Model for a Universe described by a non-minimally coupled scalar field and interacting dark
Wavy Strings Black or Bright
McGill 96–44 WATPHYS-TH-96/17
Wavy Strings: Black or Bright?
arXiv:hep-th/9612248v1 30 Dec 1996
Nemanja Kaloper,a,1 Robert C. Myersb,2 and Harold Roussedevelopments in string theory have brought forth a considerable interest in time-dependent hair on extended objects. This novel new hair is typically characterized by a wave profile along the horizon and angular momentum quantum numbers l, m in the transverse space. In this work, we present an extensive treatment of such oscillating black objects, focusing on their geometric properties. We first give a theorem of purely geometric nature, stating that such wavy hair cannot be detected by any scalar invariant built out of the curvature and/or matter fields. However, we show that the tidal forces detected by an infalling observer diverge at the ‘horizon’ of a black string superposed with a vibration in any mode with l ≥ 1. The same argument applied to longitudinal (l = 0) waves detects only finite tidal forces. We also provide an example with a manifestly smooth metric, proving that at least a certain class of these longitudinal waves have regular horizons.
The Subleading Isgur-Wise Form Factor $chi_3(vcdot v')$ to Order $alpha_s$ in QCD Sum Rules
a rXiv:h ep-ph/9212266v116Dec1992SLAC–PUB–6017WIS–92/99/Dec–PH December 1992T/E The Subleading Isgur-Wise Form Factor χ3(v ·v ′)to Order αs in QCD Sum Rules Matthias Neubert Stanford Linear Accelerator Center Stanford University,Stanford,California 94309Zoltan Ligeti and Yosef Nir Weizmann Institute of Science Physics Department,Rehovot 76100,Israel We calculate the contributions arising at order αs in the QCD sum rule for the spin-symmetry violating universal function χ3(v ·v ′),which appears at order 1/m Q in the heavy quark expansion of meson form factors.In particular,we derive the two-loop perturbative contribution to the sum rule.Over the kinematic range accessible in B →D (∗)ℓνdecays,we find that χ3(v ·v ′)does not exceed the level of ∼1%,indicating that power corrections induced by the chromo-magnetic operator in the heavy quark expansion are small.(submitted to Physical Review D)I.INTRODUCTIONIn the heavy quark effective theory(HQET),the hadronic matrix elements describing the semileptonic decays M(v)→M′(v′)ℓν,where M and M′are pseudoscalar or vector mesons containing a heavy quark,can be systematically expanded in inverse powers of the heavy quark masses[1–5].The coefficients in this expansion are m Q-independent,universal functions of the kinematic variable y=v·v′.These so-called Isgur-Wise form factors characterize the properties of the cloud of light quarks and gluons surrounding the heavy quarks,which act as static color sources.At leading order,a single functionξ(y)suffices to parameterize all matrix elements[6].This is expressed in the compact trace formula[5,7] M′(v′)|J(0)|M(v) =−ξ(y)tr{(2)m M P+ −γ5;pseudoscalar meson/ǫ;vector mesonis a spin wave function that describes correctly the transformation properties(under boosts and heavy quark spin rotations)of the meson states in the effective theory.P+=1g s2m Q O mag,O mag=M′(v′)ΓP+iσαβM(v) .(4)The mass parameter¯Λsets the canonical scale for power corrections in HQET.In the m Q→∞limit,it measures thefinite mass difference between a heavy meson and the heavy quark that it contains[11].By factoring out this parameter,χαβ(v,v′)becomes dimensionless.The most general decomposition of this form factor involves two real,scalar functionsχ2(y)andχ3(y)defined by[10]χαβ(v,v′)=(v′αγβ−v′βγα)χ2(y)−2iσαβχ3(y).(5)Irrespective of the structure of the current J ,the form factor χ3(y )appears always in the following combination with ξ(y ):ξ(y )+2Z ¯Λ d M m Q ′ χ3(y ),(6)where d P =3for a pseudoscalar and d V =−1for a vector meson.It thus effectively renormalizes the leading Isgur-Wise function,preserving its normalization at y =1since χ3(1)=0according to Luke’s theorem [10].Eq.(6)shows that knowledge of χ3(y )is needed if one wants to relate processes which are connected by the spin symmetry,such as B →D ℓνand B →D ∗ℓν.Being hadronic form factors,the universal functions in HQET can only be investigated using nonperturbative methods.QCD sum rules have become very popular for this purpose.They have been reformulated in the context of the effective theory and have been applied to the study of meson decay constants and the Isgur-Wise functions both in leading and next-to-leading order in the 1/m Q expansion [12–21].In particular,it has been shown that very simple predictions for the spin-symmetry violating form factors are obtained when terms of order αs are neglected,namely [17]χ2(y )=0,χ3(y )∝ ¯q g s σαβG αβq [1−ξ(y )].(7)In this approach χ3(y )is proportional to the mixed quark-gluon condensate,and it was estimated that χ3(y )∼1%for large recoil (y ∼1.5).In a recent work we have refined the prediction for χ2(y )by including contributions of order αs in the sum rule analysis [20].We found that these are as important as the contribution of the mixed condensate in (7).It is,therefore,worthwhile to include such effects also in the analysis of χ3(y ).This is the purpose of this article.II.DERIV ATION OF THE SUM RULEThe QCD sum rule analysis of the functions χ2(y )and χ3(y )is very similar.We shall,therefore,only briefly sketch the general procedure and refer for details to Refs.[17,20].Our starting point is the correlatord x d x ′d ze i (k ′·x ′−k ·x ) 0|T[¯q ΓM ′P ′+ΓP +iσαβP +ΓM+Ξ3(ω,ω′,y )tr 2σαβ2(1+/v ′),and we omit the velocity labels in h and h ′for simplicity.The heavy-light currents interpolate pseudoscalar or vector mesons,depending on the choice ΓM =−γ5or ΓM =γµ−v µ,respectively.The external momenta k and k ′in (8)are the “residual”off-shell momenta of the heavy quarks.Due to the phase redefinition of the effective heavy quark fields in HQET,they are related to the total momenta P and P ′by k =P −m Q v and k ′=P ′−m Q ′v ′[3].The coefficient functions Ξi are analytic in ω=2v ·k and ω′=2v ′·k ′,with discontinuities for positive values of these variables.They can be saturated by intermediate states which couple to the heavy-light currents.In particular,there is a double-pole contribution from the ground-state mesons M and M ′.To leading order in the 1/m Q expansion the pole position is at ω=ω′=2¯Λ.In the case of Ξ2,the residue of the pole is proportional to the universal function χ2(y ).For Ξ3the situation is more complicated,however,since insertions of the chromo-magnetic operator not only renormalize the leading Isgur-Wise function,but also the coupling of the heavy mesons to the interpolating heavy-light currents (i.e.,the meson decay constants)and the physical meson masses,which define the position of the pole.1The correct expression for the pole contribution to Ξ3is [17]Ξpole 3(ω,ω′,y )=F 2(ω−2¯Λ+iǫ) .(9)Here F is the analog of the meson decay constant in the effective theory (F ∼f M√m QδΛ2+... , 0|j (0)|M (v ) =iF2G 2tr 2σαβΓP +σαβM (v ) ,where the ellipses represent spin-symmetry conserving or higher order power corrections,and j =¯q Γh (v ).In terms of the vector–pseudoscalar mass splitting,the parameter δΛ2isgiven by m 2V −m 2P =−8¯ΛδΛ2.For not too small,negative values of ωand ω′,the coefficient function Ξ3can be approx-imated as a perturbative series in αs ,supplemented by the leading power corrections in 1/ωand 1/ω′,which are proportional to vacuum expectation values of local quark-gluon opera-tors,the so-called condensates [22].This is how nonperturbative corrections are incorporated in this approach.The idea of QCD sum rules is to match this theoretical representation of Ξ3to the phenomenological pole contribution given in (9).To this end,one first writes the theoretical expression in terms of a double dispersion integral,Ξth 3(ω,ω′,y )= d νd ν′ρth 3(ν,ν′,y )1Thereare no such additional terms for Ξ2because of the peculiar trace structure associated with this coefficient function.possible subtraction terms.Because of theflavor symmetry it is natural to set the Borel parameters associated withωandω′equal:τ=τ′=2T.One then introduces new variables ω±=12T ξ(y) F2e−2¯Λ/T=ω0dω+e−ω+/T ρth3(ω+,y)≡K(T,ω0,y).(12)The effective spectral density ρth3arises after integration of the double spectral density over ω−.Note that for each contribution to it the dependence onω+is known on dimensionalgrounds.It thus suffices to calculate directly the Borel transform of the individual con-tributions toΞth3,corresponding to the limitω0→∞in(12).Theω0-dependence can be recovered at the end of the calculation.When terms of orderαs are neglected,contributions to the sum rule forΞ3can only be proportional to condensates involving the gluonfield,since there is no way to contract the gluon contained in O mag.The leading power correction of this type is represented by the diagram shown in Fig.1(d).It is proportional to the mixed quark-gluon condensate and,as shown in Ref.[17],leads to(7).Here we are interested in the additional contributions arising at orderαs.They are shown in Fig.1(a)-(c).Besides a two-loop perturbative contribution, one encounters further nonperturbative corrections proportional to the quark and the gluon condensate.Let usfirst present the result for the nonperturbative power corrections.WefindK cond(T,ω0,y)=αs ¯q q TT + αs GG y+1− ¯q g sσαβGαβq√y2−1),δn(x)=1(4π)D×1dλλ1−D∞λd u1∞1/λd u2(u1u2−1)D/2−2where C F=(N2c−1)/2N c,and D is the dimension of space-time.For D=4,the integrand diverges asλ→0.To regulate the integral,we assume D<2and use a triple integration by parts inλto obtain an expression which can be analytically continued to the vicinity of D=4.Next we set D=4+2ǫ,expand inǫ,write the result as an integral overω+,and introduce back the continuum threshold.This givesK pert(T,ω0,y)=−αsy+1 2ω0dω+ω3+e−ω+/T(16)× 12−23∂µ+3αs9π¯Λ,(17)which shows that divergences arise at orderαs.At this order,the renormalization of the sum rule is thus accomplished by a renormalization of the“bare”parameter G2in(12).In the9π¯Λ 1µ2 +O(g3s).(18)Hence a counterterm proportional to¯Λξ(y)has to be added to the bracket on the left-hand side of the sum rule(12).To evaluate its effect on the right-hand side,we note that in D dimensions[17]¯Λξ(y)F2e−2¯Λ/T=3y+1 2ω0dω+ω3+e−ω+/T(19)× 1+ǫ γE−ln4π+2lnω+−ln y+12T ξ(y) F2e−2¯Λ/T=αsy+1 2ω0dω+ω3+e−ω+/T 2lnµ6+ y r(y)−1+ln y+1According to Luke’stheorem,theuniversalfunction χ3(y )vanishes at zero recoil [10].Evaluating (20)for y =1,we thus obtain a sum rule for G 2(µ)and δΛ2.It reads G 2(µ)−¯ΛδΛ224π3ω00d ω+ω3+e −ω+/T ln µ12 +K cond (T,ω0,1),(21)where we have used that r (1)=1.Precisely this sum rule has been derived previously,starting from a two-current correlator,in Ref.[16].This provides a nontrivial check of our ing the fact that ξ(y )=[2/(y +1)]2+O (g s )according to (19),we find that the µ-dependent terms cancel out when we eliminate G 2(µ)and δΛ2from the sum rule for χ3(y ).Before we present our final result,there is one more effect which has to be taken into account,namely a spin-symmetry violating correction to the continuum threshold ω0.Since the chromo-magnetic interaction changes the masses of the ground-state mesons [cf.(10)],it also changes the masses of higher resonance states.Expanding the physical threshold asωphys =ω0 1+d M8π3 22 δ3 ω032π2ω30e −ω0/T 26π2−r (y )−ξ(y ) δ0 ω096π 248T 1−ξ(y ).It explicitly exhibits the fact that χ3(1)=0.III.NUMERICAL ANALYSISLet us now turn to the evaluation of the sum rule (23).For the QCD parameters we take the standard values¯q q =−(0.23GeV)3,αs GG =0.04GeV4,¯q g sσαβGαβq =m20 ¯q q ,m20=0.8GeV2.(24) Furthermore,we useδω2=−0.1GeV from above,andαs/π=0.1corresponding to the scale µ=2¯Λ≃1GeV,which is appropriate for evaluating radiative corrections in the effective theory[15].The sensitivity of our results to changes in these parameters will be discussed below.The dependence of the left-hand side of(23)on¯Λand F can be eliminated by using a QCD sum rule for these parameters,too.It reads[16]¯ΛF2e−2¯Λ/T=9T4T − ¯q g sσαβGαβq4π2 2T − ¯q q +(2y+1)4T2.(26) Combining(23),(25)and(26),we obtainχ3(y)as a function ofω0and T.These parameters can be determined from the analysis of a QCD sum rule for the correlator of two heavy-light currents in the effective theory[16,18].Onefinds good stability forω0=2.0±0.3GeV,and the consistency of the theoretical calculation requires that the Borel parameter be in the range0.6<T<1.0GeV.It supports the self-consistency of the approach that,as shown in Fig.2,wefind stability of the sum rule(23)in the same region of parameter space.Note that it is in fact theδω2-term that stabilizes the sum rule.Without it there were no plateau.Over the kinematic range accessible in semileptonic B→D(∗)ℓνdecays,we show in Fig.3(a)the range of predictions forχ3(y)obtained for1.7<ω0<2.3GeV and0.7<T< 1.2GeV.From this we estimate a relative uncertainty of∼±25%,which is mainly due to the uncertainty in the continuum threshold.It is apparent that the form factor is small,not exceeding the level of1%.2Finally,we show in Fig.3(b)the contributions of the individual terms in the sum rule (23).Due to the large negative contribution proportional to the quark condensate,the terms of orderαs,which we have calculated in this paper,cancel each other to a large extent.As a consequence,ourfinal result forχ3(y)is not very different from that obtained neglecting these terms[17].This is,however,an accident.For instance,the order-αs corrections would enhance the sum rule prediction by a factor of two if the ¯q q -term had the opposite sign. From thisfigure one can also deduce how changes in the values of the vacuum condensates would affect the numerical results.As long as one stays within the standard limits,the sensitivity to such changes is in fact rather small.For instance,working with the larger value ¯q q =−(0.26GeV)3,or varying m20between0.6and1.0GeV2,changesχ3(y)by no more than±0.15%.In conclusion,we have presented the complete order-αs QCD sum rule analysis of the subleading Isgur-Wise functionχ3(y),including in particular the two-loop perturbative con-tribution.Wefind that over the kinematic region accessible in semileptonic B decays this form factor is small,typically of the order of1%.When combined with our previous analysis [20],which predicted similarly small values for the universal functionχ2(y),these results strongly indicate that power corrections in the heavy quark expansion which are induced by the chromo-magnetic interaction between the gluonfield and the heavy quark spin are small.ACKNOWLEDGMENTSIt is a pleasure to thank Michael Peskin for helpful discussions.M.N.gratefully acknowl-edgesfinancial support from the BASF Aktiengesellschaft and from the German National Scholarship Foundation.Y.N.is an incumbent of the Ruth E.Recu Career Development chair,and is supported in part by the Israel Commission for Basic Research and by the Minerva Foundation.This work was also supported by the Department of Energy,contract DE-AC03-76SF00515.REFERENCES[1]E.Eichten and B.Hill,Phys.Lett.B234,511(1990);243,427(1990).[2]B.Grinstein,Nucl.Phys.B339,253(1990).[3]H.Georgi,Phys.Lett.B240,447(1990).[4]T.Mannel,W.Roberts and Z.Ryzak,Nucl.Phys.B368,204(1992).[5]A.F.Falk,H.Georgi,B.Grinstein,and M.B.Wise,Nucl.Phys.B343,1(1990).[6]N.Isgur and M.B.Wise,Phys.Lett.B232,113(1989);237,527(1990).[7]J.D.Bjorken,Proceedings of the18th SLAC Summer Institute on Particle Physics,pp.167,Stanford,California,July1990,edited by J.F.Hawthorne(SLAC,Stanford,1991).[8]M.B.Voloshin and M.A.Shifman,Yad.Fiz.45,463(1987)[Sov.J.Nucl.Phys.45,292(1987)];47,801(1988)[47,511(1988)].[9]A.F.Falk,B.Grinstein,and M.E.Luke,Nucl.Phys.B357,185(1991).[10]M.E.Luke,Phys.Lett.B252,447(1990).[11]A.F.Falk,M.Neubert,and M.E.Luke,SLAC preprint SLAC–PUB–5771(1992),toappear in Nucl.Phys.B.[12]M.Neubert,V.Rieckert,B.Stech,and Q.P.Xu,in Heavy Flavours,edited by A.J.Buras and M.Lindner,Advanced Series on Directions in High Energy Physics(World Scientific,Singapore,1992).[13]A.V.Radyushkin,Phys.Lett.B271,218(1991).[14]D.J.Broadhurst and A.G.Grozin,Phys.Lett.B274,421(1992).[15]M.Neubert,Phys.Rev.D45,2451(1992).[16]M.Neubert,Phys.Rev.D46,1076(1992).[17]M.Neubert,Phys.Rev.D46,3914(1992).[18]E.Bagan,P.Ball,V.M.Braun,and H.G.Dosch,Phys.Lett.B278,457(1992);E.Bagan,P.Ball,and P.Gosdzinsky,Heidelberg preprint HD–THEP–92–40(1992).[19]B.Blok and M.Shifman,Santa Barbara preprint NSF–ITP–92–100(1992).[20]M.Neubert,Z.Ligeti,and Y.Nir,SLAC preprint SLAC–PUB–5915(1992).[21]M.Neubert,SLAC preprint SLAC–PUB–5992(1992).[22]M.A.Shifman,A.I.Vainshtein,and V.I.Zakharov,Nucl.Phys.B147,385(1979);B147,448(1979).FIGURESFIG.1.Diagrams contributing to the sum rule for the universal form factorχ3(v·v′):two-loop perturbative contribution(a),and nonperturbative contributions proportional to the quark con-densate(b),the gluon condensate(c),and the mixed condensate(d).Heavy quark propagators are drawn as double lines.The square represents the chromo-magnetic operator.FIG.2.Analysis of the stability region for the sum rule(23):The form factorχ3(y)is shown for y=1.5as a function of the Borel parameter.From top to bottom,the solid curves refer toω0=1.7,2.0,and2.3GeV.The dashes lines are obtained by neglecting the contribution proportional toδω2.FIG.3.(a)Prediction for the form factorχ3(v·v′)in the stability region1.7<ω0<2.3 GeV and0.7<T<1.2GeV.(b)Individual contributions toχ3(v·v′)for T=0.8GeV and ω0=2.0GeV:total(solid),mixed condensate(dashed-dotted),gluon condensate(wide dots), quark condensate(dashes).The perturbative contribution and theδω2-term are indistinguishable in thisfigure and are both represented by the narrow dots.11。
The entropy formula for the Ricci flow and its geometric applications
a rX iv:mat h /211159v1[ma t h.DG]11Nov22The entropy formula for the Ricci flow and its geometric applications Grisha Perelman ∗November 20,2007Introduction 1.The Ricci flow equation,introduced by Richard Hamilton [H 1],is the evolution equation d ∗St.Petersburg branch of Steklov Mathematical Institute,Fontanka 27,St.Petersburg191011,Russia.Email:perelman@pdmi.ras.ru or perelman@ ;I was partially supported by personal savings accumulated during my visits to the Courant Institute in the Fall of 1992,to the SUNY at Stony Brook in the Spring of 1993,and to the UC at Berkeley as a Miller Fellow in 1993-95.I’d like to thank everyone who worked to make those opportunities available to me.1in dimension four converge,modulo scaling,to metrics of constant positivecurvature.Without assumptions on curvature the long time behavior of the metricevolving by Ricciflow may be more complicated.In particular,as t ap-proaches somefinite time T,the curvatures may become arbitrarily large in some region while staying bounded in its complement.In such a case,it isuseful to look at the blow up of the solution for t close to T at a point where curvature is large(the time is scaled with the same factor as the metric ten-sor).Hamilton[H9]proved a convergence theorem,which implies that asubsequence of such scalings smoothly converges(modulo diffeomorphisms) to a complete solution to the Ricciflow whenever the curvatures of the scaledmetrics are uniformly bounded(on some time interval),and their injectivity radii at the origin are bounded away from zero;moreover,if the size of thescaled time interval goes to infinity,then the limit solution is ancient,thatis defined on a time interval of the form(−∞,T).In general it may be hard to analyze an arbitrary ancient solution.However,Ivey[I]and Hamilton[H4]proved that in dimension three,at the points where scalar curvatureis large,the negative part of the curvature tensor is small compared to the scalar curvature,and therefore the blow-up limits have necessarily nonneg-ative sectional curvature.On the other hand,Hamilton[H3]discovered a remarkable property of solutions with nonnegative curvature operator in ar-bitrary dimension,called a differential Harnack inequality,which allows,inparticular,to compare the curvatures of the solution at different points and different times.These results lead Hamilton to certain conjectures on thestructure of the blow-up limits in dimension three,see[H4,§26];the presentwork confirms them.The most natural way of forming a singularity infinite time is by pinchingan(almost)round cylindrical neck.In this case it is natural to make a surgery by cutting open the neck and gluing small caps to each of the boundaries,andthen to continue running the Ricciflow.The exact procedure was describedby Hamilton[H5]in the case of four-manifolds,satisfying certain curvature assumptions.He also expressed the hope that a similar procedure wouldwork in the three dimensional case,without any a priory assumptions,and that afterfinite number of surgeries,the Ricciflow would exist for all timet→∞,and be nonsingular,in the sense that the normalized curvatures ˜Rm(x,t)=tRm(x,t)would stay bounded.The topology of such nonsingular solutions was described by Hamilton[H6]to the extent sufficient to makesure that no counterexample to the Thurston geometrization conjecture can2occur among them.Thus,the implementation of Hamilton program would imply the geometrization conjecture for closed three-manifolds.In this paper we carry out some details of Hamilton program.The more technically complicated arguments,related to the surgery,will be discussed elsewhere.We have not been able to confirm Hamilton’s hope that the so-lution that exists for all time t→∞necessarily has bounded normalized curvature;still we are able to show that the region where this does not hold is locally collapsed with curvature bounded below;by our earlier(partly unpublished)work this is enough for topological conclusions.Our present work has also some applications to the Hamilton-Tian con-jecture concerning K¨a hler-Ricciflow on K¨a hler manifolds with positivefirst Chern class;these will be discussed in a separate paper.2.The Ricciflow has also been discussed in quantumfield theory,as an ap-proximation to the renormalization group(RG)flow for the two-dimensional nonlinearσ-model,see[Gaw,§3]and references therein.While my back-ground in quantum physics is insufficient to discuss this on a technical level, I would like to speculate on the Wilsonian picture of the RGflow.In this picture,t corresponds to the scale parameter;the larger is t,the larger is the distance scale and the smaller is the energy scale;to compute something on a lower energy scale one has to average the contributions of the degrees of freedom,corresponding to the higher energy scale.In other words,decreasing of t should correspond to looking at our Space through a microscope with higher resolution,where Space is now described not by some(riemannian or any other)metric,but by an hierarchy of riemannian metrics,connected by the Ricciflow equation.Note that we have a paradox here:the regions that appear to be far from each other at larger distance scale may become close at smaller distance scale;moreover,if we allow Ricci flow through singularities,the regions that are in different connected compo-nents at larger distance scale may become neighboring when viewed through microscope.Anyway,this connection between the Ricciflow and the RGflow sug-gests that Ricciflow must be gradient-like;the present work confirms this expectation.3.The paper is organized as follows.In§1we explain why Ricciflow can be regarded as a gradientflow.In§2,3we prove that Ricciflow,considered as a dynamical system on the space of riemannian metrics modulo diffeomor-phisms and scaling,has no nontrivial periodic orbits.The easy(and known)3case of metrics with negative minimum of scalar curvature is treated in§2; the other case is dealt with in§3,using our main monotonicity formula(3.4) and the Gaussian logarithmic Sobolev inequality,due to L.Gross.In§4we apply our monotonicity formula to prove that for a smooth solution on a finite time interval,the injectivity radius at each point is controlled by the curvatures at nearby points.This result removes the major stumbling block in Hamilton’s approach to geometrization.In§5we give an interpretation of our monotonicity formula in terms of the entropy for certain canonical ensemble.In§6we try to interpret the formal expressions,arising in the study of the Ricciflow,as the natural geometric quantities for a certain Riemannian manifold of potentially infinite dimension.The Bishop-Gromov relative volume comparison theorem for this particular manifold can in turn be interpreted as another monotonicity formula for the Ricciflow.This for-mula is rigorously proved in§7;it may be more useful than thefirst one in local considerations.In§8it is applied to obtain the injectivity radius control under somewhat different assumptions than in§4.In§9we consider one more way to localize the original monotonicity formula,this time using the differential Harnack inequality for the solutions of the conjugate heat equation,in the spirit of Li-Yau and Hamilton.The technique of§9and the logarithmic Sobolev inequality are then used in§10to show that Ricciflow can not quickly turn an almost euclidean region into a very curved one,no matter what happens far away.The results of sections1through10require no dimensional or curvature restrictions,and are not immediately related to Hamilton program for geometrization of three manifolds.The work on details of this program starts in§11,where we describe the ancient solutions with nonnegative curvature that may occur as blow-up limits offinite time singularities(they must satisfy a certain noncollaps-ing assumption,which,in the interpretation of§5,corresponds to having bounded entropy).Then in§12we describe the regions of high curvature under the assumption of almost nonnegative curvature,which is guaranteed to hold by the Hamilton and Ivey result,mentioned above.We also prove, under the same assumption,some results on the control of the curvatures forward and backward in time in terms of the curvature and volume at a given time in a given ball.Finally,in§13we give a brief sketch of the proof of geometrization conjecture.The subsections marked by*contain historical remarks and references. See also[Cao-C]for a relatively recent survey on the Ricciflow.41Ricciflow as a gradientflow1.1.Consider the functional F= M(R+|∇f|2)e−f dV for a riemannian metric g ij and a function f on a closed manifold M.Itsfirst variation can be expressed as follows:δF(v ij,h)= M e−f[−△v+∇i∇j v ij−R ij v ij−v ij∇i f∇j f+2<∇f,∇h>+(R+|∇f|2)(v/2−h)]= M e−f[−v ij(R ij+∇i∇j f)+(v/2−h)(2△f−|∇f|2+R)], whereδg ij=v ij,δf=h,v=g ij v ij.Notice that v/2−h vanishes identically iffthe measure dm=e−f dV is keptfixed.Therefore,the symmetric tensor −(R ij+∇i∇j f)is the L2gradient of the functional F m= M(R+|∇f|2)dm, where now f denotes log(dV/dm).Thus given a measure m,we may consider the gradientflow(g ij)t=−2(R ij+∇i∇j f)for F m.For general m thisflow may not exist even for short time;however,when it exists,it is just the Ricciflow,modified by a diffeomorphism.The remarkable fact here is that different choices of m lead to the sameflow,up to a diffeomorphism;that is, the choice of m is analogous to the choice of gauge.1.2Proposition.Suppose that the gradientflow for F m exists for t∈[0,T]. Then at t=0we have F m≤nNow we computeF t≥2n( (R+△f)e−f dV)2=2t1and t2,are called Ricci solitons.(Thus,if one considers Ricciflow as a dy-namical system on the space of riemannian metrics modulo diffeomorphism and scaling,then breathers and solitons correspond to periodic orbits and fixed points respectively).At each time the Ricci soliton metric satisfies an equation of the form R ij+cg ij+∇i b j+∇j b i=0,where c is a number and b i is a one-form;in particular,when b i=1log V=1dtV(2−n)/nλ −RdV≥n2V2/n[ |R ij+∇i∇j f−1( (R+△f)2e−f dV−( (R+△f)e−f dV)2)]≥0,nwhere f is the minimizer for F.72.4.The arguments above also show that there are no nontrivial(that is with non-constant Ricci curvature)steady or expanding Ricci solitons(on closed M).Indeed,the equality case in the chain of inequalities above requires that R+△f be constant on M;on the other hand,the Euler-Lagrange equation for the minimizer f is2△f−|∇f|2+R=const.Thus,△f−|∇f|2=const=0, because (△f−|∇f|2)e−f dV=0.Therefore,f is constant by the maximum principle.2.5*.A similar,but simpler proof of the results in this section,follows im-mediately from[H6,§2],where Hamilton checks that the minimum of RV22e−f dV,(3.1)restricted to f satisfying(4πτ)−nM,τt=−1(3.3)2τThe evolution equation for f can also be written as follows:2∗u=0,where u=(4πτ)−ng ij|2(4πτ)−n2τalong the Ricciflow.It is not hard to show that in the definition ofµthere always exists a smooth minimizer f(on a closed M).It is also clear that limτ→∞µ(g ij,τ)=+∞whenever thefirst eigenvalue of−4△+R is positive. Thus,our statement that there is no shrinking breathers other than gradient solitons,is implied by the followingClaim For an arbitrary metric g ij on a closed manifold M,the function µ(g ij,τ)is negative for smallτ>0and tends to zero asτtends to zero.Proof of the Claim.(sketch)Assume that¯τ>0is so small that Ricci flow starting from g ij exists on[0,¯τ].Let u=(4πτ)−n2τ−1g ij,fτ,12τ−1g ij”converge”to the euclidean metric,and if we couldextract a converging subsequence from fτ,we would get a function f on R n, such that R n(2π)−n2|∇f|2+f−n](2π)−n2−t)=µ(g ij(0),12)satisfiesR ij+∇i∇j f−g ij=0.Of course,this argument requires the existence of minimizer,and justification of the integration by parts;this is easy if M is closed,but can also be done with more efforts on some complete M,for instance when M is the Gaussian soliton.93.3*The no breathers theorem in dimension three was proved by Ivey[I]; in fact,he also ruled out nontrivial Ricci solitons;his proof uses the almost nonnegative curvature estimate,mentioned in the introduction.Logarithmic Sobolev inequalities is a vast area of research;see[G]for a survey and bibliography up to the year1992;the influence of the curvature was discussed by Bakry-Emery[B-Em].In the context of geometric evolution equations,the logarithmic Sobolev inequality occurs in Ecker[E1].4No local collapsing theorem IIn this section we present an application of the monotonicity formula(3.4) to the analysis of singularities of the Ricciflow.4.1.Let g ij(t)be a smooth solution to the Ricciflow(g ij)t=−2R ij on[0,T). We say that g ij(t)is locally collapsing at T,if there is a sequence of times t k→T and a sequence of metric balls B k=B(p k,r k)at times t k,such that r2k/t k is bounded,|Rm|(g ij(t k))≤r−2k in B k and r−n k V ol(B k)→0.Theorem.If M is closed and T<∞,then g ij(t)is not locally collapsing at T.Proof.Assume that there is a sequence of collapsing balls B k=B(p k,r k) at times t k→T.Then we claim thatµ(g ij(t k),r2k)→−∞.Indeed one(x,p k)r−1k)+c k,whereφis a function of one can take f k(x)=−logφ(dist tkvariable,equal1on[0,1/2],decreasing on[1/2,1],and very close to0on [1,∞),and c k is a constant;clearly c k→−∞as r−n k V ol(B k)→0.Therefore, applying the monotonicity formula(3.4),we getµ(g ij(0),t k+r2k)→−∞. However this is impossible,since t k+r2k is bounded.4.2.Definition We say that a metric g ij isκ-noncollapsed on the scaleρ,if every metric ball B of radius r<ρ,which satisfies|Rm|(x)≤r−2for every x∈B,has volume at leastκr n.It is clear that a limit ofκ-noncollapsed metrics on the scaleρis also κ-noncollapsed on the scaleρ;it is also clear thatα2g ij isκ-noncollapsed on the scaleαρwhenever g ij isκ-noncollapsed on the scaleρ.The theorem above essentially says that given a metric g ij on a closed manifold M and T<∞,one canfindκ=κ(g ij,T)>0,such that the solution g ij(t)to the Ricciflow starting at g ij isκ-noncollapsed on the scale T1/2for all t∈[0,T), provided it exists on this interval.Therefore,using the convergence theorem of Hamilton,we obtain the following10Corollary.Let g ij (t ),t ∈[0,T )be a solution to the Ricci flow on a closed manifold M,T <∞.Assume that for some sequences t k →T,p k ∈M and some constant C we have Q k =|Rm |(p k ,t k )→∞and |Rm |(x,t )≤CQ k ,whenever t <t k .Then (a subsequence of)the scalings of g ij (t k )at p k with factors Q k converges to a complete ancient solution to the Ricci flow,which is κ-noncollapsed on all scales for some κ>0.5A statistical analogyIn this section we show that the functional W ,introduced in section 3,is in a sense analogous to minus entropy.5.1Recall that the partition function for the canonical ensemble at tem-perature β−1is given by Z = exp (−βE )dω(E ),where ω(E )is a ”density of states”measure,which does not depend on β.Then one computes the average energy <E >=−∂(∂β)2log Z.Now fix a closed manifold M with a probability measure m ,and suppose that our system is described by a metric g ij (τ),which depends on the temper-ature τaccording to equation (g ij )τ=2(R ij +∇i ∇j f ),where dm =udV,u =(4πτ)−n 2)dm.(We do not discuss here what assumptions on g ij guarantee that the corre-sponding ”density of states”measure can be found)Then we compute<E >=−τ2 M(R +|∇f |2−n 2τg ij |2dmAlternatively,we could prescribe the evolution equations by replacing the t -derivatives by minus τ-derivatives in (3.3),and get the same formulas for Z,<E >,S,σ,with dm replaced by udV.Clearly,σis nonnegative;it vanishes only on a gradient shrinking soliton.<E >is nonnegative as well,whenever the flow exists for all sufficiently small τ>0(by proposition 1.2).Furthermore,if (a)u tends to a δ-function as τ→0,or (b)u is a limit of a sequence of functions u i ,such that each u i11tends to aδ-function asτ→τi>0,andτi→0,then S is also nonnegative. In case(a)all the quantities<E>,S,σtend to zero asτ→0,while in case (b),which may be interesting if g ij(τ)goes singular atτ=0,the entropy S may tend to a positive limit.If theflow is defined for all sufficiently largeτ(that is,we have an ancient solution to the Ricciflow,in Hamilton’s terminology),we may be interested in the behavior of the entropy S asτ→∞.A natural question is whether we have a gradient shrinking soliton whenever S stays bounded.5.2Remark.Heuristically,this statistical analogy is related to the de-scription of the renormalization groupflow,mentioned in the introduction: in the latter one obtains various quantities by averaging over higher energy states,whereas in the former those states are suppressed by the exponential factor.5.3*An entropy formula for the Ricciflow in dimension two was found by Chow[C];there seems to be no relation between his formula and ours.The interplay of statistical physics and(pseudo)-riemannian geometry occurs in the subject of Black Hole Thermodynamics,developed by Hawking et al.Unfortunately,this subject is beyond my understanding at the moment.6Riemannian formalism in potentially infi-nite dimensionsWhen one is talking of the canonical ensemble,one is usually considering an embedding of the system of interest into a much larger standard system of fixed temperature(thermostat).In this section we attempt to describe such an embedding using the formalism of Rimannian geometry.6.1Consider the manifold˜M=M×S N×R+with the following metric:˜g ij=g ij,˜gαβ=τgαβ,˜g00=N2N .It turns out that the components of the curvaturetensor of this metric coincide(modulo N−1)with the components of the matrix Harnack expression(and its traces),discovered by Hamilton[H3]. One can also compute that all the components of the Ricci tensor are equal12to zero(mod N−1).The heat equation and the conjugate heat equation on M can be interpreted via Laplace equation on˜M for functions and volume forms respectively:u satisfies the heat equation on M iff˜u(the extension of u to˜M constant along the S Nfibres)satisfies˜△˜u=0mod N−1;similarly,u satisfies the conjugate heat equation on M iff˜u∗=τ−N−12e−f dV).To achieve this,first apply to˜g a(small)diffeomor-phism,mapping each point(x i,yα,τ)into(x i,yα,τ(1−2fN)˜gαβ,˜g m00=˜g00−2fτ−fN)˜gαβ,g m00=˜g m00−|∇f|2=12−[τ(2△f−|∇f|2+R)+f−n]),g m i0=g mα0=g m iα=0Note that the hypersurfaceτ=const in the metric g m has the volume form τN/2e−f times the canonical form on M and S N,and the scalar curvatureof this hypersurface is12+τ(2△f−|∇f|2+R)+f)mod N−1.Thus theentropy S multiplied by the inverse temperatureβis essentially minus the total scalar curvature of this hypersurface.6.3Now we return to the metric˜g and try to use its Ricci-flatness by interpreting the Bishop-Gromov relative volume comparison theorem.Con-sider a metric ball in(˜M,˜g)centered at some point p whereτ=0.Then clearly the shortest geodesic between p and an arbitrary point q is always orthogonal to the S Nfibre.The length of such curveγ(τ)can be computedas τ(q)2τ+R+|˙γM(τ)|2dτ= √τ(R+|˙γM(τ)|2)dτ+O(N−3Thus a shortest geodesic should minimize L(γ)= τ(q)0√2Nτ(q) centered at p is O(N−1)-close to the hypersurfaceτ=τ(q),and its volume can be computed as V(S N) M( 2N L(x)+O(N−2))N dx,so the ratio of this volume to 2timesτ(q)−nMτ(R(γ(τ))+|˙γ(τ)|2)dτ(of course,R(γ(τ))and|˙γ(τ)|2are computed using g ij(τ))Let X(τ)=˙γ(τ),and let Y(τ)be any vectorfield alongγ(τ).Then the first variation formula can be derived as follows:δY(L)=14τ2τ1√τ(<Y,∇R >+2<∇X Y,X >)dτ= τ2τ1√dτ<Y,X >−2<Y,∇X X >−4Ric(Y,X ))dτ=2√τ<Y,∇R −2∇X X −4Ric(X,·)−12∇R +1τX (τ)has a limit as τ→0.From now on we fix p and τ1=0and denote by L (q,¯τ)the L -length of the L -shortest curve γ(τ),0≤τ≤¯τ,connecting p and q.In the computations below we pretend that shortest L -geodesics between p and q are unique for all pairs (q,¯τ);if this is not the case,the inequalities that we obtain are still valid when understood in the barrier sense,or in the sense of distributions.The first variation formula (7.1)implies that ∇L (q,¯τ)=2√¯τ(R +|X |2)−<X,∇L >=2√¯τ(R +|X |2)To evaluate R +|X |2we compute (using (7.2))dτR +2<∇R,X >−2Ric(X,X )−1τ(R +|X |2),(7.3)where H (X )is the Hamilton’s expression for the trace Harnack inequality (with t =−τ).Hence,¯τ32L (q,¯τ),(7.4)15where K =K (γ,¯τ)denotes the integral ¯τ0τ3¯τR −1¯τK (7.5)|∇L |2=−4¯τR +2¯τL −4¯τK (7.6)Finally we need to estimate the second variation of L.We computeδ2Y (L )=¯τ0√τ(Y ·Y ·R +2<∇X ∇Y Y,X >+2<R (Y,X ),Y,X >+2|∇X Y |2)dτNowd¯τ+¯τ0√2τY (7.8)We computed τ<Y,Y >,16so |Y (τ)|2=ττ(∇Y ∇Y R +2<R (Y,X ),Y,X >+2∇X Ric(Y,Y )−4∇Y Ric(Y,X )+2|Ric(Y,·)|2−22τ¯τ)dτTo put this in a more convenient form,observe thatdτRic(Y,Y )−2|Ric(Y,·)|2,so Hess L (Y,Y )≤1¯τ−2√τH (X,Y )dτ,(7.9)whereH (X,Y )=−∇Y ∇Y R −2<R (Y,X )Y,X >−4(∇X Ric(Y,Y )−∇Y Ric(Y,X ))−2Ric τ(Y,Y )+2|Ric(Y,·)|2−1τR +n τ−1dτ|Y |2=2Ric(Y,Y )+2<∇X Y,Y >=2Ric(Y,Y )+2<∇Y X,Y >=2Ric(Y,Y )+1¯τHess L (Y,Y )≤1√2H (X,˜Y )dτ,(7.11)where ˜Y is obtained by solving ODE (7.8)with initial data ˜Y (¯τ)=Y (¯τ).Moreover,the equality in (7.11)holds only if ˜Y is L -Jacobi and hence d √¯τ.17Now we can deduce an estimate for the jacobian J of the L-exponential map,given by L exp X(¯τ)=γ(¯τ),whereγ(τ)is the L-geodesic,starting at p and having X as the limit of√dτlog J(τ)≤n2¯τ−3√¯τg.Let l(q,τ)=1τL(q,τ)be thereduced distance.Then along an L-geodesicγ(τ)we have(by(7.4))d2¯τl+12¯τ−32exp(−l(τ))J(τ)is nonincreasing inτalongγ, and monotonicity is strict unless we are on a gradient shrinking soliton. Integrating over M,we get monotonicity of the reduced volume function ˜V(τ)= Mτ−n2¯τ≥0,(7.13) which follows immediately from(7.5),(7.6)and(7.10).Note also a useful inequality2△l−|∇l|2+R+l−nτL(q,τ),then from(7.5), (7.10)we obtain¯L¯τ+△¯L≤2n(7.15) Therefore,the minimum of¯L(·,¯τ)−2n¯τis nonincreasing,so in particular, the minimum of l(·,¯τ)does not exceed n2(τ0−τ),whenever theflow exists forτ∈[0,τ0].)7.2If the metrics g ij(τ)have nonnegative curvature operator,then Hamil-ton’s differential Harnack inequalities hold,and one can say more about the behavior of l.Indeed,in this case,if the solution is defined forτ∈[0,τ0],then H(X,Y)≥−Ric(Y,Y)(1τ0−τ)≥−R(1τ0−τ)|Y|2and18H(X)≥−R(1τ0−τ).Therefore,wheneverτis bounded away fromτ0(say,τ≤(1−c)τ0,c>0),we get(using(7.6),(7.11))|∇l|2+R≤Cldτlog|Y|2≤1n.We claim that˜V k(ǫk r2k)<3ǫn2ǫ−12k;on the otherhand,the contribution of the longer vectors does not exceed exp(−12k)by the jacobian comparison theorem.However,˜V k(t k)(that is,at t=0)stays bounded away from zero.Indeed,since min l k(·,t k−12,we can pick a point q k,where it is attained,and obtain a universal upper bound on l k(·,t k)by considering only curvesγwithγ(t k−12T].Sincethe monotonicity of the reduced volume requires˜V k(t k)≤˜V k(ǫk r2k),this is a contradiction.A similar argument shows that the statement of the corollary in4.2can be strengthened by adding another property of the ancient solution,obtained as a blow-up ly,we may claim that if,say,this solution is defined for t∈(−∞,0),then for any point p and any t0>0,the reduced volume function˜V(τ),constructed using p andτ(t)=t0−t,is bounded below byκ.7.4*The computations in this section are just natural modifications of those in the classical variational theory of geodesics that can be found in any textbook on Riemannian geometry;an even closer reference is[L-Y],where they use”length”,associated to a linear parabolic equation,which is pretty much the same as in our case.198No local collapsing theorem II8.1Let usfirst formalize the notion of local collapsing,that was used in7.3.Definition.A solution to the Ricciflow(g ij)t=−2R ij is said to be κ-collapsed at(x0,t0)on the scale r>0if|Rm|(x,t)≤r−2for all(x,t) satisfying dist t(x,x0)<r and t0−r2≤t≤t0,and the volume of the metric ball B(x0,r2)at time t0is less thanκr n.8.2Theorem.For any A>0there existsκ=κ(A)>0with the fol-lowing property.If g ij(t)is a smooth solution to the Ricciflow(g ij)t=−2R ij,0≤t≤r20,which has|Rm|(x,t)≤r−20for all(x,t),satisfying dist0(x,x0)<r0,and the volume of the metric ball B(x0,r0)at time zero is at least A−1r n0,then g ij(t)can not beκ-collapsed on the scales less than r0at a point(x,r20)with dist r20(x,x0)≤Ar0.Proof.By scaling we may assume r0=1;we may also assume dist1(x,x0)= A.Let us apply the constructions of7.1choosing p=x,τ(t)=1−t.Arguing as in7.3,we see that if our solution is collapsed at x on the scale r≤1,then the reduced volume˜V(r2)must be very small;on the other hand,˜V(1)can not be small unless min l(x,12(x,x0)≤13Kr0+r−10)(the inequality must be understood in the barrier sense,when necessary)(b)(cf.[H4,§17])Suppose Ric(x,t0)≤(n−1)K when dist t(x,x0)<r0, or dist t(x,x1)<r0.Thend3Kr0+r−10)at t=t0 Proof of Lemma.(a)Clearly,d t(x)= γ−Ric(X,X),whereγis the shortest geodesic between x and x0and X is its unit tangent vector,On the other hand,△d≤ n−1k=1s′′Y k(γ),where Y k are vectorfields alongγ,vanishing at20x0and forming an orthonormal basis at x when complemented by X,ands′′Yk (γ)denotes the second variation along Y k of the length ofγ.Take Y k to beparallel between x and x1,and linear between x1and x0,where d(x1,t0)=r0. Then△d≤n−1k=1s′′Y k(γ)= d(x,t0)r0−Ric(X,X)ds+ r00(s2r20)ds= γ−Ric(X,X)+ r00(Ric(X,X)(1−s2r20)ds≤d t+(n−1)(220),andrapidly increasing to infinity on(110),in such a way that2(φ′)2/φ−φ′′≥(2A+100n)φ′−C(A)φ,(8.1) for some constant C(A)<∞.Note that¯L+2n+1≥1for t≥12)is achieved for some y satisfying d(y,110.Now we compute2h=(¯L+2n+1)(−φ′′+(d t−△d−2A)φ′)−2<∇φ∇¯L>+(¯L t−△¯L)φ(8.2)∇h=(¯L+2n+1)∇φ+φ∇¯L(8.3) At a minimum point of h we have∇h=0,so(8.2)becomes2h=(¯L+2n+1)(−φ′′+(d t−△d−2A)φ′+2(φ′)2/φ)+(¯L t−△¯L)φ(8.4)Now since d(y,t)≥120),we can apply our lemma(a)to get d t−△d≥−100(n−1)on the set where φ′=0.Thus,using(8.1)and(7.15),we get2h≥−(¯L+2n+1)C(A)φ−2nφ≥−(2n+C(A))hThis implies that min h can not decrease too fast,and we get the required estimate.219Differential Harnack inequality for solutions of the conjugate heat equation9.1Proposition.Let g ij(t)be a solution to the Ricciflow(g ij)t=−2R ij,0≤t≤T,and let u=(4π(T−t))−ng ij|2(9.1)2(T−t)Proof.Routine computation.Clearly,this proposition immediately implies the monotonicity formula (3.4);its advantage over(3.4)shows up when one has to work locally.9.2Corollary.Under the same assumptions,on a closed manifold M,or whenever the application of the maximum principle can be justified,min v/u is nondecreasing in t.9.3Corollary.Under the same assumptions,if u tends to aδ-function as t→T,then v≤0for all t<T.Proof.If h satisfies the ordinary heat equation h t=△h with respect to the evolving metric g ij(t),then we have ddt hv≥0.Thus we only need to check that for everywhere positive h the limit of hv as t→T is nonpositive.But it is easy to see,that this limit is in fact zero.9.4Corollary.Under assumptions of the previous corollary,for any smooth curveγ(t)in M holds−d2(R(γ(t),t)+|˙γ(t)|2)−1and2(T−t)v≤0we get f t+12|∇f|2−f dt f(γ(t),t)=−f t−<∇f,˙γ(t)>≤−f t+12|˙γ|2.Summing these two inequalities, we get(9.2).9.5Corollary.If under assumptions of the previous corollary,p is the point where the limitδ-function is concentrated,then f(q,t)≤l(q,T−t),where l is the reduced distance,defined in7.1,using p andτ(t)=T−t.22。
Two-Dimensional Gas of Massless Dirac Fermions in Graphene
Two-Dimensional Gas of Massless Dirac Fermions in Graphene K.S. Novoselov1, A.K. Geim1, S.V. Morozov2, D. Jiang1, M.I. Katsnelson3, I.V. Grigorieva1, S.V. Dubonos2, A.A. Firsov21Manchester Centre for Mesoscience and Nanotechnology, University of Manchester, Manchester, M13 9PL, UK2Institute for Microelectronics Technology, 142432, Chernogolovka, Russia3Institute for Molecules and Materials, Radboud University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, the NetherlandsElectronic properties of materials are commonly described by quasiparticles that behave as nonrelativistic electrons with a finite mass and obey the Schrödinger equation. Here we report a condensed matter system where electron transport is essentially governed by the Dirac equation and charge carriers mimic relativistic particles with zero mass and an effective “speed of light” c∗ ≈106m/s. Our studies of graphene – a single atomic layer of carbon – have revealed a variety of unusual phenomena characteristic of two-dimensional (2D) Dirac fermions. In particular, we have observed that a) the integer quantum Hall effect in graphene is anomalous in that it occurs at halfinteger filling factors; b) graphene’s conductivity never falls below a minimum value corresponding to the conductance quantum e2/h, even when carrier concentrations tend to zero; c) the cyclotron mass mc of massless carriers with energy E in graphene is described by equation E =mcc∗2; and d) Shubnikov-de Haas oscillations in graphene exhibit a phase shift of π due to Berry’s phase.Graphene is a monolayer of carbon atoms packed into a dense honeycomb crystal structure that can be viewed as either an individual atomic plane extracted from graphite or unrolled single-wall carbon nanotubes or as a giant flat fullerene molecule. This material was not studied experimentally before and, until recently [1,2], presumed not to exist. To obtain graphene samples, we used the original procedures described in [1], which involve micromechanical cleavage of graphite followed by identification and selection of monolayers using a combination of optical, scanning-electron and atomic-force microscopies. The selected graphene films were further processed into multi-terminal devices such as the one shown in Fig. 1, following standard microfabrication procedures [2]. Despite being only one atom thick and unprotected from the environment, our graphene devices remain stable under ambient conditions and exhibit high mobility of charge carriers. Below we focus on the physics of “ideal” (single-layer) graphene which has a different electronic structure and exhibits properties qualitatively different from those characteristic of either ultra-thin graphite films (which are semimetals and whose material properties were studied recently [2-5]) or even of our other devices consisting of just two layers of graphene (see further). Figure 1 shows the electric field effect [2-4] in graphene. Its conductivity σ increases linearly with increasing gate voltage Vg for both polarities and the Hall effect changes its sign at Vg ≈0. This behaviour shows that substantial concentrations of electrons (holes) are induced by positive (negative) gate voltages. Away from the transition region Vg ≈0, Hall coefficient RH = 1/ne varies as 1/Vg where n is the concentration of electrons or holes and e the electron charge. The linear dependence 1/RH ∝Vg yields n =α·Vg with α ≈7.3·1010cm-2/V, in agreement with the theoretical estimate n/Vg ≈7.2·1010cm-2/V for the surface charge density induced by the field effect (see Fig. 1’s caption). The agreement indicates that all the induced carriers are mobile and there are no trapped charges in graphene. From the linear dependence σ(Vg) we found carrier mobilities µ =σ/ne, whichreached up to 5,000 cm2/Vs for both electrons and holes, were independent of temperature T between 10 and 100K and probably still limited by defects in parent graphite. To characterise graphene further, we studied Shubnikov-de Haas oscillations (SdHO). Figure 2 shows examples of these oscillations for different magnetic fields B, gate voltages and temperatures. Unlike ultra-thin graphite [2], graphene exhibits only one set of SdHO for both electrons and holes. By using standard fan diagrams [2,3], we have determined the fundamental SdHO frequency BF for various Vg. The resulting dependence of BF as a function of n is plotted in Fig. 3a. Both carriers exhibit the same linear dependence BF = β·n with β ≈1.04·10-15 T·m2 (±2%). Theoretically, for any 2D system β is defined only by its degeneracy f so that BF =φ0n/f, where φ0 =4.14·10-15 T·m2 is the flux quantum. Comparison with the experiment yields f =4, in agreement with the double-spin and double-valley degeneracy expected for graphene [6,7] (cf. caption of Fig. 2). Note however an anomalous feature of SdHO in graphene, which is their phase. In contrast to conventional metals, graphene’s longitudinal resistance ρxx(B) exhibits maxima rather than minima at integer values of the Landau filling factor ν (Fig. 2a). Fig. 3b emphasizes this fact by comparing the phase of SdHO in graphene with that in a thin graphite film [2]. The origin of the “odd” phase is explained below. Another unusual feature of 2D transport in graphene clearly reveals itself in the T-dependence of SdHO (Fig. 2b). Indeed, with increasing T the oscillations at high Vg (high n) decay more rapidly. One can see that the last oscillation (Vg ≈100V) becomes practically invisible already at 80K whereas the first one (Vg <10V) clearly survives at 140K and, in fact, remains notable even at room temperature. To quantify this behaviour we measured the T-dependence of SdHO’s amplitude at various gate voltages and magnetic fields. The results could be fitted accurately (Fig. 3c) by the standard expression T/sinh(2π2kBTmc/heB), which yielded mc varying between ≈ 0.02 and 0.07m0 (m0 is the free electron mass). Changes in mc are well described by a square-root dependence mc ∝n1/2 (Fig. 3d). To explain the observed behaviour of mc, we refer to the semiclassical expressions BF = (h/2πe)S(E) and mc =(h2/2π)∂S(E)/∂E where S(E) =πk2 is the area in k-space of the orbits at the Fermi energy E(k) [8]. Combining these expressions with the experimentally-found dependences mc ∝n1/2 and BF =(h/4e)n it is straightforward to show that S must be proportional to E2 which yields E ∝k. Hence, the data in Fig. 3 unambiguously prove the linear dispersion E =hkc∗ for both electrons and holes with a common origin at E =0 [6,7]. Furthermore, the above equations also imply mc =E/c∗2 =(h2n/4πc∗2)1/2 and the best fit to our data yields c∗ ≈1⋅106 m/s, in agreement with band structure calculations [6,7]. The employed semiclassical model is fully justified by a recent theory for graphene [9], which shows that SdHO’s amplitude can indeed be described by the above expression T/sinh(2π2kBTmc/heB) with mc =E/c∗2. Note that, even though the linear spectrum of fermions in graphene (Fig. 3e) implies zero rest mass, their cyclotron mass is not zero. The unusual response of massless fermions to magnetic field is highlighted further by their behaviour in the high-field limit where SdHO evolve into the quantum Hall effect (QHE). Figure 4 shows Hall conductivity σxy of graphene plotted as a function of electron and hole concentrations in a constant field B. Pronounced QHE plateaux are clearly seen but, surprisingly, they do not occur in the expected sequence σxy =(4e2/h)N where N is integer. On the contrary, the plateaux correspond to half-integer ν so that the first plateau occurs at 2e2/h and the sequence is (4e2/h)(N + ½). Note that the transition from the lowest hole (ν =–½) to lowest electron (ν =+½) Landau level (LL) in graphene requires the same number of carriers (∆n =4B/φ0 ≈1.2·1012cm-2) as the transition between other nearest levels (cf. distances between minima in ρxx). This results in a ladder of equidistant steps in σxy which are not interrupted when passing through zero. To emphasize this highly unusual behaviour, Fig. 4 also shows σxy for a graphite film consisting of only two graphene layers where the sequence of plateaux returns to normal and the first plateau is at 4e2/h, as in the conventional QHE. We attribute this qualitative transition between graphene and its two-layer counterpart to the fact that fermions in the latter exhibit a finite mass near n ≈0 (as found experimentally; to be published elsewhere) and can no longer be described as massless Dirac particles. 2The half-integer QHE in graphene has recently been suggested by two theory groups [10,11], stimulated by our work on thin graphite films [2] but unaware of the present experiment. The effect is single-particle and intimately related to subtle properties of massless Dirac fermions, in particular, to the existence of both electron- and hole-like Landau states at exactly zero energy [912]. The latter can be viewed as a direct consequence of the Atiyah-Singer index theorem that plays an important role in quantum field theory and the theory of superstrings [13,14]. For the case of 2D massless Dirac fermions, the theorem guarantees the existence of Landau states at E=0 by relating the difference in the number of such states with opposite chiralities to the total flux through the system (note that magnetic field can also be inhomogeneous). To explain the half-integer QHE qualitatively, we invoke the formal expression [9-12] for the energy of massless relativistic fermions in quantized fields, EN =[2ehc∗2B(N +½ ±½)]1/2. In QED, sign ± describes two spins whereas in the case of graphene it refers to “pseudospins”. The latter have nothing to do with the real spin but are “built in” the Dirac-like spectrum of graphene, and their origin can be traced to the presence of two carbon sublattices. The above formula shows that the lowest LL (N =0) appears at E =0 (in agreement with the index theorem) and accommodates fermions with only one (minus) projection of the pseudospin. All other levels N ≥1 are occupied by fermions with both (±) pseudospins. This implies that for N =0 the degeneracy is half of that for any other N. Alternatively, one can say that all LL have the same “compound” degeneracy but zeroenergy LL is shared equally by electrons and holes. As a result the first Hall plateau occurs at half the normal filling and, oddly, both ν = –½ and +½ correspond to the same LL (N =0). All other levels have normal degeneracy 4B/φ0 and, therefore, remain shifted by the same ½ from the standard sequence. This explains the QHE at ν =N + ½ and, at the same time, the “odd” phase of SdHO (minima in ρxx correspond to plateaux in ρxy and, hence, occur at half-integer ν; see Figs. 2&3), in agreement with theory [9-12]. Note however that from another perspective the phase shift can be viewed as the direct manifestation of Berry’s phase acquired by Dirac fermions moving in magnetic field [15,16]. Finally, we return to zero-field behaviour and discuss another feature related to graphene’s relativistic-like spectrum. The spectrum implies vanishing concentrations of both carriers near the Dirac point E =0 (Fig. 3e), which suggests that low-T resistivity of the zero-gap semiconductor should diverge at Vg ≈0. However, neither of our devices showed such behaviour. On the contrary, in the transition region between holes and electrons graphene’s conductivity never falls below a well-defined value, practically independent of T between 4 and 100K. Fig. 1c plots values of the maximum resistivity ρmax(B =0) found in 15 different devices, which within an experimental error of ≈15% all exhibit ρmax ≈6.5kΩ, independent of their mobility that varies by a factor of 10. Given the quadruple degeneracy f, it is obvious to associate ρmax with h/fe2 =6.45kΩ where h/e2 is the resistance quantum. We emphasize that it is the resistivity (or conductivity) rather than resistance (or conductance), which is quantized in graphene (i.e., resistance R measured experimentally was not quantized but scaled in the usual manner as R =ρL/w with changing length L and width w of our devices). Thus, the effect is completely different from the conductance quantization observed previously in quantum transport experiments. However surprising, the minimum conductivity is an intrinsic property of electronic systems described by the Dirac equation [17-20]. It is due to the fact that, in the presence of disorder, localization effects in such systems are strongly suppressed and emerge only at exponentially large length scales. Assuming the absence of localization, the observed minimum conductivity can be explained qualitatively by invoking Mott’s argument [21] that mean-free-path l of charge carriers in a metal can never be shorter that their wavelength λF. Then, σ =neµ can be re-written as σ = (e2/h)kFl and, hence, σ cannot be smaller than ≈e2/h per each type of carriers. This argument is known to have failed for 2D systems with a parabolic spectrum where disorder leads to localization and eventually to insulating behaviour [17,18]. For the case of 2D Dirac fermions, no localization is expected [17-20] and, accordingly, Mott’s argument can be used. Although there is a broad theoretical consensus [18-23,10,11] that a 2D gas of Dirac fermions should exhibit a minimum 3conductivity of about e2/h, this quantization was not expected to be accurate and most theories suggest a value of ≈e2/πh, in disagreement with the experiment. In conclusion, graphene exhibits electronic properties distinctive for a 2D gas of particles described by the Dirac rather than Schrödinger equation. This 2D system is not only interesting in itself but also allows one to access – in a condensed matter experiment – the subtle and rich physics of quantum electrodynamics [24-27] and provides a bench-top setting for studies of phenomena relevant to cosmology and astrophysics [27,28].1. Novoselov, K.S. et al. PNAS 102, 10451 (2005). 2. Novoselov, K.S. et al. Science 306, 666 (2004); cond-mat/0505319. 3. Zhang, Y., Small, J.P., Amori, M.E.S. & Kim, P. Phys. Rev. Lett. 94, 176803 (2005). 4. Berger, C. et al. J. Phys. Chem. B, 108, 19912 (2004). 5. Bunch, J.S., Yaish, Y., Brink, M., Bolotin, K. & McEuen, P.L. Nanoletters 5, 287 (2005). 6. Dresselhaus, M.S. & Dresselhaus, G. Adv. Phys. 51, 1 (2002). 7. Brandt, N.B., Chudinov, S.M. & Ponomarev, Y.G. Semimetals 1: Graphite and Its Compounds (North-Holland, Amsterdam, 1988). 8. Vonsovsky, S.V. and Katsnelson, M.I. Quantum Solid State Physics (Springer, New York, 1989). 9. Gusynin, V.P. & Sharapov, S.G. Phys. Rev. B 71, 125124 (2005). 10. Gusynin, V.P. & Sharapov, S.G. cond-mat/0506575. 11. Peres, N.M.R., Guinea, F. & Castro Neto, A.H. cond-mat/0506709. 12. Zheng, Y. & Ando, T. Phys. Rev. B 65, 245420 (2002). 13. Kaku, M. Introduction to Superstrings (Springer, New York, 1988). 14. Nakahara, M. Geometry, Topology and Physics (IOP Publishing, Bristol, 1990). 15. Mikitik, G. P. & Sharlai, Yu.V. Phys. Rev. Lett. 82, 2147 (1999). 16. Luk’yanchuk, I.A. & Kopelevich, Y. Phys. Rev. Lett. 93, 166402 (2004). 17. Abrahams, E., Anderson, P.W., Licciardello, D.C. & Ramakrishnan, T.V. Phys. Rev. Lett. 42, 673 (1979). 18. Fradkin, E. Phys. Rev. B 33, 3263 (1986). 19. Lee, P.A. Phys. Rev. Lett. 71, 1887 (1993). 20. Ziegler, K. Phys. Rev. Lett. 80, 3113 (1998). 21. Mott, N.F. & Davis, E.A. Electron Processes in Non-Crystalline Materials (Clarendon Press, Oxford, 1979). 22. Morita, Y. & Hatsugai, Y. Phys. Rev. Lett. 79, 3728 (1997). 23. Nersesyan, A.A., Tsvelik, A.M. & Wenger, F. Phys. Rev. Lett. 72, 2628 (1997). 24. Rose, M.E. Relativistic Electron Theory (John Wiley, New York, 1961). 25. Berestetskii, V.B., Lifshitz, E.M. & Pitaevskii, L.P. Relativistic Quantum Theory (Pergamon Press, Oxford, 1971). 26. Lai, D. Rev. Mod. Phys. 73, 629 (2001). 27. Fradkin, E. Field Theories of Condensed Matter Systems (Westview Press, Oxford, 1997). 28. Volovik, G.E. The Universe in a Helium Droplet (Clarendon Press, Oxford, 2003).Acknowledgements This research was supported by the EPSRC (UK). We are most grateful to L. Glazman, V. Falko, S. Sharapov and A. Castro Netto for helpful discussions. K.S.N. was supported by Leverhulme Trust. S.V.M., S.V.D. and A.A.F. acknowledge support from the Russian Academy of Science and INTAS.43µ (m2/Vs)0.8c4P0.4 22 σ (1/kΩ)10K0 0 1/RH(T/kΩ) 1 2ρmax (h/4e2)1-5010 Vg (V) 50 -10ab 0 -100-500 Vg (V)50100Figure 1. Electric field effect in graphene. a, Scanning electron microscope image of one of our experimental devices (width of the central wire is 0.2µm). False colours are chosen to match real colours as seen in an optical microscope for larger areas of the same materials. Changes in graphene’s conductivity σ (main panel) and Hall coefficient RH (b) as a function of gate voltage Vg. σ and RH were measured in magnetic fields B =0 and 2T, respectively. The induced carrier concentrations n are described by [2] n/Vg =ε0ε/te where ε0 and ε are permittivities of free space and SiO2, respectively, and t ≈300 nm is the thickness of SiO2 on top of the Si wafer used as a substrate. RH = 1/ne is inverted to emphasize the linear dependence n ∝Vg. 1/RH diverges at small n because the Hall effect changes its sign around Vg =0 indicating a transition between electrons and holes. Note that the transition region (RH ≈ 0) was often shifted from zero Vg due to chemical doping [2] but annealing of our devices in vacuum normally allowed us to eliminate the shift. The extrapolation of the linear slopes σ(Vg) for electrons and holes results in their intersection at a value of σ indistinguishable from zero. c, Maximum values of resistivity ρ =1/σ (circles) exhibited by devices with different mobilites µ (left y-axis). The histogram (orange background) shows the number P of devices exhibiting ρmax within 10% intervals around the average value of ≈h/4e2. Several of the devices shown were made from 2 or 3 layers of graphene indicating that the quantized minimum conductivity is a robust effect and does not require “ideal” graphene.ρxx (kΩ)0.60 aVg = -60V4B (T)810K12∆σxx (1/kΩ)0.4 1ν=4 140K 80K B =12T0 b 0 25 50 Vg (V) 7520K100Figure 2. Quantum oscillations in graphene. SdHO at constant gate voltage Vg as a function of magnetic field B (a) and at constant B as a function of Vg (b). Because µ does not change much with Vg, the constant-B measurements (at a constant ωcτ =µB) were found more informative. Panel b illustrates that SdHO in graphene are more sensitive to T at high carrier concentrations. The ∆σxx-curves were obtained by subtracting a smooth (nearly linear) increase in σ with increasing Vg and are shifted for clarity. SdHO periodicity ∆Vg in a constant B is determined by the density of states at each Landau level (α∆Vg = fB/φ0) which for the observed periodicity of ≈15.8V at B =12T yields a quadruple degeneracy. Arrows in a indicate integer ν (e.g., ν =4 corresponds to 10.9T) as found from SdHO frequency BF ≈43.5T. Note the absence of any significant contribution of universal conductance fluctuations (see also Fig. 1) and weak localization magnetoresistance, which are normally intrinsic for 2D materials with so high resistivity.75 BF (T) 500.2 0.11/B (1/T)b5 10 N 1/2025 a 0 0.061dmc /m00.04∆0.02 0c0 0 T (K) 150n =0e-6-3036Figure 3. Dirac fermions of graphene. a, Dependence of BF on carrier concentration n (positive n correspond to electrons; negative to holes). b, Examples of fan diagrams used in our analysis [2] to find BF. N is the number associated with different minima of oscillations. Lower and upper curves are for graphene (sample of Fig. 2a) and a 5-nm-thick film of graphite with a similar value of BF, respectively. Note that the curves extrapolate to different origins; namely, to N = ½ and 0. In graphene, curves for all n extrapolate to N = ½ (cf. [2]). This indicates a phase shift of π with respect to the conventional Landau quantization in metals. The shift is due to Berry’s phase [9,15]. c, Examples of the behaviour of SdHO amplitude ∆ (symbols) as a function of T for mc ≈0.069 and 0.023m0; solid curves are best fits. d, Cyclotron mass mc of electrons and holes as a function of their concentration. Symbols are experimental data, solid curves the best fit to theory. e, Electronic spectrum of graphene, as inferred experimentally and in agreement with theory. This is the spectrum of a zero-gap 2D semiconductor that describes massless Dirac fermions with c∗ 300 times less than the speed of light.n (1012 cm-2)σxy (4e2/h)4 3 2 -2 1 -1 -2 -3 2 44Kn7/ 5/ 3/ 1/2 2 2 210 ρxx (kΩ)-4σxy (4e2/h)0-1/2 -3/2 -5/2514T0-7/2 -4 -2 0 2 4 n (1012 cm-2)Figure 4. Quantum Hall effect for massless Dirac fermions. Hall conductivity σxy and longitudinal resistivity ρxx of graphene as a function of their concentration at B =14T. σxy =(4e2/h)ν is calculated from the measured dependences of ρxy(Vg) and ρxx(Vg) as σxy = ρxy/(ρxy + ρxx)2. The behaviour of 1/ρxy is similar but exhibits a discontinuity at Vg ≈0, which is avoided by plotting σxy. Inset: σxy in “two-layer graphene” where the quantization sequence is normal and occurs at integer ν. The latter shows that the half-integer QHE is exclusive to “ideal” graphene.。
21 物理学咬文嚼字之二十一Dimension_维度_量纲加尺度
度意义和量纲意义算能基本得到照顾 , 关于尺寸的 意思经翻译后就让人很难看出原文是 dim ension 了 . 且就算是仅关于尺寸的 ,这个 dim ension 也还包含着 不同维 度意 义 , 比 如 ,“ a man of a giant dim ension (大块头 ) ” , 这里 di m ension 是三维的 ; 而“beam s of
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大小随长度单位的立方成反比 ” 或者 “ 比热容的量纲 - 3 含L ” 都稍显不足 . 又比如 ,量子场论喜欢采用自然 单位 ,即取 c = 1; = 1. 用 c = 197MeV ・fm 就能恢复 相关物理量的正确的 dim ension ( One can easily re2 cover the right dim ension of any physical quantities by making use of c = 197MeV ・fm ) . 这里 , dim ension 既 [1] 是量纲 ,也有数值的意思 ! 对于采用自然单位的做 法 ,有人很不以为然 ,W esson 就觉得“ 这也许是数学 上可接受的省劲小把戏 ,但物理上它却意味着信息的 缺失 ,甚至会造成混乱 (Mathematically it is an accep t2 able trick which saves labour . Physically it rep resents a [2] loss of infor mation and can lead to confusion. ) ” 信 哉此言 ! 把 dim ension 翻译成中文的维度和量纲是非常 有意思的 . 汉字的维字是大绳 ,作为动词则有用绳子 约束的意思 ,推而广之有保持 、 维护的意思 . 所谓的 “ 天柱折 ,地维绝 — — — 《淮南子 ・天文训 》 ” 可见维 与柱的对应 . 所谓的“ 国有四维 , 一维绝则倾 , 二维 绝则危 ,三维绝则覆 , 四维绝则灭 ” , 四维指的是四 种互相独立的有维系功能的存在 , 比如礼 、 义、 廉、 耻 . 这里维度用来翻译 dim ension 一个方面的意思 , 是坚持了其独立 、 分立 、 linearly2noncorrelated 的性质 (线性不相关是维数的重要特征 ) . 有意思的是 , 关 于维度的动作 “ 张” 字 ,同英文的 subtend 也对应得 很妙 . D im ension 的另一译法“ 量纲 ” 中的“ 纲” 字, 和“ 维” 近似 ,也是较粗的绳子的意思 , 中文本就有 “ 维纲 ” 的说法 . 对于渔网来说 , 重要的是网纲 (图 1 ) . 一侧的网纲上挂上漂浮物 , 一侧的网纲上挂上 重物 ,这样三维水体中两条平行的纲线就使渔网造 成了一个大致垂直于水平面的或开或闭的截面 ! 对 于图 1 中的这种只在底部有网纲的渔网 , 撒网技术 的关键 (维也有关键的意思 ) 就是要做到会提纲 , 纲 举才有目 (网眼 )张 . 关于维度 许多时候 ,研究一个物理问题 ,特别是运动学或 动力学问题 , 先确定它是一个多少维空间 (包括参 数空间 )的问题是有益的 . 简单的单摆是在二维平 面内运动的 ,是一个自由度的问题 ; 傅科摆在三维空 间内运动 ,是一个两自由度的问题 . 1921 年以前的 物理学 ,其基本出发点是“ 各种物理现象是在一个
光子阻塞效应
学号：201105774题目名称: 强耦合下的光子阻塞效应研究题目类型: 研究论文学生姓名: 董昌瑞院（系）: 物理与光电工程学院专业班级: 物理11102班指导教师: 邹金花辅导教师: 邹金花时间: 2015年1月至2015年6月目录毕业论文任务书` (I)指导教师评审意见 (VIII)评阅教师评语 (IX)答辩记录及成绩评定 (X)中文摘要 (XI)外文摘要 (XII)1引言 (1)2 基础理论知识 (1)2.1 光力振子系统 (1)2.2二能级原子与光场相互作用的全量子理论 (2)2.3光场关联函数 (5)2.4 光子计数统计 (8)3 模型方程与结果分析 (10)3.1模型方程 (10)3.2 方程分析 (12)4总结与展望 (14)参考文献 (14)致谢 (16)毕业论文任务书`院（系）物理与光电工程学院专业物理班级物理11102 学生姓名董昌瑞指导教师/职称邹金花/副教授1.毕业论文(设计)题目：强耦合下的光子阻塞效应研究2.毕业论文(设计)起止时间： 2015 年1月1 日～2015 年 6月10 日3．毕业论文(设计)所需资料及原始数据（指导教师选定部分）[1] A Ridolfo, M Leib, S Savasta, M J Hartmann. Photon Blockade in the Ultrastrong CouplingRegime [J]. Phys. Rev. Lett., 2012, 109: 193602-1~193602-5[2] Jieqiao Liao, C K Law. Cooling of a mirror in cavity optomechanics with a chirped pulse [J]. Phys. Rev. A, 2011, 84: 053838-1~053838-6[3] P Komar, S D Bennett, K Stannigel, S J M Habraken, P Rabl, P Zoller, M D Lukin. Single-photon nonlinearities in two-mode optomechanics [J]. Phys. Rev. A, 2013, 87: 013839-1~013839-10[4] T Ramos, V Sudhir, K Stannigel, P Zoller, T Kippenbrg. Nonlinear quantum optomechanics viaindividual intrinsic two-level defects [J]. Phys. Rev. Lett., 2013, 110: 193602-1~193602-5 [5] G Anetsberger, O Arcizet, Q P Unterreithmeier, R Riviere, A Schliesser, E M Weig, J P Kotthaus,T Kippenberg. Near-field cavity optomechanics with nanomechanical oscillators [J]. Nat. Phys., 2009, 5: 909~914[6] S J M Habraken, W Lechner, P Zoller. Resonances in dissipative optomechanics withnanoparticles: Sorting, speed rectification, and transverse coolings [J]. Phys. Rev. A, 2013, 87: 053808-1~053808-8[7] K Qu, G S Agarwal. Fano resonances and their control in optomechanics [J]. Phys. Rev. A, 2013,87: 063813-1~063813-7[8] A Nunnenkamp, K Borkje, S M Girvin. Cooling in the single-photon strong-coupling regime ofcavity optomechanics [J]. Phys. Rev. A, 2012, 85: 051803-1~051803-4[9] Y C Liu, Y F Xiao, X S Luan, C W Wong. Dynamic Dissipative Cooling of a MechanicalResonator in Strong Coupling Optomechanics [J]. Phys. Rev. A, 2013, 110: 153606-1~153606-5[10] A Nunnekamp, K Borkie, S M Girvin. Single-photon optomechanics [J]. Phys. Rev. Lett., 2011,107: 063602-1~063602-5[11] J M Dobrindt, I Wilson-Rae, T J Kippenbeg. Parametric Normal-Mode Splitting in CavityOptomechanics [J]. Phys. Rev. Lett., 2008, 101: 263602-1~263602-4[12]樊菲菲. 光力振子与原子间量子纠缠和振子压缩的研究[D]. 华中师范大学，2014[13] 张文慧. 光机械腔系统的动力学行为[D]. 华中师范大学，2014[14]詹孝贵. 腔光机械系统中电磁诱导透明及其相关现象的理论研究[D]. 华中科技大学，20134．毕业论文(设计)应完成的主要内容在阅读大量文献的基础上，完成开题报告，并通过开题答辩。
Unravelling
with plenty of inspiration, but not many In fact, there are ample reasons why one might doubt whether Einstein’s vision is achievable, or at least achievable in the fore-seeable future. Crucial clues may be hope-lessly out of reach. When looking back at Einstein’s own work, most physicists would say that many of the most important clues for a unified field theory — involving strong andweak nuclear interactions, the roleof gauge theory and the world of elementary particles — were simplyand Newton’s gravitational constant,one can construct a natural unit of length — the Planck length. First defined by Max Planck a century ago, this length is so fantastically ror reflection— is oneof themostimportant findings ever madeabout elementary particles.In 1984, this flaw was abruptlyovercome when Green and Schwarza reasonable rough draft of particle physicsunified with gravity.And finally, string theory has proved tobe remarkably rich, more so than even theenthusiasts tend to realize. It has led topenetrating insights on topics from quarkconfinement to quantum mechanics ofblack holes, to numerous problemsin pure geometry. All this sug-gests that string theory is on theright track; otherwise, whywould it generate so many unex-pected ideas? And where criticshave had good ideas, they havetended to be absorbed as part ofstring theory, whether it was black-hole entropy, the holographicprinciple of quantum gravity,noncommutative geometry, ortwistor theory.well be the only way to reconcilegravity and quantum mechanics, but whatis the core idea behind it? Einstein under-stood the central concepts of general rela-tivity years before he developed the detailed。
黎曼曲面讲义
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本书主要内容如下：第一章基本上是关于复变函数的简单复习，我们给出了 单值化定理的简单情形，即 Riemann 映照定理的证明。这一章也得到了调和函数 的梯度估计以及 Harnack 原理，这里采用的方法可以推广到一般的黎曼流形上。 第二章引入了抽象黎曼曲面的定义，并给出了单连通黎曼曲面的分类（单值化定 理），其中，黎曼环面作为一类重要的紧致黎曼曲面也加以了分类。证明单值化定 理的方法是通过调和函数（可能带有奇点）来构造特殊的全纯映射。而调和函数 的存在性是通过经典的 Perron 方法获得的。第三章是本书核心内容之一，我们给 出了 Riemann-Roch 公式的证明，并选择了若干有意思的应用加以介绍。我们选 择的 Riemann-Roch 公式的这个证明也是经典的，它也涉及某些给定奇性的亚纯 微分的存在性，这种亚纯微分的存在性是通过 Hodge 定理获得的，为了尽快的介 绍 Riemann-Roch 公式的应用，我们把重要的 Hodge 定理的证明放在本书第二个 附录中了。通过 Riemann-Roch 公式我们知道了紧致黎曼曲面上亚纯函数的丰富 性，我们也证明了亚纯函数域是一个一元代数函数域，并且它惟一地决定了黎曼曲 面本身。作为例子我们简单介绍了黎曼环面上的亚纯函数，它们就是经典的椭圆 函数。通过适当地挑选亚纯函数，我们把黎曼曲面全纯地嵌入到了复投影空间中， 因此可以从代数曲线的角度来研究它们。我们还介绍了计算总分歧数的 RiemannHurwitz 公式，并利用它简单研究了超椭圆型的黎曼曲面。接下来我们介绍了曲面 上的 Weierstrass 点，得到了 Weierstrass 点的个数估计。这些结果又被应用于曲面 的全纯自同构群，特别地，我们证明了亏格大于 1 的紧致黎曼曲面全纯自同构群 的阶的估计。作为第二章的结束，我们还介绍了重要的双线性关系、Jacobi 簇，证
The Standard Model of Particle Physics
a r X i v :h e p -p h /9812285v 1 8 D e c 1998The Standard Model of Particle PhysicsMary K.Gaillard 1,Paul D.Grannis 2,and Frank J.Sciulli 31University of California,Berkeley,2State University of New York,Stony Brook,3Columbia UniversityParticle physics has evolved a coherent model that characterizes forces and particles at the mostelementary level.This Standard Model,built from many theoretical and experimental studies,isin excellent accord with almost all current data.However,there are many hints that it is but anapproximation to a yet more fundamental theory.We trace the development of the Standard Modeland indicate the reasons for believing that it is incomplete.Nov.20,1998(To be published in Reviews of Modern Physics)I.INTRODUCTION:A BIRD’S EYE VIEW OF THE STANDARD MODEL Over the past three decades a compelling case has emerged for the now widely accepted Standard Model of elementary particles and forces.A ‘Standard Model’is a theoretical framework built from observation that predicts and correlates new data.The Mendeleev table of elements was an early example in chemistry;from the periodic table one could predict the properties of many hitherto unstudied elements and compounds.Nonrelativistic quantum theory is another Standard Model that has correlated the results of countless experiments.Like its precursors in other fields,the Standard Model (SM)of particle physics has been enormously successful in predicting a wide range of phenomena.And,just as ordinary quantum mechanics fails in the relativistic limit,we do not expect the SM to be valid at arbitrarily short distances.However its remarkable success strongly suggests that the SM will remain an excellent approximation to nature at distance scales as small as 10−18m.In the early 1960’s particle physicists described nature in terms of four distinct forces,characterized by widely different ranges and strengths as measured at a typical energy scale of 1GeV.The strong nuclear force has a range of about a fermi or 10−15m.The weak force responsible for radioactive decay,with a range of 10−17m,is about 10−5times weaker at low energy.The electromagnetic force that governs much of macroscopic physics has infinite range and strength determined by the finestructure constant,α≈10−2.The fourth force,gravity,also has infinite range and a low energy coupling (about 10−38)too weak to be observable in laboratory experiments.The achievement of the SM was the elaboration of a unified description of the strong,weak and electromagnetic forces in the language of quantum gauge field theories.Moreover,the SM combines the weak and electromagnetic forces in a single electroweak gauge theory,reminiscent of Maxwell’s unification of the seemingly distinct forces of electricity and magnetism.By mid-century,the electromagnetic force was well understood as a renormalizable quantum field theory (QFT)known as quantum electrodynamics or QED,described in the preceeding article.‘Renormalizable’means that once a few parameters are determined by a limited set of measurements,the quantitative features of interactions among charged particles and photons can be calculated to arbitrary accuracy as a perturbative expansion in the fine structure constant.QED has been tested over an energy range from 10−16eV to tens of GeV,i.e.distances ranging from 108km to 10−2fm.In contrast,the nuclear force was characterized by a coupling strength that precluded a perturbativeexpansion.Moreover,couplings involving higher spin states(resonances),that appeared to be onthe same footing as nucleons and pions,could not be described by a renormalizable theory,nor couldthe weak interactions that were attributed to the direct coupling of four fermions to one another.In the ensuing years the search for renormalizable theories of strong and weak interactions,coupledwith experimental discoveries and attempts to interpret available data,led to the formulation ofthe SM,which has been experimentally verified to a high degree of accuracy over a broad range ofenergy and processes.The SM is characterized in part by the spectrum of elementaryfields shown in Table I.The matterfields are fermions and their anti-particles,with half a unit of intrinsic angular momentum,or spin.There are three families of fermionfields that are identical in every attribute except their masses.Thefirst family includes the up(u)and down(d)quarks that are the constituents of nucleons aswell as pions and other mesons responsible for nuclear binding.It also contains the electron and theneutrino emitted with a positron in nuclearβ-decay.The quarks of the other families are constituentsof heavier short-lived particles;they and their companion charged leptons rapidly decay via the weakforce to the quarks and leptons of thefirst family.The spin-1gauge bosons mediate interactions among fermions.In QED,interactions among elec-trically charged particles are due to the exchange of quanta of the electromagneticfield called photons(γ).The fact that theγis massless accounts for the long range of the electromagnetic force.Thestrong force,quantum chromodynamics or QCD,is mediated by the exchange of massless gluons(g)between quarks that carry a quantum number called color.In contrast to the electrically neutralphoton,gluons(the quanta of the‘chromo-magnetic’field)possess color charge and hence couple toone another.As a consequence,the color force between two colored particles increases in strengthwith increasing distance.Thus quarks and gluons cannot appear as free particles,but exist onlyinside composite particles,called hadrons,with no net color charge.Nucleons are composed ofthree quarks of different colors,resulting in‘white’color-neutral states.Mesons contain quark andanti-quark pairs whose color charges cancel.Since a gluon inside a nucleon cannot escape its bound-aries,the nuclear force is mediated by color-neutral bound states,accounting for its short range,characterized by the Compton wavelength of the lightest of these:theπ-meson.The even shorter range of the weak force is associated with the Compton wave-lengths of thecharged W and neutral Z bosons that mediate it.Their couplings to the‘weak charges’of quarksand leptons are comparable in strength to the electromagnetic coupling.When the weak interactionis measured over distances much larger than its range,its effects are averaged over the measurementarea and hence suppressed in amplitude by a factor(E/M W,Z)2≈(E/100GeV)2,where E is the characteristic energy transfer in the measurement.Because the W particles carry electric charge theymust couple to theγ,implying a gauge theory that unites the weak and electromagnetic interactions,similar to QCD in that the gauge particles are self-coupled.In distinction toγ’s and gluons,W’scouple only to left-handed fermions(with spin oriented opposite to the direction of motion).The SM is further characterized by a high degree of symmetry.For example,one cannot performan experiment that would distinguish the color of the quarks involved.If the symmetries of theSM couplings were fully respected in nature,we would not distinguish an electron from a neutrinoor a proton from a neutron;their detectable differences are attributed to‘spontaneous’breakingof the symmetry.Just as the spherical symmetry of the earth is broken to a cylindrical symmetry by the earth’s magneticfield,afield permeating all space,called the Higgsfield,is invoked to explain the observation that the symmetries of the electroweak theory are broken to the residual gauge symmetry of QED.Particles that interact with the Higgsfield cannot propagate at the speed of light,and acquire masses,in analogy to the index of refraction that slows a photon traversing matter.Particles that do not interact with the Higgsfield—the photon,gluons and possibly neutrinos–remain massless.Fermion couplings to the Higgsfield not only determine their masses; they induce a misalignment of quark mass eigenstates with respect to the eigenstates of the weak charges,thereby allowing all fermions of heavy families to decay to lighter ones.These couplings provide the only mechanism within the SM that can account for the observed violation of CP,that is,invariance of the laws of nature under mirror reflection(parity P)and the interchange of particles with their anti-particles(charge conjugation C).The origin of the Higgsfield has not yet been determined.However our very understanding of the SM implies that physics associated with electroweak symmetry breaking(ESB)must become manifest at energies of present colliders or at the LHC under construction.There is strong reason, stemming from the quantum instability of scalar masses,to believe that this physics will point to modifications of the theory.One shortcoming of the SM is its failure to accommodate gravity,for which there is no renormalizable QFT because the quantum of the gravitationalfield has two units of spin.Recent theoretical progress suggests that quantum gravity can be formulated only in terms of extended objects like strings and membranes,with dimensions of order of the Planck length10−35m. Experiments probing higher energies and shorter distances may reveal clues connecting SM physics to gravity,and may shed light on other questions that it leaves unanswered.In the following we trace the steps that led to the formulation of the SM,describe the experiments that have confirmed it,and discuss some outstanding unresolved issues that suggest a more fundamental theory underlies the SM.II.THE PATH TO QCDThe invention of the bubble chamber permitted the observation of a rich spectroscopy of hadron states.Attempts at their classification using group theory,analogous to the introduction of isotopic spin as a classification scheme for nuclear states,culminated in the‘Eightfold Way’based on the group SU(3),in which particles are ordered by their‘flavor’quantum numbers:isotopic spin and strangeness.This scheme was spectacularly confirmed by the discovery at Brookhaven Laboratory (BNL)of theΩ−particle,with three units of strangeness,at the predicted mass.It was subsequently realized that the spectrum of the Eightfold Way could be understood if hadrons were composed of three types of quarks:u,d,and the strange quark s.However the quark model presented a dilemma: each quark was attributed one half unit of spin,but Fermi statistics precluded the existence of a state like theΩ−composed of three strange quarks with total spin3A combination of experimental observations and theoretical analyses in the1960’s led to anotherimportant conclusion:pions behave like the Goldstone bosons of a spontaneously broken symmetry,called chiral symmetry.Massless fermions have a conserved quantum number called chirality,equalto their helicity:+1(−1)for right(left)-handed fermions.The analysis of pion scattering lengths andweak decays into pions strongly suggested that chiral symmetry is explicitly broken only by quarkmasses,which in turn implied that the underlying theory describing strong interactions among quarksmust conserve quark helicity–just as QED conserves electron helicity.This further implied thatinteractions among quarks must be mediated by the exchange of spin-1particles.In the early1970’s,experimenters at the Stanford Linear Accelerator Center(SLAC)analyzed thedistributions in energy and angle of electrons scattered from nuclear targets in inelastic collisionswith momentum transfer Q2≈1GeV/c from the electron to the struck nucleon.The distributions they observed suggested that electrons interact via photon exchange with point-like objects calledpartons–electrically charged particles much smaller than nucleons.If the electrons were scatteredby an extended object,e.g.a strongly interacting nucleon with its electric charge spread out by acloud of pions,the cross section would drop rapidly for values of momentum transfer greater than theinverse radius of the charge distribution.Instead,the data showed a‘scale invariant’distribution:across section equal to the QED cross section up to a dimensionless function of kinematic variables,independent of the energy of the incident electron.Neutrino scattering experiments at CERN andFermilab(FNAL)yielded similar parison of electron and neutrino data allowed adetermination of the average squared electric charge of the partons in the nucleon,and the result wasconsistent with the interpretation that they are fractionally charged quarks.Subsequent experimentsat SLAC showed that,at center-of-mass energies above about two GeV,thefinal states in e+e−annihilation into hadrons have a two-jet configuration.The angular distribution of the jets withrespect to the beam,which depends on the spin of thefinal state particles,is similar to that of themuons in anµ+µ−final state,providing direct evidence for spin-1√where G F is the Fermi coupling constant,γµis a Dirac matrix and12fermions via the exchange of spinless particles.Both the chiral symmetry of thestrong interactions and the V−A nature of the weak interactions suggested that all forces except gravity are mediated by spin-1particles,like the photon.QED is renormalizable because gauge invariance,which gives conservation of electric charge,also ensures the cancellation of quantum corrections that would otherwise result in infinitely large amplitudes.Gauge invariance implies a massless gauge particle and hence a long-range force.Moreover the mediator of weak interactions must carry electric charge and thus couple to the photon,requiring its description within a Yang-Mills theory that is characterized by self-coupled gauge bosons.The important theoretical breakthrough of the early1970’s was the proof that Yang-Mills theories are renormalizable,and that renormalizability remains intact if gauge symmetry is spontaneously broken,that is,if the Lagrangian is gauge invariant,but the vacuum state and spectrum of particles are not.An example is a ferromagnet for which the lowest energy configuration has electron spins aligned;the direction of alignment spontaneously breaks the rotational invariance of the laws ofphysics.In QFT,the simplest way to induce spontaneous symmetry breaking is the Higgs mech-anism.A set of elementary scalarsφis introduced with a potential energy density function V(φ) that is minimized at a value<φ>=0and the vacuum energy is degenerate.For example,the gauge invariant potential for an electrically charged scalarfieldφ=|φ|e iθ,V(|φ|2)=−µ2|φ|2+λ|φ|4,(3)√λ=v,but is independent of the phaseθ.Nature’s choice forθhas its minimum atspontaneously breaks the gauge symmetry.Quantum excitations of|φ|about its vacuum value are massive Higgs scalars:m2H=2µ2=2λv2.Quantum excitations around the vacuum value ofθcost no energy and are massless,spinless particles called Goldstone bosons.They appear in the physical spectrum as the longitudinally polarized spin states of gauge bosons that acquire masses through their couplings to the Higgsfield.A gauge boson mass m is determined by its coupling g to theHiggsfield and the vacuum value v.Since gauge couplings are universal this also determines the√Fermi constant G for this toy model:m=gv/2,G/2|φ|=212F=246GeV,leaving three Goldstone bosons that are eaten by three massive vector bosons:W±and Z=cosθw W0−sinθw B0,while the photonγ=cosθw B0+sinθw W0remains massless.This theory predicted neutrino-induced neutral current(NC)interactions of the typeν+atom→ν+anything,mediated by Z exchange.The weak mixing angleθw governs the dependence of NC couplings on fermion helicity and electric charge, and their interaction rates are determined by the Fermi constant G Z F.The ratioρ=G Z F/G F= m2W/m2Z cos2θw,predicted to be1,is the only measured parameter of the SM that probes thesymmetry breaking mechanism.Once the value ofθw was determined in neutrino experiments,the√W and Z masses could be predicted:m2W=m2Z cos2θw=sin2θwπα/QUARKS:S=1LEPTONS:S=13m3m Q=0m quanta mu1u2u3(2–8)10−3e 5.11×10−4c1c2c3 1.0–1.6µ0.10566t1t2t3173.8±5.0τ 1.77705/3g′,where g1isfixed by requiring the same normalization for all fermion currents.Their measured values at low energy satisfy g3>g2>g1.Like g3,the coupling g2decreases with increasing energy,but more slowly because there are fewer gauge bosons contributing.As in QED,the U(1)coupling increases with energy.Vacuum polarization effects calculated using the particle content of the SM show that the three coupling constants are very nearly equal at an energy scale around1016GeV,providing a tantalizing hint of a more highly symmetric theory,embedding the SM interactions into a single force.Particle masses also depend on energy;the b andτmasses become equal at a similar scale,suggesting a possibility of quark and lepton unification as different charge states of a singlefield.V.BRIEF SUMMARY OF THE STANDARD MODEL ELEMENTSThe SM contains the set of elementary particles shown in Table I.The forces operative in the particle domain are the strong(QCD)interaction responsive to particles carrying color,and the two pieces of the electroweak interaction responsive to particles carrying weak isospin and hypercharge. The quarks come in three experimentally indistinguishable colors and there are eight colored gluons. All quarks and leptons,and theγ,W and Z bosons,carry weak isospin.In the strict view of the SM,there are no right-handed neutrinos or left-handed anti-neutrinos.As a consequence the simple Higgs mechanism described in section IV cannot generate neutrino masses,which are posited to be zero.In addition,the SM provides the quark mixing matrix which gives the transformation from the basis of the strong interaction charge−1Finding the constituents of the SM spanned thefirst century of the APS,starting with the discovery by Thomson of the electron in1897.Pauli in1930postulated the existence of the neutrino as the agent of missing energy and angular momentum inβ-decay;only in1953was the neutrino found in experiments at reactors.The muon was unexpectedly added from cosmic ray searches for the Yukawa particle in1936;in1962its companion neutrino was found in the decays of the pion.The Eightfold Way classification of the hadrons in1961suggested the possible existence of the three lightest quarks(u,d and s),though their physical reality was then regarded as doubtful.The observation of substructure of the proton,and the1974observation of the J/ψmeson interpreted as a cp collider in1983was a dramatic confirmation of this theory.The gluon which mediates the color force QCD wasfirst demonstrated in the e+e−collider at DESY in Hamburg.The minimal version of the SM,with no right-handed neutrinos and the simplest possible ESB mechanism,has19arbitrary parameters:9fermion masses;3angles and one phase that specify the quark mixing matrix;3gauge coupling constants;2parameters to specify the Higgs potential; and an additional phaseθthat characterizes the QCD vacuum state.The number of parameters is larger if the ESB mechanism is more complicated or if there are right-handed neutrinos.Aside from constraints imposed by renormalizability,the spectrum of elementary particles is also arbitrary.As discussed in Section VII,this high degree of arbitrariness suggests that a more fundamental theory underlies the SM.VI.EXPERIMENTAL ESTABLISHMENT OF THE STANDARD MODELThe current picture of particles and interactions has been shaped and tested by three decades of experimental studies at laboratories around the world.We briefly summarize here some typical and landmark results.FIG.1.The proton structure function(F2)versus Q2atfixed x,measured with incident electrons or muons,showing scale invariance at larger x and substantial dependence on Q2as x becomes small.The data are taken from the HERA ep collider experiments H1and ZEUS,as well as the muon scattering experiments BCDMS and NMC at CERN and E665at FNAL.A.Establishing QCD1.Deep inelastic scatteringPioneering experiments at SLAC in the late1960’s directed high energy electrons on proton and nuclear targets.The deep inelastic scattering(DIS)process results in a deflected electron and a hadronic recoil system from the initial baryon.The scattering occurs through the exchange of a photon coupled to the electric charges of the participants.DIS experiments were the spiritual descendents of Rutherford’s scattering ofαparticles by gold atoms and,as with the earlier experi-ment,showed the existence of the target’s substructure.Lorentz and gauge invariance restrict the matrix element representing the hadronic part of the interaction to two terms,each multiplied by phenomenological form factors or structure functions.These in principle depend on the two inde-pendent kinematic variables;the momentum transfer carried by the photon(Q2)and energy loss by the electron(ν).The experiments showed that the structure functions were,to good approximation, independent of Q2forfixed values of x=Q2/2Mν.This‘scaling’result was interpreted as evi-dence that the proton contains sub-elements,originally called partons.The DIS scattering occurs as the elastic scatter of the beam electron with one of the partons.The original and subsequent experiments established that the struck partons carry the fractional electric charges and half-integer spins dictated by the quark model.Furthermore,the experiments demonstrated that three such partons(valence quarks)provide the nucleon with its quantum numbers.The variable x represents the fraction of the target nucleon’s momentum carried by the struck parton,viewed in a Lorentz frame where the proton is relativistic.The DIS experiments further showed that the charged partons (quarks)carry only about half of the proton momentum,giving indirect evidence for an electrically neutral partonic gluon.1011010101010FIG.2.The quark and gluon momentum densities in the proton versus x for Q 2=20GeV 2.The integrated values of each component density gives the fraction of the proton momentum carried by that component.The valence u and d quarks carry the quantum numbers of the proton.The large number of quarks at small x arise from a ‘sea’of quark-antiquark pairs.The quark densities are from a phenomenological fit (the CTEQ collaboration)to data from many sources;the gluon density bands are the one standard deviation bounds to QCD fits to ZEUS data (low x )and muon scattering data (higher x ).Further DIS investigations using electrons,muons,and neutrinos and a variety of targets refined this picture and demonstrated small but systematic nonscaling behavior.The structure functions were shown to vary more rapidly with Q 2as x decreases,in accord with the nascent QCD prediction that the fundamental strong coupling constant αS varies with Q 2,and that at short distance scales (high Q 2)the number of observable partons increases due to increasingly resolved quantum fluc-tuations.Figure 1shows sample modern results for the Q 2dependence of the dominant structure function,in excellent accord with QCD predictions.The structure function values at all x depend on the quark content;the increases at larger Q 2depend on both quark and gluon content.The data permit the mapping of the proton’s quark and gluon content exemplified in Fig.2.2.Quark and gluon jetsThe gluon was firmly predicted as the carrier of the color force.Though its presence had been inferred because only about half the proton momentum was found in charged constituents,direct observation of the gluon was essential.This came from experiments at the DESY e +e −collider (PETRA)in 1979.The collision forms an intermediate virtual photon state,which may subsequently decay into a pair of leptons or pair of quarks.The colored quarks cannot emerge intact from the collision region;instead they create many quark-antiquark pairs from the vacuum that arrange themselves into a set of colorless hadrons moving approximately in the directions of the original quarks.These sprays of roughly collinear particles,called jets,reflect the directions of the progenitor quarks.However,the quarks may radiate quanta of QCD (a gluon)prior to formation of the jets,just as electrons radiate photons.If at sufficiently large angle to be distinguished,the gluon radiation evolves into a separate jet.Evidence was found in the event energy-flow patterns for the ‘three-pronged’jet topologies expected for events containing a gluon.Experiments at higher energy e +e −colliders illustrate this gluon radiation even better,as shown in Fig.3.Studies in e +e −and hadron collisions have verified the expected QCD structure of the quark-gluon couplings,and their interference patterns.FIG.3.A three jet event from the OPAL experiment at LEP.The curving tracks from the three jets may be associated with the energy deposits in the surrounding calorimeter,shown here as histograms on the middle two circles,whose bin heights are proportional to energy.Jets1and2contain muons as indicated,suggesting that these are both quark jets(likely from b quarks).The lowest energy jet3is attributed to a radiated gluon.3.Strong coupling constantThe fundamental characteristic of QCD is asymptotic freedom,dictating that the coupling constant for color interactions decreases logarithmically as Q2increases.The couplingαS can be measured in a variety of strong interaction reactions at different Q2scales.At low Q2,processes like DIS,tau decays to hadrons,and the annihilation rate for e+e−into multi-hadronfinal states give accurate determinations ofαS.The decays of theΥinto three jets primarily involve gluons,and the rate for this decay givesαS(M2Υ).At higher Q2,studies of the W and Z bosons(for example,the decay width of the Z,or the fraction of W bosons associated with jets)measureαS at the100GeV scale. These and many other determinations have now solidified the experimental evidence thatαS does indeed‘run’with Q2as expected in QCD.Predictions forαS(Q2),relative to its value at some reference scale,can be made within perturbative QCD.The current information from many sources are compared with calculated values in Fig.4.4.Strong interaction scattering of partonsAt sufficiently large Q2whereαS is small,the QCD perturbation series converges sufficiently rapidly to permit accurate predictions.An important process probing the highest accessible Q2 scales is the scattering of two constituent partons(quarks or gluons)within colliding protons and antiprotons.Figure5shows the impressive data for the inclusive production of jets due to scattered partons in pp collisions reveals the structure of the scattering matrix element.These amplitudes are dominated by the exchange of the spin1gluon.If this scattering were identical to Rutherford scattering,the angular variable0.10.20.30.40.511010FIG.4.The dependence of the strong coupling constant,αS ,versus Q using data from DIS structure functions from e ,µ,and νbeam experiments as well as ep collider experiments,production rates of jets,heavy quark flavors,photons,and weak vector bosons in ep ,e +e −,and pt ,is sensitive not only to to perturbative processes,but reflectsadditional effects due to multiple gluon radiation from the scattering quarks.Within the limited statistics of current data samples,the top quark production cross section is also in good agreement with QCD.FIG.6.The dijet angular distribution from the DØexperiment plotted as a function ofχ(see text)for which Rutherford scattering would give dσ/dχ=constant.The predictions of NLO QCD(at scaleµ=E T/2)are shown by the curves.Λis the compositeness scale for quark/gluon substructure,withΛ=∞for no compositness(solid curve);the data rule out values of Λ<2TeV.5.Nonperturbative QCDMany physicists believe that QCD is a theory‘solved in principle’.The basic validity of QCD at large Q2where the coupling is small has been verified in many experimental studies,but the large coupling at low Q2makes calculation exceedingly difficult.This low Q2region of QCD is relevant to the wealth of experimental data on the static properties of nucleons,most hadronic interactions, hadronic weak decays,nucleon and nucleus structure,proton and neutron spin structure,and systems of hadronic matter with very high temperature and energy densities.The ability of theory to predict such phenomena has yet to match the experimental progress.Several techniques for dealing with nonperturbative QCD have been developed.The most suc-cessful address processes in which some energy or mass in the problem is large.An example is the confrontation of data on the rates of mesons containing heavy quarks(c or b)decaying into lighter hadrons,where the heavy quark can be treated nonrelativistically and its contribution to the matrix element is taken from experiment.With this phenomenological input,the ratios of calculated par-tial decay rates agree well with experiment.Calculations based on evaluation at discrete space-time points on a lattice and extrapolated to zero spacing have also had some success.With computing advances and new calculational algorithms,the lattice calculations are now advanced to the stage of calculating hadronic masses,the strong coupling constant,and decay widths to within roughly10–20%of the experimental values.The quark and gluon content of protons are consequences of QCD,much as the wave functions of electrons in atoms are consequences of electromagnetism.Such calculations require nonperturbative techniques.Measurements of the small-x proton structure functions at the HERA ep collider show a much larger increase of parton density with decreasing x than were extrapolated from larger x measurements.It was also found that a large fraction(∼10%)of such events contained afinal。
Chaos in Dicke model(Dicke模型中的混沌)
Dicke 模型中的混沌摘 要 Dicke 哈密顿函数是一种量子光学模型。
描述了N 个二能级原子与一个单模玻色子场的相互作用。
本文从Dicke 哈密顿函数的量子表达式出发，将其回推到经典表达式。
通过改变耦合参量λ的数值，绘制Poincaré截面。
结果表明，当λ值小于临界值时，Poincaré截面保持规律的周期性的轨迹。
当趋于临界值时，伴有混乱轨迹的出现。
继续增加λ的值，将破坏周期性轨迹，使得整个相空间因为λ值比临界值稍大而变得混乱无序。
同时，本文简单介绍了系统的量子混沌。
关键词 混沌 Dicke 哈密顿 Dicke 模型 经典混沌 量子混沌0 引言第一个发现混沌的是法国数学家、物理学家H.Poincaré (1854-1912)。
1903年提出庞加莱猜想，指出在三体问题中，在一定范围内，其解是随机的[1]。
三体引力相互作用有着惊人的复杂行为，确定性动力学方程有许多解有很强的不可预计性。
到了上个世纪70年代，混沌学的研究在数学、物理、生物、气象、医学等多个学科领域同时展开，形成了世界性的研究热潮。
在以后的几十年中，人们研究了大量量子系统的混沌现象，并给出了量子混沌的特征描述。
进入20世纪90年代，对混沌的研究不仅推动了其他学科的发展，而且其他学科的发展又促进了对混沌的深入研究。
进入21世纪，混沌与其他学科的相互交错、渗透、促进，使得混沌在生物学、数学、物理学、化学、电子学、信息科学、气象学等多个领域中得到了广泛的应用[2]。
近十几年来人们发现很多量子系统中存在量子混沌现象，并且研究了量子混沌与相变、纠缠、隧穿等物理现象之间的关系，得到很多有意义的结果。
本文着手于Dicke 模型，通过数值计算的方法来绘制Poincaré截面，研究系统随参量λ的变化趋势。
1 混沌介绍1.1 混沌研究简史现代科学意义上的混沌的发现，可以追溯到19世纪末20世纪初[1]。
混沌研究的第一个重大突破就是KAM 定理，KAM 定理给出了太阳系稳定性的合理解释，并使人们重新看待统计力学中一系列基本假设和观点。
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arXiv:gr-qc/0601105v1 25 Jan 2006ModelforaUniversedescribedbyanon-minimallycoupledscalarfieldandinteractingdarkmatter
J.B.Binder∗andG.M.Kremer†DepartamentodeF´ısica,UniversidadeFederaldoParan´aCaixaPostal19044,81531-990Curitiba,Brazil
AbstractInthisworkitisinvestigatedtheevolutionofaUniversewhereascalarfield,non-minimallycoupledtospace-timecurvature,playstheroleofquintessenceanddrivestheUniversetoapresentacceleratedexpansion.Anon-relativisticdarkmatterconstituentthatinteractsdirectlywithdarkenergyisalsoconsidered,wherethedarkmatterparticlemassisassumedtobeproportionaltothevalueofthescalarfield.Twomodelsfordarkmatterpressureareconsidered:theusualone,pressureless,andanotherthatcomesfromathermodynamictheoryandrelatesthepressurewiththecouplingbetweenthescalarfieldandthecurvaturescalar.Althoughthemodelhasastrongdependenceontheinitialconditions,itisshownthatthemixtureconsistedofdarkcomponentsplusbaryonicmatterandradiationcanreproducetheexpectedred-shiftbehaviorofthedecelerationparameter,densityparametersandluminositydistance.
1IntroductionThemeasurementsoftherotationcurvesofspiralgalaxies[1]aswellasotherastronomicalexperi-mentssuggestthattheluminousmatterrepresentsonlyasmallamountofthemassiveparticlesoftheUniverse,andthatthemoresignificantamountisrelatedtodarkmatter.Recentlytheastronomicalobservationswithsuper-novaeoftypeIasuggestedthatourUniverseispresentlysubmittedtoanacceleratedexpansion[2,3];thenatureoftheresponsibleentity,calleddarkenergy,stillremainsunknown.Thesimplestexplanationfortheaccelerationisacosmologicalconstant(see[4]),whichfitsthepresentdataverywellbuthassomeimportantunsolvedproblems.Anotherpossibilityistointroduceascalarfieldφ(t),whichhasbeenextensivelystudiedbythescientificcommunity.Byconsideringabarotropicequationofstateforthescalarfieldpφ=ωρφ,wehaveaconstantvalueofω=−1forthecosmologicalconstantmodelandavariableω≥−1foraminimallycoupledscalarfieldmodel.However,themeasureddatafromHubbleSpaceTelescope[5]statestherestrictionforω=−1.02+0.13−0.19,whereassomerecentobservations[6]hasdrawntheattentiontotheviabilityofmodelswhereω<−1,whichwouldinvalidatebothcosmologicalconstantandminimallycoupledscalarfielddescriptions.Somephantomfieldmodelswereproposedinordertocontemplatethishypothesis[7],buttheyfacealsosomestrongdifficulties[8].Inthisworkweconsiderascalarfieldnon-minimallycoupledtospace-timecurvature,whichwasinvestigatedin[9]andwidelystudiedrecently,amongothers,intheworks[10,11,12,13,14].Inamorerecentpaper[15]thesuitabilityofsuchmodelswasstudiedanditwasshownthattheEinsteinfieldequationsarestableundersomeconditionsfortheconstantthatcouplesthescalarfieldwiththecurvaturescalar.Asideseveralmodelsforthedarksectorinteraction[16,17,18,19]–thatcouldreplacethecosmologicalconstantmodelinthecaseoffutureexperimentaltests–weconsiderhereadirectcouplingbetweendarkmatteranddarkenergy.Inthismodel,studiedin[20]andmorerecentlyin[21],thedarkmatterparticlemassisproportionaltothevalueofthescalarfieldthatrepresentsthedarkenergy.However,inthismodelitisnecessarytohavesomeprecautions,sincebyassumingthedarkmatterparticlemasstochangewithtime,wecouldmakethedarkmatterenergydensitytobecomephysicallyinconsistentatearlystages.Weemphasizethatinthisworkthefullsetofdifferentialequationsaresolvednumerically,unliketheasymptoticapproximationofthework[21].Thebaryonicmatterandradiationaredescribedhereasusualnon-interactingcomponents,beingthefirstnon-relativisticandpressureless.However,forthedarkmattertwomodelsareconsidered:thefirstone,byassumingthedarkmatteraspressureless,andthesecondone,byusingathermodynamictheoryinordertorelatetheeffectsofthenon-minimallycouplingtothedarkmatterpressure.Allcomponentswillbedescribedbyasetoffieldequations,andtheresultingobservables,–i.e.,thedensityparameters,thedeceleratingparameterandtheluminositydistance–whichareobtainedassolutionsofthefieldequationswillbecomparedtotheavailabledatasetinordertodrawntheconclusionsabouttheviabilityofthismodel.Weshowthatphysicallyacceptablesolutionsareobtained,ifwechoseinitialconditionsrestrictingtherollingnatureofthescalarfieldatearlystages.Furthermore,thereexistsomefreedomparametersthatwillbeimportanttofitthedatafromincomingexperiments.Theworkisorganizedasfollows:insection2thebasicfieldequationsforeachcomponent,i.e.darkenergy,darkmatter,baryonsandradiation,arederived.Thesolutionsforthepressurelessdarkmatterarefoundinsection3forgiveninitialconditions,couplingandpotentialconstants,whereasinsection4wegivethecorrespondingsolutionsforthedarkmatterwithnon-vanishingpressure.Weclosetheworkwithsection5whereweaddresstosomeremarksandsummarizetheresultsofprevioussections.Unitshavebeenchosensothat8πG=c==k=1,whereasthemetrictensorhassignature(+,−,−,−).