Deformation Quantization of Certain Non-linear Poisson Structures

物理专业英语词汇 Q 物理专业英语词汇q物理专业英语词汇(q)qbranch q分支qmeter q值计qnumber q数qswitch q开关qswitchedlaser q控制器激光器qvalue q值quadrantelectrometer 象限静电计quadraticform 二次形式quadrupole 四极quadrupolecoupling 四极耦合quadrupoledeformation 四极畸变quadrupoleinteraction 四极相互酌quadrupolemagnet 四极磁铁quadrupolemoment 四极矩quadrupolemomentum 四极矩quadrupolequadrupoleforce 四极四极力quadrupoleradiation 四极电磁辐射quadrupoleresonance 四极共振quadrupoletransition 四极光子quadrupolevibration 四极振荡qualitativeanalysis 定性分析quantacon 光电倍增管quantitativeanalysis 定量分析quantitativespectralanalysis 定量光谱分析 quantity 量quantityofelectricity 电量quantityofflow 量thermocouplepyrometer热电偶高温计quantityofheat 热量quantityoflight 光量quantityofmotion 动量quantityofstate 状态量quantization 量子化quantizedfield 量子场quantizedsystem 量子化系统quantizedvortex 量子旋涡quantumanomaly 量子反常quantumbiology 量子生物学quantumchemistry 量子化学quantumchromodynamics 量子色动力学quantumcondition 量子条件quantumcosmology 量子宇宙学quantumcreationoftheuniverse 宇宙的量子产生 quantumdefect 量子筐quantumdefecttheory 量子筐理论quantumdisorderedsystem 量子无序系quantumefficiency 量子产额quantumelectrodynamics 量子电动力学quantumelectronics 量子电子学quantumenergy 量子能量quantumfield 量子场quantumfieldtheory 量子场论quantumfluctuation 量子涨落quantumfluid 量子铃quantumgravitation 量子引力quantumgravitationalfluctuation 量子引力差值 quantumgravity 量子引力quantumgroup 量子群quantumhalleffect 量子霍尔效应quantumhilbertspace 量子希耳伯空间quantumhypothesis 量子假说quantuminversescatteringmethod 量子逆反射法 quantumjump 量子跃迁quantumlatticemodel 量子图形模型quantumliquid 量子液体quantumlogic 量子逻辑quantummechanics 量子力学quantummontecarlomethod 量子蒙特卡罗法quantumnoise 量子噪声quantumnumber 量子数quantumoptics 量子光学quantumorbit 量子轨道quantumphysics 量子物理学quantumsizeeffect 量子尺寸效应quantumsolid 量子液态quantumsoliton 量子孤立子quantumstate 量子状态quantumstatisticalmechanics 量子统计力学 quantumstatistics 量子统计数据quantumtheory 量子论quantumtheoryoffield 量子场论quantumtheoryofgravity 引力场的量子论quantumtransition 量子光子quantumwell 量子势阱quantumwellopticalwaveguide 量子势阱光波导 quantumyield 量子产额quantumbrowwidth18fieldeng 量子quark 夸克quarkatom 夸克偶素quarkcondensate 夸克凝聚quarkconfinement 夸克禁锢quarkflavor 夸克味quarkgluonplasma 夸克胶子等离子体quarkleptonsymmetry 夸克轻子对称quarkmodel 夸克模型quarkstar 夸克星quarkonium 夸克偶素quarterwaveplate 四分之一波片quartet 四重态quartz 水晶quartzclock 石英钟quartzfibreelectroscope 石英丝验电器quartzglass 石英玻璃quartzmonochromator 水晶单色仪quartzoscillator 石英振荡器quartzplate 水晶片quasar 类星体thermocouplepyrometer热电偶高温计quasicrystal 科东俄晶体quasielasticforce 准弹性力quasielasticscattering 科东俄弹性散射quasiergodichypothesis 准脯历经假说quasifouriertransformhologram 科东俄傅里叶转换全息图 quasimolecularresonance 准分子共振quasimonochromaticlight 科东俄单色光quasistaticporcess 准静态过程quasistationaryelectriccurrent 准稳电流quasistationarystate 准稳态quasistellarobject 类星体quasistellarsource 类星射电源quasiviscouseffect 科东俄表面张力效应 quasianisotropy 类蛤异性quasimode 准模quasimolecule 准分子quasiparticle 科东俄粒子quasiparticletunnelling 准粒子隧道效应 quenching 点燃quenchingcircuit 熄灭电路quiescentprominence 宁静日珥quietsun 宁静太阳quintet 五重态。

quadrupole probe,四极探头quadrupole residual gas analyzer,四极残余气体分析器quadrupole rods,四极杆qualification,鉴定qualification standard,鉴定标准qualification test,鉴定试验qualitative analysis,定性分析quantitative differential thermal analysis (QDTA),定量差示热分析qualitative physical model,定性物理模型quality,质量quality assurance,质量保证quality control (QC),质量控制quality inspection,质量检验quality management,质量管理quality supervision,质量监督quantitative analysis,定量分析quantitative differential thermal analysis,定量差热分析quantitative differential thermal analyzer,定量差热（分析）仪quantity of heat,热量quantity of illumination,光照量quantity of radiant energy,辐射能量quantity of radiation,幅射能量quantity to be measured,被测量quantization,量化quantization error,量化误差quantized error,量化误差quantized noise,量化噪声quantized signal,量化信号quartet,四重布置；四重检测quarter circle orifice plate,1/4圆孔板quartz-Bourdon tube (pressure)gauge,石英弹簧管压力计quartz crystal unit humidity transducer[sensor]晶体振子式湿度传感器quartz module,石英组件quartz spring gravimeter,石英弹簧重力仪quartz thermometer,石英温度计；石英温度表quasi-equilibrium theory,准平衡理论quasi-molecular ion,准分子离子quasi-periodic vibration,准周期振动quasi-rigid rotor,准刚性转子quasi-sinusoidal quantity,准正弦量quasi-static unbalance,准静不平衡quasilinear chareacteristics,准线性特性quenching,阻塞question and answer mode,问答式queuing theory,排队论quick thermophysical property measuring apparatus,热物性快速测定仪quintet,五重布置；五重检测quotient-meter,商值表Rrack,框架radar altimeter,雷达高度计radial clearance,径向间隙radial distortion,径向畸变radial electrostatic field analyzer,径向静电场分析器radial heat shield,隔热屏radiance,幅射亮度；辐射率radiance temperature,亮度温度radiance thermometry,亮度测温法radiant element,辐射元件radiant energy,辐射能（量）radiant flux,辐射通量radiant intensity,辐射强度radiant power,辐射功率radiation,辐射radiation balance meter,辐射平衡表radiation detecting device,辐射检测器radiation dose transducer [sensor],射线齐量传感器radiation energy,辐射能量radiation exitance,辐（射）出（射）度radiation fin bonnet,散热片型上阀盖radiation flux,辐射通量radiation heatflowmeter,辐射热流计radiation intensity,辐射强度radiation monitor,辐射监测仪radiation pyrometer,辐射高温计radiation sensor,射线敏感元件radiation temperature,辐射温度radiation temperature transducer [sensor],辐射温度传感器radiation test,辐射试验radiation thermometer,辐射温度计radiation thermometry,辐射测温法radiation transducer [sensor],射线传感器radio chromatography,放射色谱法radio frequency sensor,射频敏感器radio wave borehole penetration system,井中无线电波仪radio wave penetration system,无线电波透视仪radioactive materials packaging,放射性物质包装radioactive survey,放射性测量radiodirection finder,无线电方位测定器radiometer,辐射仪；辐射表radiosonde,无线电探空仪radiotheodolite,无线电经纬仪radiowave penetration instrument,无线电波透视法仪器rain proof instrument,防溅式仪器仪表rainfall amount,雨量rainfall intensity meter,雨量强度计rainfallmeter,雨量强度计raingauge,雨量器raingauge receiver,承水器raingauge shield,雨量器防风罩Raman line,拉曼谱线Raman shift,拉曼位移Raman spectrum,拉曼光谱ramp function,斜坡函数ramp response,斜坡响应ramp response time,斜坡响应时间random access,随机存取random disturbance,随机扰动random error,随机误差random event,随机事件random failure,偶然失效random noise,随机噪声；无规噪声random process,随机过程random signal,随机信号random uncertainty,随机不确定度random vibration,随机振动range,范围range-change device,范围转换器；量限转换器range of load,负荷范围range of strain,应变范围range of stress,应力范围range response time,范围响应时间rangefinder,测距仪ranging distance,测程Rankine temperature scale,兰金温标rangeability,范围度rapid Fourier transform NMR correlation spectroscopy,快速傅立叶变换核磁共振相关波谱法rapid TG-DTA unit,快速加热差示热分析仪/热天平rate of change limiting control,变化率极限控制rate of deformation,变形速率rate of loading,加荷速率rate of salt spray precipitation,盐雾沉降率rate time,预调时间rated accuracy limit primary current (of a protective current transformer),（保护用电流互感器的）准确度限定的额定一次电流。

the intensional qualification ofquantificationIntensional qualification of quantification refers to the idea that the meaning of a quantified statement depends not only on the objects being quantified but also on the properties or qualities associated with those objects. In other words, the meaning of a statement like "all dogs have fur" is not just a matter of counting all the dogs and observing their fur, but also involves an understanding of what it means to be a dog and what it means to have fur.One example of intensional qualification in action can be seen in the statement "all bachelors are unmarried." The meaning of this statement depends not just on the fact that there are a certain number of unmarried men in the world, but also on our understanding of what it means to be a bachelor. If we were to define bachelor as "a man who has never been married and has no children," then the statement "all bachelors are unmarried" would be tautological and unsurprising. But if we were to define bachelor in adifferent way, such as "a man who is over 30 and has never been married," then the statement would be false, since there may well be unmarried men over 30 who are not considered bachelors.Intensional qualification can also come into play in more complex statements. For example, consider the statement "some cats are not black." The truth of this statement depends not just on the existence of non-black cats, but alsoon our understanding of what it means to be a cat and what it means to be black. If we were to define cat as "a small,furry animal that meows," and black as "a color associated with darkness," then the statement would be true, since there are certainly cats that are not black. But if we define cat more narrowly as "a member of the Felidae family," and black more broadly as "any color that is not white," then the statement might be false, as all members of the Felidae family are typically black, brown, or orange.Overall, intensional qualification of quantification highlights the importance of context and interpretation in understanding the true meaning of statements involving quantifiers. By considering not just the objects being quantified, but also the qualities or properties associated with those objects, we can arrive at a more nuanced and accurate understanding of the world around us.。

反量化英语Quantification has become an increasingly prevalent aspect of modern life, with data and metrics being used to measure and evaluate various facets of our existence. While this trend has brought about numerous benefits, it has also given rise to a growing concern regarding the potential drawbacks of an over-reliance on quantification. In this essay, we will explore the concept of "anti-quantification" and examine the arguments for a more balanced approach to the role of data and metrics in our lives.One of the primary arguments against the excessive use of quantification is the inherent reductionism inherent in the process. By reducing complex phenomena to numerical values, we risk oversimplifying the nuances and contextual factors that contribute to the richness and depth of human experience. This can lead to a distorted understanding of reality, where the quantifiable aspects are prioritized at the expense of the qualitative and intangible elements that are equally, if not more, important.Moreover, the reliance on quantification can foster a culture of obsession with metrics and a fixation on numerical targets, often at the expense of deeper, more meaningful goals. This can manifest invarious domains, from education, where test scores become the primary measure of success, to healthcare, where patient outcomes are reduced to a series of statistics. In such scenarios, the true purpose of these institutions – to nurture well-rounded individuals and promote holistic well-being – can become obscured.Another concern with the overuse of quantification is the potential for unintended consequences and the distortion of behavior. When individuals and organizations are evaluated primarily based on numerical targets, they may be tempted to manipulate or game the system in order to achieve those targets, even if it means sacrificing integrity or ethical considerations. This can lead to a culture of mistrust, where the reliability and validity of the data become increasingly questionable.Furthermore, the reliance on quantification can contribute to a sense of dehumanization, where individuals are reduced to mere data points, stripped of their unique experiences, emotions, and personal narratives. This can have profound implications for our social interactions, decision-making processes, and the way we perceive and value one another.In response to these concerns, the concept of "anti-quantification" advocates for a more balanced and nuanced approach to the use of data and metrics. This perspective recognizes the value ofquantification in certain contexts, such as scientific research, policy-making, and decision-support systems, but also acknowledges the need to temper its application with a deeper understanding of the qualitative and contextual factors that shape human experiences and societal dynamics.At the heart of the anti-quantification movement is a call for a greater emphasis on the subjective, the experiential, and the intangible aspects of life. This includes a focus on narrative, storytelling, and the exploration of the human condition through the arts, humanities, and social sciences. By embracing these alternative modes of understanding, we can cultivate a richer, more nuanced perspective on the world around us, one that acknowledges the inherent complexity and diversity of human experiences.Moreover, the anti-quantification approach encourages a more critical and reflective stance towards the use of data and metrics. This involves questioning the underlying assumptions, methodologies, and potential biases that shape the collection and interpretation of data, as well as a willingness to challenge the dominant narratives and preconceptions that often drive the quantification agenda.In practical terms, the implementation of anti-quantification principles could involve a range of strategies, such as the incorporation of qualitative assessments alongside quantitativemeasures, the emphasis on contextual factors in decision-making processes, and the fostering of interdisciplinary collaborations that bridge the divide between the quantitative and the qualitative.Ultimately, the call for anti-quantification is not a rejection of the value of data and metrics, but rather a recognition of the need to strike a balance between the quantifiable and the intangible, the objective and the subjective, in order to create a more holistic and meaningful understanding of the human experience. By embracing this approach, we can work towards a future where the richness and complexity of our lives are not reduced to mere numbers, but instead celebrated and understood in all their nuanced glory.。

a rX iv:mat h /33183v2[mat h.QA ]26Ma y24FLABBY STRICT DEFORMATION QUANTIZATIONS AND K -GROUPS HANFENG LI Abstract.We construct examples of flabby strict deformation quantizations not preserving K -groups.This answers a question of Rieffel negatively.1.Introduction In the passage from classical mechanics to quantum mechanics,one replaces smooth functions on symplectic manifolds (more generally,Poisson manifolds)by operators on Hilbert spaces,and replaces the Poisson bracket of smooth functions by commutators of operators.Thinking of classical mechanics as limits of quantum mechanics,one requires that the Poisson brackets becomes limits of commutators.There is an algebraic way of studying such process using formal power series,called deformation quantization [1,13].In order to study it in a stricter way,Rieffel introduced [6]strict deformation quantization of Poisson manifolds,within the framework of C ∗-algebras.He showed that noncommutative tori arise naturally as strict deformation quantizations of the ordinary torus in the direction of certain Poisson bracket.After that,a lot of interesting examples of strict deformation quantizations have been constructed.See [8,9]and the references therein.We refer the reader to [2,Sections 10.1–10.3]for the basic information about continuous fields of C ∗-algebras.Recall the definition of strict deformation quanti-zation [6,9]:Definition 1.1.[9,Definition 1]Let M be a Poisson manifold,and let C ∞(M )be the algebra of C -valued continuous functions on M vanishing at ∞.By a strict deformation quantization of M we mean a dense ∗-subalgebra A of C ∞(M )closed under the Poisson bracket,together with a continuous field of C ∗-algebras A over a closed subset I of the real line containing 0as a non-isolated point,and linear maps π :A →A for each ∈I ,such that(1)A 0=C ∞(M )and π0is the canonical inclusion of A into C ∞(M ),(2)the section (π (f ))is continuous for every f ∈A ,(3)for all f,g ∈A we havelim →0 [π (f ),π (g )]/(i )−π ({f,g }) =0,(4)π is injective and π (A )is a dense ∗-subalgebra of A for every ∈I .If A ⊇C ∞c (M ),the space of compactly supported C -valued smooth functions on M ,we say that the strict deformation quantization is flabby .2HANFENG LICondition(4)above enables us to define a new∗-algebra structure and a new C∗-norm on A at each by pulling back the∗-algebra structure and norm of π (A)⊆A to A viaπ .Condition(2)means that this deformation of the∗-algebra structure and norm on A is continuous.Given a strict deformation quantization,a natural question is whether the de-formed C∗-algebras A have the same”algebraic topology”,in particular,whether they have isomorphic K-groups.Rieffel’s quantization of Poisson manifolds in-duced from actions of R d[7]and many other examples[5]are known to preserve K-groups.Rieffel showed examples of non-flabby strict deformation quantizations not preserving K-groups,and asked[9,Question18]:Are the K-groups of the de-formed C∗-algebras of anyflabby strict deformation quantization all isomorphic?A nice survey of various positive results on related problems may be found in[10].Shim[11]showed that above question has a negative answer if one allows orb-ifolds.But it is not clear whether one can adapt the method there to get smooth examples.Rieffel also pointed out[9,page321]that in any strict deformation quantization of a non-zero Poisson bracket if one reparametrizes by replacing by 2one obtains a strict deformation quantization of the0Poisson bracket.Thus to answer Rieffel’s question it suffices to consider strict deformation quantizations of the0Poisson bracket.The main purpose of this paper is to answer above question.In Section2we give a general method of constructingflabby strict deformation quantization for the0 Poisson bracket.In particular,we proveTheorem1.2.Let M be a smooth manifold with dim M≥2,equipped with the0 Poisson bracket.If dim M is even(odd,resp.),then for any integers n0≥n1≥0 (n1≥n0≥0resp.)there is aflabby strict deformation quantization{A ,π } ∈I of M over I=[0,1]with A=C∞c(M)such that K i(A )∼=K i(C∞(M))⊕Z n i for all0< ≤1and i=0,1.Theorem1.2is far from being the most general result one can obtain using our construction in Section2.However,it illustrates clearly that a lot of manifolds equipped with the0Poisson bracket haveflabby strict deformation quantizations not preserving K-groups.In order to accommodate some other interesting examples such as Berezin-Toeplitz quantization of K¨a hler manifolds,Landsman introduced a weaker notion strict quantization[3,Definition II.1.1.1][9,Definition23].This is defined in a way similar to a strict deformation quantization,but without requiring the condition(4) in Definition1.1.Ifπ is injective for each ∈I we say that the strict quantization is faithful.It is natural to ask for the precise relation between strict quantizations and strict deformation quantizations.Rieffel also raised the question[9,Question 25]:Is there an example of a faithful strict quantization such that it is impossible to restrictπ to a dense∗-subalgebra B⊆A to get a strict deformation quantization of M?Adapting our method in Section2we also give such an example for every manifold M equipped with the0Poisson bracket.In[4]strict quantizations are constructed for every Poisson manifold,and it is impossible to restrict the strict quantizations constructed there to dense∗-subalgebras to get strict deformation quantizations unless the Poisson bracket is0[4,Corollary5.6].Thus we get a complete answer to Rieffel’s question.FLABBY STRICT DEFORMATION QUANTIZATIONS AND K-GROUPS3 Acknowledgments.I am grateful to Marc Rieffel for many helpful discussions and suggestions,and I also thank the referee for pointing out the reference[10].2.Strict deformation quantizations for the0Poisson bracketWe start with a general method of deforming a C∗-algebra.Let A be a C∗-algebra and A⊆A a dense∗-subalgebra.Let I(A)={b∈M(A):bA,Ab⊆A} be the idealizer of A in the multiplier algebra M(A)of A.Then I(A)is a∗-algebra containing A as an ideal,and for every b∈(I(A))sa clearly bAb is a∗-subalgebra of A.If furthermore the multiplication by b is injective on A,that is, b∈Ann:={b′∈M(A):b′a=0for some0=a∈A},then we can pull back the multiplication and norm on bAb to define a new multiplication×b and a new norm · b on A via the bijection A→bAb.Explicitly,a×b a′=ab2a′anda b= bab .The completion of(A,×b, · b)is isomorphic tob x Ab x at x∈X,as a subfield of the trivial continuousfield of C∗-algebras over X withfibres A,and it contains(b x ab x) as a continuous section for every a∈A.Now we specialize to the commutative case.Let M be a smooth manifold,and let A=C∞(M),A=C∞c(M).Then M(A)is the space C b(M)consisting of all C-valued bounded continuous functions on M,and the strict topology on C b(M)is determined by uniform convergence on every compact subset of M.The idealizer I(A)is the space C∞b(M)consisting of all C-valued bounded smooth functions on M.Given b∈I(A),it is not in Ann exactly if the zero set Z b of b is nowhere dense.Clearly C∞(M\Z b)⊇bAb⊇C∞c(M\Z b),and hence4HANFENG LI158]for every0< ≤1and i=0,1.Thus when n is odd K i(A )∼=K i(C∞(R n)) for all0< ≤1and i=0,1(see for instance[12,page123]for the K-groups of R n and S n).When n is even,K1(A )∼=K1(C∞(R n))for all0< ≤1.When M is compact,in Proposition2.1the element b has to be invertible in C(M)for small and consequently A =C(M).Thus in order to construct strict deformation quantizations for compact M such that the K-groups of A are not isomorphic to those of C(M)for any =0,we have to modify the construction in Proposition2.1.Notice that if we setπ′ (a+λ)=b ab +λfor a∈C∞c(R n),λ∈C in Example2.2,then we get a strict deformation quantization of S n equipped with the0Poisson bracket.This leads to Proposition2.4below.Notation2.3.We denote by F m the space of smooth real-valued functions F on R m such that F is equal to1outside a compact subset of R m and the zero set Z F of F is nowhere dense.Proposition2.4.Let M be a smooth manifold equipped with the0Poisson bracket. Let U be an open subset of M with a diffeomorphismϕ:U→R m.For any F∈F m there is aflabby strict deformation quantization{A ,π } ∈I of M over I=[0,1]with A=C∞c(M)such that A ∼=C∞(M/Y)for every0< ≤1,where Y=ϕ−1(Z F∪{0}).Proof.Set F0=1and F (x)=F(x/ )for all0< ≤1and x∈R m.Then F ∈F m for each ∈I and we can extend the pull-back F ◦ϕ∈C∞(U)to a smooth function b on M by setting it to be1outside U.Clearly b A′b is a∗-subalgebra of A′.Notice that there is a compact set W⊂U such that b =1on M\W for all ∈I,and W containsϕ−1(0).Take an H∈(C∞c(M))R such that H=1 on W.Denote by A′the space of functions in C∞c(M)vanishing atϕ−1(0).Then C∞c(M)=A′⊕C H as complex vector spaces,and H2−H=b (H2−H)b ∈b A′b . It is easy to see that b A′b +C H is a∗-subalgebra of C∞c(M)and the linear map π :C∞c(M)→b A′b +C H defined byπ (a′+λH)=b a′b +λH for a′∈A′andλ∈C is bijective.For each a′∈A′clearly the map →b a′b ∈C∞(M)is continuous on I=[0,1].Thus for each a∈A=C∞c(M),(π (a))is a continuous section in the continuous subfield{A =b A′b +C H=b A′b +C H∼=C∞(M/Y)as desired.Next we describe a case in which we can relate the K-groups of C∞(M/Y)to those of C∞(M)easily:Lemma 2.5.Let D be the subset of R m consisting of points(x1,···,x m)with 0<x1,···,x m<1.Let M,ϕ,F and Y be as in Proposition2.4.Suppose that ∂D⊆Z F⊆¯D.ThenK i(C∞(M\Y))∼=K i(C∞(M))⊕K i(C∞(D\Z F))for i=0,1.FLABBY STRICT DEFORMATION QUANTIZATIONS AND K-GROUPS5 Proof.Letφ:M→M/Y be the quotient map,and let W=φ(M\ϕ−1(D)).Then W is a closed subset of M/Y,and the complement is homeomorphic to D\Z F. Define a mapψ:M/Y→W as the identity map on W andψ((M/F)\W)=φ(Y). Thenψis continuous and proper,i.e.the inverse image of every compact subset of W is compact.Thus the exact sequence0→C∞(D\Z F)→C∞(M/Y)→C∞(W)→0splits.Therefore K i(C∞(M\Y))∼=K i(C∞(W))⊕K i(C∞(D\Z F))for i=0,1. Now Lemma2.5follows from the fact that W is homeomorphic to M.Notice that if a compact set Z⊆R m is the zero set of some non-negative f∈C∞(M),then it is also the zero set of some F∈F m(for instance,take a non-negative g∈C∞c(M)with g|Z=1and set F(x)=f(x)6HANFENG LI[5]G.Nagy,Deformation quantization and K-theory.In:Perspectives on Quantization,(SouthHadley,MA,1996),111–134,Contemp.Math.,214,Amer.Math.Soc.,Providence,RI,1998.[6]M.A.Rieffel,Deformation quantization of Heisenberg m.Math.Phys.122(1989),no.4,531–562.[7]M.A.Rieffel,K-groups of C∗-algebras deformed by actions of R d.J.Funct.Anal.116(1993),no.1,199–214.[8]M.A.Rieffel,Quantization and C∗-algebras.In:C∗-algebras:1943-1993(San Antonio,TX,1993),66–97,Contemp.Math.,167,Amer.Math.Soc.,Providence,RI,1994.[9]M. A.Rieffel,Questions on quantization.In:Operator algebras and operator theory(Shanghai,1997),315–326,Contemp.Math.,228,Amer.Math.Soc.,Providence,RI,1998.arXiv:quant-ph/9712009.[10]J.Rosenberg,Behavior of K-theory under quantization.In:Operator algebras and quantumfield theory(Rome,1996),404–415,Internat.Press,Cambridge,MA,1997.[11]J.K.Shim,A negative answer to a Rieffel’s question on the behavior of K-groups understrict deformation quantization.J.Geom.Phys.44(2003),no.4,475–480.[12]N.E.Wegge-Olsen,K-theory and C∗-algebras.A Friendly Approach.Oxford Science Pub-lications.The Clarendon Press,Oxford University Press,New York,1993.[13]A.Weinstein,Deformation Quantization.S´e minaire Bourbaki Vol.1993/94.Ast´e risque No.227(1995),Exp.No.789,5,389–409.Department of Mathematics,University of Toronto,Toronto ON M5S3G3,CANADA E-mail address:hli@。

a r X i v :m a t h /9802102v 3 [m a t h .D G ] 2 F eb 1999Connes’tangent groupoid and strictquantizationJos´e F.Cari˜n ena a ,Jes´u s Clemente-Gallardo a ,Eduardo Follana a ,Jos´e M.Gracia-Bond´ıa a ,b ,Alejandro Rivero a ,and Joseph C.V´a rilly c a Departamento de F´ısica Te´o rica,Universidad de Zaragoza,50009Zaragoza,Spain b Departamento de F´ısica,Universidad de Costa Rica,2060San Pedro,Costa Rica c Departamento de Matem´a ticas,Universidad de Costa Rica,2060San Pedro,Costa Rica 1Introduction1.1MotivationRecently,Rieffel has published a list of open problems [1]in quantization.The main aim of this paper is to address Question 20of Rieffel’s list:“...in what ways can a suitable Riemannian metric on a manifold M be used to obtain a strict deformation quantization of T ∗M ?”We do that by giving proofs for (a slightly improved variant of)a construction sketched by Connes in Section II.5of his book [2]on noncommutative geometry,and elsewhere [3].The paper can be considered as an introduction to the subject of quantization from Connes’point of view and thus serves too a pedagogical purpose.Rieffel’s requirements are stronger that those of formal deformation theory,ex-tant for any Poisson manifold[4].To our mind,however,the fact that T∗M,for M Riemannian,possesses a strict quantization,was indeed proved by Lands-man in the path-breaking paper[5].Nevertheless,the noncommutative geo-metry approach presents several advantages,not least that the C∗-theoretical aspects come in naturally.The procedure was suggested by Landsman himself, even before[2]was in print,at the end of his paper.For the sake of simplicity, we deal here with the nonequivariant case only.The plan of the article is as follows.In thisfirst Section,after introducing groupoids and the tangent groupoid construction,we give an elementary dis-cussion of Connes’recipe for quantization.Wefind the intersection,for the case M=R n,of the family of what could be termed“Connes’quantization rules”with the ordinary quantum-mechanical ordering prescriptions.Groupoids are here regarded set-theoretically,questions of smoothness being deferred to Sec-tion2.We show that Moyal’s quantization rule belongs to the collection of Connes’quantization rules and in fact is singled out by natural conditions in strict deformation theory.In Section2we spell out our variant of Connes’tangent groupoid construction in full detail.The heart of the matter is the continuity of the groupoid product, for which we give two different proofs.In Section3,by means of the mathematical apparatus of Section2,we restate Landsman’s partial answer to Rieffel’s question.In particular,we rework the existence proof for a strict quantization of the Moyal type(in the sense of being both real and tracial)on Riemannian manifolds.Moreover,using a strong form of the tubular neighbourhood theorem,we show that there exists what Rieffel calls aflabby quantization[1].The paper concludes with a discussion on the C∗-algebraic aspect of the tangent groupoid construction and its relation to the index theorem.1.2Basic facts on groupoidsThe most economical way to think of a groupoid is as a pair of sets G0⊂G, and to regard elements of G as arrows and elements of G0as nodes.Definition1A groupoid G⇉G0is a small category in which every mor-phism has an inverse.Its set of objects is G0,its set of morphisms is G.A group is of course a groupoid with a single object.The gist of the definition is conveyed by the following example.1.2.0.1Partial isometries Consider a complex Hilbert space H.The col-lection of unitary arrows between closed subspaces of H obviously defines a groupoid,for which G is the set of partial isometries{w∈L(H):ww∗w=w} and G0is the set of orthogonal projectors in L(H).Recall that,given w,we can writeH=ker w⊕im w∗=ker w∗⊕im w.Hence w∗w is the orthogonal projector with range im w∗,while ww∗is the orthogonal projector with range im w;of course,w∗is the inverse of w.We naturally identify two maps r,s(respectively“range”and“source”)from G to G0:r(w):=ww∗and s(w):=w∗w.Also,it is natural to considerG(2):={(u,v)∈G×G:u∗u=vv∗}.This defining equation is a sufficient condition for the operator product uv to be a partial isometry,as follows from the simple calculationuv(uv)∗uv=uvv∗u∗uv=uvv∗v=uv.The example motivates a more cumbersome restatement,that includes all the practical elements of the definition.Definition2A groupoid G⇉G0consists of:a set G,a set G0of“units”with an inclusion G0֒→G,two maps r,s:G→G0,and a composition law G(2)→G with domainG(2):={(g,h):s(g)=r(h)}⊆G×G,subject to the following rules:(1)if g∈G0then r(g)=s(g)=g;(2)r(g)g=g=gs(g);(3)each g∈G has an“inverse”g−1,satisfying gg−1=r(g)and g−1g=s(g);(4)r(gh)=r(g)and s(gh)=s(h)if(g,h)∈G(2);(5)(gh)k=g(hk)if(g,h)∈G(2)and(gh,k)∈G(2).The examples of groupoids we shall use have a differential geometricflavour instead.1.2.0.2A vector bundle Eπ−→M Here G=E is the total space,G0= M is the base space,r=s=πso that G(2)= x∈M E x×E x(the total space of the Whitney sum E⊕E),and the composition law isfibrewise addition.1.2.0.3The double groupoid of a set Given a set M,take G=M×M and G0=M,included in M×M as the diagonal subset∆(M):={(x,x): x∈M}.Define r(x,y):=x,s(x,y):=y.Then(x,y)−1=(y,x)and the composition law is(x,y)·(y,z)=(x,z).We shall generally call G0the diagonal of G.1.3Connes’tangent groupoidsWe recall some basic concepts of differential geometry,particularly sprays and normal bundles,that we shall later use.Given a symmetric linear connection on a differentiable manifold M,one can define a vectorfieldΓon T M whose value at v∈T M is its horizontal lift to T v(T M).This vectorfield is called the geodesic spray of the connection and its integral curves are just the natural lift of geodesics in M.The geodesic spray is a second-order differential equation vectorfield satisfying an addi-tional condition of degree-one homogeneity which corresponds to the affine reparametrization property of geodesics.More generally,a second-order differ-ential equation vectorfield is said to be a spray if the set of its integral curves is invariant under any affine reparametrization:these curves are called geodesics of the spray.Given a sprayΓ,there is a symmetric connection whose geodesic spray isΓ;and conversely,the connection is fully determined by its geodesic spray,that can also be used to construct the exponential map exp:T x M→M.A Riemannian structure on M determines one symmetric linear connection, the Levi-Civita connection,and consequently a Riemannian spray.Now let Y0be a submanifold of a manifold Y.The normal bundle N YY0toY0in Y is defined as the vector bundle N YY0:=T Y/T Y0,where the notationmeans that its base is Y0and itsfibre is given by the equivalence classes of the elements of the tangent bundle T Y under the relation:X1∼X2,for X1,X2∈T q Y with q∈Y0,if and only if X1=X2+V for some V∈T q Y0.The usual way to work with such a structure is to choose a representative in each class, thereby forming a complementary bundle to T Y0in T Y restricted to Y0.There is no canonical choice for the latter,in general.When a Riemannian metric is provided on Y,there is a natural definition of the complementary bundle as the orthogonal complement of the tangent space T q Y0in T q Y(although other choices may be convenient,even in the Riemannian case).Once we have chosena suitable representative of each class,the bundle N YY0becomes a subbundleof T Y and we can consider the exponential map exp of T Y restricted to N YY0.We recall also that a tubular neighbourhood of Y0in Y is a vector bundle E→Y0,an open neighbourhood Z of its zero section and a diffeomorphism of Z onto an open set(the tube)U⊂Y containing Y0,which restricts over the zero section to the inclusion of Y0in Y[6].We say that the tubular neighbourhood is total when Z=E.The main theorem in this context establishes that,given a spray on Y,one can always construct a tubular neighbourhood by making use of the corresponding exponential map.When the spray is associated to a Riemannian metric,one can always have a total tubular neighbourhood, because a Euclidean bundle is compressible,i.e.,isomorphic as afibre bundle to an open neighbourhood of the zero section:see[6].For suitable 0>0,one can then define the normal cone deformation(a sort of blowup in the differentiable category)of the pair(Y,Y0),denotedM YY0,by gluing together Y×(0, 0]with N YY0as a boundary with the help ofthe tubular neighbourhood construction[7].The construction is particularly interesting when(Y,Y0)is a groupoid,in that it gives a“normal groupoid”with diagonal Y0×[0, 0].We consider a particular case of this construction,the tangent groupoid.Let M now be an orientable Riemannian manifold.Denote by N∆,rather thanN M×MM ,the normal bundle associated to the diagonal embedding∆:M→M×M.We can identify T M⊕T M with the restriction of T(M×M)to ∆(M),and the tangent bundle over M is identified to{(∆(q),X q,X q):(q,X q)∈T M};thereby the normal bundle N∆to M in M×M can a priori be identified with{(∆(q),ϕ1q X q,ϕ2q X q):(q,X q)∈T M},(1) whereϕ1,ϕ2∈End(T M)are any two bundle endomorphisms(i.e.,continuous vector bundle maps from T M into itself)such that the linear mapϕ1q−ϕ2q on T q M is invertible for all q∈M;this we write asϕ1−ϕ2∈GL(T M).We shall assume,for definiteness,that eachϕ1q−ϕ2q is homotopic to the iden-tity:in other words,that there is an isomorphism of oriented vector bundles between T M and N∆.The tangent groupoid G M⇉G0M,according to Connes,is essentially thenormal groupoid M M×MM modulo that isomorphism[2].That is to say,wethink of the disjoint union G M=G1⊎G2of two groupoidsG1:=M×M×(0, 0],G2:=T M,where G1is the disjoint union of copies M×M×{ }of the double groupoidof M,parametrized by0< ≤ 0.The compositions are(x,y, )·(y,z, )=(x,z, ) (q,X1)·(q,X2)=(q,X1+X2)with0< ≤ 0, if X1,X2∈T q M.The topology on G M is such that,if(x n,y n, n)is a sequence of elements of G1with n↓0,then it converges to a tangent vector(q,X)iffx n→q,y n→qandx n−y nLet a(q,p)be a function on T∗R n.Its inverse Fourier transform in the second variable gives us a function on T R n:F−1a(q,X)= R n e i Xp a(q,p)d p.To the function a,Connes’prescription associates then the following family of kernels:k a(x,y; ):=F−1a(Φ−1 (x,y, )).The map a→k a is thus linear.We get the dequantization rule by Fourier inversion:a(q,p)=1∂q +∂∂yK=0,(4)i.e.,K depends only on the combinations x−y and q−1satisfying(4)and(5)is of the formK(q,p,x,y; )=12(x+y))/ )dθ,where f should have no zeros,on account of invertibility;that can be rewritten asK(q,p,x,y; )=1(2π )nR nδ(x−q− ϕ1X)δ(y−x+ X)exp(−i pX)d X.This integral can be rewritten as(2π )−nδ(x−q−ϕ1(x−y))e−i p(x−y)/ , and further transformed into12(x+y)−q+(1which is of the previously given form,withf (θ,τ)=exp(i θ(1(2π )nR n δ(x −q − ϕ1q X )δ(y −x + (ϕ1q −ϕ2q )X )exp(−i pX )d X,we would have had two difficulties:equivariance under translation of the posi-tion would be lost since ϕ1q =ϕ1q ′in general,and equivariance under transla-tion of the momentum would be lost since ϕ1q −ϕ2q =id T q R n in general;this bears on the physical meaning of Connes’limit condition (2).It would be tempting,also in view of (2),to add the time variable to the mathematical apparatus of this paper,in the spirit of Feynman’s formalism,and to study the interchangeability of the limits t ↓0and ↓0.Happily,Moyal’s rule is included among Connes’rules:it follows from themost natural choice ϕ1=12(see the discussion in [12]).We arrive at the following conclusion.Proposition 3Moyal quantization rule is the only real quantization of the Connes type,in the case M =R n .Connes’own choice in [2]is ϕ1=0,ϕ2=−1;this corresponds to the “standard”ordering prescription,in which the quantization of q n p m is Q n P m ;whereas the choice ϕ1=1,ϕ2=0leads to the “antistandard”ordering,in which the quantization of p m q n is P m Q n .Note that all of Connes’prescriptions are tracial.In order to obtain the “normal”and “antinormal”prescriptions of use in field theory,which are real but not tracial,one would have to complexify Connes’construction;we shall not go into that.Following Landsman,we will strengthen the definition of strict quantization in order to remain in the real context (see Section 3).Our elementary discussion in this subsection leads to presume that real strict quantization can be done in the framework of tangent groupoids;this is tantamount to the generalization of the Moyal rule to arbitrary manifolds endowed with sprays.All the pertinent differential geometric constructions for that purpose are taken up in the next section.2The tangent groupoid construction:the Moyal versionDefinition4A smooth groupoid is a groupoid(G,G0)together with differ-entiable structures on G and G0such that the maps r and s are submersions, and the inclusion map G0֒→G is smooth as well as the product G(2)→G.Note the dimension count:if dim G=n,dim G0=m,then dim G(2)=2n−m.The definition of smooth groupoid is taken from[2].Now we establish his Proposition II.5.4,there left unproven by Connes.Proposition5The tangent groupoid G M to a smooth manifold M is a smooth groupoid.2.1G M as a manifold with boundaryThe groupoid G1=M×M×(0, 0],that will be the interior of G M(plus a trivial“outer”boundary,which we neglect to mention in our following argu-ments)is given the usual product manifold structure.To complete the defi-nition of the manifold structure of G,consider the isomorphism from T M toN∆given by(q,X q)→(q,12X q).Consider also the product manifoldT M×[0, 0].We choose a spray on M—this is provided,for instance,by a choice of Riemannian metric on M—and define a mapT M×[0, 0]⊃UΦ−→G M,where U is open in T M×[0, 0]and includes T M×{0},as follows:Φ(q,X q, ):=(exp q( X q/2),exp q(− X q/2), )for >0,andΦ(q,X q,0):=(q,X q)for =0.Here exp denotes the exponential map associated to the spray:we know that, for afixed ,the exponential map defines a diffeomorphism of an open neigh-bourhood V of the zero section of T M onto an open neighbourhood of the diagonal in M×M;and we decide that a point(q,X q, )is in U ifΦ(q,X q, ) is contained in V.Therefore,both the existence and,for a suitable choice of U,the bijectivity of the mapΦfollow from the tubular neighbourhood theorem.As U is anopen(sub)manifold with boundary,we can carry the structure of manifold with boundary to G M,obtaining that T M is the boundary of the groupoid G M in the topology associated to that structure.The diagonal is obviously M×[0, 0].We remind the reader that Connes uses instead the chart given by(q,X q, )→(q,exp q(− X q), )for >0,and(q,X q,0)→(q,X q)for =0.From now on,a particular Riemannian structure is assumed chosen,and when convenient we shall also assume,as we then may,that U is the whole of T M×[0, 0].To continue the proof of smoothness of G M,we need to check that the various mappings have the required smoothness properties.The basic idea is to pull all maps on G M back to T M×[0, 0]and prove smoothness there.Considerfirst the inclusion map i:G0M→G M.It is obvious that the restriction of i to M×(0, 0]is smooth in its domain.If we now consider its restriction to i−1(Φ(U))and compose it withΦ−1,we obtain a map which can be written as:Φ−1◦i(x, )=Φ−1(x,x, )=(x,0x, )for >0,andΦ−1◦i(x,0)=Φ−1(x,0x,0)=(x,0x,0)for =0.This map is obviously smooth in its domain and,asΦis a diffeomorphism, i is smooth in its domain.We shall consider now the range and source maps.The smoothness of both maps when restricted to M×M×(0, ]is again obvious,as is the fact that they are of maximum rank(hence submersions when restricted to this domain).The composition ofΦwith the restriction of r toΦ(U)is expressed as: r◦Φ(q,X q, )=r(exp q( X q/2),exp p(− X p/2), )=(exp q( X q/2), )for >0andr◦Φ(q,X q,0)=(q,0)for =0.Again this map is smooth and of maximum rank in its domain,so that r must also be a submersion.The corresponding proof for the source map is analogous.2.2The geometrical structure of G(2)M⊂G M×G MTo define a differentiable structure for G(2)M,we proceed as in the previous case,by defining a bijection between an open set in G(2)M and an open set in a manifold with boundary,and transporting the differential structure via the bijection.What kind of manifold is G(2)M?The product has to be a smooth mapping between two manifolds with boundary,mapping the boundary on the bound-ary and the interior on the interior.Thus,from the definition of the product it is clear that the boundary of the manifold should be the Whitney sum T M⊕T M,arising as the pullback with respect to the diagonal injection of M in M×M of the product bundle T M×T M.Points in the interior of the manifold are pairs of the form:((x,y, ),(y,z, )). Therefore,we need to define a differential structure in G(2)M in such a way that it becomes a manifold with boundary T M⊕T M,its interior being diffeomorphic to M×M×M×(0, 0].Consider now T M×T M×[0, 0].This is a manifold with boundary,with a natural differential structure.Let(q′,q,X q′,Y q, )be a point in this mani-fold;we use our construction for G M on each T M separately,i.e.,we set the bijection:(q′,q,X q′,Y q, )↔{(e X q′/2q′,e− X q′/2q′, ),(e Y q/2q,e− Y q/2q, )}for >0—with an obvious notation for exp.Those points that interest usresult just from imposing that e− X q′q′=e Y q q.We shall see that this constraintdefines a regular submanifold of T M×T M×[0, 0].For that,for afixed value of ,consider the sequence of mapsΨ:T M×T MΦ−→M×M×M×Mπ23−→M×MΦ−1−→T M→R n given by(q′,q,X q′,Y q)→(e X q′/2q′,e− X q′/2q′,e Y q/2q,e− Y q/2q)→(e− X q′/2q′,e Y q/2q)→(r,X r)→X r,where the(r,X r),that depend on ,are found soΦ(r,X r)=(e− X q′/2q′,e Y q/2q).This composition defines a differentiable mapping of constant rank(it is com-posed of two bijections and two projections onto factors of a product).We then extendΨto a map from T M×T M×[0, 0]to R n×[0, 0]and have then thatΨ−1({0}×[0, 0])defines a regular submanifold S of T M×T M×[0, 0], whose differentiable structure we use to define the structure of manifold with boundary on G(2).[It is perhaps not entirely clear that the boundary of S is T M⊕T M,as we wish.But remember that r depends on through the above manipulations.If we have a sequence(q′n,q n,X q′n ,Y qn, n)in S,with n↓0,its limit is a point(s,s,X,Y)in T M×T M where s= lim q′n=lim q n=lim r( n).]2.3Continuity of the productA very simple argument with Riemannianflavour allows one to prove at least continuity of the product operation in the tangent groupoid.Consider a se-quence in G(2):{(x n,y n, n),(y n,z n, n)}=Φ(q′n,q n,X n,Y n, n)with limit on the boundary.We need to check that the limit of the products coincides with the product of the limits in G M.For elements close enough to the boundary,we know that(s(n),Z s(n))exists,sox n=e Z s(n)/2s(n),z n=e− Z s(n)/2s(n).Assume thatq′n→s,q n→s,X n→A,Y n→B as ↓0. Now,we have(e Z s(n)/2s(n),e− Z s(n)/2s(n))=(e X n/2q′n,e− Y n/2q n)→(s,s)as ↓0,hencelim s(n)=s.It remains to show that lim Z s(n)=A+B.This will follow if we prove Z n= X n+Y n+o( n).But that follows from consideration of the small triangle with vertices x n,y n,z n,formed by geodesics through q′n,q n,s(n)with directions ±X n,±Y n,±Z s(n).One sees that2(X n+Y n−Z s(n))is approximately a circuit around this triangle; by the Gauss–Bonnet theorem,we conclude thatX n+Y n−Z s(n)≈O( 2).2.4A functorial proof of smoothnessLemma6Let there be given two closed submanifolds X0֒→X and Y0֒→Y and a smooth mapping f:X→Y which satisfies f(X0)⊂Y0.Then theinduced mapping˜f:M XX0→M YY0between the corresponding normal conedeformations is also smooth.This is Lemma2.1in[13],where no proof is offered.We give some details of the lemma and then of its application.First of all,if f is a smooth mapping from X to Y such that f(X0)⊂Y0,the image under f∗of the tangent bundle to X0is a subbundle of T Y0,implying the existence of an induced mapping between the respective normal bundles,that we shall continue to call f∗.Now we define˜f by˜f(x, ):=(f(x), )for >0,and˜f(a,Xa):=(f(a),f∗(X a))for(a,X a)an element of the normal bundle and =0.Now,differentiability of˜f follows from the limitlim ↓0 −1exp−1f(a)[f(exp a( X a))]=f∗(X a).The application to the groupoid operations in our context is plain.We discuss the product operation,the only relatively tricky one.Let m:G(2)→G be the groupoid multiplication.Now G(2)∩(G0×G0)=∆(G0)and m(G(2)∩(G0×G0))=G0;indeed,if(u,v)∈G(2)∩(G0×G0)then u=s(u)=r(v)=v by property(i)of Definition2,and so uv=us(u)=u by property(ii).Therefore˜m:M G(2)G0→M GG0is smooth.It remains to prove that M G(2)G0is diffeomorphicto(M GG0)(2).But this is clear on examining the definitions:indeed,M G(2)G0=G(2)×(0, 0]⊎N G(2)G0={(g,h, ):s(g)=r(h)}⊎{(u,X u,Y u):X u,Y u∈N G G0},whereas(M G G0)(2)={(g, 1;h, 2):s(g)=r(h), 1= 2}⊎{(u,X u;v,Y v):u=v}.For G=G M,that boils down to T M⊕T M≈N M×M×MM .Notefinally thedimension count:the dimension of(M GG0)(2)is2(dim G+1)−dim(G0+1)=2dim G−dim G0+1,which is clearly the same as the dimension of M G(2)G0. This completes the proof of Proposition5.3Tangent groupoids and strict quantization3.1The deformation conditionsIn our definition of quantization,we actually strengthen some of Rieffel’s re-quirements.Regard T∗M as a Poisson manifold and consider the classical C∗-algebra A0:=C0(T∗M)of continuous functions vanishing at infinity.We choose a dense subalgebra A0(there is considerable freedom in that,but,to fix ideas,we think of the functions whose Fourier transform in the second argument has compact support),and we search for a family of mappings Q into noncommutative C∗-algebras A such that the following relations hold for arbitrary functions in A0:(1)the map → Q (f) is continuous on[0,h0)with Q0=I;(2)lim →0 Q (f1)Q (f2)−Q (f2)Q (f1)−i Q ({f1,f2}) =0;Those are Rieffel’s strict quantization conditions(Question23of[1]),to which we add:(3)the asymptotic morphism condition lim →0 Q (f1)Q (f2)−Q (f1f2) =0;(4)the reality condition Q (f∗)=Q (f)∗;and also(5)the traciality condition Tr[Q (f1)Q (f2)]= T∗M f1(q,p)f2(q,p)dµ (q,p);where we use the same symbols∗(a bit overworked,admittedly)and · for the adjoint and norm in every C∗-algebra.Axioms(1)to(4)are a slight variant of Landsman’s axioms.That the tangent groupoid construction provides an answer to the twentieth query by Rieffel fol-lows indeed from Landsman’s calculations in[5].There is no point in repeatingthem here,and we limit ourselves to the necessary remarks tofit them in the tangent groupoid framework.Axiom(5)is employed to further select a unique recipe.In[1],Rieffel introduces the important concept of“flabbiness”:a deformation quantization isflabby if it contains the algebra of smooth functions of compact support on M.The constructions performed in this paper require in princi-ple only the existence of sprays,in order to use the tubular neighbourhood theorem.However,in that caseflabbiness is not guaranteed(see below).3.2The C∗-algebra of a groupoidThe natural operation on functions of a groupoid is convolution:(a∗b)(g):= {hk=g}a(h)b(k)= {h:r(h)=r(g)}a(h)b(h−1g)but for this to make sense we need a measure to integrate with.We can either define a family of measures on thefibres of the map r,G x:={g∈G: r(g)=x}for x∈G0(see the detailed treatments given by Kastler[14]and Renault[15])or we canfinesse the issue by ensuring that the integrand is always a1-density on each G x.In the second approach,one uses half-densities,rather than functions.We summarize it here,for completeness.Denote the typicalfibre of s by G y:= {g∈G:s(g)=y}for y∈G0.Since r and s are submersions,thefibres G x and G y are submanifolds of G of the same dimension,say k.If x=r(g)and y=s(g),thenΛk T g G x andΛk T g G y are lines.LetΩ1/2g be the set of maps ρ:Λk T g G x⊗Λk T g G y→C such thatρ(tα)=|t|1/2ρ(α)for t∈R.This is a(complex)line,and it forms thefibre at g of a line bundleΩ1/2→G, called the“half-density bundle”.Let C∞c(G,Ω1/2)be the space of smooth, compactly supported sections of this bundle.For a,b in this space,the con-volution formula makes sense and a∗b∈C∞c(G,Ω1/2)also.The C∗-algebra of the smooth groupoid G⇉G0is the algebra C∗(G)obtained by completing C∞c(G,Ω1/2)in the norm a :=sup y∈G0 πy(a) ,whereπy is the representa-tion of C∞c(G,Ω1/2)on the Hilbert space L2(G y,Ω1/2)of half-densities on the s-fibre G y:πy(a)ξ:g→ G y a(h)ξ(h−1g)where one notices that the integrand is a1-density on G y.If G=M×M,we get just the convolution of kernels:(a∗b)(x,z):= M a(x,y)b(y,z)d ywhere d y denotes integration of a1-density on M parametrized by y.This business of half-densities is very canonical and independent of preassigned measures.However,in our case,if M is an oriented Riemannian manifold,we may use the volume form dν(x):=g(q)d Ywhere we may take f1(x,·)and f2(x,·)in C∞c(T x M).The Fourier transformF a(q,p)=1g(q)d Xreplaces convolution by the ordinary product on the total space T∗M of the cotangent bundle.This extends to the isomorphism also called F:C∗(T M)→C0(T∗M),with inverse:F−1b(q,X)= T∗q M e i pX b(q,p)d n p g(q).3.3The quantization and dequantization recipesLetγq,X be the geodesic on M starting at q with velocity X,with an affine parameter s,i.e.,γq,tX(s)≡γq,X(ts).Locally,we may writex:=γq,X(s), y:=γq,X(−s),with Jacobian matrix∂(x,y)Then one has the change of variables formula:M×MF(x,y)dν(x)dν(y)= M T q M F(γq,X(12))J(q,X;1g(q)d X dν(q), where we introduceJ(q,X;s):=s−n det g(γq,X(−s))∂(q,X)(s).This object can be computed from the equations of geodesic deviation[5].The crucial estimate isJ(q,X,1∂qν;the second one using the variation of the tangent vector,˜hµν(q,X,s)=∂xµ1(q,X,s)Here J−1/2,J1/2mean the corresponding multiplication operators.It is pos-sible to rewrite the formulae so J only appears once,but then we would lose property(5).So there is actually a family of“Moyal”quantizations in the general case,unless we demand traciality(as we do).This is a feature of the nonflat case:note that J=1in the framework of our Section1.In the main development in[5]J actually only appears in the dequantization formula,but again Landsman gives indication on how to modify the formulae to get tracial-ity.Those factors are analogous to the preexponential factors that appear in the semiclassical expression for the path integral:see,e.g.,[16,p.95].One can insert them in several ways without altering the axioms,precisely on account of(6).A bit more explicitly,the previous formulae arek a(x,y; )=J−1/2(q,X,1)k a(x,y; )],(7)2where(x,y)and(q,X)are related by x=γq,X(1).2The long but relatively straightforward verifications in[5]then show that we have defined an(obviously real)preasymptotic morphism which moreover is a strict quantization from C∞c(T∗M)to K(L2(M)).In addition,the tracial property(5)is satisfied.In the Riemannian context we have total tubular neighbourhoods,and it is clear that Fourier transforms of smooth functions of compact support in T∗M decay fast enough that our formulae make sense for them;therefore the quantization isflabby.We make afinal comment on the uniqueness,or lack of it,of the quantiza-tion considered here.Among our choices are those we make to establish the isomorphism between T M and N∆on which the whole construction hinges. Regarding only the differential structure,those choices are parametrized by, say,the pair(ϕ1+ϕ2,ϕ1−ϕ2)∈End(T M)×GL(T M).However,the identifi-cation of the normal bundle with the orthogonal bundle to the diagonal leads to the equationϕ1+ϕ2=0,if we take the natural metric on M×M.To the same equation leads the reality constraint(4),as a simple calculation from(7) shows.After so narrowing the freedom of parameter choice to GL(T M),semitraciality implies at least det(ϕ1−ϕ2)=1.Connes’condition(2)and our discussion in Section1strongly suggest to adopt the restrictionϕ1−ϕ2=1.Let us agree to call all the quantizations(parametrized by End(T M))for which the last equation holds quantizations of the Connes type.Then we have proved: Theorem7For any Riemannian manifold,the only real quantization rules of the Connes type are Moyal quantizations.。

量化金融分析师（AQF®）全国统一考试模拟题适用场次：2018年3月使用本模拟题，您应该遵守：1.本模拟题仅提供给参加2018年3月份AQF全国统一考试的考生，考生仅可以出于准备个人考试的目的查阅和打印本模拟题；2.严禁出于任何目的的复制、网络发布和传播、抄袭本模考题内容，如有违反，可能导致违纪或违法行为；©版权所有，侵权必究。

量化金融标准委员会Standard Committee of Quantitative Finance量化金融分析师（AQF®）全国统一考试模拟题说明：本场考试中的代码都应采用Python 3.X版本作答。

1.单选题（每题1分，本部分共20分）:只有一个正确答案，选对得1分，选错或不选得0分。

1.1 技术分析是重要的投资分析方法之一。

其中，起源于日本德川幕府时代的“K线图”（又称蜡烛图、阴阳线）是常用的技术分析方法。

当我们发现某交易日的K线为无下影线阴线时，那么该K线实体的上边线表示（）？A. 最高价B. 收盘价C. 最低价D. 开盘价1.2 以下关于各大量化投资交易策略的描述中，不正确的是（）？A. 在多因子策略中，一般而言，所选因子的相关性越低越好B. 一般而言，资金流对个股的短期波动影响更大C. 动量反转策略利用的是价格均值回归的特性D. 趋势跟踪策略本质上是一种追涨杀跌的策略，因此并不具有任何盈利的可能性1.3 李明，AQF，某量化基金经理，他在量化投资交易的过程中，发现回测收益往往会大幅高于实盘收益，研究发现造成这种现象的原因有很多，其中，常见的一种原因是在回测的过程中使用了未来数据，未来数据会使得回测收益虚高。

那么在下列量化策略研究过程中，哪个选项最有可能没有使用到未来数据（）？A. 在策略回测的过程中，采用某天的最低价作为当天的买入成本价B. 使用整个样本数据对策略参数进行寻优后，使用该优化后的参数进行策略回测，并对该策略进行有效性评估C. 以发出交易信号后的下一天的开盘价作为策略的成交价，未设置滑点D. 以当前沪深300成分股为研究对象，研究过去10年沪深300成分股的选股策略1.4 李明，AQF，某量化基金经理，正在研究事件驱动型套利策略，以下哪个选项描述了事件驱动型套利策略（）？A. 基金经理持有目前或者预期会发生诸如以下交易事项的公司的金融产品：（包括但不限于）合并、重组、财务危机、股权收购、股东回购、发行债务交换、证券发行或其他资本结构调整B. 基金经理根据潜在的宏观经济变量及其对股票、固定资产、货币和大宗商品市场的影响，进行相关交易C. 基金经理基于多个证券估值差异及其关系的理论进行交易D. 基金经理在现货市场和衍生品市场进行方向相反的操作1.5 字符串格式化是量化投资策略编写过程中常用的方法，那么以下哪种代码可以用来实现浮点数格式化（）？A．%cB. %dC. %fD. %s1.6 李明，AQF，某量化基金经理，在1.2308做空欧元/美元的差价合约，之后欧元/美元汇率跌至1.2133，李明可以通过以下哪个类型的委托单来实现继续持有空头仓位的同时控制回撤风险（）？A. 市价买单B. 市价卖单C. 限价买单D. 止损买单1.7 当横线处填入（）时，代码打印输出的结果是列表中所有的深交所上市的股票代码？（注意：上交所代码以6开头，深交所代码以0或3开头）for stock_code in ['002003', '600015', '300001', '002300']: if stock_code.startswith('6'):_____print(stock_code)A．raiseB. continueC. passD. break1.8 李明，AQF，某量化基金经理，他在进行策略研究时需要从DataFrame数据类型stock_base_data中提取2018-01-03至2018-01-05（含01-05）时间段中的股票PE、CLOSE数据，该DataFrame如下：则李明提取数据时可以使用的代码为（）？A. stock_base_data.iloc['2018-01-03':'2018-01-05', ['PE', 'CLOSE']]B. stock_base_data.loc['2018-01-03':'2018-01-05', ['PE', 'CLOSE']]C. stock_base_data.loc[['PE', 'CLOSE'], '2018-01-03':'2018-01-05']D. stock_base_data.iloc[2:4, ['PE', 'CLOSE']]1.9 李明，AQF，某量化基金经理，想要评估长期持有的某只股票在过去三天的总体表现。