粒子物理与对称性

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,Particle and Symmetry,Lecturer,：,Wei-Zhou Jiang,References,1,L.Ryder, Elementary particles and symmetries,(粒子物理与对称性，宋孝同等译）,2W.M. Gibson, B.R.Pollard,Symmetry principles in elementary particle physics,（基本粒子物理学中的对称性原理， 丁里译）,3李政道，场论与粒子物理,1979,4黄克孙，夸克、轻子与规范场,1982,5,许咨宗，课件：粒子物理中的守恒定律（中科大）,;,核与粒子物理导论,中国科技大学出版社,6戴又善，粒子物理讲义（浙大）,6,R. Casalbuoni , Quantum Field Theory,(1997),7来自Wikipedia的资料,9郑恒阳，粒子物理中的对称性和对称性破缺（山大，PPT报告）,10陆埮,罗辽复,物质探微 从电子到夸克,(2005,科普读物),11Ian Aitchison, Supersymmetry in Particle Physics: An Elementary Introduction(2007,Cambridge),Outline,1.,Introduction to fundamental particles,2.,Classic and quantum symmetry principles,3.,Lorentz invariance (covariance),4.,Parity,（）,space inflection, angular momentum,5.,Charge conjugate symmetry (C), CP symmetry and its breaking,6.,Time inversion symmetry,（）、,theorem,7.,Isospin, gauge, chiral symmetries and their spontaneous breakings,Below are possible lectures:,夸克模型、标准模型、强子共振态，介子衰变、G宇称(?),超对称Ads/CFT（？）,iggs 粒子与重正化（？）,手征对称性与标量场（？）,超对称,粒子物理的基本内容,与量子力学和量子场论相关,微观粒子（原子、分子）性质；薛定谔方程；定态及散射,基本粒子遵循之对称性,基本粒子及其运动规律,基本粒子及其相互作用,物质世界的基本构成(结构与相互作用）、宇宙学等，beyond quantum mechanics,1.,基本粒子简介,1.1,发展简史,物质的基本单元和基本结构的思辨探索,殷（前,1600,年）周（前,1027,）时期的古代五行说，金木水火土,战国 墨子：“端”是无同也，莫能破,古印度：风火水土，四大皆空,古希腊：恩培多克勒（约公元前,490430,）柏拉图（公元前,427347,） 水、气、火、土四元素。,宋朝,:,朱熹,理气说,理在气先,古希腊：德谟克利特（前四世纪），原子,近代：,1807,年道尔顿， 提出原子论,1869,年 门捷列夫， 元素周期律,现代粒子物理与核物理的发展,1895年，放射性： X射线（W.C.Rontgen伦琴）,1896、1898年 天然放射性（A.H. Becquerel贝克勒尔；M. Curie 居里夫妇 ),1897年， 电子（J.J. Tohmson汤姆生）,1911年， Rutherford：,散射实验-原子核式结构；R. A. Millikan 油滴实验,1926年， P.A. Dirac: 相对论量子力学，真空，反粒子,1930年，正电子发现：,赵忠尧,；,1932年，Anderson（1936年，Nobel Prize),1932年，中子的发现：Chadwick,1930年，中微子：泡利 Pauli，费米Fermi（理论），,王淦昌,（实验建议），莱茵斯（1955年, Nobel Prize）,1935年，介子：汤川秀树 H. Yukawa,1950年代，强子共振态, 200多种 （宇宙线、加速器）,1954年，,杨,-Mills规范场， W, Z介子, 胶子: 弱、强作用,1957年,李政道，杨振宁,：Nobel Prize, 弱作用宇称不守恒,1964,年，,M. Gell-Mann,强子结构的夸克模型,1967,年，,S. Wemberg, A.Salam, S.L. Glashow,弱电统一理论，,W,，,Z,波色子,1983,年，西欧核子中心（,CERN,）发现,W,，,Z,波色子,1970,年代，量子色动力学（,QCD,）,亚夸克，前子（,Preon,），毛子,(Maon,1977 Glashow,基于,Onion-layer straton model,的提议,),？ 公孙龙：一尺之棰，日取其半，万世不竭,物质从哪里来？真空与物质，,心经,，色不异空，空不异色，色即是空，空即是色。,1.2,粒子分类,轻子,e,; ； 及其反粒子,强子,a) 重子：,b) 介子：,规范波色子,Higgs粒子 ?,共振态,右图：Kerson Huang, Quarks，,Leptons & Gauge Fields, p5,(1992, World Scientific Publishing), Denoted as,Huang1992,later on.,高能物理实验,宇宙线：,其中包括约87%质子，12%粒子（氦核子），其余大部分是原子核、电子、,射线以及超高能中微子也构成一小部分宇宙射线。右图来自Wiki,pedia。,中国：云南落雪山 3180米云室， 在川3222米 大型磁云室；西藏甘巴拉山5500米，高山乳胶室，羊八井4300米，国际宇宙线观测站（中日、中意）；高空热气球，卫星（AMS，东南大学）,地球实验室,加速器,1932,年，高压加速器，质子,0.7MeV,1939,年，回旋加速器，质子,20MeV,1946,年，稳相加速器（变电场），,190MeV,1953,年，质子同步加速器（变磁场），,23GeV,现有著名加速器，高真空，强流，超导强磁,右图：张闯，漫谈对撞机，,现代物理知识,2007,年第二期,1.3,相互作用,From,Huang1992,四种相互作用,一些例子,以交换虚粒子实现相互作用,A),力程,B) 衰变寿命,Interactions and scattering cross sections,Rutherford,scattering,S,matrix in general cases,：,All reactions including,have the,electromagnetic interaction,Reactions including,come from the weak interaction,With,e,included are reactions of the weak or electric interactions,All mesons decay eventually into pairs of the electron an d positron, neutrino, an diphotons,；,All baryons, except protons, will decay.,From,Huang1992,Questions,：,1,。,How to produce more particles?,2,。,How to classify these particles?,3,。,How are particles arranged into the structure of bulk matter that is determined by the interactions?,4,。,More fundamental interactions? new physics?,0,到双光子的衰变？,+g,q +qbar?,1.4,Natural unit,Small energy scale for microscopic particles short time scale in the process of interactions. Its more convenient to use the natural unit:,1.5,Particles, celestrial bodies and cosmos,Big bang, inflation, dark matter, dark energy,温度（K）,Energy,（eV）,Time,（,s,）,Epoch,物理过程,10,32,10,28,10,-44,Planck,10,28,10,24,10,-36,Grand unif.,10,-35,-33,Inflation,暴胀过程,10,13,10,9,10,-6,Quark,10,11,10,7,10,-2,Lepton,10,10,10,6,1, decoupling,中微子脱耦,5,10,9,5,10,5,5,Photon,电子对湮灭,10,9,10,5,3分,Proton,轻核素生成,3,10,3,0.3,38万年,Atom,微波背景辐射,4亿年,Stars,再电离,Galaxy,大尺度结构形成,2.7,3,10,-4,137亿年,Present,表格来自 陆埮教授,What are dark matter and energy?,Dark Matter,23%,Dark Energy,73%,“Normal Matter”,4%,Are they related to large-mass celestial body?,From Big Bang to Black Holes,10,-10,-100,s,Proton,neutron,After 1 Billion years,Neutrson Star,Proton,Neutron,Black hole,(R10km),The energy resources in,nature are from the,combustion of PROTONS!,习题,掌握粒子物理发展的主要事件。,加速器在粒子物理发展中的作用是什么？,简述四种相互作用。,说明相互作用与散射截面、寿命的关系。,基本粒子是如何分类的？,举例说明自然单位制的时空和物质度量单位及与国际单位制的关系。,Symmety,Conservation law,Rotational symmetry,Angular momentum,Baryon number,Lepton number,Isospin,Spatial translation,Momentum conservation,Temporal translation,Energy conservation,Global symmetry,Conservation law,Chapter 2: Classical and quantum symmetry princples,Symmetry,:,Invariance of the physical laws under the transformation of interior and exterior degrees of freedom,(in,distinguishability,),2.1,Symmetry and conservation laws,，Noether,theorem,U（1），,Charge,Electromagnetic interaction,SU（2），,Weak Charge,weak interaction,*,SU（3），,color (charge),Strong interaction,Local symmetry,Interaction,*SU(2) 对称性首先是杨-Mills规范场中引入,以研究强相互作用的同位旋对称性的，见,C.N.Yang and R.L.Mills, Phys.Rev. 96,191(1954).,Above is the continuous symmetry,Concrete symmetry,：,Space reflection,（P）,Time inversion,（T）,Charge conjugate,（C）,particle and antiparticle,Explicit breaking of symmetry, Breakings of parity conservation and charge conjugate symmetry in weak interaction, (,弱作用的宇称不守恒、电荷共轭对称破坏等,),；,Related to the spontaneous symmetry breaking,？,(,与自发破缺的关系？,),Spontaneous breaking of symmetry （caused by vacuum asymmetry）breakings of gauge and chiral symmetries.,Two modes: Goldstone Theorem,J. Goldstone, Nuovo Cimento, 19,154(1961),；Higgs mechanism,。,2.2,Examples of conservation lawsBaryon number and lepton number,In any process, the baryon and lepton numbers should be conserved.,Because Fermions are real particles，the stability of matter requires,that the creation and annihilation be in pairs.,2.3,Symmetry in classical mechanics,1。,Invariance of spatial translation, namely,an absolute position is unmeasurable,Momentum conservation,1）,Example: two-particle system,Interaction potential:,displacement:,a,？,2）General case: Lagrange equation in Analytical Mechanics,2。,Energy conservation,，,namely, absolute time is unmeasurable,?,2.3,Symmetry in quantum mechanics,在量子力学中已有讨论，如见曾谨言量子力学上册,Generally speaking, Conserved quanta are the expectation value of operators, different from the classical mechanics,From,Schroedinger,equation, it has,1。,Translational transformation,Infinitesimal transformation,Finite transformation,2。,Temporal translation,3。,Invariance of rotation,4. Invariance of gauge transformation,Here, we are limited to the global symmetry that is independent of space and time.,习题,1,。在经典情形证明时间平移不变性导致能量守恒；在量子情形下，无穷下变换和有限变换下分别证明时间平移不变性导致能量守恒。,2,。如,1,题证明转动不变性导致角动量守恒。,3.,Lorentz invariance (covariance),Lorentz Group,Equation of relativistic quantum mechanics,Requirements for,Lagrangian,3.,1,Lorentz Group,Concept of the group,：The operations or the set that have closeness.,Lorentz transformation,The event interval is a Lorentz scalar：,Lorentz group,Lorentz group also includes the space-time reflection for the invariance of s,2 .,If not included, it is dubbed Lorentz boost洛仑兹换速。,If the translation is added to the Lorentz rotation, then it forms the Poincare group.,3.,2 Simple relativistic mechanics,Lorentz Scalar,3.,3 Requirements for Lagrangian,All equations of motion are derived from the principle of least action, i.e., Euler-Lagrange equaiton。,Lagrangian must be the Lorentz scalar，so that not changed with the transformation of the inertial frame:,Free fields and interactions,Is the mass the Lorentz scalar？,3.,4 More content of covariance,CPT （discuss later on）,Generalised covariance: covariant derivative,习题,1.,2.,3.,About learning method: to have goals,苍龙：粒子物理， Sky dragon,地龙：量子力学, Earth dragon,Self-confidence to dig out new physics,beyond quantum mechanics,元音老人,碧岩录,讲座,昔五祖演会下有一僧请益五祖：如何是末后句？ 祖云：你师兄会末后句，问他去。僧问师兄，适逢游山回，僧为打水洗脚次，进问云：如何是末后句？ 师兄以脚挑水洒其面斥云：什么末后句？ ！ 僧哭诉祖，祖云：我向你道，他会末后句！ 僧于言下大,悟。,陈献章、王守仁,4.Parity,（） 、,Space reflection symmetry, angular momentum,4.1,Space reflection,（夫思妻）枯眼望遥山隔水，往来曾见几心知。壶空怕酌一杯酒，笔下难成和韵诗。途路阻人离别久，讯音无雁寄回迟。孤灯夜守长寥寂，夫忆妻兮父忆儿。,（,妻想夫）,儿忆父兮妻忆夫，寂寥长守夜灯孤。迟回寄雁无音讯，久别离人阻路途。诗韵和成难下笔，酒杯一酌怕空壶。知心几见曾来往，水隔山遥望眼枯。,夫妻相思宋李禺,（夫思妻）枯眼望遥山隔水，往来曾见几心知。壶空怕酌一杯酒，笔下难成和韵诗。途路阻人离别久，讯音无雁寄回迟。孤灯夜守长寥寂，夫忆妻兮父忆儿。,DNA,shows the breaking of mirror image,.,Chirality,长瓣兜兰paphiopedilum,：,The helixes at both sides have the left-right symmetry: the right is left-handed helix and the left right-handed.,DNA,自然界的许多分子，手性分子占去了很大的比例。构成蛋白质的,氨基酸,都是,L 型,氨基酸，,多糖,和核酸的,单糖,是 R,型,糖。人们甚至发现，1969年坠落在澳大利亚默奇森的陨石,(,aerolite,),中的氨基酸,(amino acid),也主要是L 型的。,Chirality of biological molecules,4.,2 Angular momentum and Parity,x,y,z,x,y,z,xyz,xyz,-,+,Spatial wave,function:,习题：,An example,：,Electric dipole transition,Probability,4.,3 Identification of the intrinsic parity of (static) particles,Single particle with given momentum is not the eigenstate of parity(?why),1,。,Identification of intrinsic parity of,meson,-Deuteron reaction,，,the final state produces two neutrons. This reaction can be accompanied by the radiation of KX-ray from the deuteron-pion atom,，,-,L=0,KX,-ray,i=L=0,Odevity of angular momentum from the relative motion of two neutrons,All possible states of two identical neutrons,0， 1， 2,Only one state meets,the conservation of,angular momentum,and the antisymmetry of,for fermionic system,：,Initial state,：,Possible final states of,（n,n）,，,0,belong to the same isospin mulplet and they should have the same parity:,meson: pseudoscalar meson,2,。,Identification of photon intrinsic parity,P,R,y,(,),x,y,z,z,y,x,y,z,x,Spin & parity quantum numbers of particles,4.,3 Parity violation in the weak interaction,We should know that the parity conservation remains in the strong and electromagnetic interactions .,（1）,-,puzzle,Angular momentum conservation of identification of parity,J,P,0,-,0,-,0,-,0,-,0,-,0,-,0,-,L L,+,L,-,L,L,+,L,-,+,+,L,+,0,J,f,=,L,+,+,L,-,=0,L,+,=,L,-,=L,J,f,=L=0,(f)=,P,3,(,)(-1),2L,=-1,(f)=,P,2,(,)(-1),L,=+1,A.M.,Parity,Dalitz,diagram analysis,Lee-Yangs work,If the weak interaction conserves parity,must be two kinds of particles having different parity.,But they share the same spin, mass, yield and lifetime. So,should be like the same particle, while the parity is violated then. This is the,puzzle.,After reviewing the experimental data of particles and nuclides before,1956, Lee and Yang found that the parity is conserved in the strong and electromagnetic interaction, while in the process of the weak interaction, like the decay of the particles,-decay of nuclides,，,no data can elaborate that the parity is conserved.,is thus the final-state particles of some particle which is produced through the strong interaction, and decey into different final states via the weak interaction that violates the parity.,-,就是粒子K,+,Experimental verification,Theoretic probe,Measurement of longitudinal polarization of,particle,The spin orientation of final-state electron and anti-neutrino should be along the orientation of polarization:,Longitudinal polarization of e is,defined as,：,I,+,I,-,For positron,+1，,of right-hand priority,；,for electron,1，,of left-hand priority,.,e-,e+,v/c,P,*,*,*,*,*,-0.5,0.5,3,H,60,Co,32,P,（,2,）,Neutrino helicity in the weak interaction,1958年M.Goldhaber等人利用,核素的 K电子俘获特殊的衰变方式,巧妙地从实验证明了中微子的螺旋度。,衰变级联过程为, &,Sm*,have opposite momentum,and angular momentum (AM) for,AM conservation,，,h of, thus,conforms with that of,Sm*,，,while,the latter is determined by the,helicity of,.,实验上测得左旋光子。,（3）,production & decay: parity conservation or not?,强产生的,粒子是横向极化的。横向极化态的空间反射具有不变性,产生平面法线,z,：,（4）,For parity conservation,peudoscalar must be zero,习题,5.Charge conjugation,（）,、,CP symmetry and its breaking,5.1Charge conjugate symmetry,the operator C,：,transformation between the particle and antiparticle,，,changes all intrinsic quanta, e.g., Charge Q,，,baryon no. B,，,lepton no. L,，,strange no. S,，,hypercharge G,，,the 3rd component of isospin,I,3,.,C does not change the spin,;,自旋虽然是内禀自由度,但与时空性质相关,.,内部自由度对称性给出,additive quantum numbers,.,Charge parity of photons:,charge parity of,0,Generalized Pauli principle,5.3 Verification of,C conjugate symmetry,1. C-parity conservation in electromagnetic interaction:,C,(-1),L+S,(-1),n,Oddevity of photon number is decided by the (L+S) of positionium,In the electromagnetic process, the C-parity conservation is verified up to,a quite high precision.,Compare two mutually conjugate process:,Meson and anti-meson in the f.s. feature a perfect symmetry.,Especially for the exclusive process,+,（,-,），,C,-parity symmetry gives,symmetric angular distribution,of cross section:,and,0,produced by,P,-,P,bar,annhilation is symmetric about,=90,。,。,Strong decay of vector meson:,J,PC,1,0,-,0,-,0,-+,L 1 1,C,1 -1 +1,P,-1 -1 +1,2. C,-parity conservation in strong interaction process,3. C-parity breaking in the weak interaction process,，,conservation of the joint CP transformtion,P：,l.h. r.h.,C：,particle-antiparticle,作业：分析下列反应的荷称是否守恒,J,PC,1,0,-,0,-,0,-+,L 1 1,5.4 CP,symmetry breaking and neutral Kaon decay,1.,Background,However, the difference comes up for the neutral K decay.,-,纯,K,0,K,0,K,0,+,-,S（Y） +1 0 1,Strangeness is not a conserved number of the weak interaction, while K,0 &,K,0,bar are eigenstates of the strong interaction, instead of the weak interaction,（见,M. Gell-Mann and A. Pais,的工作（,Phys. Rev. 97, 1387, 1955,.）,The weak interaction eigenstates should come from the mixing of them:,The oscillation observation：in pure K,0,beam there is the reaction arising,from K,0,bar,-,+ p,-,纯,K,0,p,纯,K,0,K,2,S=-1,3. CP,breaking,In,1964,，,Christenson,Cronin, Fitch,Turlay,found in the neutral Kaon decays the symmetry breaking of the joint CP operation,(,Phys. Rev. Lett., 13, 138 (1964).),The same meson decays into states with opposite CPCP breaking！,Solution：,（,1,）Mixing leads to,CP,breaking,Phenomenologic theo,r,y,may use 3 parameters related to the amplitude and width to describe to describe the CP breaking,of the decay of the long-lived K,L,如果K,L,中K,0,K,0,Bar等幅混合，,0,K,1,&K2 have the definite CP oddevity:,Odd and even CP mixing states: K,S ,K,L,The unequal mixing of that leads to the CP breaking is the,so-called superweak theory for CP breaking,p=q ，equal mixing，CP symmetric,K,0,K,0,+,-,G,w,G,w,G,sw,K,0,K,0,（,2,）Direct,CP,breaking,直接比较K,0,反K,0,到达同位旋为,I,的介子末态的衰变振幅的差异,the final state phase shift,。For the CP invariance， must be real.,The observable of CP breaking , mixing parameters and decay amplitude,meet following relations,The data indicate：,1,Extraction from experiments：,In the neutral K decay, direct CP breaking is unimportant.,If has a tiny imaginary part, of 2008 Nobel Prize,CP& 3,rd,generation of quarks,3.,Microscopic model for,CP,breaking,CKM model,CabibboKobayashiMaskawa,In 1963，Cabibbo introduced the Cabibbo angle to keep the universality of,the quark model in the weak interaction，After Charm quark found, the,strength of the flavor changing decay can be expressed by the 2X2 matrix,1973年KM发现2X2的四夸克模型不能解释CP破缺，把Cabibbo矩阵,一般化为3X3的如下形式：,三代夸克。,带撇的是弱作用的本征态，,右边不带撇的是质量本征,态，弱作用里它们不一样,From Wikipedia,But in KM theory, CP breaking is not enough to interpret the matter-antimatter asymmetry.,Must look for new physics beyond the standard model,Why in the Universe, still no anti-matter found,？CP,symmetry breaking,，,e.g.,Who is the future Hopeful of the Nobel Prize?,6. Time reversal symmetry,（）、,theorem,Time reversal:,The macroscopic time direction can be defined by the irreversible process, like the antropy evolution,Time reversal is, in fact, the reversal of the motion. Time reversal invariance is just the invariance of the motion.,6.1,Time reversal in Classical mechanics,注意：,F,T,的自变量为,-t,.,角动量,L,与动量,p,一致.,Time reversal,Motion reversal,见 W.M. 吉布森，B.R.波拉德，(丁里译）基本粒子物理学中的对称性原理，p197.,6.2 Time reversal in quantum mechanics,Expectation value after time reversal,：,T,anti-unitary operator,：,可令,T=UK，K,为取复共轭且t-t，,T transformation for operator,A,:,6.3 Detailed balance principle,This indicates that for the motion reversal, probability is invariant:,Time reversal invariance.,2. The principle of detailed balance,For the space-time reflection symmetry, the transition matrix elements for the forward and backward processes should be equal.,detailed balance principle,.,3. Examples,C,is phase space factor,cm,the center of mass. One should,sum all the final states and average over the initial state.,1950s, Marshak, Cheston, Cartwright, Durbin 等人用来确定,介子的自旋.,4. Electric diploe (ED) of neutrons,Given a definite parity for the particle then its expectation value must be zero, in that the electric dipole has the odd parity.,For a spin particle, the ED should also be strictly zero due to the T symmetry.,Note,：,Parity violation can also leads to nonzero ED, so we need consider,（1）,Weak interaction that violates the parity, with magnitude,G,F,M,p,2,（2）,EM interaction that violates T symmetry with the magnitude,g=o/e，,the,ratio of T-odd and T-even matrix element.,对于弱作用，强子的非轻子衰变(K,0,等)破坏T 对称性。,CPT联合变换的不变性？,6.4 CPT,定理,所有相互作用过程，在CP和T变换的联合作用下具有不变性，,不管它们的顺序如何放置。这是相对论协变场论的内在性质，,比如正反粒子的对称性蕴含在Dirac方程中。,但CPT联合变换，没有守恒量,此外，可以证明：正反粒子具有相同的总寿命、电荷、旋磁比,(gyromagnetic ratio),g.,7.,同位旋,(Isospin),、规范对称性,(gauge symmetry),、手征对称性,(chiral symmetry),相互作用,(interaction),7.1,同位旋,1),质子数和中子数互易的镜像核,(mirror nuclei),具有几乎相同的结合,(binding energy),能,(,其结合能差仅为库仑能差,),、相似的能级,(energy level),分布和相同的自旋,2),在轨角量子数,l,和自旋量子数,s,分别相同时，,pp,散射与,np,散射的相角相同、散射长度,(scattering length),相近、有效力程,(force range),相同；在,l=0,，,s,0,的,1s,态，相互作用能都近似为,11.9 MeV,3),入射核子能量相等时，在所有散射角下，,p,p,和,n,n,弹性散射的微分截面和总截面分别相等,1932,年，海森伯指出：在原子核内，状态相同的质子和质子、质子和中子、中子和中子之间的核力相等，与其是否带电无关,;1932,年，,Wigner,采用了类似于自旋的表述，,1937,年取名为同位旋（,Isotopic spin, Isobaric spin,今天为,Isospin).,这在,1937,年为实验证实，,19461955,年间为更精确的实验证实,Vpp,Vnn,和,Vpp=Vnn=Vnp,的准确性分别大于,99,和,98,Nucleon Isospin,介,子同位旋,超荷与奇异性,粒子电荷与同位旋第三分量,Quantities of mesons and baryons,SU,（,2,）对称性与生成元,Symmetry and generator,Isospin conservation,：,I,H,=0.,7.,2,gauge symmetry,A typical instance for gauge transformation,Invariance of,Maxwell,equations,Equations have gauge symmetry,Global gauge symmetry,：,因此，拉氏量,0,在局域规范变换下不是不变的.,电荷守恒要求拉氏量电中性,这里，e可看作U（1）群的生成元，电磁场A,即为遵循U(1)规范变换的规范场。,Yang-Mills,场,规范不变：,标准模型（弱电统一、QCD）的基础.,7.,3 Chiral symmetry,手征对称性,Noether theorem: 给定系统的Lagrangian，有,对于内部对称性，,7.,4,对称性自发破缺（,spontaneous symmetry breaking),物理真空,(vacuum),能量最低态,(lowest-energy state),模型真空,(modeling vacuum,，破缺的真空,对称性自发破缺通常是指在对称操作下系统能量最低态不是不变的。,例子：,Photon mass in superconductor,Mass of gauge bosons in the unified weak-electromagnetic theory,宇宙大暴涨,(inflation),的标量场真空破缺，,Spontaneous breaking of chiral model,From Abers & Lee, Phys. Rept. 9 (1973)1-141,破缺生成元,n,2,个矩阵元，n(n-1)/2个独立,生成元,Two modes for spontaneous breaking,Goldstone mode,For global