non-trivial solution for ψ. Furthermorethat is ψ approaches zero as T gets closer to T c from below. Such a behaviour is typical for a second order phase transition.In Ginzburg–Landau theory the electrons that contribute to superconductivity were proposed to forma superfluid.[1] In this interpretation, |ψ|2 indicates the fraction of electrons that have condensed into a superfluid.[1] Coherence length and penetration depth[edit]The Ginzburg–Landau equations predicted two new characteristic lengths in a superconductor which wastermed coherence length, ξ. For T > T c (normal phase), it is given bywhile for T < T c (superconducting phase), where it is more relevant, it is given byIt sets the exponential law according to which small perturbations of density of superconducting electrons recover their equilibrium value ψ0. Thus this theory characterized all superconductors by two length scales. The second one is the penetration depth, λ. It was previously introduced by the London brothers in their London theory. Expressed in terms of the parameters of Ginzburg-Landau model it iswhere ψ0 is the equilibrium value of the order parameter in the absence of an electromagnetic field. The penetration depth sets the exponential law according to which an external magnetic field decays inside the superconductor. The original idea on the parameter "k" belongs to Landau. The ratio κ = λ/ξ is presently known asthe Ginzburg–Landau parameter. It has been proposed by Landau that Type I superconductors are those with 0 < κ< 1/√2, and Type II superconductors those with κ> 1/√2.The exponential decay of the magnetic field is equivalent with the Higgs mechanism in high-energy physics. Fluctuations in the Ginzburg–Landau model[edit]Taking into account fluctuations. For Type II superconductors, the phase transition from the normal state is of second order, as demonstrated by Dasgupta and Halperin. While for Type I superconductors it is of first order as demonstrated by Halperin, Lubensky and Ma.Classification of superconductors based on Ginzburg–Landau theory[edit]In the original paper Ginzburg and Landau observed the existence of two types of superconductors depending on the energy of the interface between the normal and superconducting states.The Meissner state breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs. In Type I superconductors, superconductivity is abruptly destroyed when the strength of the applied field rises above a critical value H c. Depending on the geometry of the sample, one may obtain an intermediate state[2] consisting of a baroque pattern[3] of regions of normal material carrying a magnetic field mixed with regions of superconducting material containing no field. In Type II superconductors, raising the applied field past a critical value H c1 leads to a mixed state (also known as the vortex state) in which an increasing amount of magnetic flux penetrates the material, but there remains no resistance to the flow of electric current as long as the current is not too large. At a second critical field strength H c2, superconductivity is destroyed. The mixed state is actually caused by vortices in the electronic superfluid, sometimes called fluxons because the flux carried by these vortices is quantized. Most pure elemental superconductors, except niobium and carbon nanotubes, are Type I, while almost all impure and compound superconductors are Type II.The most important finding from Ginzburg–Landau theory was made by Alexei Abrikosov in 1957. He used Ginzburg–Landau theory to explain experiments on superconducting alloys and thin films. He found that in a type-II superconductor in a high magnetic field, the field penetrates in a triangular lattice of quantized tubes offlux vortices.[citation needed]Landau–Ginzburg theories in string theory[edit]In particle physics, any quantum field theory with a unique classical vacuum state and a potential energy witha degenerate critical point is called a Landau–Ginzburg theory. The generalization to N=(2,2) supersymmetric theories in 2 spacetime dimensions was proposed by Cumrun Vafa and Nicholas Warner in the November 1988 article Catastrophes and the Classification of Conformal Theories, in this generalization one imposes thatthe superpotential possess a degenerate critical point. The same month, together with Brian Greene they argued that these theories are related by a renormalization group flow to sigma models on Calabi–Yau manifolds in thepaper Calabi–Yau Manifolds and Renormalization Group Flows. In his 1993 paper Phases of N=2 theories intwo-dimensions, Edward Witten argued that Landau–Ginzburg theories and sigma models on Calabi–Yau manifolds are different phases of the same theory. A construction of such a duality was given by relating the Gromov-Witten theory of Calabi-Yau orbifolds to FJRW theory an analogous Landau-Ginzburg "FJRW" theory in The Witten Equation, Mirror Symmetry and Quantum Singularity Theory. Witten's sigma models were later used to describe the low energy dynamics of 4-dimensional gauge theories with monopoles as well as brane constructions. Gaiotto, Gukov & Seiberg (2013)See also[edit]•Domain wall (magnetism)•Flux pinning•Gross–Pitaevskii equation•Husimi Q representation•Landau theory•Magnetic domain•Magnetic flux quantum•Reaction–diffusion systems•Quantum vortex•Topological defectReferences[edit]1.^ Jump up to:a b Ginzburg VL (July 2004). "On superconductivity and superfluidity (what I have and havenot managed to do), as well as on the 'physical minimum' at the beginning of the 21 st century". Chemphyschem.5 (7): 930–945. doi:10.1002/cphc.200400182. PMID15298379.2.Jump up^ Lev D. Landau; Evgeny M. Lifschitz (1984). Electrodynamics of Continuous Media. Course ofTheoretical Physics8. Oxford: Butterworth-Heinemann. ISBN0-7506-2634-8.3.Jump up^ David J. E. Callaway (1990). "On the remarkable structure of the superconductingintermediate state". Nuclear Physics B344 (3): 627–645. Bibcode:1990NuPhB.344..627C.doi:10.1016/0550-3213(90)90672-Z.Papers[edit]•V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz.20, 1064 (1950). English translation in: L. D. Landau, Collected papers (Oxford: Pergamon Press, 1965) p. 546• A.A. Abrikosov, Zh. Eksp. Teor. Fiz.32, 1442 (1957) (English translation: Sov. Phys. JETP5 1174 (1957)].) Abrikosov's original paper on vortex structure of Type-II superconductors derived as a solution of G–L equations for κ > 1/√2•L.P. Gor'kov, Sov. Phys. JETP36, 1364 (1959)• A.A. Abrikosov's 2003 Nobel lecture: pdf file or video•V.L. Ginzburg's 2003 Nobel Lecture: pdf file or video•Gaiotto, David; Gukov, Sergei; Seiberg, Nathan (2013), "Surface Defects and Resolvents" (PDF), Journal of High Energy Physics。
微积分专有名词中英文对照
微积分专有名词中英文对照absolutely convergent 绝对收敛absolute value 绝对值algebraic function 代数函数analytic geometry 解析几何antiderivative 不定积分approximate integration 近似积分approximation 近似法、逼近法arbitrary constant 任意常数arithmetic series/progression (AP)算数级数asymptotes (vertical and horizontal)(垂直/水平)渐近线average rate of change 平均变化率base 基数binomial theorem 二项式定理，二项展开式Cartesian coordinates 笛卡儿坐标(一般指直角坐标) Cartesian coordinates system 笛卡儿坐标系Cauch’s Mean Value Theorem 柯西均值定理chain rule 链式求导法则calculus 微积分学closed interval integral 闭区间积分coefficient 系数composite function 复合函数conchoid 蚌线continuity (函数的)连续性concavity (函数的)凹凸性conditionally convergent 有条件收敛continuity 连续性critical point 临界点cubic function 三次函数cylindrical coordinates 圆柱坐标decreasing function 递减函数decreasing sequence 递减数列definite integral 定积分derivative 导数determinant 行列式differential coefficient 微分系数differential equation 微分方程directional derivative 方向导数discontinuity 不连续性discriminant (二次函数)判别式disk method 圆盘法divergence 散度divergent 发散的domain 定义域dot product 点积double integral 二重积分ellipse 椭圆ellipsoid 椭圆体epicycloid 外摆线Euler's method (BC)欧拉法expected valued 期望值exponential function 指数函数extreme value heorem 极值定理factorial 阶乘finite series 有限级数fundamental theorem of calculus 微积分基本定理geometric series/progression (GP)几何级数gradient 梯度Green formula 格林公式half-angle formulas 半角公式harmonic series 调和级数helix 螺旋线higher derivative 高阶导数horizontal asymptote 水平渐近线horizontal line 水平线hyperbola 双曲线hyper boloid 双曲面implicit differentiation 隐函数求导implicit function 隐函数improper integral 广义积分、瑕积分increment 增量increasing function 增函数indefinite integral 不定积分independent variable 自变数inequality 不等式ndeterminate form 不定型infinite point 无穷极限infinite series 无穷级数infinite series 无限级数inflection point (POI) 拐点initial condition 初始条件instantaneous rate of change 瞬时变化率integrable 可积的integral 积分integrand 被积分式integration 积分integration by part 分部积分法intercept 截距intermediate value of Theorem ：中间值定理inverse function 反函数irrational function 无理函数iterated integral 逐次积分Laplace transform 拉普拉斯变换law of cosines 余弦定理least upper bound 最小上界left-hand derivative 左导数left-hand limit 左极限L'Hospital's rule 洛必达法则limacon 蚶线linear approximation 线性近似法linear equation 线性方程式linear function 线性函数linearity 线性linearization 线性化local maximum 极大值local minimum 极小值logarithmic function 对数函数MacLaurin series 麦克劳林级数maximum 最大值mean value theorem (MVT)中值定理minimum 最小值method of lagrange multipliers 拉格朗日乘数法modulus 绝对值multiple integral 多重积分multiple 倍数multiplier 乘子octant 卦限open interval integral 开区间积分optimization 优化法，极值法origin 原点orthogonal 正交parametric equation (BC)参数方程partial derivative 偏导数partial differential equation 偏微分方程partial fractions 部分分式piece-wise function 分段函数parabola 抛物线parabolic cylinder 抛物柱面paraboloid ：抛物面parallelepiped 平行六面体parallel lines 并行线parameter ：参数partial integration 部分积分partiton ：分割period ：周期periodic function 周期函数perpendicular lines 垂直线piecewise defined function 分段定义函数plane 平面point of inflection 反曲点point-slope form 点斜式polar axis 极轴polar coordinates 极坐标polar equation 极坐标方程pole 极点polynomial 多项式power series 幂级数product rule 积的求导法则quadrant 象限quadratic functions 二次函数quotient rule 商的求导法则radical 根式radius of convergence 收敛半径range 值域(related) rate of change with time (时间)变化率rational function 有理函数reciprocal 倒数remainder theorem 余数定理Riemann sum 黎曼和Riemannian geometry 黎曼几何right-hand limit 右极限Rolle's theorem 罗尔(中值)定理root 根rotation 旋转secant line 割线second derivative 二阶导数second derivative test 二阶导数试验法second partial derivative 二阶偏导数series 级数shell method (积分)圆筒法sine function 正弦函数singularity 奇点slant 母线slant asymptote 斜渐近线slope 斜率slope-intercept equation of a line 直线的斜截式smooth curve 平滑曲线smooth surface 平滑曲面solid of revolution 旋转体symmetry 对称性substitution 代入法、变量代换tangent function 正切函数tangent line 切线tangent plane 切(平)面tangent vector 切矢量taylor's series 泰勒级数three-dimensional analytic geometry 空间解析几何total differentiation 全微分trapezoid rule 梯形(积分)法则。
关于自伴算子的奇异连续谱(英文)
关于自伴算子的奇异连续谱(英文)
张显文;ZHANG;Xian-wen
【期刊名称】《应用数学》
【年(卷),期】2005(18)2
【摘要】本文建立了自伴算子奇异连续谱的三个定理,它们推广了Barry Simon近期所获得的一些重要结果.
【总页数】6页(P188-193)
【关键词】奇异连续谱;正则空间;Baire典型集
【作者】张显文;ZHANG;Xian-wen
【作者单位】华中科技大学数学系,武汉,430074
【正文语种】中文
【中图分类】O175
【相关文献】
1.一类Schr dinger算子的连续谱测度(英文) [J], Shang Pengjian (Dept. of Appl. Math., Northern Jiaotong University, Beijing 100044)
2.广义Morrey空间中奇异积分算子和极大算子的加权不等式(英文) [J], 刘岚
3.一类不连续奇异Strum-Liouville算子的渐近估计(英文) [J], 周立广;王万义;索建青
4.带变量Calderón-Zygmund核的奇异积分算子在加权Morrey空间上的有界性（英文） [J], 潘亚丽;李昌文;温宗良
5.奇异2n阶J-自伴向量微分算子的预解算子 [J], 刘肖云;王忠
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a r X i v :h e p -t h /9209118v 1 29 S e p 1992NDA-FP-8/92,OCHA–PP-30September 1992Dilatonic Supergravity in Two Dimensions and the Disappearance of Quantum Black HoleShin’ichi NojiriDepartment of Mathematics and PhysicsNational Defense Academy Hashirimizu,Yokosuka 239,JAPANIchiro OdaFaculty of Science,Department of PhysicsOchanomizu University1-1Otsuka 2,Bunkyo-ku,Tokyo 112,JAPANABSTRACTWe analyze a supergravity theory coupled to a dilaton and superconformal matters in two dimensions.This theory is classically soluble and we find all the solutions appeared in Callan,Giddings,Harvey and Strominger’s dilatonic gravity also satisfy the constraints and the equations of motion in this supersymmetric theory.We quantize this theory by following the procedure of Distler,Hlousek and Kawai.In the quantum action,the cosmological term is renormalized to vanish.As a result,any solution corresponding to classical black hole does not appear in the quantum theory,which should be compared with the non-supersymmetric case.1.IntroductionThe dilaton gravity theory proposed by Callan,Giddings,Harvey and Stro-minger[1](CGHS)is very instructive for the understanding of black hole physics. Especially,the problems associated with Hawking radiation[2]have been discussed by using this toy model.3−17In the original paper by CGHS,the quantum effects, such as the Hawking radiation and its back reaction of the metric,were expected to be described by adding correction term,which only comes from the conformal anomaly,18to the classical action.Several authors,7−9however,have claimed that the procedure of David[19]and of Distler and Kawai[20]is necessary when we quan-tize this theory consistently.By using this procedure,it has been found[12,13]that the quantum theory has no lower bound in energy and it has been conjectured that this problem will be resolved by supersymmetrizing the theory.In this paper, we propose a supergravity theory coupled to a dilaton in two dimensions.This theory is also classically soluble and wefind all the solutions found in Callan,Gid-dings,Harvey and Strominger’s dilatonic gravity also satisfy the constraints and the equations of motion in this supersymmetric theory.We quantize this theory by following the procedure of Distler,Hlousek and Kawai.21In the quantum action, the cosmological term is renormalized to vanish.As a result,any solution cor-responding to classical black hole does not appear in the quantum theory.This might tell that supersymmetry would forbid the existence of black hole in quan-tum theory even in higher dimensions.In the next section,we propose the classical action of dilatonic supergravity by using the tensor calculus by Higashijima,Ue-matsu,Yu,22,23which is based on conformal supergravity.23−25We show that all the solutions,including black hole solutions,in CGHS theory are also solution of this supersymmetric theory.Here it is interesting to observe that the cosmolog-ical constant is always positive semi-definite as in CGHS model.This situation is analogous to that of four dimensional supergravity,which can be constructed only in the anti-de Sitter space.In Section3,we quantize this action following the procedure of Distler,Hlousek and Kawai.The cosmological term cannot appear if we require that the quantum action has superconformal symmetry.This tells that,in quantum theory,there is not any solution corresponding to black hole solutions in the classical theory.By bosonizing the fermionfields,wefind that the equations of motion which are obtained from the effective action of the quantum theory are Liouville equations.The last section is devoted to summary and discussion.2.Classical Black HoleWe start from the following action.This action will describe the effective action of superstring in two dimensional black hole background.26We use the notations and the tensor calculus in the papers by Higashijima,Uematsu,Yu.22,23⋆S=12[˜Σi⊗T(˜Σi)]inv.=12¯ζζ¯ψνγµγνψµ+4λ 2ϕf′−¯ζζ+ϕ¯ψµγµζ+12gµν∂µa i∂νa i−12G′2i+116¯ξiξi¯ψνγµγνψµ(2.1)HereΣi’s are matter scalar multipletsΣi=(a i,ξi,G′i),i=1,···,N.A scalar multiplet˜Φ=(ϕ,ζ,f′)is given in terms of a dilaton multipletΦ=(φ,χ,F′)by˜Φ≡e−Φ=(e−φ,−e−φχ,−e−φ(F+1⋆The definition of the scalar curvature R in Refs.22and23is different from that in Ref.1by sign;R=−R CGHS.W is a curvature multiplet,W=(S,η,−S2+12¯ψµγνψµν+12Sγµψµ+1 2¯ψµγµξ+12ωµγ5+12S¯ǫγµψµ+12¯ψµχ) ǫδF′=¯ǫγµ ∂µ+12γν ∂νA−12F′ψµ δa i=¯ǫξiδξi= G′i+γµ(∂µa i−12ωµγ5 ξi−12¯ψνξi ψµ−1tion andωµis the spin connection and given byωµ=−ie−1e aµǫλν∂λe aν−12π d2x√2N i gµν∂µa i∂νa i .(2.8) This tells that all the classical solutions found in Ref.1,including the solutions describing the formation of a black hole by collapsing matter,are also solutions of this supersymmetric theory.We note that we cannot construct a supersymmetric model of the dilaton gravity when the cosmological constantλ2is negative.3.Quantum EffectsIn the original paper by CGHS,the quantum effects were expected to be described by adding correction term,which only comes from the conformal anomaly,18to the classical action.In the supersymmetric model proposed here, this corresponds to add the following term,27⋆S anomaly=κ2∂µρ∂µρ−12eS2 ,(3.1)when we choose the following superconformal gaugefixing condition,1g∓±=−is a constant,which should be determined by the conformal anomaly.4We need more counterterms since the quantum action should have superconformal symmetry when we choose the superconformal gauge(3.2).21By following de Alwis’paper,12we assume the kinetic term is given by1S kin=¯θθF′,21ˆρ≡ρ+¯θψ+2(1+¯h(t))2+κe2t(1+h(t)),ˆY= κ+22π d2x d2θ[∓¯DˆXDˆX±¯DˆY DˆY](3.6) Here upper/lower signs correspond toκ>0/κ<0,respectively.From now on, we will consider the case with lower signs in Eq.(3.6)since another case can be treated in a similar way.If we assume that there is any interaction term with respect toˆX andˆY,the energy momentum tensor T has the following form,T=T X+T Y+TΣ+T ghost,T X=−12(∂Y∂Y−χY∂χY)+ 2∂2Y(3.7)Here we have writtenˆX andˆYfields in the componentsˆX=X+¯θχX+12¯θθFY.TΣand T ghost are energy momentum tensors of matterfields and ghostfields and they contribute to the central charge by32and that by T Y is given byc Y=32(−N+9).(3.8)Therefore the total central charge c vanishes:c=32+c Y=0.Wenow introduce interaction term V so that the term does not violate the super-conformal symmetry.This requires that V should be given by a vertex operatorV=:eαˆX+βˆY:whose conformal dimension is(12),i.e.,12β(β+ 2.(3.9)If we impose the condition that T is proportional to eρ−φin the weak coupling limit,wefindα=β,i.e.,α=β= 2|κ|= |N−8|.(3.10)Therefore wefind the quantum theory is described by the following effective action:12S q=2π d2x − ∂µX∂µX−i¯χXγµ∂µχX−F2X+ ∂µY∂µY−i¯χYγµ∂µχY−F2Y(3.11) +˜λ |N−8|e |N−8|(X+Y) F X+F Y−122 −∂µa i∂µa i+iξiγµ∂µξi+G2iHereˆΣi’s are matter superfields:ˆΣi=a i+¯θξi+12π d2x − ∂µX∂µX−i¯χXγµ∂µχX+ ∂µY∂µY−i¯χYγµ∂µχY˜λ22−2The equations of motion for X and Y are given by0=∂µ∂µX−˜λ|N−8| 322 22e|N−8|(X+Y)(¯χX+¯χY)(χX+χY)(3.13)If we consider solutions where all the fermionfields vanish,X and Y are given by the sums of holomorphic and anti-holomorphic functions.This tells that there is not any solution corresponding to black hole solution in the classical theory. Supersymmetry forbids the existence of black hole in the quantum theory.If we bosonize the fermionfieldsχX andχY:⋆χX±χY∼±:e±ϑ:,we obtain the following equations of motion,0=∂µ∂µX−˜λ|N−8| 322 22e|N−8|(X+Y)+ϑ0=∂µ∂µϑ−˜λ|N−8| 322 22eϑ.(3.15)The equations(3.14)tell that X and Y are given in terms ofϑ,X=−Y=ϑ+f+(x+)+f−(x−).(3.16)Here f±are arbitrary functions.Note that there are static solutions:f±=0,e−ϑ=−16{1−C(x+x−)n}2.(3.17)Here A=˜λ 22,C is an arbitrary constant and n is an integer.4.Summary and DiscussionWe have analyzed a supergravity theory coupled to a dilaton and supercon-formal matters in two dimensions.This theory is classically soluble and we have found all the solutions appeared in Callan,Giddings,Harvey and Strominger’s dilatonic gravity also satisfy the constraints and the equations of motion in this supersymmetric theory.When we quantize this theory following the procedure of Distler,Hlousek and Kawai,the cosmological term is renormalized to vanish in the quantum action.As a result,any solution corresponding to classical black hole does not appear in the quantum theory,which should be compared with the non-supersymmetric case.It should be amazing that supersymmetry forbids the existence of quantum black hole although classical black hole is allowed to exist.One of the motivations of the present work was to build a theory of dilaton gravity where the Bondi mass of black hole is bounded from below and the theory has a ground state.The above motivation becomes,however,irrelevant since there is no quantum black hole in our model.However,from the alternative viewpoint,our model gives us an interesting conjecture.Let us assume that the quantum black hole disappears even in four dimensions although it is difficult to take account of quantum effects owing to the non-renormalizability of gravity.In four dimensions,the Schwarzschild radius of black holes and the Compton wave length of elementary particles become com-parable at the Planck length scale.This might suggest that we should includethe black hole–like states in the Hilbert space.From the observation done in the present article,however,we might conjecture that,if supersymmetry is realized at the Planck scale,such black hole–like states need not to be included as quantum states in quantum gravity where only quantum states expressing smooth space-time structure are admitted.We wish to thank K.Higashijima,M.Kato,K.Odaka and A.Sugamoto for valuable discussion.The work of I.O.is supported by the Japan Society for the Promotionof Science.REFERENCES1.C.G.Callan,S.B.Giddings,J.A.Harvey and A.Strominger,Phys.Rev.D45(1992),R10052.S.W.Hawking,Comm.Math.Phys.43(1975),1993.J.G.Russo,L.Susskind and L.Thorlacius,preprint SU-ITP-92-4(1992)4.L.Susskind and L.Thorlacius,preprint SU-ITP-92-12(1992)5.S.W.Hawking,Phys.Rev.Lett.69(1992),6386.T.Banks,A.Dabholkar,M.R.Douglas and M.O’Loughlin,Phys.Rev.D45(1992),36077.A.Bilal and C.G.Callan,preprint PUPT-1320(1992)8.S.P.de Alwis,Phys.Lett.B289(1992),2789.K.Hamada,preprint UT-Komaba92-710.J.G.Russo,L.Susskind and L.Thorlacius,preprint SU-ITP-92-17(1992)11.S.W.Hawking and S.A.Stewart,Univ.of Cambridge preprint(1992)12.S.P.de Alwis,preprint COLO-HEP-288(1992)13.S.B.Giddings and A.Strominger,preprint UCSBTH-92-28(1992)14.J.G.Russo and L.Susskind and L.Thorlacius,preprint UTTG-19-92,SU-ITP-92-24(1992)15.O.Lechtenfeld and C.Nappi,Phys.Lett.B288(1992),7216.E.Elizalde and S.D.Odintsov,preprint UB-ECM-PF92/(1992)17.S.Nojiri and I.Oda,preprint NDA-FP-5/92,OCHA-PP-26(1992),to bepublished in Phys.Lett.B,NDA-FP-6/92,OCHA-PP-27(1992)18.A.M.Polyakov,Phys.Lett.103B(1981),20719.F.David,Mod.Phys.Lett.A3(1988),165120.J.Distler and H.Kawai,Nucl.Phys.B321(1989),50921.J.Distler,Z.Hlousek and H.Kawai,Int.J of Mod.Phys.A5(1990),39122.K.Higashijima,T.Uematsu and Y.Z.Yu,Phys.Lett.139B(1984),16123.T.Uematsu,Z.Phys.C29(1985),14324.M.Hayashi,S.Nojiri and S.Uehara,Z.Phys.C31(1986),56125.T.Uematsu,Z.Phys.C32(1986),3326.S.Nojiri,Phys.Lett.B274(1992),4127.A.M.Polyakov,Phys.Lett.103B(1981),29128.M.M.Nojiri and S.Nojiri,Prog.Theor.Phys.76(1986),73329.S.Nojiri,Prog.Theor.Phys.77(1987),159,Phys.Rev.D35(1987),2466。