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1、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、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
3、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 acti
4、on：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 th
5、e 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：2010-11-9Examp
6、les：(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 t
7、he 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
8、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 Lagran
9、gian 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 Oscil
10、lators （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 p
11、resence 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 eas
12、y 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 Hermit
13、ian 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 Tr
14、ansformation： 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
