Non-commutative Geometry Beyond the Standard Model

FC-投射模和Gorenstein FC-投射模FC-投射模和Gorenstein FC-投射模引言：在代数学和同调代数领域，模论是一个重要的研究方向。

在模的研究中，FC-投射模和Gorenstein FC-投射模是两个经典概念。

本文将介绍这两个概念的定义、性质以及它们的关系。

一、FC-投射模1.1 定义首先，我们来介绍FC-投射模的定义。

给定一个环R和一个左R-模M，如果存在一个函子F: Mod(R) -> Ab，使得对于任意的左R-模N，都有一个自然的同构Hom_R(N, F(M)) ≅ Hom_R(M, N)那么称M为一个FC-投射模。

1.2 性质FC-投射模的一个重要性质是：对于任意的左R-模N和M，都存在一个自然的同构Ext_R^i(N, F(M)) ≅ Ext_R^i(M, N)其中，Ext_R^i表示R-模的i-次扩张群。

此外，如果模M是一个FC-投射模，那么任意的短正合列0 → A → B → C → 0在应用Hom_R(-, F(M))函子后，仍然是正合的。

二、Gorenstein FC-投射模2.1 定义接下来，我们来介绍Gorenstein FC-投射模的定义。

给定一个环R和一个左R-模M，如果存在一个函子F: Mod(R) -> Ab，以及对于任意的左R-模N，都有一个自然的同构G_Ext_R^i(N, F(M)) ≅ Ext_R^{n-i}(G, N)其中，G = Hom_R(-, F(R))G_Ext_R^i表示G-模的i-次Ext函子。

那么称M为一个Gorenstein FC-投射模。

2.2 性质Gorenstein FC-投射模与FC-投射模的关系也是非常紧密的，它们满足以下性质：（1）FC-投射模是Gorenstein FC-投射模。

（2）如果模M是一个Gorenstein FC-投射模，并且在G-模范畴上是满足AB5条件的，那么M是一个FC-投射模。

a r X i v :h e p -t h /0002210v 2 3 M a r 2000SISSA 16/00/EP/FMhep-th/0002210A note on consistent anomalies in noncommutative YM theoriesL.Bonora ,M.Schnabl and A.TomasielloInternational School for Advanced Studies (SISSA/ISAS),Via Beirut 2,34014Trieste,Italy and INFN,Sezione di Trieste bonora,schnabl,tomasiel@sissa.it Abstract.Via descent equations we derive formulas for consistent gauge anomalies in noncommutative Yang–Mills theories.The growing interest in noncommutative YM theories is calling our attention upon old problems.We would like to know if or to what extent old problems,solutions or algorithms in commutative YM theories fit in the new noncommutative framework.One of these is the question of chiral anomalies .We would like to know what form chiral anomalies take in the new noncommutative setting.This has partially already been answered via a direct computation [1,2].In commutative YM theories there is another way to compute the explicit form of consistent chiral anomalies,i.e.by solving the Wess–Zumino consistency conditions (WZcc),[3],in particular via the descent equations,[4,5,6].Now it is easy to see that in noncommutative YM theories WZcc still characterize chiral Ward identities,therefore one may wonder whether one can find the form of consistent anomalies bysolving them,in particular whether a powerful algorithm like the descent equations is still at work.The answer is yes,as we briefly illustrate below.In this note we have in mind a YM theory in a noncommutative R D ,with Moyal deformation parameters θij .We use the form notation for all the expression.Therefore we have a matrix–valued one–form gauge potential A ,with gauge field strength two–form F =dA +A ∗A .We also introduce the gauge transformation parameter C ,in the form of an anticommuting Faddeev–Popov ghost (using anticommuting gauge parameters simplifies a lot anomaly formulas,as is well–known).Next we introduce the following gauge transformation conventions:δA =dC +A ∗C −C ∗A,δC =−C ∗C (1)The differential d is defined in the general non–commutative geometry setting as follows:it is the exterior derivative of the universal differential graded algebra Ω(B )associated to any algebra B [7,8,9];B is for us the algebra generated by A and C .In simple words,this means that we deal with forms as usual,but never use the relation:ω1ω2=(−)k 1k 2ω2ω1,for any k i –form ωi .d andδare assumed to commute.As a consequence the transformations(1)are nilpotent as in the commutative case.They are noncommutative BRST transformations.If one tries to derive for commutative YM theories descent equations similar to those of the commutative case,atfirst sight this seems to be impossible.In fact the standard expression one starts with,Tr(F...F),in the commutative case should be replaced by Tr(F∗...∗F)(Tr denotes throughout the paper the trace over matrix indices1;but the latter is neither closed nor invariant,as one may easily realize.However one notices that it would be both closed and invariant if we were allowed to permute cyclically the terms under the trace symbol.In fact,terms differing by a cyclic permutation differ by a total derivative of the formθij∂i....Such terms could of course be discarded upon integration. However,the spirit of the descent equations requires precisely to work with unintegrated objects.The way out is then to define a bi–complex which does the right job.It is defined as follows.Consider the space of(A–valued,where A is the algebra defining our non–commutative space)traces of∗products of such objects as A,dA,C,dC.The space of cochains is now this space,modulo the circular relationTr(E1∗E2∗...∗E n)≈Tr(E n∗E1∗...∗E n−1)(−1)k n(k1+...+k n−1)(2) where E i is any of A,dA,C,dC,and k i is the order form of E i.The definition of the bi–complex,let us call it C,is completed by introducing two differential operators.Thefirst is d,as defined above.The second differential isδ,the BRST cohomology operator.We define it to commute with d.Both preserve the relation (2).We can now start the usual machinery of consistent anomalies,reducing the problem to a cohomological one.In a noncommutative even D–dimensional space we start with Tr(F∗F∗...∗F)with n entries,n=D/2+1.In the complex C this expression is closed and BRST–invariant.Then it is easy to prove the descent equations:Tr(F∗F∗...∗F)=dΩ02n+1δΩ02n+1=dΩ12n(3)δΩ12n=dΩ22n−1and so on.Here the Chern–Simons term can be represented in C byΩ02n+1=n 10dt Tr(A∗F t∗F t∗...∗F t)(4)where we have introduced a parameter t,0≤t≤1,and the traditional notation F t= tdA+t2A∗A.The anomaly can instead be represented byΩ12n=n 10dt(t−1)Tr(dC∗A∗F t∗...∗F t+dC∗F t∗A∗...∗F t+...+dC∗F t∗F t∗...∗A)(5) where the sum under the trace symbol includes n−1terms.FinallyΩ22n−1=n 10dt(t−1)2dt ,and then integrate by parts.In four dimensions the anomaly takes the form Ω14=−12Tr(T a{T b,T c})+1[3]J.Wess and B.Zumino,Phys.Lett.B37(1971)95.[4]R.Stora,in New development in quantumfield theories and statistical mechanics,(eds.H.Levy and P.Mitter),New York1977.and in Recent progress in Gauge Theories,ed.H.Letmann et al.,New York1984.[5]L.Bonora and P.Cotta–Ramusino,Phys.Lett.B107(1981)87.L.Bonora,Acta Phys.Pol.B13(1982)799.[6]B.Zumino,Wu Yong–shi and A.Zee,Nucl.Phys.B239(1984)477.[7]A.Connes,Noncommutative geometry,Academic Press,1994.[8]J.Madore,An introduction to noncommutative differential geometry and its physicalapplications,Cambridge University Press1995.[9]ndi,An introduction to noncommutative spaces and their geometries,SpringerVerlag1997.[10]ngmann,Descent equations of Yang–Mills anomalies in noncommutative geom-etry,J.Geom.Phys.22(1997)259-279.4。

高等数学英语词汇高等数学英语词汇引导语：高等数学指相对于初等数学而言，数学的对象及方法较为繁杂的'一部分。

以下是店铺分享给大家的高等数学英语词汇，欢迎阅读!Aabelian group：阿贝尔群; absolute geometry：绝对几何; absolute value：绝对值; abstract algebra：抽象代数; addition：加法; algebra：代数; algebraicclosure：代数闭包; algebraic geometry：代数几何;algebraic geometry and analytic geometry：代数几何和解析几何; algebraic numbers：代数数; algorithm：算法; almost all：绝大多数; analytic function：解析函数; analytic geometry：解析几何; and：且;angle：角度; anticommutative：反交换律; antisymmetric relation：反对称关系; antisymmetry：反对称性; approximately equal：约等于; Archimedean field：阿基米德域; Archimedean group：阿基米德群; area：面积; arithmetic：算术; associative algebra：结合代数; associativity：结合律; axiom：公理; axiom of constructibility：可构造公理; axiom of empty set：空集公理;axiom of extensionality：外延公理; axiom of foundation：正则公理; axiom of pairing：对集公理; axiom of regularity：正则公理; axiom of replacement：代换公理; axiom of union：并集公理; axiom schema of separation：分离公理; axiom schema of specification：分离公理;axiomatic set theory：公理集合论; axiomatic system：公理系统;BBaire space：贝利空间; basis：基; Bézout's identity：贝祖恒等式; Bernoulli's inequality：伯努利不等式 ; Big O notation：大O符号; bilinear operator：双线性算子; binary operation：二元运算; binary predicate：二元谓词; binary relation：二元关系; Booleanalgebra：布尔代数;Boolean logic：布尔逻辑; Boolean ring：布尔环; boundary：边界; boundary point：边界点;bounded lattice：有界格;Ccalculus：微积分学; Cantor's diagonal argument：康托尔对角线方法; cardinal number：基数;cardinality：势; cardinality of the continuum：连续统的势; Cartesian coordinate system：直角坐标系; Cartesian product：笛卡尔积; category：范畴; Cauchy sequence：柯西序列; Cauchy-Schwarz inequality：柯西不等式; Ceva's Theorem：塞瓦定理; characteristic：特征;characteristic polynomial：特征多项式; circle：圆; class：类; closed：闭集; closure：封闭性或闭包; closure algebra：闭包代数; combinatorial identities：组合恒等式; commutativegroup：交换群; commutative ring：交换环; commutativity:：交换律; compact：紧致的;compact set：紧致集合; compact space：紧致空间; complement：补集或补运算; completelattice：完备格; complete metric space：完备的度量空间; complete space：完备空间; complexmanifold：复流形; complex plane：复平面; congruence：同余; congruent：全等; connectedspace：连通空间; constructible universe：可构造全集; constructions of the real numbers：实数的构造; continued fraction：连分数; continuous：连续; continuum hypothesis：连续统假设;contractible space：可缩空间; convergence space：收敛空间; cosine：余弦; countable：可数;countable set：可数集; cross product：叉积; cycle space：圈空间; cyclic group：循环群;Dde Morgan's laws：德·摩根律; Dedekind completion：戴德金完备性; Dedekind cut：戴德金分割;del：微分算子; dense：稠密; densely ordered：稠密排列; derivative：导数; determinant：行列式; diffeomorphism：可微同构; difference：差; differentiablemanifold：可微流形;differential calculus：微分学; dimension：维数; directed graph：有向图; discrete space：离散空间; discriminant：判别式; distance：距离; distributivity：分配律; dividend：被除数;dividing：除; divisibility：整除; division：除法; divisor：除数; dot product：点积;Eeigenvalue：特征值; eigenvector：特征向量; element：元素; elementary algebra：初等代数;empty function：空函数; empty set：空集; empty product：空积; equal：等于; equality：等式或等于; equation：方程; equivalence relation：等价关系; Euclidean geometry：欧几里德几何;Euclidean metric：欧几里德度量; Euclidean space：欧几里德空间; Euler's identity：欧拉恒等式;even number：偶数; event：事件; existential quantifier：存在量词; exponential function：指数函数; exponential identities：指数恒等式; expression：表达式; extended real number line：扩展的实数轴;Ffalse：假; field：域; finite：有限; finite field：有限域; finite set：有限集合; first-countablespace：第一可数空间; first order logic：一阶逻辑; foundations of mathematics：数学基础;function：函数; functional analysis：泛函分析; functional predicate：函数谓词;fundamental theorem of algebra：代数基本定理; fraction：分数;Ggauge space：规格空间; general linear group：一般线性群; geometry：几何学; gradient：梯度;graph：图; graph of a relation：关系图; graph theory：图论; greatest element：最大元;group：群; group homomorphism：群同态;HHausdorff space：豪斯多夫空间; hereditarily finite set：遗传有限集合; Heron's formula：海伦公式; Hilbert space：希尔伯特空间;Hilbert's axioms：希尔伯特公理系统; Hodge decomposition：霍奇分解; Hodge Laplacian：霍奇拉普拉斯算子; homeomorphism：同胚; horizontal：水平;hyperbolic function identities：双曲线函数恒等式; hypergeometric function identities：超几何函数恒等式; hyperreal number：超实数;Iidentical：同一的; identity：恒等式; identity element：单位元; identity matrix：单位矩阵;idempotent：幂等; if：若; if and only if：当且仅当; iff：当且仅当; imaginary number：虚数;inclusion：包含; index set：索引集合; indiscrete space：非离散空间; inequality：不等式或不等; inequality of arithmetic and geometric means：平均数不等式; infimum：下确界; infiniteseries：无穷级数; infinite：无穷大; infinitesimal：无穷小; infinity：无穷大; initial object：初始对象; inner angle：内角; inner product：内积; inner product space：内积空间; integer：整数; integer sequence：整数列; integral：积分; integral domain：整数环; interior：内部;interior algebra：内部代数; interior point：内点; intersection：交集; inverse element：逆元;invertible matrix：可逆矩阵; interval：区间; involution：回旋; irrational number：无理数;isolated point：孤点; isomorphism：同构;JJacobi identity：雅可比恒等式; join：并运算;K格式： Kuratowski closure axioms：Kuratowski 闭包公理;Lleast element：最小元; Lebesgue measure：勒贝格测度; Leibniz's law：莱布尼茨律; Liealgebra：李代数; Lie group：李群; limit：极限; limit point：极限点; line：线; line segment：线段; linear：线性; linear algebra：线性代数; linear operator：线性算子; linear space：线性空间; linear transformation：线性变换; linearity：线性性; list of inequalities：不等式列表; list oflinear algebra topics：线性代数相关条目; locally compact space：局部紧致空间; logarithmicidentities：对数恒等式; logic：逻辑学; logical positivism：逻辑实证主义; law of cosines：余弦定理; L??wenheim-Skolem theorem：L??wenheim-Skolem 定理; lower limit topology：下限拓扑;Mmagnitude：量; manifold：流形; map：映射; mathematical symbols：数学符号; mathematicalanalysis：数学分析; mathematical proof：数学证明; mathematics：数学; matrix：矩阵;matrix multiplication：矩阵乘法; meaning：语义; measure：测度; meet：交运算; member：元素; metamathematics：元数学; metric：度量; metric space：度量空间; model：模型; modeltheory：模型论; modular arithmetic：模运算; module：模; monotonic function：单调函数;multilinear algebra：多重线性代数; multiplication：乘法; multiset：多样集;Nnaive set theory：朴素集合论; natural logarithm：自然对数; natural number：自然数; naturalscience：自然科学; negative number：负数; neighbourhood：邻域; New Foundations：新基础理论; nine point circle：九点圆; non-Euclidean geometry：非欧几里德几何; nonlinearity：非线性; non-singular matrix：非奇异矩阵; nonstandard model：非标准模型; nonstandardanalysis：非标准分析; norm：范数; normed vector space：赋范向量空间; n-tuple：n 元组或多元组; nullary：空; nullary intersection：空交集; number：数; number line：数轴;Oobject：对象; octonion：八元数; one-to-one correspondence：一一对应; open：开集; openball：开球; operation：运算; operator：算子; or：或; order topology：序拓扑; ordered field：有序域;ordered pair：有序对; ordered set：偏序集; ordinal number：序数; ordinarymathematics：一般数学; origin：原点; orthogonal matrix：正交矩阵;Pp-adic number：p进数; paracompact space：仿紧致空间; parallel postulate：平行公理;parallelepiped：平行六面体; parallelogram：平行四边形; partial order：偏序关系; partition：分割; Peano arithmetic：皮亚诺公理; Pedoe's inequality：佩多不等式; perpendicular：垂直;philosopher：哲学家; philosophy：哲学; philosophy journals：哲学类杂志; plane：平面; pluralquantification：复数量化; point：点; Point-Line-Plane postulate：点线面假设; polarcoordinates：极坐标系; polynomial：多项式; polynomial sequence：多项式列; positive-definitematrix：正定矩阵; positive-semidefinite matrix：半正定矩阵; power set：幂集; predicate：谓词; predicate logic：谓词逻辑; preorder：预序关系; prime number：素数; product：积;proof：证明; proper class：纯类; proper subset：真子集; property：性质; proposition：命题; pseudovector：伪向量; Pythagorean theorem：勾股定理;QQ.E.D.：Q.E.D.; quaternion：四元数; quaternions and spatial rotation：四元数与空间旋转;question：疑问句; quotient field：商域; quotient set：商集;Rradius：半径; ratio：比; rational number：有理数; real analysis：实分析; real closed field：实闭域; real line：实数轴; real number：实数; real number line：实数线; reflexive relation：自反关系; reflexivity：自反性; reification：具体化; relation：关系; relative complement：相对补集;relatively complemented lattice：相对补格; right angle：直角; right-handed rule：右手定则;ring：环;Sscalar：标量; second-countable space：第二可数空间; self-adjoint operator：自伴随算子;sentence：判断; separable space：可分空间; sequence：数列或序列; sequence space：序列空间; series：级数; sesquilinear function：半双线性函数; set：集合; set-theoretic definitionof natural numbers：自然数的集合论定义; set theory：集合论; several complex variables：一些复变量; shape：几何形状; sign function：符号函数; singleton：单元素集合; social science：社会科学; solid geometry：立体几何; space：空间; spherical coordinates：球坐标系; squarematrix：方块矩阵; square root：平方根; strict：严格; structural recursion：结构递归;subset：子集; subsequence：子序列; subspace：子空间; subspace topology：子空间拓扑;subtraction：减法; sum：和; summation：求和; supremum：上确界; surreal number：超实数; symmetric difference：对称差; symmetric relation：对称关系; system of linearequations：线性方程组;Ttensor：张量; terminal object：终结对象; the algebra of sets：集合代数; theorem：定理; topelement：最大元; topological field：拓扑域; topological manifold：拓扑流形; topological space：拓扑空间; topology：拓扑或拓扑学; total order：全序关系; totally disconnected：完全不连贯;totally ordered set：全序集; transcendental number：超越数; transfinite recursion：超限归纳法; transitivity：传递性; transitive relation：传递关系; transpose：转置; triangleinequality：三角不等式; trigonometric identities：三角恒等式; triple product：三重积; trivialtopology：密着拓扑; true：真; truth value：真值;Uunary operation：一元运算; uncountable：不可数; uniform space：一致空间; union：并集;unique：唯一; unit interval：单位区间; unit step function：单位阶跃函数; unit vector：单位向量;universal quantification：全称量词; universal set：全集; upper bound：上界;Vvacuously true：??; Vandermonde's identity：Vandermonde 恒等式; variable：变量;vector：向量; vector calculus：向量分析; vector space：向量空间; Venn diagram：文氏图;volume：体积; von Neumann ordinal：冯·诺伊曼序数; von Neumann universe：冯·诺伊曼全集;vulgar fraction：分数;ZZermelo set theory：策梅罗集合论; Zermelo-Fraenkel set theory：策梅罗-弗兰克尔集合论; ZF settheory：ZF 系统; zero：零; zero object：零对象;下载全文。

a r X i v :h e p -t h /0006167v 1 21 J u n 2000QUANTUM GROUPS AND NONCOMMUTATIVE GEOMETRYShahn MajidSchool of Mathematical Sciences,Queen Mary and Westfield College University of London,Mile End Rd,London E14NS,UK November,1999Abstract Quantum groups emerged in the latter quarter of the 20th century as,on the one hand,a deep and natural generalisation of symmetry groups for certain integrable systems,and on the other as part of a generalisation of geometry itself powerful enough to make sense in the quantum domain.Just as the last century saw the birth of classical geometry,so the present century sees at its end the birth of this quantum or noncommutative geometry,both as an elegant mathematical reality and in the form of the first theoretical predictions for Planck-scale physics via ongoing astronomical measurements.Noncommutativity of spacetime,in particular,amounts to a postulated new force or physical effect called cogravity.I Introduction Now that quantum groups and their associated quantum geometry have been around for more than a decade,it is surely time to take stock.Where did quantum groups come from,what have they achieved and where are they going?This article,which is addressed to non-specialists (but should also be interesting for experts)tries to answer this on two levels.First of all on the level of quantum groups themselves as mathematical tools and building blocks for physical models.And,equally importantly,quantum groups and their associated noncommutative geometry in terms of their overall significance for mathematics and theoretical physics,i.e.,at a more conceptual level.Obviously this latter aspect will be very much my own perspective,which is that of a theoretical physicist who came to quantum groups a decade ago as a tool to unify quantum theory and gravity in an algebraic approach to Planck scale physics.This is in fact only one of the two main origins in physics of quantum groups;the other being integrable systems,which I will try to cover as well.Let me also say that noncommutative geometry has other approaches,notably the one of A.Connes coming out of operator theory.I will say something about this too,although,until recently,this has largely been a somewhat different approach.We start with the conceptual significance for theoretical physics.It seems clear to me that future generations looking back on the 20th century will regard the discovery of quantum mechanics in the 1920s,i.e.the idea to replace the coordinates x,p of classical mechanics bynoncommuting operators x,p,as one of its greatest achievements in our understanding of Nature, matched in its significance only by the unification of space and time as a theory of gravity.But whereas the latter was well-founded in the classical geometry of Newton,Gauss,Riemann and Poincar´e,quantum theory was something much more radical and mysterious.Exactly which variables in the classical theory should correspond to operators?They are local coordinates on phase space but how does the global geometry of the classical theory look in the quantum theory,what does it fully correspond to?The problem for most of this century was that the required mathematical structures to which the classical geometry might correspond had not been invented and such questions could not be answered.As I hope to convince the reader,quantum groups and their associated noncommutative geometry have led in the last decades of the20th century to thefirst definitive answers to this kind of question.There has in fact emerged a more or less systematic generalisation of geometry every bit as radical as the step from Euclidean to non-Euclidean,and powerful enough not to break down in the quantum domain.I do doubt very much that what we know today will be thefinal formulation,but it is a definitive step in a right and necessary direction and a turning point in the future development of mathematical and theoretical physics.For example,any attempt to build a theory of quantum gravity with classical starting point a smooth manifold–this includes loop-variable quantum gravity,string theory and quantum cosmology,is necessarily misguided except as some kind of effective approximation:smooth manifolds should come out of the algebraic structure of the quantum theory and not be a starting point for the latter. There is no evidence that the real world is any kind of smooth continuum manifold except as a macroscopic approximation and every reason to think that it is fundamentally not.I therefore doubt that any one of the above could be a‘theory everything’until it becomes an entirely algebraic theory founded in noncommutative geometry of some kind or other.Of course,this is my personal view.At any rate,I do not think that the fundamental importance of noncommutative geometry can be overestimated.First of all,anyone who does quantum theory is doing noncommutative geometry whether wanting to admit it or not,namely noncommutative geometry of the phase space.Less obvious but also true,we will see in Section II that if the position space is curved then the momentum space is by itself intrinsically noncommutative.If one gets this far then it is also natural that the position space or spacetime by itself could be noncommutative,which would correspond to a curved or nonAbelian momentum group.This is one of the bolder predictions coming out of noncommutative geometry.It has the simple physical interpretation as what I call cogravity,i.e.curvature or‘gravity’in momentum space.As such it is independent of i.e. dual to curvature or gravity in spacetime and would appear as a quite different and new physical effect.Theoretically cogravity can,for example,be detected as energy-dependence of the speed8185909510000500981200yearpapersFigure 1:Growth of research papers on quantum groupsof light.Moreover,even if cogravity was very weak,of the order of a Planck-scale effect,it could still in principle be detected by astronomical measurements at a cosmological level.Therefore,just in time for the new millennium,we have the possibility of an entirely new physical effect in Nature coming from fresh and conceptually sound new mathematics .Where quantum groups precisely come into this is as follows.Just as Lie groups and their associated homogeneous spaces provided definitive examples of classical differential geometry even before Riemann formulated their intrinsic structure as a theory of manifolds,so quantum groups and their associated quantum homogeneous spaces,quantum planes etc.,provide large (i.e.infinite)classes of examples of proven mathematical and physical worth and clear geomet-rical content on which to build and develop noncommutative differential geometry.They are noncommutative spaces in the sense that they have generators or ‘coordinates’like the non-commuting operators x ,p in quantum mechanics but with a much richer and more geometric algebraic structure than the Heisenberg or CCR algebra.In particular,I do not believe that one can build a theory of noncommutative differential geometry based on only one example such as the Heisenberg algebra or its variants (however fascinating)such as the much-studied noncommutative torus.One needs many more ‘sample points’in the form of natural and varied examples to obtain a valid general theory.By contrast,if one does a search of BIDS one finds,see Figure 1,vast numbers of papers in which the rich structure and applications of quantum groups are explored and justified in their own right (data complied from BIDS:published pa-pers since 1981with title or abstract containing ‘quantum group*’,‘Hopf alg*’,‘noncommutative geom*’,‘braided categ*’,‘braided group*’,‘braided Hopf*’.)This is the significance of quantum groups.And of course something like them should be needed in a quantum world where there is no evidence for a classical space such as the underlying set of a Lie group.Finally,it turns out that noncommutative geometry,at least of the type that we shall de-scribe,is in many ways cleaner and more straightforward than the special commutative limit. One simply does not need to assume commutativity in most geometrical constructions,including differential calculus and gauge theory.The noncommutative version is often less infinite,dif-ferentials are often more regularfinite-differences,etc.And noncommutative geometry(unlike classical geometry)can be specialised without effort to discrete spaces or tofinite-dimensional algebras.It is simply a powerful and natural generalisation of geometry as we usually know it. So my overall summary and prediction for the next millennium from this point of view is:•All geometry will be noncommutative(or whatever comes beyond that),with conventional geometry merely a special case.•The discovery of quantum theory,its correspondence principle(and noncommutative ge-ometry is nothing more than the elaboration of that)will be considered one of the century’s greatest achievement in mathematical physics,commensurate with the discovery of clas-sical geometry by Newton some centuries before.•Quantum groups will be viewed as thefirst nontrivial class of examples and thereby point-ers to the correct structure of this noncommutative geometry.•Spacetime too(not only phase space)will be known to be noncommutative(cogravity will have been detected).•At some point a future Einstein will combine the then-standard noncommutative geomet-rical ideas with some deep philosophical ideas and explain something really fundamental about our physical reality.In the fun spirit of this article,I will not be above putting down my own thoughts on this last point.These have to do with what I have called for the last decade the Principle of representation-theoretic self-duality[1].In effect,it amounts to extending the ideas of Born reciprocity,Mach’s principle and Fourier theory to the quantum domain.Roughly speaking, quantum gravity should be recast as gravity and cogravity both present and dual to each other and with Einstein’s equation appearing as a self-duality condition.The longer-term philosophical implications are a Kantian or Hegelian view of the nature of physical reality,which I propose in Section V as a new foundation for next millennium.We now turn to another fundamental side of quantum groups,which is at the heart of their other origin in physics,namely as generalised symmetry groups in exactly solvable lattice models. It leads to diverse applications ranging from knot theory to representation theory to Poisson geometry,all areas that quantum groups have revolutionised.What is really going on here in my opinion is not so much the noncommutative geometry of quantum groups themselves as a different kind of noncommutativity or braid statistics which certain quantum groups induce onany objects of which they are a symmetry.The latter is what I have called‘noncommutativity of the second kind’or outer noncommutativity since it not so much a noncommutativity of one algebra as a noncommutative modification of the exchange law or tensor product of any two independent algebras or systems.It is the notion of independence which is really being deformed here.Recall that the other great‘isation’idea in mathematical physics in this century(after ‘quantisation’)was‘superisation’,where everything is Z2-graded and this grading enters into how two independent systems are interchanged.Physics traditionally has a division into bosonic or force particles and fermionic or matter particles according to this grading and exchange behaviour.So certain quantum groups lead to a generalisation of that as braided geometry[2]or a process of braidification.These quantum groups typically have a parameter q and its meaning is a generalisation of the−1for supersymmetry.This in turn leads to a profound generalisation of conventional(including super)mathematics in the form of a new concept of algebra wherin one‘wires up’algebraic operations much as the wiring in a computer,i.e.outputs of one into inputs of another.Only,this time,the under or over crossings are nontrivial(and generally distinct)operations depending on q.These are the so-called‘R-matrices’.Afterwards one has the luxury of both viewing q in this way or expanding it around1in terms of a multiple of Planck’s constant and calling it a formal‘quantisation’–q-deformation actually unifies both ‘isation’processes.For example,Lorentz-invariance,by the time it is q-deformed[3],induces braid statistics even when particles are initially bosonic.In summary,•The notion of symmetry or automorphism group is an artifact of classical geometry and ina quantum world should naturally be generalised to something more like a quantum groupsymmetry.•Quantum symmetry groups induce braid statistics on the systems on which they act.In particular,the notion of bose-fermi statistics or the division into force and matter particles is an artifact of classical geometry.•Quantisation and the departure from bosonic statistics are two limits of the same phe-nomenon of braided geometry.Again,there are plenty of concrete models in solid state physics already known with quantum group symmetry.The symmetry is useful and can be viewed(albeit with hindsight)as the origin of the exact solvability of these models.These two points of view,the noncommutative geometrical and the generalised symmetry, are to date the two main sources of quantum groups.One has correspondingly two mainflavours or types of quantum groups which really allowed the theory to take off.Both were introduced at the mid1980s although the latter have been more extensively studied in terms of applicationsto date.They include the deformationsU q(g)(1)of the enveloping algebra U(g)of every complex semisimple Lie algebra g[4][5].These have as many generators as the usual ones of the Lie algebra but modified relations and,additionally, a structure called the‘coproduct’.The general class here is that of quasitriangular quantum groups.They arose as generalised symmetries in certain lattice models but are also visible in the continuum limit quantumfield theories(such as the Wess-Zumino-Novikov-Witten model on the Lie group G with Lie algebra g).The coordinate algebras of these quantum groups are further quantum groups C q[G]deforming the commutative algebra of coordinate functions on G.There is again a coproduct,this time expressing the group law or matrix multiplication.Meanwhile, the type coming out of Planck scale physics[6]are the bicrossproduct quantum groupsC[M]◮⊳U(g)(2)associated to the factorisation of a Lie group X into Lie subgroups,X=GM.Here the in-gredients are the conventional enveloping algebra U(g)and the commutative coordinate algebra C[M].The factorisation is encoded in an action and coaction of one on the other to make a semidirect product and coproduct◮⊳.These quantum arose at about the same time but quite independently of the U q(g),as the quantum algebras of observables of certain quantum spaces. Namely it turns out that G acts on the set M(and vice-versa)and the quantisation of those orbits are these quantum groups.This means that they are literally noncommutative phase spaces of honest quantum systems.In particular,every complex semisimple g has an associated complexification and its Lie group factorises G C=GG⋆(the classical Iwasawa decomposition) so there is an exampleC[G⋆]◮⊳U(g)(3)built from just the same data as for U q(g).In fact the Iwasawa decomposition can be understood in Poisson-Lie terms with g⋆the classical‘Yang-Baxter dual’of g.In spite of this,there is,even after a decade of development,no direct connection between the two quantum groups:gւց(4)U q(g)←?→C[G⋆]◮⊳U(g).They are both‘exponentiations’of the same classical data but apparently of completely different type(this remains a mystery to date.)Figure2:The landscape of noncommutative geometry todayAssociated to these twoflavours of quantum groups there are corresponding homogeneous spaces such as quantum spheres,quantum spacetimes,etc.Thus,of thefirst type there is a q-Minkowski space introduced in[7]as a q-Lorentz covariant algebra,and independently about a year later in[8]as2×2braided hermitian matrices.It is characterised by[x i,t]=0,[x i,x j]=0.(5) Meanwhile,of the second type there is a noncommutativeλ-Minkowski space with[x i,t]=λx i,[x i,x j]=0(6)which is the one that provides thefirst known predictions testable by astronomical measurements (by gamma-ray bursts of cosmological origin[9]).This kind of algebra was proposed as spacetime in[10]and in the4-dimensional case it was shown in[11]to be covariant under a Poincar´e quantum group of bicrossproduct form.These are clearly in sharp contrast.There are of course many more objects than these.q-spheres,q-planes etc.In Section IV we turn to the notion of‘quantum manifold’that is emerging from all these examples.Riemann was able to formulate the notion of Riemannian manifold as a way to capture known examples like spheres and tori but broad enough to formulate general equations for the intrinsic structure of space itself(or after Einstein,space-time).We are at a similar point now and what this ‘quantum groups approach to noncommutative geometry’is is more or less taking shape.It has the same degree of‘flabbiness’as Riemannian geometry(it is not tied to specific integrable systems etc.)while at the same time it includes the‘zoo’of already known naturally occurring examples,mostly linked to quantum groups.Such things as Ricci tensor and Einstein’s equation are not yet understood from this approach,however,so I would not say it is the last word.This approach is in fairly sharp contrast to‘traditional’noncommutative geometry as it was done before the emergence of quantum groups.That theory was developed by mathematiciansand mathematical physicists also coming from quantum mechanics but being concerned more with topological completions and Hilbert spaces.Certainly a beautiful theory of von-Neumann and C∗algebras emerged as an analogue of point-set topology.Some general methods such as cyclic cohomology were also developed in the1970s,with remarkable applications throughout mathematics[12].However,for concrete examples with actual noncommutative differential geometry one usually turned either to an actual manifold as input datum or to the Weyl algebra (or noncommutative torus)defined by relationsvu=e2πıθuv.(7)This in turn is basically the usual CCR or Heisenberg algebra[x,p]=ı (8)in exponentiated form.And at an algebraic level(i.e.until one considers the precise C∗-algebra completion)this is basically the usual algebra B(H)of operators on a Hilbert space as in quantum mechanics.Or at roots of unity it is M n(C)the algebra of n×n matrices.So at some level these are all basically one example.Unfortunately many of the tricks one can pull for this kind of example are special to it and not a foundation for noncommutative differential geometry of the type we need.For example,to do gauge theory Connes and M.Rieffel[13] used derivations for two independent vectorfields on the torus.The formulation of‘vector field’as a derivation of the coordinate algebra is what I would call the traditional approach to noncommutative geometry.For quantum groups such as C q[G]one simply does not have those derivations(rather,they are in general braided derivations).Similarly,in the traditional approach one defines a‘vector bundle’as afinitely-generated projective module without any of the infrastructure of differential geometry such as a principal bundle to which the vector bundle might be associated,etc.All of that could not emerge until quantum groups arrived(one clearly should take a quantum group asfiber).This is how the quantum groups approach differs from the work of Connes,Rieffel,Madore and others.It is also worth noting that string theorists have recently woken up to the need for a noncommutative spacetime but,so far at least,have still considered only this‘traditional’Heisenberg-type algebra.In the last year or two there has been some success in merging these approaches,however;a trend surely to be continued. By now both approaches have a notion of‘noncommutative manifold’which appear somewhat different but which have as point of contact the Dirac operator.Preliminaries.A full text on quantum groups is[14].To be self-contained we provide here a quick defiter on we will see many examples and various justifications for this concept. Thus,a quantum group or Hopf algebra is•A unital algebra H,1over thefield C(say)•A coproduct∆:H→H⊗H and counitǫ:H→C forming a coalgebra,with∆,ǫalgebra homomorphisms.•An antipode S:H→H such that·(S⊗id)∆=1ǫ=·(id⊗S)∆.Here a coalgebra is just like an algebra but with the axioms written as maps and arrows on the maps reversed.Thus the coassociativity and counity axioms are(∆⊗id)∆=(id⊗∆)∆,(ǫ⊗id)∆=(id⊗ǫ)∆=id.(9)The antipode plays a role that generalises the concept of group inversion.Other than that the only new mathematical structure that the reader has to contend with is the coproduct∆and its associated counit.There are several ways of thinking about the meaning of this depending on our point of view.If the quantum group is like the enveloping algebra U(g)generated by a Lie algebra g,one should think of∆as providing the rule by which actions extend to tensor products.Thus,U(g)is trivially a Hopf algebra with∆ξ=ξ⊗1+1⊗ξ,∀ξ∈g,(10)which says that when a Lie algebra elementξacts on tensor products it does so byξin the first factor and thenξin the second factor.Similarly it says that when a Lie algebra acts on an algebra it does so as a derivation.On the other hand,if the quantum group is like a coordinate algebra C[G]then∆expresses the group multiplication andǫthe group identity element e. Thus,if f∈C[G]the coalgebra is(∆f)(g,h)=f(gh),∀g,h∈Gǫf=f(e)(11)at least for suitable f(or with suitable topological completions).In other words it expresses the group product G×G→G by a map in the other direction in terms of coordinate algebras. From yet another point of view∆simply makes the dual H∗also into an algebra.So a Hopf algebra is basically an algebra such that H∗is also an algebra,in a compatible way,which makes the axioms‘self-dual’.For everyfinite-dimensional H there is a dual H∗.Similarly in the infinite-dimensional case.It said that in the Roman empire,‘all roads led to Rome’.It is remarkable that several different ideas for generalising groups all led to the same axioms.The axioms themselves werefirst introduced(actually in a super context)by H.Hopf in1947in his study of group cohomology but the subject only came into its own in the mid1980s with the arrival from mathematical physics of the large classes of examples(as above)that are neither like U(g)nor like C[G],i.e.going truly beyond Lie theory or algebraic group theory.Acknowledgements.An announcement of this article appears in a short millennium article[15] and a version more focused on the meaning for Planck scale physics in[16].II Quantum groups and Planck scale physicsThis section covers quantum groups of the bicrossproduct type coming out of Planck-scale physics[6]and their associated noncommutative geometry.These are certainly less well-developed than the more familiar U q(g)in terms of their concrete applications;one does not have inter-esting knot invariants etc.On the other hand,these quantum groups have a clearer physical meaning as models of Planck scale physics and are also technically easier to construct.Therefore they are a good place to start.Obviously if we want to unify quantum theory and geometry then a necessaryfirst step is to cast both in the same language,which for us will be that of algebra.We have already mentioned that vectorfields can be thought of classically as derivations of the algebra of functions on the manifold,and if one wants points they can be recovered as maximal ideals in the algebra, etc.This is the more of less standard idea of algebraic geometry dating from the late19th century and early on in the20th.It will certainly need to be modified before it works in the noncommutative case but it is a starting point.The algebraic structure on the quantum side will need more attention,however.II.A CogravityWe begin with some very general considerations.In fact there are fundamental reasons why one needs noncommutative geometry for any theory that pretends to be a fundamental one.Since gravity and quantum theory both work extremely well in their separate domains,this comment refers mainly to a theory that might hope to unify the two.As a matter of fact I believe that, through noncommutative geometry,this‘holy grail’of theoretical physics may now be in sight.Thefirst point is that we usually do not try to apply or extend our geometrical intuition to the quantum domain directly,since the mathematics for that has traditionally not been known. Thus,one usually considers quantisation as the result of a process applied to an underlying classical phase space,with all of the geometrical content there(as a Poisson manifold).But demanding any algebra such that its commutators to lowest order are some given Poisson bracket is clearly an illogical and ill-defined process.It not only does not have a unique answer but also it depends on the coordinates chosen to map over the quantum operators.Almost always one takes the Poisson bracket in a canonical form and the quantisation is the usual CCR or canonical commutation relations algebra.Maybe this is the local picture but what of the global geometry of the classical phase space?Clearly all of these problems are putting the cart before the horse:the real world is to our best knowledge quantum so that should comefirst.We should build models guided by the intrinsic(noncommutative)geometry at the level of noncommutative algebras and only at the end consider classical limits and classical geometry(and Poisson brackets)as emerging from a choice,where possible,of‘classical handles’in the quantum system.In more physical terms,classical observables should come out of quantum theory as some kind of limit and not really be the starting point;in quantum gravity,for example,classical geometry should appear as an idealisation of the expectation value of certain operators in certain states of the system.Likewise in string theory one starts with strings moving in classical spacetime, defines Lagrangians etc.and tries to quantise.Even in more algebraic approaches,such as axiomatic quantumfield theory,one still assumes an underlying classical spacetime and classical Poincar´e group etc.,on which the operatorfields live.Yet if the real world is quantum then phase space and hence probably spacetime itself should be‘fuzzy’and only approximately modeled by classical geometrical concepts.Why then should one take classical geometrical concepts inside the functional integral except other than as an effective theory or approximate model tailored to the desired classical geometry that we hope to come out.This can be useful but it cannot possibly be the fundamental‘theory of everything’if it is built in such an illogical manner.There is simply no evidence for the assumption of nice smooth manifolds other than now-discredited classical mechanics.And in certain domains such as,but not only,in Planck scale physics or quantum gravity,it will certainly be unjustified even as an approximation.Next let us observe that any quantum system which contains a nonAbelian global symmetry group is already crying out for noncommutative geometry.This is in addition to the more obvious position-momentum noncommutativity of quantisation.The point is that if our quantum system has a nonAbelian Lie algebra symmetry,which is usually the case when the classical system does, then from among the quantum observables we should be able to realise the generators of this Lie algebra.That is,the algebra of observables A should contain the algebra generated by the Lie algebra,A⊇U(g).(12)Typically,A might be the semidirect product of a smaller part with external symmetry g by the action of U(g)(which means that in the bigger algebra the action of g is implemented by the commutator).This may soundfine but if the algebra A is supposed to be the quantum analogue of the‘functions on phase space’,then for part of it we should regard U(g)‘up side down’not as an enveloping algebra but as a noncommutative space with g the noncommutative coordinates.In other words,if we want to elucidate the geometrical content of the quantum algebra of observables then part of that will be to understand in what sense U(g)is a coordinate algebra,U(g)=C[?].(13)Here?cannot be an ordinary space because its supposed coordinate algebra U(g)is noncom-mutative.。

a rXiv:h ep-th/0358v22M a y23A note about the quantum of area in a non-commutative space Juan M.Romero,J.A.Santiago and J.David Vergara Instituto de Ciencias Nucleares,U.N.A.M.,Apdo.Postal 70-543,M´e xico D.F.,M´e xico sanpedro,santiago,vergara@nuclecu.unam.mx Abstract In this note we show that in a two-dimensional non-commutative space the area operator is quantized,this outcome is compared with the result obtained by Loop Quantum Gravity methods.PACS numbers:02.40.Gh,04.60-m,03.65.Bz One of the most relevant results of quantum mechanics is that physical quantities as the energy and the angular momenta,can have a discrete spec-tra.In fact,this theory begins with the Planck idea about the quantum nature of light.In contrast,general relativity modifies our understanding of gravity not as a force but as geometry.In recent years,there are severalattempts to reconcile the ideas of quantum mechanics and general relativ-ity.This implies in principle to change the geometrical properties of the space-time at the Planck length scales [1].One of this attempt is the loop quantum gravity [2].A remarkable result of this theory,is that geometrical operators as area or volume have discrete spectra at the Planck length order.For instance,one can define an operator for the area which has the spectrumA (j i )=8πγl 2pparameter[3].The spectrum of the area operator is a very important result of loop quantum gravity.However,considering the recent numerical analysis of spin foam models[4],it is possible that the correct area formula is given not by equation(1)but byA(j i)=γl2p(j i+1/2)(2) This result was proposed in Ref.[5]and our analysis for a two dimensional non-commutative space agrees with this spectra.On the other hand,the horizon area of black holes has been a relevant geometrical object.The quantization of black holes was proposed in the pioneering work of Bekenstein[6](see the reviews[7],[8]).He conjectured that the horizon area of a non extremal black hole should have a discrete and uniformly spaced spectrum,A n=˜γl2p n,n=1,2, (3)where˜γis a dimensionless constant.Bekenstein idea was to consider that the horizon area behaves as a classical adiabatic invariant,and then use the Ehrenfest principle[9],which does correspond to any classical adiabatic invariant a quantum operator with discrete spectrum.Recently,Hod[10] gave arguments based in the Bohr’s Correspondence Principle,to determine the constant˜γ=4ln3.Another proposal to explain the physics of small scales is to consider that the space is non commutative,i.e.,that the commutation relations between spacial coordinates is of the formx i, x j =i¯hΘij(4) whereΘij is an antisymmetric matrix with real and constant parameters.The seminal idea of this proposal has been attributed to Heisenberg[11],although thefirst report was given by Snyder[12].Non-commutative spaces appear in a natural way in the context of string theory under some backgrounds[13]. Nevertheless,one can also construct in a independent manner afield theory in a non-commutative space[14].The physic associated to this spaces is very interesting.For instance,infield the theory build with these commutators exist a relationship between ultraviolet and infrared divergences(mixture UV/IR)[15].In fact,this property could be required for the formulation of a Field Theory valid at small and large scales[16].In agreement with2the bounds found for¯hΘij,exist the possibility that the space may be non-commutative before reaching the Planck scale[17],[18].In this note we do a brief observation about the quantization of the area operator by using non-commutative quantum theory in a two dimensional Euclidean space.We begin with the commutation relation[ x1, x2]=i¯hΘΘ>0.(5) Defining the operatorsa=12¯hΘ( x1+i x2), a†=12¯hΘ( x1−i x2).(6) These operators satisfy the following commutation relationa, a† =1.(7) Using a and a†we build the number operator asN= a† a,(8) and thus wefindN, a† = a†, N, a =− a.(9) In this way,if N|n =n|n ,we identify to a†and a as creation and annihilation operators respectively.In terms of the coordinate operators we can writeN=1π≡ x21+ x22,it now follows from Eq.(10)thatATaking into account that the operator A/πis positive by definition,the operator N has a minimal eigenvalue.In consequence A/πis quantized with levels equally spaced by the interval2¯hΘ,Aπ|n =2¯hΘ(n+1/2)|n ,n=0,1, (12)with|n =( a†)n n!|0 (13) It is tempting consider toReferences[1]B.DeWitt,in Gravitation,edited by L.Witten,(1962)266.[2]C.Rovelli,Loop Quantum Gravity,Living Rev.Rel,1(1998)1,gr-qc/9710008;T,Thiemann,Introduction to Modern Canonical Quan-tum General Relativity gr-qc/0110034;S.Carlip,Quantum Gravity:a Progress Report,Rept.Prog.Phys.64(2001)885,gr-qc/0108040 [3]R.Capovilla,M.Montesinos,V.A.Prieto and E.Rojas,BF gravityand the Immirzi parameter Class.Quantum Grav.18(2001)L49. [4]J.C.Baez,J.D.Christensen,T.R.Halford,D.C.Tsang,Spin FoamModels of Riemannian Quantum Gravity,Class.Quantum Grav.19 (2002)4627,gr-qc/0202017.[5]A.Alekseev,A.P.Polychronakos,M.Smedbck,On area and entropy ofa black hole,hep-th/0004036.[6]J.D.Bekenstein,Lett.Nuovo Cimento11,467(1974).[7]J.D.Bekenstein,in Proceedings of the VIII Marcel Grossmann Meet-ing on General Relativity,edited by T.Piran and R.Ruffini(World Scientific,Singapore,1998),gr-qc/9710076.[8]J.D.Bekenstein,in Proceedings of the XVII Brazilian National Meetingon Particles and Fields,edited by A.J.da Silva et al.(Brazilian Physical Society,Sao Paulo1996),gr-qc/9808028.[9]P.Ehrenfest,Collected scientific papers,edited by M.J.Klein,(North-Holland Pub.Co.1959).[10]S.Hod,Bohr’s Correspondence Principle and the Area Spectrum ofQuantum Black Holes,Phys.Rev.Lett.81(1998)4293,gr-qc/9812002.[11]R.Jackiw,Physical Instances of Noncommuting Coordinates,Nucl.Phys.Proc.Suppl.108(2002)30-36;Phys.Part.Nucl.33(2002)S6-S11,hep-th/0110057;J.Polchinski,M Theory:Uncertainty and Unifi-cation,hep-th/0209105.[12]H.Snyder,Quantized Space-Time,Phy.Rev.71,38(1947).5[13]N.Seiberg and E.Witten,String Theory and Noncommutative Geome-try,JHEP9909(1999)032,hep-th/9908142.[14]R.J.Szabo,Quantum Field Theory on Noncommutative Spaces,hep-th/0109162[15]S.Minwalla,M.V.Raamsdonk,N.Seiberg Noncommutative Perturba-tive Dynamics,JHEP0002(2000)020,hep-th/9912072.[16]A.G.Cohen,D.B.Kaplan,A.E.Nelson,Effective Field Theory,BlackHoles,and the Cosmological Constant,Phys.Rev.Lett.82(1999)4971-4974,hep-th/9803132.[17]C.E.Carlson,C.D.Carone and R.F.Lebed,Bounding noncommutativeQCD,Phys.Lett.B518(2001)201,hep-ph/0107291.[18]I.Mocioiu,M.Pospelov,R.Roiban,Low-energy Limits on the An-tisymmetric Tensor Field Background on the Brane and on the Non-commutative Scale,Phys.Lett.B489(2000)390-396,hep-ph/0005191.[19]C.Rovelli,Spectral Noncommutative Geometry and Quantization,Phys.Rev.Lett.83,1079(1999).6。

英语词汇-数学词汇abbreviation 简写符号；简写abscissa 横坐标absolute complement 绝对补集absolute error 绝对误差absolute inequality 绝不等式absolute maximum 绝对极大值absolute minimum 绝对极小值absolute monotonic 绝对单调absolute value 绝对值accelerate 加速acceleration 加速度acceleration due to gravity 重力加速度; 地心加速度accumulation 累积accumulative 累积的accuracy 准确度act on 施于action 作用; 作用力acute angle 锐角acute-angled triangle 锐角三角形add 加addition 加法addition formula 加法公式addition law 加法定律addition law(of probability) （概率）加法定律additive inverse 加法逆元; 加法反元additive property 可加性adjacent angle 邻角adjacent side 邻边adjoint matrix 伴随矩阵algebra 代数algebraic 代数的algebraic equation 代数方程algebraic expression 代数式algebraic fraction 代数分式；代数分数式algebraic inequality 代数不等式algebraic number 代数数algebraic operation 代数运算algebraically closed 代数封闭algorithm 算法系统; 规则系统alternate angle （交）错角alternate segment 内错弓形alternating series 交错级数alternative hypothesis 择一假设; 备择假设; 另一假设altitude 高；高度；顶垂线；高线ambiguous case 两义情况；二义情况amount 本利和；总数analysis 分析；解析analytic geometry 解析几何angle 角angle at the centre 圆心角angle at the circumference 圆周角angle between a line and a plane 直与平面的交角angle between two planes 两平面的交角angle bisection 角平分angle bisector 角平分线；分角线angle in the alternate segment 交错弓形的圆周角angle in the same segment 同弓形内的圆周角angle of depression 俯角angle of elevation 仰角angle of friction 静摩擦角; 极限角angle of greatest slope 最大斜率的角angle of inclination 倾斜角angle of intersection 相交角；交角angle of projection 投射角angle of rotation 旋转角angle of the sector 扇形角angle sum of a triangle 三角形内角和angles at a point 同顶角angular displacement 角移位angular momentum 角动量angular motion 角运动angular velocity 角速度annum(X% per annum) 年（年利率X%）anti-clockwise direction 逆时针方向；返时针方向anti-clockwise moment 逆时针力矩anti-derivative 反导数; 反微商anti-logarithm 逆对数；反对数anti-symmetric 反对称apex 顶点approach 接近；趋近approximate value 近似值approximation 近似；略计；逼近Arabic system 阿刺伯数字系统arbitrary 任意arbitrary constant 任意常数arc 弧arc length 弧长arc-cosine function 反余弦函数arc-sin function 反正弦函数arc-tangent function 反正切函数area 面积Argand diagram 阿根图, 阿氏图argument (1)论证; (2)辐角argument of a complex number 复数的辐角argument of a function 函数的自变量arithmetic 算术arithmetic mean 算术平均；等差中顶；算术中顶arithmetic progression 算术级数；等差级数arithmetic sequence 等差序列arithmetic series 等差级数arm 边array 数组; 数组arrow 前号ascending order 递升序ascending powers of X X 的升幂assertion 断语; 断定associative law 结合律assumed mean 假定平均数assumption 假定；假设asymmetrical 非对称asymptote 渐近asymptotic error constant 渐近误差常数at rest 静止augmented matrix 增广矩阵auxiliary angle 辅助角auxiliary circle 辅助圆auxiliary equation 辅助方程average 平均；平均数；平均值average speed 平均速率axiom 公理axiom of existence 存在公理axiom of extension 延伸公理axiom of inclusion 包含公理axiom of pairing 配对公理axiom of power 幂集公理axiom of specification 分类公理axiomatic theory of probability 概率公理论axis 轴axis of parabola 抛物线的轴axis of revolution 旋转轴axis of rotation 旋转轴axis of symmetry 对称轴back substitution 回代bar chart 棒形图；条线图；条形图；线条图base （1）底；（2）基；基数base angle 底角base area 底面base line 底线base number 底数；基数base of logarithm 对数的底basis 基Bayes′theorem 贝叶斯定理bearing 方位（角）；角方向（角）bell-shaped curve 钟形图belong to 属于Bernoulli distribution 伯努利分布Bernoulli trials 伯努利试验bias 偏差；偏倚biconditional 双修件式; 双修件句bijection 对射; 双射; 单满射bijective function 对射函数; 只射函数billion 十亿bimodal distribution 双峰分布binary number 二进数binary operation 二元运算binary scale 二进法binary system 二进制binomial 二项式binomial distribution 二项分布binomial expression 二项式binomial series 二项级数binomial theorem 二项式定理bisect 平分；等分bisection method 分半法；分半方法bisector 等分线；平分线Boolean algebra 布尔代数boundary condition 边界条件boundary line 界（线）；边界bounded 有界的bounded above 有上界的；上有界的bounded below 有下界的；下有界的bounded function 有界函数bounded sequence 有界序列brace 大括号bracket 括号breadth 阔度broken line graph 折线图calculation 计算calculator 计算器；计算器calculus (1) 微积分学; (2) 演算cancel 消法；相消canellation law 消去律canonical 典型; 标准capacity 容量cardioid 心脏Cartesian coordinates 笛卡儿坐标Cartesian equation 笛卡儿方程Cartesian plane 笛卡儿平面Cartesian product 笛卡儿积category 类型；范畴catenary 悬链Cauchy sequence 柯西序列Cauchy′s principal value 柯西主值Cauchy-Schwarz inequality 柯西- 许瓦尔兹不等式central limit theorem 中心极限定理central line 中线central tendency 集中趋centre 中心；心centre of a circle 圆心centre of gravity 重心centre of mass 质量中心centrifugal force 离心力centripedal acceleration 向心加速度centripedal force force 向心力centroid 形心；距心certain event 必然事件chain rule 链式法则chance 机会change of axes 坐标轴的变换change of base 基的变换change of coordinates 坐标轴的变换change of subject 主项变换change of variable 换元；变量的换characteristic equation 特征(征)方程characteristic function 特征(征)函数characteristic of logarithm 对数的首数; 对数的定位部characteristic root 特征(征)根chart 图；图表check digit 检验数位checking 验算chord 弦chord of contact 切点弦circle 圆circular 圆形；圆的circular function 圆函数；三角函数circular measure 弧度法circular motion 圆周运动circular permutation 环形排列; 圆形排列; 循环排列circumcentre 外心；外接圆心circumcircle 外接圆circumference 圆周circumradius 外接圆半径circumscribed circle 外接圆cissoid 蔓叶class 区；组；类class boundary 组界class interval 组区间；组距class limit 组限；区限class mark 组中点；区中点classical theory of probability 古典概率论classification 分类clnometer 测斜仪clockwise direction 顺时针方向clockwise moment 顺时针力矩closed convex region 闭凸区域closed interval 闭区间coaxial 共轴coaxial circles 共轴圆coaxial system 共轴系coded data 编码数据coding method 编码法co-domain 上域coefficient 系数coefficient of friction 摩擦系数coefficient of restitution 碰撞系数; 恢复系数coefficient of variation 变差系数cofactor 余因子; 余因式cofactor matrix 列矩阵coincide 迭合；重合collection of terms 并项collinear 共线collinear planes 共线面collision 碰撞column (1)列；纵行；(2) 柱column matrix 列矩阵column vector 列向量combination 组合common chord 公弦common denominator 同分母；公分母common difference 公差common divisor 公约数；公约common factor 公因子；公因子common logarithm 常用对数common multiple 公位数；公倍common ratio 公比common tangent 公切commutative law 交换律comparable 可比较的compass 罗盘compass bearing 罗盘方位角compasses 圆规compasses construction 圆规作图compatible 可相容的complement 余；补余complement law 补余律complementary angle 余角complementary equation 补充方程complementary event 互补事件complementary function 余函数complementary probability 互补概率complete oscillation 全振动completing the square 配方complex conjugate 复共轭complex number 复数complex unmber plane 复数平面complex root 复数根component 分量component of force 分力composite function 复合函数; 合成函数composite number 复合数；合成数composition of mappings 映射构合composition of relations 复合关系compound angle 复角compound angle formula 复角公式compound bar chart 综合棒形图compound discount 复折扣compound interest 复利；复利息compound probability 合成概率compound statement 复合命题; 复合叙述computation 计算computer 计算机；电子计算器concave 凹concave downward 凹向下的concave polygon 凹多边形concave upward 凹向上的concentric circles 同心圆concept 概念conclusion 结论concurrent 共点concyclic 共圆concyclic points 共圆点condition 条件conditional 条件句；条件式conditional identity 条件恒等式conditional inequality 条件不等式conditional probability 条件概率cone 锥；圆锥（体）confidence coefficient 置信系数confidence interval 置信区间confidence level 置信水平confidence limit 置信极限confocal section 共焦圆锥曲congruence (1)全等；(2)同余congruence class 同余类congruent 全等congruent figures 全等图形congruent triangles 全等三角形conic 二次曲; 圆锥曲conic section 二次曲; 圆锥曲conical pendulum 圆锥摆conjecture 猜想conjugate 共轭conjugate axis 共轭conjugate diameters 共轭轴conjugate hyperbola 共轭(直)径conjugate imaginary / complex number 共轭双曲conjugate radical 共轭虚/复数conjugate surd 共轭根式; 共轭不尽根conjunction 合取connective 连词connector box 捙接框consecutive integers 连续整数consecutive numbers 连续数；相邻数consequence 结论；推论consequent 条件；后项conservation of energy 能量守恒conservation of momentum 动量守恒conserved 守恒consistency condition 相容条件consistent 一贯的；相容的consistent estimator 相容估计量constant 常数constant acceleration 恒加速度constant force 恒力constant of integration 积分常数constant speed 恒速率constant term 常项constant velocity 怛速度constraint 约束；约束条件construct 作construction 作图construction of equation 方程的设立continued proportion 连比例continued ratio 连比continuity 连续性continuity correction 连续校正continuous 连续的continuous data 连续数据continuous function 连续函数continuous proportion 连续比例continuous random variable 连续随机变量contradiction 矛盾converge 收敛convergence 收敛性convergent 收敛的convergent iteration 收敛的迭代convergent sequence 收敛序列convergent series 收敛级数converse 逆(定理)converse of a relation 逆关系converse theorem 逆定理conversion 转换convex 凸convex polygon 凸多边形convexity 凸性coordinate 坐标coordinate geometry 解析几何；坐标几何coordinate system 坐标系系定理；系；推论coplanar 共面coplanar forces 共面力coplanar lines 共面co-prime 互质; 互素corollary 系定理; 系; 推论correct to 准确至；取值至correlation 相关correlation coefficient 相关系数correspondence 对应corresponding angles (1)同位角；(2)对应角corresponding element 对应边corresponding sides 对应边cosecant 余割cosine 余弦cosine formula 余弦公式cost price 成本cotangent 余切countable 可数countable set 可数集countably infinite 可数无限counter clockwise direction 逆时针方向；返时针方向counter example 反例counting 数数；计数couple 力偶Carmer′s rule 克莱玛法则criterion 准则critical point 临界点critical region 临界域cirtical value 临界值cross-multiplication 交叉相乘cross-section 横切面；横截面；截痕cube 正方体；立方；立方体cube root 立方根cubic 三次方；立方；三次(的)cubic equation 三次方程cubic roots of unity 单位的立方根cuboid 长方体；矩体cumulative 累积的cumulative distribution function 累积分布函数cumulative frequecy 累积频数；累积频率cumulative frequency curve 累积频数曲cumulative frequcncy distribution 累积频数分布cumulative frequency polygon 累积频数多边形；累积频率直方图curvature of a curve 曲线的曲率curve 曲线curve sketching 曲线描绘(法)curve tracing 曲线描迹(法)curved line 曲线curved surface 曲面curved surface area 曲面面积cyclic expression 输换式cyclic permutation 圆形排列cyclic quadrilateral 圆内接四边形cycloid 旋输线; 摆线cylinder 柱；圆柱体cylindrical 圆柱形的damped oscillation 阻尼振动data 数据De Moivre′s theorem 棣美弗定理De Morgan′s law 德摩根律decagon 十边形decay 衰变decay factor 衰变因子decelerate 减速decelaration 减速度decile 十分位数decimal 小数decimal place 小数位decimal point 小数点decimal system 十进制decision box 判定框declarative sentence 说明语句declarative statement 说明命题decoding 译码decrease 递减decreasing function 递减函数；下降函数decreasing sequence 递减序列；下降序列decreasing series 递减级数；下降级数decrement 减量deduce 演绎deduction 推论deductive reasoning 演绎推理definite 确定的；定的definite integral 定积分definition 定义degenerated conic section 降级锥曲线degree (1) 度; (2) 次degree of a polynomial 多项式的次数degree of accuracy 准确度degree of confidence 置信度degree of freedom 自由度degree of ODE 常微分方程次数degree of precision 精确度delete 删除; 删去denary number 十进数denominator 分母dependence (1)相关; (2)应变dependent event(s) 相关事件; 相依事件; 从属事件dependent variable 应变量; 应变数depreciation 折旧derivable 可导derivative 导数derived curve 导函数曲线derived function 导函数derived statistics 推算统计资料; 派生统计资料descending order 递降序descending powers of x x的降序descriptive statistics 描述统计学detached coefficients 分离系数(法) determinant 行列式deviation 偏差; 变差deviation from the mean 离均差diagonal 对角线diagonal matrix 对角矩阵diagram 图; 图表diameter 直径diameter of a conic 二次曲线的直径difference 差difference equation 差分方程difference of sets 差集differentiable 可微differential 微分differential coefficient 微商; 微分系数differential equation 微分方程differential mean value theorem 微分中值定理differentiate 求...的导数differentiate from first principle 从基本原理求导数differentiation 微分法digit 数字dimension 量; 量网; 维(数)direct impact 直接碰撞direct image 直接像direct proportion 正比例direct tax, direct taxation 直接税direct variation 正变(分)directed angle 有向角directed line 有向直线directed line segment 有向线段directed number 有向数direction 方向; 方位direction angle 方向角direction cosine 方向余弦direction number 方向数direction ratio 方向比directrix 准线Dirichlet function 狄利克来函数discontinuity 不连续性discontinuous 间断(的);连续(的); 不连续(的) discontinuous point 不连续点discount 折扣discrete 分立; 离散discrete data 离散数据; 间断数据discrete random variable 间断随机变数discrete uniform distribution 离散均匀分布discriminant 判别式disjoint 不相交的disjoint sets 不相交的集disjunction 析取dispersion 离差displacement 位移disprove 反证distance 距离distance formula 距离公式distinct roots 相异根distincr solution 相异解distribution 公布distributive law 分配律diverge 发散divergence 发散(性)divergent 发散的divergent iteration 发散性迭代divergent sequence 发散序列divergent series 发散级数divide 除dividend (1)被除数；(2)股息divisible 可整除division 除法division algorithm 除法算式divisor 除数；除式；因子divisor of zero 零因子dodecagon 十二边形domain 定义域dot 点dot product 点积double angle 二倍角double angle formula 二倍角公式double root 二重根dual 对偶duality (1)对偶性; (2) 双重性due east/ south/ west /north 向东/ 南/ 西/ 北dynamics 动力学eccentric angle 离心角eccentric circles 离心圆eccentricity 离心率echelon form 梯阵式echelon matrix 梯矩阵edge 棱；边efficient estimator 有效估计量effort 施力eigenvalue 本征值eigenvector 本征向量elastic body 弹性体elastic collision 弹性碰撞elastic constant 弹性常数elastic force 弹力elasticity 弹性element 元素elementary event 基本事件elementary function 初等函数elementary row operation 基本行运算elimination 消法elimination method 消去法；消元法ellipse 椭圆ellipsiod 椭球体elliptic function 椭圆函数elongation 伸张；展empirical data 实验数据empirical formula 实验公式empirical probability 实验概率；经验概率empty set 空集encoding 编码enclosure 界限end point 端点energy 能; 能量entire surd 整方根epicycloid 外摆线equal 相等equal ratios theorem 等比定理equal roots 等根equal sets 等集equality 等(式)equality sign 等号equation 方程equation in one unknown 一元方程equation in two unknowns (variables) 二元方程equation of a straight line 直线方程equation of locus 轨迹方程equiangular 等角(的)equidistant 等距(的)equilateral 等边(的)equilateral polygon 等边多边形equilateral triangle 等边三角形equilibrium 平衡equiprobable 等概率的equiprobable space 等概率空间equivalence 等价equivalence class 等价类equivalence relation 等价关系equivalent 等价(的)error 误差error allowance 误差宽容度error estimate 误差估计error term 误差项error tolerance 误差宽容度escribed circle 旁切圆estimate 估计；估计量estimator 估计量Euclidean algorithm 欧几里德算法Euclidean geometry 欧几里德几何Euler′s formula 尤拉公式；欧拉公式evaluate 计值even function 偶函数even number 偶数evenly distributed 均匀分布的event 事件exact 真确exact differential form 恰当微分形式exact solution 准确解；精确解；真确解exact value 法确解；精确解；真确解example 例excentre 外心exception 例外excess 起exclusive 不包含exclusive disjunction 不包含性析取exclusive events 互斥事件exercise 练习exhaustive event(s) 彻底事件existential quantifier 存在量词expand 展开expand form 展开式expansion 展式expectation 期望expectation value, expected value 期望值；预期值experiment 实验；试验experimental 试验的experimental probability 实验概率explicit function 显函数exponent 指数exponential function 指数函数exponential order 指数阶; 指数级express…in terms of…以………表达expression 式；数式extension 外延；延长；扩张；扩充extension of a function 函数的扩张exterior angle 外角external angle bisector 外分角external point of division 外分点extreme point 极值点extreme value 极值extremum 极值face 面factor 因子；因式；商factor method 因式分解法factor theorem 因子定理；因式定理factorial 阶乘factorization 因子分解；因式分解factorization of polynomial 多项式因式分解fallacy 谬误FALSE 假(的)falsehood 假值family 族family of circles 圆族family of concentric circles 同心圆族family of straight lines 直线族feasible solution 可行解；容许解Fermat′s last theorem 费尔马最后定理Fibonacci number 斐波那契数；黄金分割数Fibonacci sequence 斐波那契序列fictitious mean 假定平均数figure (1)图(形)；(2)数字final velocity 末速度finite 有限finite dimensional vector space 有限维向量空间finite population 有限总体finite probability space 有限概率空间finite sequence 有限序列finite series 有限级数finite set 有限集first approximation 首近似值first derivative 一阶导数first order differential equation 一阶微分方程first projection 第一投影; 第一射影first quartile 第一四分位数first term 首项fixed deposit 定期存款fixed point 定点fixed point iteration method 定点迭代法fixed pulley 定滑轮flow chart 流程图focal axis 焦轴focal chord 焦弦focal length 焦距focus(foci) 焦点folium of Descartes 笛卡儿叶形线foot of perpendicular 垂足for all X 对所有Xfor each /every X 对每一Xforce 力forced oscillation 受迫振动form 形式；型formal proof 形式化的证明format 格式；规格formula(formulae) 公式four leaved rose curve 四瓣玫瑰线four rules 四则four-figure table 四位数表fourth root 四次方根fraction 分数；分式fraction in lowest term 最简分数fractional equation 分式方程fractional index 分数指数fractional inequality 分式不等式free fall 自由下坠free vector 自由向量; 自由矢量frequency 频数；频率frequency distribution 频数分布；频率分布frequency distribution table 频数分布表frequency polygon 频数多边形；频率多边形friction 摩擦; 摩擦力frictionless motion 无摩擦运动frustum 平截头体fulcrum 支点function 函数function of function 复合函数；迭函数functional notation 函数记号fundamental theorem of algebra 代数基本定理fundamental theorem of calculus 微积分基本定理gain 增益；赚；盈利gain perent 赚率；增益率；盈利百分率game （1）对策；（2）博奕Gaussian distribution 高斯分布Gaussian elimination 高斯消去法general form 一般式；通式general solution 通解；一般解general term 通项generating function 母函数; 生成函数generator (1)母线; (2)生成元geoborad 几何板geometric distribution 几何分布geometric mean 几何平均数；等比中项geometric progression 几何级数；等比级数geometric sequence 等比序列geometric series 等比级数geometry 几何；几何学given 给定；已知global 全局; 整体global maximum 全局极大值; 整体极大值global minimum 全局极小值; 整体极小值golden section 黄金分割grade 等级gradient (1)斜率；倾斜率；(2)梯度grand total 总计graph 图像；图形；图表graph paper 图表纸graphical method 图解法graphical representation 图示；以图样表达graphical solution 图解gravitational acceleration 重力加速度gravity 重力greatest term 最大项greatest value 最大值grid lines 网网格线group 组；grouped data 分组数据；分类数据grouping terms 并项；集项growth 增长growth factor 增长因子half angle 半角half angle formula 半角公式half closed interval 半闭区间half open interval 半开区间harmonic mean (1) 调和平均数; (2) 调和中项harmonic progression 调和级数head 正面（钱币）height 高（度）helix 螺旋线hemisphere 半球体；半球heptagon 七边形Heron′s formula 希罗公式heterogeneous (1)参差的; (2)不纯一的hexagon 六边形higher order derivative 高阶导数highest common factor(H.C.F) 最大公因子；最高公因式；最高公因子Hindu-Arabic numeral 阿刺伯数字histogram 组织图；直方图；矩形图Holder′s Inequality 赫耳德不等式homogeneous 齐次的homogeneous equation 齐次方程Hooke′s law 虎克定律horizontal 水平的；水平horizontal asymptote 水平渐近线horizontal component 水平分量horizontal line 横线；水平线horizontal range 水平射程hyperbola 双曲线hyperbolic function 双曲函数hypergeometric distribution 超几何分布hypocycloid 内摆线hypotenuse 斜边hypothesis 假设hypothesis testing 假设检验hypothetical syllogism 假设三段论hypotrochoid 次内摆线idempotent 全幂等的identical 全等；恒等identity 等（式）identity element 单位元identity law 同一律identity mapping 恒等映射identity matrix 恒等矩阵identity relation 恒等关系式if and only if/iff 当且仅当；若且仅若if…, then 若….则；如果…..则illustration 例证；说明image 像点；像image axis 虚轴imaginary circle 虚圆imaginary number 虚数imaginary part 虚部imaginary root 虚根imaginary unit 虚数单位impact 碰撞implication 蕴涵式；蕴含式implicit definition 隐定义implicit function 隐函数imply 蕴涵；蕴含impossible event 不可能事件improper fraction 假分数improper integral 广义积分; 非正常积分impulse 冲量impulsive force 冲力incentre 内力incircle 内切圆inclination 倾角；斜角inclined plane 斜面included angle 夹角included side 夹边inclusion mapping 包含映射inclusive 包含的；可兼的inclusive disjunction 包含性析取；可兼析取inconsistent 不相的(的)；不一致(的) increase 递增；增加increasing function 递增函数increasing sequence 递增序列increasing series 递增级数increment 增量indefinite integral 不定积分idenfinite integration 不定积分法independence 独立；自变independent equations 独立方程independent event 独立事件independent variable 自变量；独立变量indeterminate (1)不定的；(2)不定元；未定元indeterminate coefficient 不定系数；未定系数indeterminate form 待定型；不定型index,indices 指数；指index notation 指数记数法induced operation 诱导运算induction hypothesis 归纳法假设inelastic collision 非弹性碰撞inequality 不等式；不等inequality sign 不等号inertia 惯性；惯量infer 推断inference 推论infinite 无限；无穷infinite dimensional 无限维infinite population 无限总体infinite sequence 无限序列；无穷序列infinite series 无限级数；无穷级数infinitely many 无穷多infinitesimal 无限小；无穷小infinity 无限(大)；无穷(大)inflection (inflexion) point 拐点；转折点inherent error 固有误差initial approximation 初始近似值initial condition 原始条件；初值条件initial point 始点；起点initial side 始边initial value 初值；始值initial velocity 初速度initial-value problem 初值问题injection 内射injective function 内射函数inner product 内积input 输入input box 输入inscribed circle 内切圆insertion 插入insertion of brackets 加括号instantaneous 瞬时的instantaneous acceleration 瞬时加速度instantaneous speed 瞬时速率instantaneous velocity 瞬时速度integer 整数integrable 可积integrable function 可积函数integral 积分integral index 整数指数integral mean value theorem 积数指数integral part 整数部份integral solution 整数解integral value 整数值integrand 被积函数integrate 积；积分；......的积分integrating factor 积分因子integration 积分法integration by parts 分部积分法integration by substitution 代换积分法；换元积分法integration constant 积分常数interaction 相互作用intercept 截距；截段intercept form 截距式intercept theorem 截线定理interchange 互换interest 利息interest rate 利率interest tax 利息税interior angle 内角interior angles on the same side of the transversal 同旁内角interior opposite angle 内对角intermediate value theorem 介值定理internal bisector 内分角internal division 内分割internal energy 内能internal force 内力internal point of division 内分点interpolating polynomial 插值多项式interpolation 插值inter-quartile range 四分位数间距intersect 相交intersection (1)交集；(2)相交；(3)交点interval 区间interval estimation 区间估计；区域估计intuition 直观invalid 失效；无效invariance 不变性invariant (1)不变的；(2)不变量；不变式inverse 反的；逆的inverse circular function 反三角函数inverse cosine function 反余弦函数inverse function 反函数；逆函数inverse cosine function 反三角函数inverse function 反函数；逆映射inverse mapping 反向映射；逆映射inverse matrix 逆矩阵inverse problem 逆算问题inverse proportion 反比例；逆比例inverse relation 逆关系inverse sine function 反正弦函数inverse tangent function 反正切函数inverse variation 反变(分)；逆变(分) invertible 可逆的invertible matrix 可逆矩阵irrational equation 无理方程irrational number 无理数irreducibility 不可约性irregular 不规则isomorphism 同构isosceles triangle 等腰三角形iterate (1)迭代值; (2)迭代iteration 迭代iteration form 迭代形iterative function 迭代函数iterative method 迭代法jet propulsion 喷气推进joint variation 联变(分)；连变(分) kinetic energy 动能kinetic friction 动摩擦known 己知L.H.S. 末项L′Hospital′s rule 洛必达法则Lagrange interpolating polynomial 拉格朗日插值多项代Lagrange theorem 拉格朗日定理Lami′s law 拉密定律Laplace expansion 拉普拉斯展式last term 末项latent root 本征根; 首通径lattice point 格点latus rectum 正焦弦; 首通径law 律；定律law of conservation of momentum 动量守恒定律law of indices 指数律；指数定律law of inference 推论律law of trichotomy 三分律leading coefficient 首项系数leading diagonal 主对角线least common multiple, lowest common multiple (L.C.M) 最小公倍数；最低公倍式least value 最小值left hand limit 左方极限lemma 引理lemniscate 双纽线length 长(度)letter 文字；字母like surd 同类根式like terms 同类项limacon 蜗牛线limit 极限limit of sequence 序列的极限limiting case 极限情况limiting friction 最大静摩擦limiting position 极限位置line 线；行line of action 作用力线line of best-fit 最佳拟合line of greatest slope 最大斜率的直；最大斜率line of intersection 交线line segment 线段linear 线性；一次linear convergence 线性收敛性linear differeantial equation 线性微分方程linear equation 线性方程；一次方程linear equation in two unknowns 二元一次方程；二元线性方程linear inequality 一次不等式；线性不等式linear momentum 线动量linear programming 线性规划linearly dependent 线性相关的linearly independent 线性无关的literal coefficient 文字系数literal equation 文字方程load 负荷loaded coin 不公正钱币loaded die 不公正骰子local maximum 局部极大(值)local minimum 局部极小(值)locus, loci 轨迹logarithm 对数logarithmic equation 对数方程logarithmic function 对数函数logic 逻辑logical deduction 逻辑推论；逻辑推理logical step 逻辑步骤long division method 长除法loop 回路loss 赔本；亏蚀loss per cent 赔率；亏蚀百分率lower bound 下界lower limit 下限lower quartile 下四分位数lower sum 下和lower triangular matrix 下三角形矩阵lowest common multiple(L.C.M) 最小公倍数machine 机械Maclaurin expansion 麦克劳林展开式Maclaurin series 麦克劳林级数magnitude 量；数量；长度；大小major arc 优弧；大弧major axis 长轴major sector 优扇形；大扇形major segment 优弓形；大弓形mantissa 尾数mantissa of logarithm 对数的尾数；对数的定值部many to one 多个对一个many-sided figure 多边形many-valued 多值的map into 映入map onto 映上mapping 映射marked price 标价Markov chain 马可夫链mass 质量mathematical analysis 数学分析mathematical induction 数学归纳法mathematical sentence 数句mathematics 数学matrix 阵; 矩阵matrix addition 矩阵加法matrix equation 矩阵方程matrix multiplication 矩阵乘法matrix operation 矩阵运算maximize 极大maximum absolute error 最大绝对误差maximum point 极大点maximum value 极大值mean 平均(值)；平均数；中数mean deviation 中均差；平均偏差mean value theorem 中值定理measure of dispersion 离差的量度measurement 量度mechanical energy 机械能median (1)中位数；(2)中线meet 相交；相遇mensuration 计量；求积法method 方法method of completing square 配方法method of interpolation 插值法; 内插法method of least squares 最小二乘法; 最小平方法method of substitution 代换法；换元法method of successive substitution 逐次代换法; 逐次调替法method of superposition 迭合法metric unit 十进制单位mid-point 中点mid-point formula 中点公式mid-point theorem 中点定理million 百万minimize 极小minimum point 极小点minimum value 极小值Minkowski Inequality 闵可夫斯基不等式minor (1)子行列式；(2)劣；较小的minor arc 劣弧；小弧minor axis 短轴minor of a determinant 子行列式minor sector 劣扇形；小扇形minor segment 劣弓形；小弓形minus 减minute 分mixed number(fraction) 带分数modal class 众数组mode 众数model 模型modulo (1)模; 模数; (2)同余modulo arithmetic 同余算术modulus 模; 模数modulus of a complex number 复数的模modulus of elasticity 弹性模(数) moment arm (1)矩臂; (2)力臂moment of a force 力矩moment of inertia 贯性矩momentum 动量monomial 单项式monotone 单调monotonic convergence 单调收敛性monotonic decreasing 单调递减monotonic decreasing function 单调递减函数monotonic function 单调函数monotonic increasing 单调递增monotonic increasing function 单调递增函数motion 运动movable pulley 动滑轮multinomial 多项式multiple 倍数multiple angle 倍角multiple-angle formula 倍角公式multiple root 多重根multiplicand 被乘数multiplication 乘法multiplication law (of probability) (概率)乘法定律multiplicative inverse 乘法逆元multiplicative property 可乘性multiplicity 重数multiplier 乘数；乘式multiply 乘multi-value 多值的mulually disjoint 互不相交mutually exclusive events 互斥事件mutually independent 独立; 互相独立mutually perpendicular lines 互相垂直n factorial n阶乘n th derivative n阶导数n th root n次根；n次方根n the root of unity 单位的n次根Napierian logarithm 纳皮尔对数; 自然对数natural logarithm 自然对数natural number 自然数natural surjection 自然满射necessary and sufficient condition 充要条件necessary condition 必要条件negation 否定式negative 负negative angle 负角negative binomial distribution 负二项式分布negative index 负指数negative integer 负整数negative number 负数negative vector 负向量; 负矢量neighborhood 邻域net 净(值)net force 净力Newton-Cote′s rule 牛顿- 高斯法则Newton-Raphson′s method 牛顿- 纳逊方法Newton′s formula 牛顿公式。

a rXiv:h ep-th/9411158v122N ov1994ASITP-94-61November,1994.N =2and N =4SUSY Yang-Mills action on M 4×(Z 2⊕Z 2)non-commutative geometry Bin Chen 1Hong-Bo Teng 2Ke Wu 3Institute of Theoretical Physics,Academia Sinica,P.O.Box 2735,Beijing 100080,China.Abstract We show that the N =2and N =4SUSY Yang-Mills action can be reformulated in the sense of non-commutative geometry on M 4×(Z 2⊕Z 2)in a rather simple way.In this way thescalars or pseudoscalars are viewed as gauge fields along two directions in the space of one-forms on Z 2⊕Z 2.1IntroductionSince A.Connes introduced his non-commutative geometry into particle physics[1,2],a successful geometrical interpretation of the Higgs mechanism and Yukawa coupling has been acquired.In this interpretation,the Higgsfields are regarded as gaugefields on the discrete gauge group. The bosonic part of the action is just the pure YM action containing the gaugefields on both continuous and discrete gauge group,and the Yukawa coupling is viewed as a kind of gauge interactions of fermions.At the same time,applying non-commutative geometry to SUSY theories has encountered many difficulties.A natural way is to introduce a non-commutative space which is the product of the superspace and a set of discrete points,similar to those which have been done in non-SUSY theories.However,it proved that such an extension of superspace is rather difficult to accomplish.In[3],A.H.Chamseddine gave an alternative way in which the supersymmetry theories in their component forms were considered.He showed that N=2and N=4SUSY Yang-Mills actions could be derived as action functionals for non-commutative spaces,and in some cases the coupling of N=1and N=2super Yang-Mills to N=1and N=2matter could also be reformulated.This paper is thefirst one in which the non-commutative geometry is successfully applied to SUSY theories.Intrigued by A.H.Chamseddine’s work,we show the N=2and N=4SUSY Yang-Mills actions can be derived with non-commutative geometry on M4×(Z2⊕Z2)in a rather simple way.We use the differential calculus on discrete group developed by A.Sitarz[4].With a specific case of differential algebra on Z2⊕Z2,we take the scalar components of the super Yang-Mills field as gaugefields on discrete symmetry group.In N=2case,the scalar and pseudoscalar fields are set to two directions of the space of one-forms.In case of N=4,the complex scalar fields which belong to6of SU(4)are regarded as components of the connection one-form.First in section2we show the specific case of the differential algebra on Z2⊕Z2.Then in section3we derive the N=2and N=4super Yang-Mills Lagrangian with this algebra.Finally we end with conclusions.2Differential Calculus on Z2⊕Z2In this section,we briefly describe the differential calculus on Z2⊕Z2and introduce a specific case of this algebra.For a detailed description of differential calculus on discrete group,we refer to[4].Let’s write the four elements of Z2⊕Z2as(e1,e2),(r1,e2),(e1,r2),(r1,r2).And the group multiplication is(g1,g2)(h1,h2)=(g1h1,g2h2).(1)Let A be the algebra of complex valued functions on Z2⊕Z2.The derivative on A is defined as∂g f=f−R g f g∈Z2⊕Z2,f∈A(2)with R g f(h)=f(hg).We will write∂i and R i for convenience where i=1,2,3refers to (r1,e2),(e1,r2),(r1,r2)respectively.{∂i,i=1,2,3}forms the basis of F,the space of left invariant vectorfields over A.One canfind that the following relations hold∂1∂2=∂1+∂2−∂3∂2∂3=∂2+∂3−∂1(3)∂3∂1=∂3+∂1−∂2∂1∂1=2∂1∂2∂2=2∂2(4)∂3∂3=2∂3∂1∂2=∂2∂1∂1∂3=∂3∂1(5)∂2∂3=∂3∂2.The Haar integral on Z2⊕Z2is defined asZ2⊕Z2f=1Next we introduce the space of one-forms,Ω1,which is the dual space of F.Haven chosen the basis of F,we automatically have the dual basis ofΩ1,{χi,i=1,2,3}.It is defined withχi(∂j)=δi j i,j=1,2,3.(7) The definition of higher forms is natural,Ωn is taken to be tensor product of n copies ofΩ1,,Ωn=Ω1⊗···⊗Ω1n copiesandΩ0=A.AndΩ= ⊕Ωn is the tensor algebra on Z2⊕Z2.As usual,the external derivative satisfies the graded Leibniz rule and is nilpotentd(ab)=d(a)b+(−1)dega a(db)a,b∈Ω(8)d2=0and for f∈Ω0=Ad f=∂1fχ1+∂2fχ2+∂3fχ3.(9)From(8),(9),onefindsχi f=(R i f)χi i=1,2,3f∈A(10)anddχ1=−χ1⊗χ2−χ1⊗χ3+χ2⊗χ3−2χ1⊗χ1−χ2⊗χ1−χ3⊗χ1+χ3⊗χ2dχ2=−χ2⊗χ3−χ2⊗χ1+χ3⊗χ1−2χ2⊗χ2−χ3⊗χ2−χ1⊗χ2+χ1⊗χ3(11)dχ3=−χ3⊗χ1−χ3⊗χ2+χ1⊗χ2−2χ3⊗χ3−χ1⊗χ3−χ2⊗χ3+χ2⊗χ1We also need the involution¯on our differential algebra,which agrees with the complex conjugation on A and(graded)commutes with d,i.e.d(da.For one-forms,then we getdχi=0i=1,2,3.(14) It is a specific case of the above algebra.Obviously the postulation induces an antisymmetric tensor algebraΛwhich is a subalgebra ofΩ.In this subalgebra,we denote the product of forms with∧.And we will use this subalgebra to derive the N=2and N=4SUSY Yang-Mills action in next section.3N=2and N=4super Yang-Mills actionLet’sfirst consider the N=2case.The N=2super Yang-Mills Lagrangian is given byL=T r(−12DµSDµS+12[S,P]2)(15)whereψis the Dirac4−spinor which is the combination of two2−spinors,S is the scalar and P is the pseudoscalar.Allfields are valued in the adjoint representation of the gauge group,i.e.,f=f a T a f=Aµ,S,P,ψ(16)where T a is the generator of gauge group.AndFµν=∂µAν−∂νAµ+i[Aµ,Aν]DµS=∂µS+i[Aµ,S]DµP=∂µP+i[Aµ,P]Dµψ=∂µψ+i[Aµ,ψ].(17)From a simple observation,we see that it is possible to choose the generalized connection one-form on M4×(Z2⊕Z2)A(x,g)=iAµ(x,g)dxµ−iS(x,g)χ1−iγ5P(x,g)χ2g∈Z2⊕Z2.(18)And we setAµ(x,g)=Aµ(x);S(x,g)=S(x);P(x,g)=P(x);g∈Z2⊕Z2.(19)And later in fermionic sector we will also letψ(x,g)=ψ(x).Since all quantities take the same values on different points of Z2⊕Z2,we will ignore the variable g.Please note in this assignment the scalar and pseudoscalar are assigned to two directions of the connection one-form,and we have manifestly set the component inχ3direction of A to zero.We will see it is sufficient for our construction.With(13),(14),(18),(19)the curvature two form is easily foundF=dA(x)+A(x)∧A(x)=i(∂µAν+iAµAν)dxµ∧dxν−i(∂µS+i[Aµ,S])dxµ∧χ1−iγ5(∂µP+i[Aµ,P])dxµ∧χ2−γ5[S,P]χ1∧χ2=iF(h)>(21) we get exactly the bosonic sector of the LagrangianL B=T r(1dxρ∧dxσ>+DµS(DνS)∗<dxµ∧χ1,dxν∧χ2>+[S,P][S,P]∗<χ1∧χ2,4FµνFµν+12DµP DµP+1dxρ∧dxσ>=<dxµ∧dxν,dxσ∧dxρ>=gµσgνρ<dxµ∧χ1,2gµν<dxµ∧χ2,2gµν<χ1∧χ2,2.(23)As to the fermionic sector of the Lagrangian,we haveL F= Z2⊕Z2T r(i¯ψ[γµDµ+D1+D2+D3,ψ])=T r(i¯ψγµDµψ+¯ψ[S+γ5P,ψ])(24)where[D1,ψ]=∂1ψ−i[SR1,ψ]=−i[S,ψ][D2,ψ]=∂2ψ−i[γ5P R2,ψ]=−i[γ5P,ψ][D3,ψ]=∂3ψ=0.(25) So the N=2super Yang-Mills Lagrangian is reformulated.We next consider N=4case,the Lagrangian isL=T r[−12DµM ij DµM ij+iλi[M ij,λj]+i¯λi[M ij,¯λj]+12ǫijkl M kl=M ij,and transforms as6of SU(4), andλi is the2−spinor transforming as4of SU(4)whereas¯λi transforming as¯4.So L is SU(4) invariant.In order to write the spinor in four component form,we introduce Majorana spinorsλi= λαi¯λ˙αi ;¯λi= λαi,¯λi˙α (27) andφij= M ij M ijI ij= δj iδi j (28)α=1,2and˙α=˙1,˙2are spinor indices.So the Lagrangian can be written asL=T r[−12¯λiγµDµλi+18[φij,φkl][φ†ij,φ†kl]].(29)here the trace also takes on the2×2matrix in whichfields in two2−component form are arranged into4−component form.We set the connection one-form on M4×(Z2⊕Z2)A(x,g)=iAµ(x,g)I ij dxµ+φij(x,g)χ1+φij(x,g)χ2g∈Z2⊕Z2.(30)Again we will let allfields take the same values on different points of Z2⊕Z2,so we will ignore the variable g,and as in the N=2case,we let the component inχ3direction be zero.Then we haveF=dA(x)+A(x)∧A(x)=i2 Z2⊕Z2T r<F(h),2T r(1dxρ∧dxσ>+Dµφij(Dνφij)∗<dxµ∧χ1,dxν∧χ2>+[φij,φkl][φij,φkl]∗<χ1∧χ2,8FµνFµνI2+18[φij,φkl][φ†ij,φ†kl])(32)here again in order to give appropriate normalization constants,we have chosen the metric <dxµ∧dxν,dxν∧χ1>=<dxµ∧χ1,−χ1∧dxν>=1dxν∧χ2>=<dxµ∧χ2,−χ2∧dxν>=1χ1∧χ2>=<χ1∧χ2,χ2∧χ1>=12 Z2⊕Z2T r(i¯λi[γµ(Dµ)I ij+(D1)ij+(D2)ij+(D3)ij,λj])=T r(i4ConclusionsWe show that the N=2and N=4super Yang-Mills Lagrangian can be reformulated on M4×(Z2⊕Z2)non-commutative geometry.We use the differential calculus on discrete group developed by A.Sitarz and regard the scalar or pseudoscalarfields as connection one-forms on the discrete symmetry group.This reformulation is different from that of A.H.Chamseddine and seems much simpler and clearer.References[1]A.Connes:Non-commutative Geometry English translation ofGeometrie Non-commutative,IHES Paris,Interedition.[2]A.Connes and J.Lott:Nucl.Phys.(Proc.Suppl.)B18,44(1990).[3]A.H.Chamseddine:Phys.Lett.B332,(1994)349-357.[4]A.Sitarz:To appear in J.Geo.Phys.[5]A.H Chamseddine,G Felder and J.Frohlich:Phys.Lett.296B(1993)109.[6]R.Coquereaux,G.Esposito-Farese and G.V aillant:Nucl.Phys.B353689(1991).。