量子场论笔记与习题(Ⅰ).doc

Chapter 1 Introduction教材：M.E.Peskin ,D.V.Schroeder ,An Introduction to Quantum Field Theory参考书：L.H.Ryder,Quantum Field Theory1、为什么要提出QFT?Q.M（Quantum Mechanics）以 Schrodinger Eq为中心：可以描述（局限）：（1）非相对论量子力学（NRQM：None Relativistic Quantum Mechanics） （2）可以描述多体（3）粒子数守恒S.R（Special Relativity）以质能方程为中心：(1)质能转换；(2)高能过程；(3)粒子数目、种类均不守恒；Schrodinger Eq.+S.R.=RQMRQM（1）Klein-Gordon Eq.（局限）Negative Energy Negative Probobality （2）Dirac Eq. （局限） Negative ProbobalityRQM是不能自洽的理论解决办法：对比电磁场方程：Maxwell Eqs，将K-G Eq.与Dirac Eq.改造成场方程RQF波粒二象性的体现：波动性场量子化粒子性 （均为参数） 场算符The first physical QFT：Quantum Electrodynamics （QED：）微扰量子场论为弱耦合理论，要求耦合常数是小量。QFT计算方式：Feynman Diagram展开。四种基本作用：S、W、EM QFT；（S作用的渐进自由性使得它可以被QFT描述）；G GR约定标记：“God-Given”Units：Length=Time=Energy-1=Mass-1引力能标：Mpl=1.221019GeV,故不用QFT描述。其它标记：（教材xixxxi页）2、A Brife Review of Classical Field Theory(1)Basic Lagrangian Mechanics：Lagrangian： Action：The Principle of least action：Dynamics(2)Lagrangian Field Theory：广义坐标：， 场量：，视为独立的广义坐标。Lagrangian Density：； Action：；QFT为定域场论，要求：；【原则上可以有：】Euler Lagrange Eq. （From the principle of least action）：The second term can be turned into a surface integral over the boundary of the four-dimension spactime region of intergration.Since the initial and final field configurations are assumed given, is zero at the temporal beginning and end of this region.Therefore it vanished. Euler Lagrange Eq：(3)Hamiltonian Field Theory：Conjugate momentum： ；Conjugate momentum density：Hamiltonian and Lagrangian：Hamiltonian Density： 整理于：2010-11-9Examples：(Find the Lagrangian of the system；通过动力学方程找出体系的Lagrangian)（1）、Newtonian Mechanics： （2）、Klein Gordon Field： The First term is a surface integral,therefore it is vanished. In QFT, should be Lorentz scalar.3、Noethers TheoremDefination of symmetry：We call the transformation a symmetry if it leaves the equations of motion invariant. By Euler-Lagrange Eq,the second and third term is vanished.Def： Therefore： ； is conserved.Conserved Charge： Example：Find the conserved Noether current by .（注：Complex scalar field thoery：自由度 2n；和为独立的两个自由度： 独立，所以）Lagrangian is unchanged under the transformation： ， is a const. Neother Theorem applied in spacetime transformation： Def： ； is Energy-Momentum Tensor.The conserved Charge：Physical Momentum：整理于2010-11-10Chapter 2 The Quantization of Klein-Gordon FieldK-G Field ：Real Scalar Field： Complex Scalar Field： Quantization In N-particles QM：Step 1：Find the Lagrangian L of the system；Step 2：Give the conjuate momentum p：Step 3：Classical Possion parentheses transform into Quantised Possion parentheses： 把量子化程式用到K-G场：1、 Lagrangian：2、 Conjuate momentum density：3、 Give the Commutation：Equal time Commutation Relations： ； 与QM的情况对比： For real K-G Field： Harmonic Oscillators （Classical Field）经典场中的量子谐振子：正则变换： (对比) In K-G Field：；Fourier Transformation：（解为平面波） and is annihilation and creation operator. Execise： Check this： Zero Point Energy：The second term is proportional to ,an infinite c-number.It is simply the sum over all modes of the Zero-Point Energy ,so its presence is completely expected,if somewhat disturbing.Fortunately,this infinite energy shift cannot be detected experimentally.We will therefore ignore the infinite constant term in all of our calculation.Physical Momentum：（零点动量不要， 空间各项同性取平均为零。）K-G Field In Spacetime： is Lorentz invariant .It is easy to check this with the identity of the delta function：In the Heisenberg Picture：The Heisenberg equation of motion：At last, andcan be written as：Which is Lorentz invariant.整理于2010-11-11 is always positive(解决RQM中Negative Energy Problem)A negative-frequency solution of the field equation,being Hermitian conjugate of a positive-frequency solution,has as its coefficient the operator that creates a particle in that positive-energy single particle wavefunction.Causality in Klein Gordon Fieldif (spacelike) 无因果 The Propagator of K-G FieldFreedom Particles： Source： Green Function of K-G Eq： Fourier Transformation： 有奇性：“T” is the“time-ordering”symbol； is called Feynman Propagation for a Klein Gordon Field particle.“time-ordering”symbol T：for any functions：A(t1) and B(t2)Problems：From Peskins Book2.1Solutions:(a) Lagrangian Density：Treat as the field ： By Euler-Lagrange Eq：() By Lorentz Gauge：；Therefore the Maxwell Eqs are： and (b) is defined by： ； ； Therefore：Energy-Momentun Tensor is：By Maxwell Eqs：；the first term is vanished,therefore： can be writen as： 2.2Solutions：(a) Lagrangian Density： Conjugate Momentum： and Hamiltonian：In Heisenberg Picture，the equation of motion：Therefore the two results above gives：It is nothing but the K-G Eq.整理于2010-11-12(b)For complex scalar field： and Def： Therefore and can be writen as：and：Then,the Hamiltonian should be： and stand for two sets of particles of mass m.(c)According to (a) (b),we have：And we can check this：It means that is the exact charge operator,and its eigenvalues are .(d)For two complex K-G field,note the field function as a matrix: and The Lagrangian under the transformation of SU(2)： is unchanged.We can rewrite the Lagrangian as follows: and Therefore by Noether Theorem,we have： With we have:And：From：We can come to this conclution，that is：For N components filed,we can write this in general： 整理于2010-11-13Chapter 2 The Quantization of Klein-Gordon Field1、 Lorentz invariance and Lorentz GroupLorentz transformation ：， 非退化、44矩阵；Lorentz invariance ：That is：Scalar Field：，for K-G Eq： Lorentz transformation ；坐标变换下形式不变，要求：Example：K-G Eq： is Lorentz invariance by checking this：Vector Field under Lorentz Transformation： N-components Field： ；(M为Lorentz群表示)Lorentz Group(封闭性)：(多次连续变换等价于一次变换)SO(3) Group：群变换保内积：；生成元：；群代数：；dimension=2J+1；Angular momentum commutation relations： Algebra：For the 4-dim：We can check the algebra with of this is：Every component of is matrix because is an operator.Therefore,define： is the matrixs components of each ；we can easily check this：At last the Lorentz Transformation can be written as： （无穷小变换下）：整理于2010-11-14Review of Dirac Equation：Dirac Algebra：；n must be an even.Def： And： therefore： ； Dirac Spinor：Where：We can verify： or equivalently：By： ； that is： is the spinor representation of the Lorentz Transformation Dirac Eq is Lorentz invariant： Dirac Eq can imply the K-G Eq： is not Lorentz Scalar because： where：However： ；therefore： ；in fact：Therefore，def：，we can check is Lorentz Scalar by：With： and： ；the result can be writen as：At last the Lagrangian of Dirac Field can be writen as：整理于2010-11-15Weyl Spinors： Because Lorentz Group is reducible,we can form two 2-dimensional representations by considering each block separately, and writing： and are called Left-hand spinor and Righ-thand spinor；their transformation laws under infinitesimal rotations and boost are：In terms of and ,the Dirac Equation is：For the particle of ，the equations for and decouple： def： and So that： and the Weyl Equations become： Free-Particle Solutions of the Dirac Eq.The solution of free paticle can be writen as a linear combination of plane wave： where Just concentrate on solutions with positive frequency,that is .The Dirac Equation should be：Analyze this equation in the rest frame,where：；the equation becomes：The solutions are：Normalize so that： when the particle has spin up along the 3-direction and when the particle has spin down along the 3-direction.Then obtain in any other frame by boosting.Consider a boost along the 3-direction in infinitesimal form： where is some infinitesimal parameter.Then：Apply the same boost to ：We can check this：Therefore the result can be simplified give： with the identity：We can check that is not Lorentz scalar similarly：Def： therefore： is Lorentz invariance.Note： and as the basis spinor which are orthogonal.The two linearly independent solutions for ： for s=1,2In the same way,we can find the negative frequency solutions： for s=1,2 and Where is another basis of two-component spinors.These solutions are normalized according to： or They are also orthogonal to each other：Helicity operator：Def：the helicity operator：A particle with is called right-handed,while one with is called left-handed.The helicity of a massive particle depends on the frame of reference,since one can always boost to a frame in which its momentum is in the opposite direction (but its spin is unchanged).For a massless particle,which travels at the speed of light,one cannot perform such a boost.Spin Sums： 整理于2010-11-16Dirac Matrixes and Dirac Field Bilinears：Introducing an additonal gamma matrix：And the anti-commutation relations：；Rewrite 44 Dirac Matrixes，and introduce some standard terminology：They are totally 16 matrixes.The two currents out of Dirac field bilinears： ； And we can check this：In the same way，we can compute：If m=0,this current called the axial vector current is also conserved.It is then useful to form the linear combinations： ； When m=0,these are the electric current densities of left-handed and right-handed particles,respectively,and are separately conserved.The two currents and are the Noether currents corresponding to the two transformations： and 整理于2010-11-172、Quantization of the Dirac FieldThe canonical commutation relations： (equal time)； Expanding with the linear combination of and ：Problems：We can check the Hamiltonian can be rewriten in the terms of a,b：That mean the energy of the system is not always positive.And：However：It means that the second term is always Zero.And it can not be writen as：Solutions：Rewrite the field operators as follows：Suppose the creation and annihilation operator obey the anticommutation： And then：Therefore the Hamiltonian can be writen as：The Energy of the system is always positive.And we can check this(Exersice)：（注：Dirac场并非可观测量，可观测量为和的双线性组合；可以简单的验证如下情况：Def：) and：利用反对易关系，可以很容易验证： when： spacelike.）Spin of Dirac Particles：绕z轴转动： The conserved Noether current is then：The angular momentum should be：We can check this (Exersice)： The propagator of Dirac Field： 分别代表正反粒子的传播子。整理于2010-11-193、Discrete Symmetries of the Dirac Theory：Parity：Parity operator： and which：Therefore：Setting：，then：In the same way：Time Reveral：Time Reveral operator： Charge Conjugation：Charge Conjugation operator： Therefore： and That means：In the same way,we have：The Lagrangian of the free Dirac Field is invariant under C,P and T separately,and all scalar combinations of and are invariant under the comblined symmetry CPT.整理于2010-11-20Problems：From Peskins BookSolutions：(a)From the commutation relations and we have：Since： and Therefore：In the same way,we have：Therefore： (b) In the representation of ： ； By： and In the representation of ：By： and While： and ,that would be (3.37)整理于2010-11-21(c)With the commutation relations： and ,we have：Which：；Therefore： that is exactly like a 4-Vector.Solutions：Accroding to Dirac Equation,we have： and Where： and Accroding to the commutation relations： and：，we have：Therefore：Solutions：(a)For 16 Dirac Matrix：，they should obtain： and this can be easily verified： ； and Therefore the set of should be：(b)Because can be any Dirac Spinor，and from this：We have： Multiplying both sides by： and Sum of ,we have： From (a) we have：，therefore：It means：(C)Accroding to the conclusion of (b) and Set： That：Therefore：Where：In the same way,we can compute that：Solutions：First of all,we should make sure that for Dirac Field： and (a)With the iden： ； therefore：By： ，therefore：整理于2010-11-23(b)Since： and： Therefore：We set：，therefore：In the same way we have：With the results above and these： ； ； We have：(c)Up to now,the Lorentz Scalar which we meet are only： and From the transformation as follows： and We have：That means： for Lorentz Scalar.整理于2010-11-25Chapter 4 Interacting Field and Feynman Diagrams4.1 Perturbation Theory Philosophy and ExamplesInteracting Lagrangian should be：（要求）(1) Lorentz Scalar；(2) C.P.T不变，即CPT=+1(3) 规范不变性；(4) 可重整化。场论可重整性：有发散，截断；物理结构不依赖于截断 可重整化 Others 不可重整化可重整化要求耦合常数不能为质量量纲的负次幂。NRQM对相互作用无约束RQFT对相互作用有约束，RQFT中相互作用的Lagrangian不能随便写。QED Gauge Invariance：Def：the Gauge Derivative：The QED Lagrangian is invariance under the transformation： and Which is called Gauge Transformation from Electrodynamics；and QFT is a Localized Field Theory，that is why is a function of .And we can check that the Lagrangian：is invariant under the gauge transformation.That means these terms： can not exist if the Lagrangian is gauge invariance.That means the particle from this gauge transformation should be massless.It is just the photon.However the Lagrangian of Maxwell Field has not been Quantized.The Quantization of Maxwell Field is not a travel work.Since does not appear in the ,the momentum conjugate to is identically Zero.This contradicts the canonical commutation relation：.One solution is to quantize thr field in Lorentz Gauge：,adding an term.One obtains the commutation relations .However in this way states created by have negative form.Yukawa Theory： theory： （Toy model）整理于2010-11-264.2 Perturbation Expansion of correlation Functions (Theory)Two-point correlation function： is the ground state of the interacting theory.And in the free theory：The Hamiltonian of is：In the definition of the Heisenberg Field：For ,becomes and this reduces to：It is easy to construct explicitly：With initial condition of a simple differential equation：The solution of the equation above is：A similar identity holds for the higher terms：Therefore：整理于2010-11-27In general,we can write：satisfies the same equation：We can find that：Since is the ground state of H,we can isolate it by the following procedure：We can get rid of all the terms in the series by sending T to in a slightly imaginary direction：，then the exponential factor dies slowest for and we have：Since T now is very large,we can shift it by a small constant：Similarly,we can express as：For the moment that：，then：Divide by 1 in the form：Finally we write it as：The virtue of considering the time-ordered product is clear：It allows us to put everything inside one large T-operator.A similar formula holds for higher correlation functions of arbitrarily many fields；for each extra factor of on the left,put an extra factor of on the right.整理于2010-11-284.3 Wicks TheoremTo calculate correlation functions to that of evaluating expressions of the form：Firstly,consider the case of two fields：；decompose into positive and negative frequency parts：Where： ； Consider the caes ：Define the normal ordering：The order of and on the right hand side makes no difference since they commute.And define one more quantity：This quantity is exactly the Feynman propagator：The relation between time-ordering and normal-ordering is now extremely simple to express,at least for two fields：The generalization to arbitrarily many fields is also easy to write down：And：整理于2010-11-294.4 Feynman DiagramsWith the Wicks Theorem,the equation can be represented as the sum of three diagrams(called Feynman Diagram)：The numerator,with the exponential expanded as a power series is：The first term gives the free-field result,.The second term,in theory，is：The external points x and y and the internal point z ；each internal point is associated with a foctor of .The above expression is equal to the sum of two diagrams：A more complicated contraction,from the term in the expansion of the correlation function：Furthermore,the generic vertex has four lines coming in form four different places,so the various placements of these contractions into generates a factor of 4! (as in the w vertex above),which cancels the denominator in .It is therefore conventional to associate the expression with the each vertex.(This was the reason for the factor of 4! In the coupling.)Summarize the rules for calculating the numerator of our expression for ：Feynman rules in theory are：They are sometimes called th