量子场论笔记与习题(Ⅱ).doc

教材：M.E.Peskin ,D.V.Schroeder ,An Introduction to Quantum Field Theory参考书：L.H.Ryder,Quantum Field TheoryA Brife Review and Introduction、Review1、经典力学 其中：； 其中：正则框架：2、量子力学3、相对论量子力学过渡理论 K-G Eq： 描述spin-zero Dirac Eq： 描述 spin-1/2 Maxwell Eq： 描述 spin-14、量子场论基础Action： 其中： Euler-Lagrange Eq：Momentum Density Conjugate：Hamiltonian：；正则量子化：Real Scalar Field： ；其中：；Hamiltonian：场粒子性5、量子电动力学 Def：the Gauge Derivative：Local Gauge Transformation： and 5、微扰量子场论；为弱耦合Feynman Diagram：Feynman Rule for QED：S-Matrix： QED过程：(1) (2)Compton Scattering Spin Sums：Wald Identity： 、Introductions7、圈图 发散 重整化8、非阿贝尔规范场理论Weak Interactions and Strong InteractionsWeak Interactions：Beta Decay： Four Fermion Theory 不可重整Strong Interactions：介子理论：(Yukawa Theory)弱电统一理论（Weinberg-Salam Model）：整理与2011-2-26Chapter 6 Functional MethodsPath Intergral Methods(1-dimensional)时间演化算符： 满足：Classical Path： 猜想：双缝实验：Path 1： ； ；Path 1： ； ；联立两式， 可得德布罗意关系： 验证：计算积分： 展开：利用积分公式： ； ； ； ； 得：取极限： 故而：可使之满足同样的方程和初始条件，因此：整理于2011-3-1推广到多自由度的情况： ；插入中间态：分析两种情况：Functional Quantization of Scalar Field Correlation Functions：Consider the functional formula：If then we have：With the completeness relation：Thus we obtain the simple formula：整理于2011-3-6Functional Derivatives and Generating FunctionalThe functional derivative obeys the basic axiom (In four dimensions)： Example： （表面项相当于变分常数）Generationg Functional of correlation：Def：So that：Therefore the two-point function is：For free scalar field： Therefore：With the Gaussian intergration formulae： Where are matrixes. Therefore the two-point function should be：Where：def：We can check these： is nothing but the the Green Function of the Klein-Gordon operator.In another way,we can complete the square by introducing a shifted field：Using these we have：Free Field：For theory：Where we make：For the vertex：整理于2011-3-7Quantization of the Electromagnetic FieldThe difficutlies of questing gauge fieldTransformation of the gauge field ：Lagrangian of electromagnetic：Therefore the conjugate momentum：However： so we cannot write down the commutation relations like：Path intergral formula：Fourier Transformation： However if we define： That means： Therefore： 不可逆。Therefore the propagator of electromegnetic field is not well defined.Faddeev Popov TrickLorenz Gauge：（Attention：It is “Lorenz Gauge” not “Lorentz Gauge”）With the identity：We have： ；therefore： 把插入路径积分表达式：Where is just a C-Number .And with the transformation of the gauge field：，we have：.We choose the general class of functions：The functional intergral would be：对上式中取Gauss平均：插入： ，得：From the conclusion before,we know that：The propagator： or：The solutions of the Eq. Above is：We define the projector operator：；We have the identities： ；With these above,we have：；therefore：At last the generating functional is：S-Matrix 与 取值无关：Example：The propagator：With the Wald Identity： is independent of ：Gauge Inviriance ：S-Matrix is Gauge Independence.Two choices that are often convenient are： Laudau Gauge； Feynman GaugeExersice：Consider the massive vector field：With the gauge condition： we could find the path intergral functional： is well defined.Compute the Feynman propagator of this massive vector field.整理于2011-3-12The solutions of the exersice：Fourier Transformation：The propagator：Which：Grassmann Numer 反对易数Grassmann Numbers： ； shift ： Taylor series： ； 、 is normal number. Mutiple intergral：Complex Grassmann Number： ；And：Where b is a normal number.Generally：Exersice：for i=1,2；check these above.Functional Quantization of Spinor Field：Lagrangian Density：Generating functional：Where： are Grassmann Numbers.Then in the same way,introduce the shifted fields：Therefore：We define：，the correlation function would be：Because：，therefore this term： is positive instead of negative.In QED：Lagrangian Density：The generating functional：The Feynman Rules would be given as：Renormalization (Intro)Lowest order in QED：Next to lead order：Basic 1-loop diagrams： （1） （2） （3）（1）电子自能（抛出一个光子再回收）（2）光子自能（衰变成正反电子对再湮灭）（3）顶角修正 平方发散因此最后一张图 不发散整理于2011-3-15Chapter 7 An Introduction to RenormalizationCounting of Ultraviolet DivergencesLarge momentum region：A typical diagram： is loop 4-momentumLoop：增加发散 ； propogators：减少发散Some notation （in QED）：Superficial degree of divergence ：D（表观发散度）D=(power of kin numerator) (power of k in denominator) D0 发散；D0 不发散；For some case： 但发散故D的判别只对单粒子不可约图（One-partical ineducible diagrams）有效。One-partical ineducible diagrams：Any diagrams cannot be split in two by removing one single line. ； 发散 QED中发散图有7种：(a) 真空平移，无贡献，可去掉；(b) ，电荷变换下： ； 奇数个：都无贡献（Farrys Therem）(c) 光子自能。把中间实心球部分记作：由Ward identity： （Gauge Symmetry） 在处无奇点。展开：由：只能从开始系数不为零，故而实际上对于光子自能图：(d) 由(c)的分析可知：(d)为零，无贡献。(e) (Photon-photon scattering)由Ward identity： 每一条外线出一个这样的项，因此上图的振幅将从开始，所以其发散度应为：(f) 电子自能将几率振幅展开：其中： 以及： ； 同时：所以： （f）Summary：（有贡献的发散图） Spacetime dimention = d ； 整理于2011-3-19For Theory：外线 ：传播子 ； ；：def： 不可重整 可重整/超可重整One loop diagrams in QED and Dimesional Regularization发散基本Feynman Diagrams： 以光子自能为例，演示高能截断的基本方法以及对光子自能情况的局限性：光子自能，费曼图：； should be a scalar function.利用积分关系：，将上式视作： ；，上式中分母化为： 其中：对的分子部分有：所以：Def： ； 当，其中是趋于零的实小量。 用Wick Rotation： 可以得到：将上式在某一大动量位置处截断，然后再考察的行为： 则有：由于Ward恒等式要求：，因此上式的结果不能保证Ward等式的成立，因此会导致光子存在质量。故而高能截断的方法对此无法适用。Demensional Regularization （维数正规化）基本思想：积分测度：，在d维时空上做积分，让d取某一个值使得最后积分的结果不发散，然后令最终结果表达式里面的d=4，将其诠释为该积分在4维时空上的结果。之前的结果有：利用Gauss积分： 其中函数为：所以有：定义： 有：其中：因此最后可得：整理于2011-3-23Table of d-dimensional intergrals inMinkowski Space：In d-dimensional space Matrixes：Example：光子散射二次发散自动消除 d-dimensional space： Wick Rotation： By table of d-dimensional intergrals inMinkowski SpaceThe final step is based on this：Compute function：def： Taylor Expansion of function： ，def： Therefore：Where we use：对于二次发散的情况： 当：Example：Scalar Particle Self-Energy （Higgs）： （No Ward Identity）有二次发散，对于截断很敏感；猜想存在SuperSymmetry使得这些发散互相消掉。Feynman Parameters：（Very general Feynman Intergral Formula）And：To finish the calculation of （光子自能二级修正完结）： In d-dimensional Wick Rotation We mark：Therefore：Def：整理于2011-3-26The electron Self-Energy：其中，让光子存在一个小质量：来处理红外发散，但最后的物理结果不依赖于。上面过程中已经令：；电子两点项(In momentum Representation)： 无相互作用的传播子有相互作用的情况： The pole at m must be shifted by but infinity.不考虑相互作用时候，Lagrangian中是裸质量，考虑自相互作用以后存在log发散，但可以通过重新定义质量的办法将发散吸收到质量参数之中。Photon Self-Energy：(极点不变)The Photon remains massless at all orders of peturbation theory.顶角修正： Gorden Indetity (Peskins book Problem 3.2) For Lowest order：With Feynman ParametersDef：通过重新定义电荷参数，将发散的吸收进来： QED中光子自能贡献一个发散量，电子自能贡献两个发散量，顶角修正贡献一个发散量，可以通过对Lagrangian中的质量参数、电荷参数的重新定义以及场量的重新归一将发散吸收掉。整理于2011-3-30Renormalized Perturbation TheoryExamples：For Theory ,One Loop structureDefine：第二行视作“抵消项”。Feynman Rules：（加入抵消项减除发散）除了原有的Feynman规则以外，后面两个新的“顶角”为“抵消项”给出的Feynman规则。“抵消项”给出的Feynman规则全部视作顶角。这里解释一下第三个图是怎么算的：其中第二步的全微分对作用量无贡献，故略去。最后一步做了Fourier变换：Renormalized Conditions：根据物理上的需要，写出重整化条件。第一个条件来自于前面对自能修正的讨论，第二个条件来自于把耦合系数定义为外动量为零的时候的振幅。下面计算单圈的图：几率振幅记作：利用第二个重整化条件，代入可得：利用Feynman参数法以及之前的积分公式计算：然后再把代回到之前的里面就可以将发散的部分减除掉：补充：如下此类情况在理论里面无需考虑： 自能情况记作：自能修正：单圈情况即为：第二项为其抵消项。对展开，要求其在处留数为1，则有：于是：上式变回：，这就要求：，所以从第一个重整化条件里面可以得到：这样一来：可以得出：可以得出这两项的贡献总是互相抵消，因此无需考虑，同样的：这些情况也无需考虑。整理于2011-4-3Renormalized of QEDWe def： The Lagrangian becomes：Recall the notation before：We give the Renormalized Feynman Rules：注：第一个抵消项的由来：最后一步利用Fourier变换。Renormalized Conditions：One Loop Structure of QEDElectron self-energy:According to the first of the renormalized conditions：（）Similarly,the second of the renormalized conditions determines ：Photon self-energy：With the third of the renormalized conditions：The eletron vertex function：One can show explicitly from the conclution above that ,that is to all orders in QED perturbation theory.The relation between the bare and renormalized charge：The relation between the bare and renormalized electric charge depends only on the photon field strengh renormalization.(电荷是朴实的物理量。)电子磁矩：Dirac理论：电磁场中的Dirac Eq： ； 取其共轭与之相乘： 取如下势场： ； ； 其它为零 此处不求和In Dirac Representation：(NR) 取非相对论极限： 分离静能： 代入Dirac Eq可得电磁场中的Schrodinger Eq：自旋磁矩： Lande g-factor： 电子反常磁矩：(QED解释)QED：电子与电磁场通过交换光子传递相互作用：通过习题4.4(From Peskins Book/第一学期笔记)可知上图振幅可写成：即：，其中由Gordon identity(From Peskins Book problem3.2/第一学期笔记)：以及： ；可以改写为：其中与是重新定义的两个系数；对于只存在电场的情况：取非相对论极限：代入：对于光子：这要求对于只存在磁场的情况：取非相对论极限：代入可得：利用等式：代入上式可得：上式第一项与自旋无关，源于动量与势场之间相互耦合，故只考虑第二项：另外一项：所以顶角修正下(非相对论极限及极限)：故几率振幅：其中： 非相对论量子力学中，几率振幅源于粒子与势场之间的相互作用，故而(动量表象)： 其中 ：由自旋磁矩：自旋磁矩： 可知Lande g-factor：利用之前对于顶角修正的讨论，计算得：(From Peskins Book Page：196)可以算得：即：Lande g-factor：注：Exercise :Check the reault of and 整理于2011-4-7Chapter 8 The Renormalization Group(Introduction)1、Wilsons Approach to Renormalization TheoryGenerating Functional：A cutoff ； is large momentum.Def：Where b is a number from 0 to 1.In theory：（注：是由于将时间虚化（Wick Rotation）,以便在欧式空间里面做场论。）代入得：上式由于正交条件：，故而没有与次幂相同的项 （角标表示积分从零到）定义自由场部分拉氏量：通过计算，有：这一节的应用主要物理思想是在低能领域内，将积分时候的大动量（高能部分）冻结（即先积分掉），然后考虑低能部分的等效拉氏量的贡献。The Callan-Symanzik Equation：In theory：（详细过程见Peskins Book Chapter 12 12,2）通过解C-S方程与重整化条件，可以逐阶的求出Green Function与、这些量。在重整化的意义下，如果把拉氏量里面的耦合系数称作，那么由重整化所定义的重整化后的耦合系数记作，而是依赖于理论所选取的能标范围的（即耦合系数是（随着能标）跑动的），而函数：是反映这种跑动行为的函数。对于的情况，是所谓的“渐进自由”，正是QCD中的情况。（详细内容见Peskins Book Chapter 12 12.3）整理于2011-4-12Chapter 9 Non-Abelian Gauge InvarianceA brief Review of Lie Group and Differential Geometry1、 Group2、Topology Space3、Differential Manifold4、Lie Group5、Lie Algebras结构常数：Carton Metric：Bianchi Identity：Casimir Operator (SU(2):Yang-Mills Field(注：做如下约定：若有这种形式的量，其中大写字母的指标代表取第几个，即群空间的指标；希腊字母代表时空指标，即在时空标架上的分量。的含义即为第个的第个分量。)Under Gauge Transformation and where ；（locally）, is Dirac Spinor.Therefore： is not a covariant vector;because:；The “true”derivative of is given by taking the difference between these above：Consider ,it is sensible to assume that is proportional to itself, and also to ,the distance over which the vector is carried ,so we put：Therefore we define the covariant dervative operator：We could find：It leaves that： Where： Define: Compute the commutation relations,we have:Where is the gauge field.In general case：The corresponding classical equation of motion is：Where：Campare with General Relativity：1、 Covariant Derivative：Gauge Field: General Relativity: is called connection coefficients,so it is clear that play a similar role.2、Vector Potentials and Connection Coefficients：Gauge Field: General Relativity: The first term is “same”, would transform like a tensor.3、The Gauge Field and Reiman Tensor：Gauge Field: General Relativity: The first two terms are the derivatives of potential vectors or connections and the last terms are the linear combinations of potential vectors or connections.The Gauge Field can reflect the geometric structure just like Rieman Tensor.Consider the example：We perform a series of four infinitesimal displacesments round the close path ABCDA.We start at A with a spinor ,denoted ,and transport the vector round the closed path using the rule for parallel transport,involing the covariant derivative,then compare the final value of the spinor at A, with its initial value .Transporting to B will give ,working to the second order in and Thence to C：Thence to D：Finally Back to A：Where： Can relect the geometric structure as Rieman Tensor.4、Bianchi Identity：Gauge Field: General Relativity: 整理于2011-4-20Chapter 10 Quantization of Non-Abelian Gauge TheoriesThe Faddeev-Popov LagrangianThe Yang-Mills LagrangianThe Gauge Field：We use the functional intergral to quantized the Non-Abelian gauge field.The functional intergral can be defined：Apply a gauge-fixing condition at each point .Following Faddeev-Popov,can we introduce this constraint by inserting into the functional intergral the identity：A finite gauge transformation as： And： We have：Where： We choose the generalized Lorenz Gauge condition： And with the Gaussian weight for as we did in Chapter 6： Lead to the class of gauge field propagators：Choose ,called the Feynman-t Hooft Gauge.However,there is one more nontrivial ingredient.The determinant： With the exersice we did in Chapter 6 where we introduce Grassmann Number：The determinant can be rewriten as:The factor 1/g is absorbed into the normalization of the field c and .That implies there is something more in the Lagrangian density,this new fields c and and their “particle” excitations are called Faddeev-Popov Ghost.The Ghost Lagrangian is：Ghost is strange,they are Grassmann Numbers,so it seems that they should be Fermion,however their equation of motion is Klein-Gordon Equation,instead of Dirac Equation.The final Lagrangian,including all of the effects of Faddeev-Popov gauge fixing is：The Feynman Rules For Fermions、Gauge Bosen and GhostFeynman Rules for Fermion and Gauge Bosen：Feynman Rules for Ghost：Exersice：Derive the Feynman Rules all above.Equality of Coupling ConstantWith all external Particles on shell,the amplitude for production of a gauge boson should obey(like the Ward Identity in QED)：Lets check the Ward Identity in a simple case： （1） （2） （3）Replace by ：Since: and Therefore：That is：And：We replace by with ：With momentum ,is on shell ,and that it has transverse polarization ;then the third and fourth terms vanish.Therefore：Therefore:Where: That is just what we want.整理于2011-4-25A Flaw in the ArgumentIt is most convenient to work with the two lightlike linear combinations of these states,with polarization vectors parallel to the vectors and .The two unphysical polarization states of a massless vector particle can be written as： ；They obey the orthogonality relations: , ; , Where is the transverse polarization state；for i=1,2 .They also satisfy the completeness relation：For this case,we substitute and for the two polarization vectors.Then the term proportional to no longer vanishes( may not always be zero),it now yields：在做树图计算的时候，似乎可以忽略这一项给出的额外的贡献。但在做圈图计算的时候，由于中间圈动量未必对应在壳粒子，所以两个极化矢量未必是类光的，所以这会导致几率幅里面存在一些非物理的因素,进而破坏理论的幺正性。Ghost and UnitaryTo solve the problem,we have introduced the ghost which will help to cancel the anomalous term：The amplitude from the left diagram is：Where stands for the amplitude for this part：And is the symmetry factor for the Feynman Diagrams.With these relations：;We could understand the non-zero terms in would be these terms：Using the identity：We could see that the two terms added above are equal.Now add the contribution from th Faddeev-Popov Ghost(The right diagrams).With the Feynman Rules for the ghost,we have：Similary,the amplitude for the ghost-antighost pair to annihilate into fermions is equal to the second half of the amplitude from the left diagram.Finally,since Faddeev-Popov ghost fields anticommute,we must supply a factor of -1 for each ghost loop.Thus the ghost contribution exactly cancels the contribution of unphysical gauge bosen polarizations.整理于2011-4-26One-Loop Divergences of Non-Abelian Gauge TheoryThe Gauge Bosen Self-Energy：Ward Identity：The value of this diagram is the same as in QED,therefore：If there are species of fermions all in the same representation r, then the total contribution of fermion loop diagrams takes from：Where：;Three Gauge Bosen vertex：Where：The overall factor is a symmetry factor.And：,and:Where and ,therefore：Four gauge bosen vertex：With:We find simply：Multiplying the intergrand by 1 in the form:,and define： that is(dropping the term linear in P)Ghost Loop:Using the dimensional regularization,the result is:We record the ultraviolet divergent part of all above:To