Invariant Homology on Standard Model Manifolds

南开大学硕士学位论文光滑闭流形上的Lefschetz不动点定理姓名：***申请学位级别：硕士专业：基础数学指导教师：***2012-05中文摘要中文摘要这是一篇读书笔记，参照文献【ｌ】，用热方程的方法证明光滑流形上的Ｌｅｆｓｅｈｅｔｚ不动点定理，并给出计算Ｌｅｆｓｃｈｅｔｚ数的公式。

文章先介绍基本概念，包括傅立叶变换，拟微分算子与象征，椭圆链复形；利用Ｈｏｄｇｅ分解定理将Ｌｅｆｓｃｈｅｔｚ数与椭圆算子的零特征值特征子空间相联系，再联系到固定椭圆算子的所有特征子空间，从而转化为热方程的解的积分核的迹，此热方程有椭圆算子，解由椭圆算子决定，最后利用椭圆算子的象征变形，再积分，得到结果。

关键词：拟微分算子，象征，椭圆算子，热方程，Ｌｅｆｓｃｈｅｔｚ不动点定理ＡｂｓｔｒａｃｔＡｂｓｔｒａｃｔＴｈｉｓｐａｓｓａｇｅｉｓａｒｅａｄｉｎｇｎｏｔｅｏｆｔｈｅｂｏｏｋ【１】ｂｙｔｈｅｍｅｔｈｏｄｏｆｈｅａｔｅｑｕｍｉｏｎ，ｇｉｖｅａｐｒｏｏｆｏｆｔｈｅｃｌａｓｓｉｃａｌＬｅｆｓｃｈｅｔｚＦｉｘｅｄＰｏｉｎｔＴｈｅｏｒｅｍ，ａｎｄｇｉｖｅｔｈｅＬｅｆｓｃｈｅｔｚＦｉｘｅｄＰｏｉｎｔＴｒａｎｓｆｏｒｍ，Ｐｓｅｕｄｏ．Ｆｏｒｍｕｌａ．Ｆｉｒｓｔｗｅｉｎｔｒｏｄｕｃｅｔｈｅｂａｓｉｃｃｏｎｃｅｐｔｓ，ｉｎｃｌｕｄｉｎｇＦｏｕｒｉｅｒＤｉｆｆｅｒｅｎｔｉａｌＯｐｅｒａｔｏｒａｎｄｓｙｍｂｏｌ，ｅｌｌｉｐｔｉｃｃｏｍｐｌｅｘ；ａｓｓｏｃｉａｔｅＬｅｆｓｃｈｅｔｚｎｕｍｂｅｒｔｏｔｈｅｚｅｒｏｅｉｇｅｎｖａｌｕｅｅｉｇｅｎｓｐａｃｅｏｆｅｌｌｉｐｔｉｃｏｐｅｒａｔｏｒｓｂｙＨｅｄｇｅＤｅｃｏｍｐｏｓｉｔｉｏｎＴｈｅｒｅｏｍ，ｍｏｒｅｏｖｅｒ，ｔｏａｌｌｔｈｅｅｉｇｅｎｓｐａｃｅｏｆｅｌｌｉｐｔｉｃｏｐｅｒａｔｏｒｓ，ｔｈｅｎｃｈａｎｇｅｉｎｔｏｔｈｅｔｒａｃｅｏｆｉｎｔｅ－ｇｒａｌｋｅｒｎｅｌｏｆｓｏｌｕｔｉｏｎｏｆｔｈｅｈｅａｔｅｑｕｍｉｏｎ，ｗｈｉｃｈａｒｅｄｅｔｅｒｍｉｎｅｄｂｙｅｌｌｉｐｔｉｃｏｐｅｒａｔｏｒｓｒｅｓｐｅｃｔｉｖｅｌｙ；ａｔｌａｓｔ，ｕｓｅｔｈｅｓｙｍｂｏｌｏｆｔｈｅｅｌｌｉｐｔｉｃｏｐｅｒａｔｏｒ，ｗｈｉｃｈｗｅｄｉｆｆｅｒｅｎｔｉａｔｅａｎｄｃｏｍｂｉｎｅ，ｔｈｅｎｍｕｔｌｉｐｌｙｃｏｎｖｅｒｇｅｎｃｅｆａｃｔｏｒａｎｄｉｎｔｅｇｒａｌｔｈｅｍｔｏｇｅｔｔｈｅｒｅｓｕＲ．ＫｅｙＷｏｒｄｓ：Ｐｓｅｕｄｏ－ＤｉｆｆｅｒｅｎｔｉａｌＯｐｅｒａｔｏｒｓ，ｓｙｍｂｏｌ，ｅｌｌｉｐｔｉｃｏｐｅｒａｔｏｒ，ｈｅａｔｅｑｕａｏＴｈｅｏｒｅｍｔｉｏｎ，ＬｅｆｓｃｈｅｔｚＦｉｘｅｄＰｏｉｎｔｎＣｈａｐｔｅｒ１ＩｎｔｒｏｄｕｃｔｉｏｎＣｈａｐｔｅｒ１ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＭｂｅａｃｏｍｐａｃｔｔｏｐｏｌｏｇｙｓｐａｃｅ，ａｓｓｕｍｅＭｈｏｍｅｏｍｏｒｐｈｉｓｍｆｉｎｉｔｅｓｉｍｐｌｅｃｏｍ·ｐｌｅｘｅｓ，ａｍａｐｆ：Ｍ＿÷Ｍｉｎｄｕｃｅｓａｌｉｎｅａｒｍａｐｏｆｔｈｅｈｏｍｏｌｏｇｙｇｒｏｕｐ＾：绋（肘，Ｑ）－－ｔ，炜（肘，Ｑ），ｔｈｅｎＬｅｆｓｃｈｅｔｚｎｕｍｂｅｒｉｓｄｅｆｉｎｅｄｂｙＬ（，）＝∑（一１）ｐＴｒ（ｙ，ｏｎＨｐ（Ｍ，Ｑ））ＰＴｈｅＬｅｆｓｃｈｅｔｚｆｉｘｅｄｐｏｉｎｔｔｈｅｏｒｅｍｉｓ：ｉｆＬ（ｆ）≠０，ｔｈｅｎｆｈａｓｆｉｘｅｄｐｏｉｎｔ．Ｉｎ【７】，ｔｈｅｒｅｉｓａｐｒｏｏｆｂｙｔｏｐｏｌｏｇｙｍｅｔｈｏｄ．Ｉｎｔｈｉｓｐａｐｅｒ＇ｗｅａｓｓｕｍｅＭｔｏｂｅａｓｍｏｏｔｈｃｌｏｓｅｄｍａｎｉｆｏｌｄ，ｍｅａｎｓａｓｍｏｏｔｈｃｏｍｐａｃｔｍａｎｉｆｏｌｄｗｉｔｈｏｕｔｂｏｕｎｄａｒｙ，ａｎｄｆ：Ｍ－－－｝Ｍｂｅａｓｍｏｏｔｈｍａｐ．广ｄｅｆｉｎｅｓａｍａｐｏｎｔｈｅｃｏｈｏｍｏｌｏｇｙｏｆＭａｎｄｔｈｅＬｅｆｓｃｈｅｔｚｎｕｍｂｅｒｏｆｆｉｓｄｅｆｉｎｅｄｂｙ：Ｌ（，）＝∑，（－］）ＰＴｒ（ｆ＂ｏｎｎｐ（肘；ｃ））ＷｅｇｅｔｔｈｅｍａｉｎｒｅｓｕｌｔＴｈｅｏｒｅｍ１．１．（ＬｅｆｉｃｈｅｔｚＦｉｘｅｄＰｏｉｎｔＴｈｅｏｒｅｍ）矿己（力≠Ｏ，ｔｈｅｎｆｈａｓｆｏｃｅｄｐｏｉｎｔ．ｉ．ｅｔｈｅｒｅｅｘｉｓｔｘ∈ｍ，ｓｕｃｈｔｈａｔｆ（ｘ）＝工Ｔｈｅｏｒｅｍ１．２．（ＣｌａｓｓｉｃａｌＬｅｆｓｃｈｅｔｚＦｉｘｅｄＰｏｉｎｔＦｏｒｍｕｌａ）Ｌｅｔｆ：Ｍ—＋Ｍｂｅｓｍｏｏｔｈｗｉｔｈｉｓｏｌａｔｅｄｎｏｎ－ｄｅｇｅｎｅｒａｔｅｆｉｘｅｄｐｏｉｎｔｓ．Ｔｈｅｎ：￡（，）＝Ｅ（－１）ｐＴｒ（ｆ。

a rXiv:h ep-th/98781v11J ul1998UICHEP-TH/98-7,February 1,2008Algebraic Shape Invariant Models S.Chaturvedi a,1,R.Dutt b,2,A.Gangopadhyaya c,3,P.Panigrahi a,4,C.Rasinariu d,5and U.Sukhatme d,6a)Department of Physics,University of Hyderabad,Hyderabad,India;b)Department of Physics,Visva Bharati University,Santiniketan,India;c)Department of Physics,Loyola University Chicago,Chicago,USA;d)Department of Physics,University of Illinois at Chicago,Chicago,USA.Abstract Motivated by the shape invariance condition in supersymmetric quantum mechanics,we develop an algebraic framework for shape invariant Hamiltonians with a general change of pa-rameters.This approach involves nonlinear generalizations of Lie algebras.Our work extends previous results showing the equivalence of shape invariant potentials involving translational change of parameters with standard SO (2,1)potential algebra for Natanzon type potentials.In recent years,there has been considerable interest in studying exactly solvable quantum mechanical problems using algebraic approaches[1-9].In this respect,supersymmetric quantum mechanics(SUSYQM)has been found to be an elegant and useful prescription for obtaining closed analytic expressions both for the energy eigenvalues and eigenfunctions for a large class of one dimensional(or spherically symmetric three dimensional)problems[3].The main ingredients in SUSYQM are the supersymmetric partner Hamiltonians H−≡A†A and H+≡AA†.The A and A†operators used in this factorization are expressed in terms of the real superpotential W as follows:A(x,a)=ddx+W(x,a).(1)Here,a is a parameter(or a set of parameters),which will play an important role in the approach of this paper.An interesting feature of SUSYQM is that for a shape invariant system[1-3],i.e.a system satisfyingH+(x,a0)=H−(x,a1)+R(a0);a1=f(a0),(2) the entire spectrum can be determined algebraically without ever referring to underlying differ-ential equations.Recently,it has been shown that shape invariant potentials in which the change of parameters a1=f(a0)is of translational type a1=a0+k,k=constant,possess a SO(2,1)potential algebra and are solvable by group theoretical techniques[4-7].Thus,for such potentials,there appear to be two seemingly independent algebraic methods,SUSYQM and potential algebras,for obtaining the complete spectra.One may naturally ask the question whether there is any connection between the two methods.Indeed,the equivalence between the two algebraic approaches has been established[6].More specifically,for all shape invariant potentials of the Natanzon type [10],the underlying potential algebra is SO(2,1).Other similar approaches are discussed in refs. [4,11].The purpose of this letter is to examine whether such an equivalence is also valid for a larger class of shape invariant potentials.We consider a much more general class of change of param-eters,which contains both translations as well as scalings.In particular,for shape invariant potentials generated by a pure scaling ansatz[12]a1=qa0,q=constant(0<q<1),wefind that the associated potential algebra is a nonlinear deformation of SO(2,1).In fact,coordinate representations of these nonlinear algebras are an interesting area of research in their own right from a mathematical point of view[13].We also show that the energy spectrum resulting from2this potential algebra agrees with the results of SUSYQM.To begin the construction of the operator algebra,let us express eq.(2)in terms of A and A†:A(x,a0)A†(x,a0)−A†(x,a1)A(x,a1)=R(a0).(3) This relation resembles a commutator structure.To obtain a closed SO(2,1)type Lie algebra, we introduce an auxiliary variableφand defineJ+=Q(φ)A†(x,χ(i∂φ)),J−=A(x,χ(i∂φ))Q⋆(φ),(4)where Q(φ)is a function to be determined andχis an arbitrary,real function.The operator A(x,χ(i∂φ))is given by eq.(1)with the substitutiona→χ(i∂φ).(5) From eq.(4),one obtains[J+,J−]=Q(φ)A†(x,χ(i∂φ))A(x,χ(i∂φ))Q⋆(φ)−A(x,χ(i∂φ))Q⋆(φ)Q(φ)A†(x,χ(i∂φ)).(6)Choosing Q(φ)=e ipφ,where p is an arbitrary real constant,eq.(6)can be easily cast into the following form[J+,J−]=− A(x,χ(i∂φ))A†(x,χ(i∂φ))−A†(x,χ(i∂φ+p))A(x,χ(i∂φ+p)) .(7) In analogy with eq.(3),we impose the algebraic shape invariance conditionA(x,χ(i∂φ))A†(x,χ(i∂φ))−A†(x,χ(i∂φ+p))A(x,χ(i∂φ+p))=R(χ(i∂φ)).(8) Note that the quantity J+J−corresponds to the HamiltonianH−(x,i∂φ+p)=A†(x,χ(i∂φ+p))A(x,χ(i∂φ+p)).(9)Identifyingi∂φJ3=−Hereξ(J3)≡−R(χ(i∂φ)).The algebraic shape invariance condition(3)is the main tool of our analysis.If we identifya0→χ(i∂φ);a1→χ(i∂φ+p),(12) then depending on the choice of theχfunction in eq.(12)we have:1.translational models:a1=a0+p forχ(z)=z,2.scaling models:a1=e p a0≡qa0forχ(z)=e z.One may note that in the translational reparametrization(χ(z)=z)the commutation rela-tions(11)describe a standard,undeformed SO(2,1)Lie algebra withξ(J3)=−2J3,as found in ref.[6].Other changes of parameters follow from more complicated choices forχ(z).For example, if one takesχ(z)=e e z,one gets the change of parameters a1=a20.Thus,with judicious choices forχ(z),various other reparametrizations can be emulated.In this letter,we will analyze the potential algebra of scaling type reparametrizations.Different models with such multiplicative reparametrizations are characterized by the form of the remainder function R(a0).As afirst example of scaling,we choose R(a0)=R1a0.This choice generates self-similar potentials studied in refs.[12,14].In this case,eqs.(11)become:[J3,J±]=±J±;[J+,J−]=ξ(J3)≡−R1exp(−pJ3).(13)Thus,we see that the resulting potential algebra is no longer given by a SO(2,1)Lie algebra,but is deformed.Tofind the energy spectrum of the Hamiltonian H−of eq.(9),wefirst construct the unitary representations of this deformed Lie algebra[15].We define,up to an additive constant, a function g(J3)such thatξ(J3)=g(J3)−g(J3−1).(14) The Casimir of this algebra is then given C2=J−J++g(J3).In a basis in which J3and C2are diagonal,J+and J−play the role of raising and lowering operators,respectively.Operating on an eigenstate|h of J3,we haveJ3|h =h|h ,J−|h =a(h)|h−1 ,J+|h =a⋆(h+1)|h+1 .(15)4Using eqs.(11)and (15)we obtain |a (h )|2−|a (h +1)|2=g (h )−g (h −1).(16)The profile of g (h )determines the dimension of the unitary representation.To illustrate how this mechanism works,let us consider the two cases presented in fig.1.One obtains finite(b)(a)Figure 1:Generic behaviors of g (h ).dimensional representations fig.1a,by starting from a point on the g (h )vs.h graph corresponding to h =h min ,and moving in integer steps parallel to the h -axis till the point corresponding to h =h max .Only those discrete values of h min are allowed for which at the end points a (h min )=a (h max +1)=0,and we get a finite representation.This is the case of SU (2)for example,where g (h )is given by the parabola h (h +1).For a monotonically decreasing function g (h ),fig.1b,there exists only one end point at h =h min .Starting from h min the value of h can be increased in integer steps till infinity.In this case we have an infinite dimensional representation.As in the finite case,h min labels the representation.However,the difference is that here h min takes continuous values.Similar arguments apply for a monotonically increasing function g (h ).A more detailed discussion of these techniques is given in ref.[15].In our case,from eqs.(13)and (14)one getsg (h )=R 11−q q −h ;q =e p ,(17)represented graphically in fig. 2.Note that for scaling problems [12],one requires 0<q <1,which leads to p <0.From the monotonically decreasing profile of the function g (h ),it follows that the unitary representations of this algebra are infinite dimensional.If we label the lowest weight state of the operator J 3by h min ,then a (h min )=0.Without loss of generality5we can choose the coefficients a(h)to be real.Then one obtains from(16)for an arbitrary h=h min+n,n=0,1,2,...q n−1a2(h)=g(h−n−1)−g(h−1)=R1q1−h|h .(19)q−1Therefore,the eigenenergy is:q n−1E n(h)=α(h)+W2(x,e i∂φ+p)−W′(x,e i∂φ+p)−E e ipφhψh min,n(x)=0,dx2or −d21h. . .Figure 2:Schematic diagram showing the correspondence between potential algebra and SUSYQM with scaling type reparametrization.For a more general case,we assume R (a 0)=∞j =1R j a j 0.In this caseg (h )=∞ j =1R j 1−q j ,(23)where αj (h )=R j e −j (h −1).This result agrees with those obtained in ref.[12].In conclusion,in this paper we have developed an algebraic approach involving nonlinear generalizations of Lie algebras for treating general shape invariant Hamiltonians.A.G.acknowledges a research leave and a grant from Loyola University Chicago which made his involvement in this work possible.R.D.would also like to thank the Physics Department of the University of Illinois at Chicago for warm hospitality.Partial financial support from the U.S.Department of Energy is gratefully acknowledged.7References[1]L.Infeld and T.E.Hull,Rev.Mod.Phys.23(1951)21.[2]L.E.Gendenshtein,JETP Lett.38(1983)356;L.E.Gendenshtein and I.V.Krive,Sov.p.28(1985)645.[3]F.Cooper,A.Khare,and U.Sukhatme,Phys.Rep.251(1995)268and references therein.[4]Y.Alhassid,F.G¨u rsey and F.Iachello,Ann.Phys.148(1983)346.[5]J.Wu and Y.Alhassid,Phys.Rev.A31(1990)557;H.Y.Cheung,Nuovo Cimento B101(1988)193.[6]A.Gangopadhyaya,J.V.Mallow and U.P.Sukhatme,Proceedings of Workshop on Super-symmetric Quantum Mechanics and Integrable Models,June1997,Editor:H.Aratyn et al., Springer-Verlag.[7]A.Gangopadhyaya,J.V.Mallow and U.P.Sukhatme,“Shape Invariant Natanzon Potentialsfrom Potential Algebra”;to appear in Phys.Rev.A.[8]A.B.Balantekin,Los Alamos e-print quant-ph/9712018.[9]A.Barut,A.Inomata and R.Wilson,Jour.Phys.A20(1987)4075and4083.[10]G.A.Natanzon,Teor.Mat.Fiz.,38(1979)219.[11]P.Cordero and S.Salamo,Foundations of Physics23(1993)675;Jour.Math.Phys.35(1994)3301.[12]D.Barclay,R.Dutt,A.Gangopadhyaya,A.Khare,A.Pagnamenta and U.Sukhatme,Phys.Rev.A48(1993)2786.[13]J.Beckers,Y.Brihaye,and N.Debergh,Los Alamos preprint hep-th/9803253[14]V.P.Spiridonov,Phys.Rev.Lett.69(1992)298.[15]M.Rocek,Phys.Lett.B255(1991)554.8。

a r X i v :h e p -p h /9902228v 1 2 F eb 1999DEVIATIONS FROM LORENTZ INV ARIANCE FORULTRAHIGH-ENERGY FERMIONSR.E.ALLENCenter for Theoretical Physics,Texas A&M UniversityCollege Station,Texas 77843,USAe-mail:allen@In a new theory,local Lorentz invariance is a low-energy symmetry which no longer holds when a fermion energy E is well above 1TeV.Here we find that the modified E (p )relation is consistent with observation,and is in fact nearly the same as in Einstein relativity.On the other hand,there is a strong modification of the fermion equation of motion and propagator at ultrahigh energy,which should lead to observable effects.Recently a new fundamental theory was proposed.1One of its predictions is a failure of Lorentz invariance at ultrahigh energies.Here the implications of this feature will be considered in more detail.According to (4.4),(4.5),and (4.7)of Ref.1,the Euclidean Lagrangian density for a massless fermion has the form2Mh µν∂ν∂µ−i 12Mηµν∂ν∂µ−iv µασα∂µψ(2)where ηµν=diag (−1,1,1,1)is the Minkowski metric tensor.This is the generalization of (9.5)of Ref.1,in which the first term is neglected.In (4.15)of Ref.1,v µαis interpreted as the gravitational vierbein e µα,which determines the gravitational metric tensor g µνthrough the relationsg µν=ηαβe αµe βν,e µαe αν=δµν,e µα=v µα.(3)Then the expression for L yields the equation of motion1which can also be written12M−λ−20e 0αe 0α∂0∂0+λ−2e k αe lα∂k ∂l +ie µασα∂µψ=0.(8)After transforming to a locally inertial frame of reference,in which e µα=δµα,we have12M ηµν∂ν∂µ+ie µασα∂µψR (12)+ψ†L 1σα∂µ ψL (13)−mψ†R ψL −mψ†L ψR(14)where σk =−σk .2In a locally inertial frame,the resultingequation of motion is(−β∂0∂0+α∂k ∂k )+iσ0∂0+σk ∂k ψR −mψL =0(15) (−β∂0∂0+α∂k ∂k )+i σ0∂0−σk ∂k ψL −mψR =0(16)2or,atfixed energy E and momentum−→p,βE2−αp2 +E−−→σ·−→p ψR−mψL=0(17)βE2−αp2 +E+−→σ·−→p ψL−mψR=0.(18)Solution of these coupled equations givesβE2−αp2 +E=±p2+m2 1/2(20) where the±signs are independent.Then the lowest-order corrections to Ein-stein relativity are given byE≈∂p x ∆p x+∂E∂m2∆m2(23)with∂Ep ,∂E[1+4βp(1+αp)]1/21∂p=1+2αp∂pcosθ=∂Eω(26)3and the threshold is for a head-on collision at the momentum p satisfying p [1+4βp(1+αp)]1/2+(1+2αp) =∆m2∂p x ∆p x+∂E∂p(−ωcosθ)(28)so this process can occur ifv=1/cosθ≥1.(29) For m2/p2≪1,the expression(25)implies thatv>1forβ<α,v<1forβ>α.(30) If we were to haveβ<α,the particle velocity at high momentum would be greater than the velocity of light,and there would be a radiation of photons in vacuum which is in conflict with observation.4Next consider the process photon→e+e−,which will occur if2E(p)=ω=2p cosθ.(31) It is reasonable to assume the threshold is belowα−1,so that(21)is valid and the condition becomesThis is inconsistent with the theory of Ref.1,whereλandλ0are regarded as small compared to unity.On the other hand,ifβ=α,orγ=0,(21)reduces toE≈c=c.Their contribution to the vacuum energy is then −45c)p.In the cosmological model of Ref.1,the value ofαin the late universe is fixed by topology,since it corresponds to SU(2)rotations of the order param-eterΨs in3-space that are due to a cosmological instanton.(Recall thatΨs5describes an SU(2)×U(1)×SO(10)GUT Higgsfield which condenses in the very early universe.1)The value ofβ,however,is notfixed by topology and is free to vary,since it is associated with U(1)oscillations of the order parameter in time.(A more detailed cosmological picture is given elsewhere.8)If the assumptions listed above are valid,then the frequency of oscillation v00must adjust itself so as to achieve a cancellation of the largest negative and positive contributions to the vacuum energy.This means thatp2+m2(34) except for very small terms involving the mass,so the present theory is consis-tent with experimental tests of this relation.At ultrahigh energies p∼α−1= 2λ2M,however,the fermion equation of motion is changed,and the propa-gator becomes(−αpµpµ+/p−m+iǫ)−1with the present conventions.This change will affect the running coupling constants and other properties as one approaches the GUT scale.AcknowledgementThis work was supported by the Robert A.Welch Foundation.References1.R.E.Allen,Int.J.Mod.Phys.A12,2385(1997);hep-th/9612041.2.P.Ramond,Field Theory:A Modern Primer(Addison-Wesley,RedwoodCity,California,1990),pp.18-19.3.K.Greisen,Phys.Rev.Lett.16,748(1966);G.T.Zatsepin and V.A.Kuz’min,JETP Lett.41,78(1966).4.S.Coleman and S.L.Glashow,hep-ph/9812418and references therein;Phys.Lett.B405,249(1997).5.G.R.Farrar and P.L.Biermann,Phys.Rev.Lett.81,3579(1998).6.D.Colladay and V.A.Kosteleck´y,Phys.Rev.D58,116002(1998),andearlier work referenced therein;hep-ph/9809521.7.J.W.Cronin,K.G.Gibbs,and T.C.Weekes,Annu.Rev.Nucl.Part.Sci.43,883(1993).8.R.E.Allen,to be published in the proceedings of the Cosmo-98Inter-national Workshop on Particle Physics and the Early Universe,edited by D.Caldwell(American Institute of Physics,New York,1999);astro-ph/9902042.6。

数学mathematics, maths(BrE), math(AmE) 公理axiom定理theorem计算calculation运算operation证明prove假设hypothesis, hypotheses(pl.)命题proposition算术arithmetic加plus(prep.), add(v.), addition(n.)被加数augend, summand加数addend和sum减minus(prep.), subtract(v.), subtraction(n.) 被减数minuend减数subtrahend差remainder乘times(prep.), multiply(v.), multiplication(n.) 被乘数multiplicand, faciend乘数multiplicator积product除divided by(prep.), divide(v.), division(n.) 被除数dividend除数divisor商quotient等于equals, is equal to, is equivalent to大于is greater than小于is lesser than大于等于is equal or greater than小于等于is equal or lesser than运算符operator数字digit数number自然数natural number整数integer小数decimal小数点decimal point分数fraction分子numerator分母denominator比ratio正positive负negative零null, zero, nought, nil十进制decimal system二进制binary system十六进制hexadecimal system权weight, significance进位carry截尾truncation四舍五入round下舍入round down上舍入round up有效数字significant digit无效数字insignificant digit代数algebra公式formula, formulae(pl.)单项式monomial多项式polynomial, multinomial系数coefficient未知数unknown, x-factor, y-factor, z-factor 等式，方程式equation一次方程simple equation二次方程quadratic equation三次方程cubic equation四次方程quartic equation不等式inequation阶乘factorial对数logarithm指数，幂exponent乘方power二次方，平方square三次方，立方cube四次方the power of four, the fourth power n次方the power of n, the nth power开方evolution, extraction二次方根，平方根square root三次方根，立方根cube root四次方根the root of four, the fourth rootn次方根the root of n, the nth root集合aggregate元素element空集void子集subset交集intersection并集union补集complement映射mapping函数function定义域domain, field of definition 值域range常量constant变量variable单调性monotonicity奇偶性parity周期性periodicity图象image数列，级数series微积分calculus微分differential导数derivative极限limit无穷大infinite(a.) infinity(n.)无穷小infinitesimal积分integral定积分definite integral不定积分indefinite integral有理数rational number无理数irrational number实数real number虚数imaginary number复数complex number矩阵matrix行列式determinant几何geometry点point线line面plane体solid线段segment射线radial平行parallel相交intersect角angle角度degree弧度radian锐角acute angle直角right angle钝角obtuse angle平角straight angle周角perigon底base高height三角形triangle锐角三角形acute triangle直角三角形right triangle直角边leg斜边hypotenuse勾股定理Pythagorean theorem钝角三角形obtuse triangle不等边三角形scalene triangle等腰三角形isosceles triangle等边三角形equilateral triangle四边形quadrilateral平行四边形parallelogram矩形rectangle长length宽width菱形rhomb, rhombus, rhombi(pl.), diamond 正方形square梯形trapezoid直角梯形right trapezoid等腰梯形isosceles trapezoid五边形pentagon六边形hexagon七边形heptagon八边形octagon九边形enneagon十边形decagon十一边形hendecagon十二边形dodecagon多边形polygon正多边形equilateral polygon圆circle圆心centre(BrE), center(AmE)半径radius直径diameter圆周率pi弧arc半圆semicircle扇形sector环ring椭圆ellipse圆周circumference周长perimeter轨迹locus, loca(pl.)相似similar全等congruent四面体tetrahedron五面体pentahedron六面体hexahedron平行六面体parallelepiped 立方体cube七面体heptahedron八面体octahedron九面体enneahedron十面体decahedron十一面体hendecahedron 十二面体dodecahedron二十面体icosahedron多面体polyhedron棱锥pyramid棱柱prism棱台frustum of a prism旋转rotation轴axis圆锥cone圆柱cylinder圆台frustum of a cone球sphere半球hemisphere底面undersurface表面积surface area体积volume空间space坐标系coordinates坐标轴x-axis, y-axis, z-axis 横坐标x-coordinate纵坐标y-coordinate原点origin双曲线hyperbola抛物线parabola三角trigonometry正弦sine余弦cosine正切tangent余切cotangent正割secant反正弦arc sine反余弦arc cosine反正切arc tangent反余切arc cotangent反正割arc secant反余割arc cosecant相位phase周期period振幅amplitude内心incentre(BrE), incenter(AmE)外心excentre(BrE), excenter(AmE)旁心escentre(BrE), escenter(AmE)垂心orthocentre(BrE), orthocenter(AmE)重心barycentre(BrE), barycenter(AmE)内切圆inscribed circle外切圆circumcircle统计statistics平均数average加权平均数weighted average方差variance标准差root-mean-square deviation, standard deviation 比例propotion百分比percent百分点percentage百分位数percentile排列permutation组合combination概率，或然率probability分布distribution正态分布normal distribution非正态分布abnormal distribution图表graph条形统计图bar graph柱形统计图histogram折线统计图broken line graph曲线统计图curve diagram扇形统计图pie diagramracah algebra 拉卡代数radial planimeter 径向面积仪radial slit domain 径向裂纹域radian 弧度radian measure 弧度radical 根radical axis 根轴radical center 根心radical function 根函数radical of a ring 环的根radical plane 根平面radical sign 根号radicand 被开方数radius 半径radius of a circle 圆的半径radius of convergence 收敛半径radius of curvature 曲率半径radius of gyration 回转半径radius of holomorphy 正则半径raise 乘方raising to a power 自乘random coincidence 偶然符合random deviation 随机偏差range 值域range of a variable 变量范围range of variation 变差范围range test 范围检验rank correlation 等级相关rank correlation coefficient 等级相关系数rank function 等级函数rank of linear mapping 线性映射的秩rank of matrix 阵的秩rankine cycle 兰金循环rational number 有理数rational root 有理根rational space 有理空间rational space curve 有理空间曲线rational tranformation 有理映射rational value 有理值rationality 有理性rationalization 有理化rationalize 有理化rationally equivalent 有理等价的ray 半直线ray center 射线中心ray representation 射影表示rays 射线reaction 反应real 实值real axis 实轴real domain 实域real function 实值函数real function of a real variable 实一变数的实函数real number 实值real number system 实数系real part 实部real quadratic number field 实二次数域real root 实根real value 实值real valued function 实值函数realizability 可实现性realization 实现rearrange 重新整理rearrangement 重排reason 根据reasoning 推论reciprocal 相互的reciprocal correspondence 反对应rectifying plane 从切平面rectifying surface 伸长曲面rectilinear 直线的rectilinear figure 直线图形rectilinear generator 直绞母线rectilinear motion 直线运动rectilinearity 直线性rectilinearly accessible point 直线可达点recurrence 循环recurrence formula 递推公式recurrence relation 递推关系recurrence theorem 循环定理recurrence time 递归时间recurrent 循环的recurrent event 递归事件recurrent sequence 递归序列recurring chain fraction 循环连分数recurring continued fraction 循环连分数recurring series 循环级数reducible differential equation 可约微分方程reducible element 可约元素reducible equation 可约方程reducible polynomial 可约多项式reducible representation 可约表示reduction 简化reduction formula 归约公式reduction of a fraction 约分reduction to a common denominator 通分reductive 可简约的reentrant angle 凹角reference ellipsoid 参考椭球refinement 加细reflected code 反演码reflecting barrier 反射壁reflection 反射reflexive 自反的regular permutation group 正则置换群regular point 正则点regular point of the boundary 边界的正则点regular point system 正则点系regular polygon 正多边形regular polyhedron 正多面体regular prime 正则素数regular pyramid 正棱锥regular rational mapping 正则有理映射regular right ideal 正则右理想regular ring 正则环regular set 正则集regular set function 正则集函数regular sheaf 正则层regular singular point 正则奇点regular singularity 正则奇点regular solution 正则积分regular space 正则拓扑空间regular surface 正则曲面regular terms 正则项regular tetrahedron 正四面体regular topological space 正则拓扑空间regular transformation 正则变换regular triangle 等边三角形regular value 正则值regular variety 正则簇regularity 正则性regularity condition 正则条件regularization 正则化regularizing operator 正则化算子regularly branched riemann surface 正则分歧黎曼面regularly connected complex 正则连通复形regularly convergent series 正则收敛级数regularly convex function 正则凸函数regulator 单元基准relative motion 相对运动relative neighborhood 相对邻域relative norm 相对范数relative product 相关乘积relative singular homology group 相对奇异同岛relative stability 相对稳定性relative surface 相对曲面relative tensor 相对张量relative topology 相对拓扑relative uniform convergence 相对一致收敛relative uniformity 相对一致性relative velocity 相对速度relatively bounded space 相对有界空间relatively closed set 相对闭集relatively compact set 相对紧集relatively compact space 相对紧空间relatively complemented lattice 相对补格relatively conjugate number 相对共轭数本文来自relatively countably compact subset 相对可数紧子集relatively dense set 相对稠密集relatively general position 相对普通位置relatively invariant measure 相对不变测度relatively minimal model 相对极小模型relatively open set 相对开集relatively perfect set 相对完备集relatively prime elements 互素元relatively prime functions 互素函数relatively prime polynomials 互素多项式relatively quasicompact set 相对拟紧集relativity 相对性relativization 相对化relaxation method 松弛法relaxation parameter 松弛参数relay computer 继电平计算机reliability 可靠性reliable ciphers 可靠性数字reliable digit 可靠性数字remainder 剩余remainder of an infinite series 无穷级数的余项remainder type 剩余型remote control 摇控removable 可去的removable discontinuity 可消不连续点èwww．cIhui．BizPJremovable singularity 可消奇点remove 消除renewal density 更新密度renewal function 更新函数renewal theory 更新理论renormalization 重正规化renormalization group 重正规化群renumbering 重新编号reparametrization 再参量化repeat 重复repeated combination 有复组合repeated permutation 重复排列repeating decimal 循环十进小数repetition 重复repetitive error 重复误差replace 替换replacement 替换replacement theorem 替换定理replica 仿样representability 可表示性representable 可表示的representable function 可表示的函数representable functor 可表的函子representation 表示representation by arrows 射表示representation in components 分量表示representation module 表示模representation of a group 群的表示representation space 表示空间representation theory 表示论0Xm Www.cIHui．bIZㄠZ5キカけ‖representative 代表representative domain 表示域representative sample 代表样本representing function 表示函数representing graph 表示图reproducer 复制穿孔机reproducing kernel 再生核reproducing property 再生性reproducing punch 复制穿孔机requirement 必要residual 剩余的residual correlation of k th order k级剩余相关residual errors 剩余误差residual matrix 剩余阵residual set 剩余集residual spectrum 剩余谱residual sum 残差和residual term 剩余项residual variance 剩余方差residuals 剩余误差residue 残数residue class 剩余类residue class module 剩余类模residue class modulo m 模m剩余类residue of a function 函数的残数residue system 剩余系residue theorem 残数定理residueclass field 剩余类域residueclass ring 剩余类环resistance 抵制ぉㄣWwW．ciHUi．BiZresolution 分解resolution into partial fractions 部分分数分解resolution of singularities 奇点的消解resolvent 豫解式resolvent equation 豫解方程resolvent kernel 豫解式resolvent operator 豫解算子resolvent set 豫解集resolvent transformation 豫解变换resonance condition 共振条件response curve 反应曲线response surface 反应面rest 剩余restrict 限制。

标准模型简化版本The Standard Model is a fundamental theory in particle physics that describes how elementary particles interact with three fundamental forces: electromagnetism, the weak nuclear force, and the strong nuclear force. 标准模型是粒子物理中的一个基础理论，描述了基本粒子如何与三种基本力相互作用：电磁力、弱核力和强核力。

The Standard Model is a mathematical framework that has successfully predicted the existence of various particles, such as the Higgs boson, which was discovered at CERN in 2012. 标准模型是一个成功预测了各种粒子存在的数学框架，比如在2012年在欧洲核子研究中心发现的希格斯玻色子。

One of the key components of the Standard Model is the classification of elementary particles into two categories: fermions and bosons. Fermions are the building blocks of matter and include quarks and leptons, while bosons are force carriers, such as the photon and the W and Z bosons. 标准模型的关键组成部分之一是将基本粒子分为两类：费米子和玻色子。

(留给这次的数学建模美国赛吧)Aabsolute value 绝对值accept 接受 acceptable region 接受域additivity 可加性 adjusted 调整的 alternative hypothesis 对立假设analysis 分析analysis of covariance 协方差分析analysis of variance 方差分析arithmetic mean 算术平均值association 相关性assumption 假设assumption checking 假设检验availability 有效度average 均值Bbalanced 平衡的band 带宽bar chart 条形图beta-distribution 贝塔分布between groups 组间的bias 偏倚binomial distribution 二项分布binomial test 二项检验Ccalculate 计算case 个案category 类别center of gravity 重心central tendency 中心趋势chi-square distribution 卡方分布chi-square test 卡方检验classify 分类cluster analysis 聚类分析c oefficient 系数coefficient of correlation 相关系数collinearity 共线性column 列compare 比较comparison 对照components 构成，分量compound 复合的confidence interval 置信区间con sistency 一致性constant 常数continuous variable 连续变量control charts 控制图correlation 相关covariance 协方差covariance matrix 协方差矩阵critical point 临界点critical value 临界值crosstab 列联表cubic 三次的，立方的cubic term 三次项cumulative distribution function 累加分布函数curve estimation 曲线估计Ddata 数据default 默认的definition 定义deleted residual 剔除残差density function 密度函数d ependent variable 因变量description 描述design of experiment 试验设计deviations 差异df. (degree of freedom) 自由度diagnostic 诊断dimension 维discrete variable 离散变量discrimin ant function 判别函数discriminatory analysis 判别分析distance 距离distribution 分布D-optim al design D-优化设计Eeaqual 相等effects of interaction 交互效应efficiency 有效性eigenvalue 特征值equal size 等含量equation 方程error 误差estimate 估计estimation of parameters 参数估计estimations 估计量evaluate 衡量exact value 精确值expectation 期望expected value 期望值exponential 指数的exponential distributon 指数分布extreme value 极值F factor 因素，因子factor analysis 因子分析factor score 因子得分factorial designs 析因设计factorial experiment 析因试验fit 拟合fitted line 拟合线fitted value 拟合值fixed model 固定模型fixed variable 固定变量fractional factorial design 部分析因设计frequency 频数F-test F 检验full factorial design 完全析因设计function 函数Ggamma distribution 伽玛分布geometric mean 几何均值group 组Hharmomic mean 调和均值heterogeneity 不齐性histogram 直方图homogeneity 齐性homogen eity of variance 方差齐性hypothesis 假设hypothesis test 假设检验Iindependence 独立independent variable 自变量independent-samples 独立样本index 指数ind ex of correlation 相关指数interaction 交互作用interclass correlation 组内相关interval estimat e 区间估计intraclass correlation 组间相关inverse 倒数的iterate 迭代Kkernal 核Kolmogorov-Smirnov test柯尔莫哥洛夫-斯米诺夫检验kurtosis 峰度Llarge sample problem 大样本问题layer 层least-significant difference 最小显著差数least-squar e estimation 最小二乘估计least-square method 最小二乘法level 水平level of significance 显著性水平leverage value 中心化杠杆值life 寿命life test 寿命试验likelihood function 似然函数likelihood ratio test 似然比检验linear 线性的linear estimator 线性估计linear model 线性模型linear regression 线性回归linear relation 线性关系linear term 线性项logarithmic 对数的log arithms 对数logistic 逻辑的lost function 损失函数Mmain effect 主效应matrix 矩阵maximum 最大值maximum likelihood estimation 极大似然估计mean squared deviation(MSD) 均方差mean sum of square 均方和measure 衡量media 中位数M-estimator M估计minimum 最小值missing values 缺失值mixed model 混合模型mo de 众数model 模型Monte Carle method 蒙特卡罗法moving average 移动平均值multicollineari ty 多元共线性multiple comparison 多重比较multiple correlation 多重相关multiple correlation coefficient 复相关系数multiple correlation coefficient 多元相关系数multiple regression analysis 多元回归分析multiple regression equation 多元回归方程multiple response 多响应multivariat e analysis 多元分析Nnegative relationship 负相关nonadditively 不可加性nonlinear 非线性nonlinear regression 非线性回归noparametric tests 非参数检验normal distribution 正态分布null hypothesis 零假设n umber of cases 个案数Oone-sample 单样本one-tailed test 单侧检验one-way ANOVA 单向方差分析one-way classificat ion 单向分类optimal 优化的optimum allocation 最优配制order 排序order statistics 次序统计量origin 原点orthogonal 正交的outliers 异常值Ppaired observations 成对观测数据paired-sample 成对样本parameter 参数parameter estimatio n 参数估计partial correlation 偏相关partial correlation coefficient 偏相关系数partial regressio n coefficient 偏回归系数percent 百分数percentiles 百分位数pie chart 饼图point estimate 点估计poisson distribution 泊松分布polynomial curve 多项式曲线polynomial regression 多项式回归polynomials 多项式positive relationship 正相关power 幂P-P plot P-P概率图predict 预测predicted value 预测值prediction intervals 预测区间principal component analysis 主成分分析proability 概率probability density function 概率密度函数probit analysis 概率分析proportion 比例Qqadratic 二次的Q-Q plot Q-Q概率图quadratic term 二次项quality control 质量控制quantitati ve 数量的，度量的quartiles 四分位数Rrandom 随机的random number 随机数random number 随机数random sampling 随机取样ra ndom seed 随机数种子random variable 随机变量randomization 随机化range 极差rank 秩r ank correlation 秩相关rank statistic 秩统计量regression analysis 回归分析regression coefficie nt 回归系数regression line 回归线reject 拒绝rejection region 拒绝域relationship 关系reliab ility 可*性repeated 重复的report 报告，报表residual 残差residual sum of squares 剩余平方和response 响应risk function 风险函数robustness 稳健性root mean square 标准差row 行run 游程run test 游程检验Sample 样本sample size 样本容量sample space 样本空间sampling 取样sampling inspection 抽样检验scatter chart 散点图S-curve S形曲线separately 单独地sets 集合sign test 符号检验significance 显著性significance level 显著性水平significance testing 显著性检验significant 显著的，有效的significant digits 有效数字skewed distribution 偏态分布skewness 偏度small s ample problem 小样本问题smooth 平滑sort 排序soruces of variation 方差来源space 空间s pread 扩展square 平方standard deviation 标准离差standard error of mean 均值的标准误差standardization 标准化standardize 标准化statistic 统计量statistical quality control 统计质量控制std. residual 标准残差stepwise regression analysis 逐步回归stimulus 刺激strong assumpt ion 强假设stud. deleted residual 学生化剔除残差stud. residual 学生化残差subsamples 次级样本sufficient statistic 充分统计量sum 和sum of squares 平方和summary 概括，综述Ttable 表t-distribution t分布test 检验test criterion 检验判据test for linearity 线性检验test o f goodness of fit 拟合优度检验test of homogeneity 齐性检验test of independence 独立性检验test rules 检验法则test statistics 检验统计量testing function 检验函数time series 时间序列tolerance limits 容许限total 总共，和transformation 转换treatment 处理trimmed mean 截尾均值true value 真值t-test t检验two-tailed test 双侧检验Uunbalanced 不平衡的unbiased estimation 无偏估计unbiasedness 无偏性uniform distribution 均匀分布Vvalue of estimator 估计值variable 变量variance 方差variance components 方差分量varianc e ratio 方差比various 不同的vector 向量Wweight 加权，权重weighted average 加权平均值within groups 组内的ZZ score Z分数2. 最优化方法词汇英汉对照表Aactive constraint 活动约束active set method 活动集法analytic gradient 解析梯度approximate 近似arbitrary 强制性的argument 变量attainment factor 达到因子Bbandwidth 带宽be equivalent to 等价于best-fit 最佳拟合bound 边界Ccoefficient 系数complex-value 复数值component 分量constant 常数constrained 有约束的co nstraint 约束constraint function 约束函数continuous 连续的converge 收敛cubic polynomial interpolation method三次多项式插值法curve-fitting 曲线拟合Ddata-fitting 数据拟合default 默认的，默认的define 定义diagonal 对角的direct search metho d 直接搜索法direction of search 搜索方向discontinuous 不连续Eeigenvalue 特征值empty matrix 空矩阵equality 等式exceeded 溢出的Ffeasible 可行的feasible solution 可行解finite-difference 有限差分first-order 一阶GGauss-Newton method 高斯-牛顿法goal attainment problem 目标达到问题gradient 梯度gradi ent method 梯度法Hhandle 句柄Hessian matrix 海色矩阵Independent variables 独立变量inequality 不等式infeasibility 不可行性infeasible 不可行的init ial feasible solution 初始可行解initialize 初始化inverse 逆invoke 激活iteration 迭代iteration 迭代JJacobian 雅可比矩阵LLagrange multiplier 拉格朗日乘子large-scale 大型的least square 最小二乘least squares sens e 最小二乘意义上的Levenberg-Marquardt method 列文伯格-马夸尔特法line search 一维搜索li near 线性的linear equality constraints 线性等式约束linear programming problem 线性规划问题local solution 局部解M medium-scale 中型的minimize 最小化mixed quadratic and cubic polynomial interpolation and extrapolation method 混合二次、三次多项式内插、外插法multiobjective 多目标的Nnonlinear 非线性的norm 范数Oobjective function 目标函数observed data 测量数据optimization routine 优化过程optimize 优化optimizer 求解器over-determined system 超定系统Pparameter 参数partial derivatives 偏导数polynomial interpolation method 多项式插值法Qquadratic 二次的quadratic interpolation method 二次内插法quadratic programming 二次规划Rreal-value 实数值residuals 残差robust 稳健的robustness 稳健性，鲁棒性S scalar 标量semi-infinitely problem 半无限问题Sequential Quadratic Programming method 序列二次规划法simplex search method 单纯形法solution 解sparse matrix 稀疏矩阵sparsity pattern 稀疏模式sparsity structure 稀疏结构starting point 初始点step length 步长subspace trust region method 子空间置信域法sum-of-squares 平方和symmetric matrix 对称矩阵Ttermination message 终止信息termination tolerance 终止容限the exit condition 退出条件the method of steepest descent 最速下降法transpose 转置Uunconstrained 无约束的under-determined system 负定系统Vvariable 变量vector 矢量Wweighting matrix 加权矩阵3 样条词汇英汉对照表Aapproximation 逼近array 数组a spline in b-form/b-spline b样条a spline of polynomial piece /ppform spline 分段多项式样条Bbivariate spline function 二元样条函数break/breaks 断点Ccoefficient/coefficients 系数cubic interpolation 三次插值/三次内插cubic polynomial 三次多项式cubic smoothing spline 三次平滑样条cubic spline 三次样条cubic spline interpolation 三次样条插值/三次样条内插curve 曲线Ddegree of freedom 自由度dimension 维数Eend conditions 约束条件input argument 输入参数interpolation 插值/内插interval 取值区间Kknot/knots 节点Lleast-squares approximation 最小二乘拟合Mmultiplicity 重次multivariate function 多元函数Ooptional argument 可选参数order 阶次output argument 输出参数P point/points 数据点Rrational spline 有理样条rounding error 舍入误差（相对误差）Sscalar 标量sequence 数列（数组）spline 样条spline approximation 样条逼近/样条拟合spline function 样条函数spline curve 样条曲线spline interpolation 样条插值/样条内插spline surface 样条曲面smoothing spline 平滑样条Ttolerance 允许精度Uunivariate function 一元函数Vvector 向量Wweight/weights 权重4 偏微分方程数值解词汇英汉对照表Aabsolute error 绝对误差absolute tolerance 绝对容限adaptive mesh 适应性网格Bboundary condition 边界条件Ccontour plot 等值线图converge 收敛coordinate 坐标系Ddecomposed 分解的decomposed geometry matrix 分解几何矩阵diagonal matrix 对角矩阵Diri chlet boundary conditions Dirichlet边界条件Eeigenvalue 特征值elliptic 椭圆形的error estimate 误差估计exact solution 精确解Ggeneralized Neumann boundary condition 推广的Neumann边界条件geometry 几何形状geome try description matrix 几何描述矩阵geometry matrix 几何矩阵graphical user interface（GUI）图形用户界面Hhyperbolic 双曲线的Iinitial mesh 初始网格Jjiggle 微调LLagrange multipliers 拉格朗日乘子Laplace equation 拉普拉斯方程linear interpolation 线性插值loop 循环Mmachine precision 机器精度mixed boundary condition 混合边界条件NNeuman boundary condition Neuman边界条件node point 节点nonlinear solver 非线性求解器normal vector 法向量PParabolic 抛物线型的partial differential equation 偏微分方程plane strain 平面应变plane stres s 平面应力Poisson's equation 泊松方程polygon 多边形positive definite 正定Qquality 质量Rrefined triangular mesh 加密的三角形网格relative tolerance 相对容限relative tolerance 相对容限residual 残差residual norm 残差范数Ssingular 奇异的。

the standard model in a nutshell -回复【标准模型在壳中】标准模型是现代物理学的基石之一，它描述了自然界中的基本粒子以及它们之间的相互作用。

这篇文章将逐步介绍标准模型的基本概念和重要特性。

1. 什么是标准模型？标准模型是一个理论框架，用于描述物质的基本构成单元和它们如何相互作用。

它是量子场论的一个分支，其中包含了所有已知的基本粒子和力的理论。

2. 标准模型包括哪些粒子？标准模型中有两种基本类型的粒子：费米子和玻色子。

费米子是构成物质的基本粒子，包括夸克和轻子。

夸克有六种类型（上、下、奇、魅、顶、底），而轻子则有六种类型（电子、μ子、τ子、电子中微子、μ子中微子、τ子中微子）。

玻色子则是传递力的粒子，包括光子、W和Z玻色子、胶子和希格斯玻色子。

3. 标准模型描述了哪些力？标准模型描述了四种基本力：强核力、弱核力、电磁力和引力。

其中，强核力由胶子传递，负责维持原子核内部的稳定；弱核力由W和Z玻色子传递，负责某些放射性衰变过程；电磁力由光子传递，负责电磁相互作用；而引力虽然在广义相对论中有很好的描述，但在标准模型中尚未得到完全整合。

4. 标准模型是如何解释粒子的质量的？在标准模型中，粒子的质量主要通过希格斯机制来解释。

根据这一机制，希格斯场处于一个非零的真空期望值状态，当其他粒子在这种状态下运动时，会与希格斯场发生相互作用，从而获得质量。

这种质量获取机制也被称为“吃掉希格斯场”。

5. 标准模型有哪些成功之处？标准模型取得了许多重大成功。

例如，它成功预测了W和Z玻色子的存在，这些粒子后来在实验中被发现。

此外，标准模型还成功地解释了质子和中子的结构，以及它们之间的强相互作用。

标准模型的预言也被多次验证，其精确度达到了惊人的程度。

6. 标准模型有哪些局限性？尽管标准模型取得了很大的成功，但它仍然存在一些重要的局限性。

首先，标准模型无法解释引力。

其次，标准模型不能解释暗物质的存在。

the standard model in a nutshell什么是标准模型？标准模型（Standard Model）是物理学中一个非常重要的理论框架，它描述了微观世界中基本粒子的组成和相互作用。

标准模型是粒子物理学领域中目前最为广泛接受的理论，它能够解释和预测经过实验证实的多种现象。

标准模型的构建经过了几十年的努力，是许多科学家和研究人员合作努力的成果。

它的基本框架由四个基本力和三代基本粒子构成，其中包括了强力、弱力、电磁力和引力。

那么，让我们逐步来了解标准模型的主要内容。

1. 基本粒子的分类标准模型将所有基本粒子分为两类：费米子和玻色子。

费米子是构成物质的基本粒子，如电子、夸克等。

它们遵循费米-狄拉克统计，具有自旋1/2，obey the Pauli Exclusion Principle 端口and therefore follow the Fermi-Dirac statistics 费米-狄拉克统计. 费米子根据电荷不同又分为带电荷的和不带电荷的：- 带电子的费米子包括：电子、muon、tau子和它们对应的中微子。

- 不带电子的费米子包括：6种夸克和其对应的反粒子以及3种中微子。

玻色子是能量传递的基本粒子，如光子、W和Z弦子以及引力子。

它们遵循玻色-爱因斯坦统计，具有自旋为整数（0、1、2等）。

玻色子是荷的载体，通过它们粒子之间的相互作用才得以实现。

2. 基本力的相互作用标准模型涵盖了四种基本力的描述：强力、弱力、电磁力和引力。

- 强力（Strong Force）是负责夸克之间的相互作用，以及质子和中子之间的结合。

它通过交换玻色子-胶子（gluon）来传递力量，这源于夸克带有"色荷"，而胶子负责夸克之间的"色力"相互作用。

- 弱力（Weak Force）是负责一些基本粒子的衰变，如β衰变。

它通过交换W和Z弦子来传递力量。

- 电磁力（Electromagnetic Force）负责电子和原子核之间的相互作用。