量子场论笔记与习题(Ⅱ)

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 Ⅰ、Review 1、经典力学)x (x212V m L -=其中：),(q q L L =；x∂∂=Lp ⇒ L p xH -= 其中：),(q p H H = 正则框架：],[],[H p qH p H q pHq=∂∂-==∂∂=2、量子力学ij j i i p q δ=],[3、相对论量子力学 过渡理论① K-G Eq ： ()022=+∂φm 描述spin-zero ② Dirac Eq ：()0=-∂ψγμμm i 描述 spin-1/2 ③ Maxwell Eq ：0=∂μνμF 描述 spin-14、量子场论基础Action ：⎰⎰==L x d dtL S 4 其中：),(φφμ∂=L L222121φφφμμm scalarreal -∂∂=-L0=S δ ⇒ Euler-Lagrange Eq ：()0=∂∂-∂∂∂∂φφμμL LMomentum Density Conjugate ：)()(x x φπ ∂∂=L Hamiltonian ：L H -=)()(x x φπ ；正则量子化：)()](),([)3(y x i y x -=δπφ Real Scalar Field ：()[])exp()exp(2)(p p 33ipx a ipx a p d x ++-=⎰πφ ； 其中：())'p p (2],[)3(3p'p -=+δπa a ；Hamiltonian ：())(2p p p 33零点能C a a E p d +=⎰+πH>+0|p a 场粒子性5、量子电动力学Int Maxwell Dirac QEDL L L L++= ⇒ ()μμμνμνμμψγψψγψA e F F m p QED---=41LDef ：the Gauge Derivative ：μμμieA D +∂=()μνμνμμψγψF F m D QED 41--=L Local Gauge Transformation ：()ψαψ)(ex p x i → and )(1x eA A αμμμ∂-→5、微扰量子场论IHH H +=0；I H 为弱耦合Feynman Diagram：Feynman Rule for QED：S-Matrix：>-=<⎰∞∞-i x d i T f S I if |}exp{|4H iT S +=1 1=+SS()M i p k k k k iT p p i i ⎪⎭⎫ ⎝⎛-+>=<∑21)4(421212,||...,δπQED 过程：(1) -+-+→μμe e(2)Compton ScatteringSpin Sums ：∑∑==-⋅=+⋅=2,12,1)()()()(s ss s s s mp p vp v m p p u p u γγ)()()(221*k k k g k k +-→∑μνλλνλμεε Wald Identity ：⇒ 0)(=k k μμMⅡ、Introductions7、圈图()⎰24412~p p d π 发散 ⇒ 重整化8、非阿贝尔规范场理论Weak Interactions and Strong InteractionsWeak Interactions ：Beta Decay ：e e p n ν++→- → Four Fermion Theoryψψψψ~IL不可重整Strong Interactions ：π介子理论：(Yukawa Theory)弱电统一理论（Weinberg-Salam Model ）：)1()2(U SU ⨯0≠w mWNπ整理与2011-2-26Chapter 6 Functional MethodsPath Intergral Methods (1-dimensional))x (2p 2V mH +=时间演化算符：>>=<-=<a a b b a b b a t x t x x iH x T x x U ,|,|)/ex p(|);,( 满足：HU U Ti =∂∂⎰∑⋅=⋅=)](exp[)]([)](exp[);,(phase i t x phase i T x x U pathAllb a DClassical Path ：0=S δ 猜想：⎰=)/exp()]([);,( iS t x T x x U b a D 双缝实验：x bDetectorPath 1：T mv S 21121= ；TDv =1 ；T mD S 221=Path 1：T mv S 22221= ；TdD v +=2 ；T d D m S 2)(22+=联立两式，D d << ⇒ 21v v ≅ 可得德布罗意关系： d T mDd p = ⇒ λh=p验证：⎰=)/exp()]([);,( iS t x T x x U b a D 计算积分：⎰-=TV m dt S 02))x (x 21( 离散化 ∑⎥⎦⎤⎢⎣⎡+--++k k k k k x x V x x m )2()(2121εεεεkk x x dt T -=≅→+1x0 ；；∏⎰⎰⎰⎰⎰∞∞--=⋯⋯=k k n C dx C C dx C dx C dx C t x D )()(1)()()()(1)]([121εεεεεε ),',()]2'(2)'(exp[)(' )]2(2)(exp[)()(1),,(22εεεεεεεε-+--=+--=⎰∏⎰∞∞-∞∞-T x x U x x V ix x m i C dx x x V ix x m i C dx C T x x U a b b k k b k b k b a展开：);(])'(21)'(1[ ])(1][2)(exp[)('),,(2222εεεε-⋯⋯∂∂-+∂∂-+⨯⋯⋯+--=⎰∞∞-T x x U x x x x x x x V ix x m i C dx T x x U b a bb b b b k b b a利用积分公式：b b d πξξ=-⎰)ex p(2；0)exp(2=-⎰ξξξb d ；bb b d πξξξ21)ex p(22=-⎰ εim b -= ； im b-=πεπ2 ； mi b ε=21);,()](2)(1[2)(1);,(222εεεεεπε-+∂∂+-⎪⎪⎭⎫ ⎝⎛-=T x x U x m i x V i im C T x x U b a b b b a O 得：imC -=επε 2)(),,()](2[),,(),,(222T x x U x V x m i T x x U T x x U b a b bb a b a +∂∂-=--⇒εε 取极限：0→ε ⇒ ),,(),,(T x x HU T x x U Ti b a b a =∂∂故而：⎰=)/exp()]([);,( iS t x T x x U b a D 可使之满足同样的方程和初始条件，因此：⎰=)/exp()]([);,( iS t x T x x U b a D整理于2011-3-1推广到多自由度的情况：}{i q q = ；}{i p p =>-=<a b b a q iHT q T q q U |)ex p(|);,(；插入中间态：1||=><⎪⎪⎭⎫ ⎝⎛∏⎰k k i i k q q dq 分析两种情况： ①：)(q f H =])(exp[22)()(|)(|1111∑∏⎰∏-⎪⎪⎭⎫ ⎝⎛⎪⎪⎭⎫ ⎝⎛+=->=<++++i ik i k i k i i k i k i k ii k ikk k k q q p i dp q q f qq q f q q f q πδ②：)(p f H =])(exp[)(2|)(|11∑∏⎰-⎪⎪⎭⎫ ⎝⎛>=<++i i k i k i k k iikk k q q p i p f dp q p f q π⎥⎦⎤⎢⎣⎡⎪⎭⎫ ⎝⎛+--⎪⎪⎭⎫ ⎝⎛=∑∑∏⎰⎰++k i k k k ik i k i k ki ik ik b a p q q H q q p i dp dq T q q U ),2()(exp 2);,(11,επFunctional Quantization of Scalar Field)()()(x x p x q ii φπφ∂∂=→→L)()(2121)(2122φφπφπφφφμμV V +∇+=-=-∂∂=L HL⎰⎰⎰⎰⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡⎪⎭⎫ ⎝⎛-∇-->=-<T T a b x d i V x d i iHT x 04224exp )()(2121exp |)exp(|)(L D D D φφφπφππφφφCorrelation Functions ：Consider the functional formula ：⎥⎦⎤⎢⎣⎡>ΩΩ<⎰⎰-T TH H x d i x x x x T LD 42121exp )()(~|)()(|φφφφφ ⎰⎰⎰⎰===)x ()x ,()x ()x ,(21202101)x ()x ()x ()(φφφφφφφφx x x D D D DIf 0201x x < then we have ：[][][]>+-<⨯>--><--<⎰⎰a b T x iH x x iH x T iH D D φφφφφφφφφφ|)(ex p ||)(ex p ||)(ex p |)x ()x ()x ()x (011101022202221121With the completeness relation ：1||=><⎰i i i D φφφ：[][][]{}>--=<>+-----<a H H b a s s b iHT x x T iHT T x iH x x iH x T iH φφφφφφφφ|)exp()()()exp(||)(exp )x ()(exp )x ()(exp |210110102202Thus we obtain the simple formula ：{}⎰⎰⎰⎰⎥⎦⎤⎢⎣⎡⎥⎦⎤⎢⎣⎡>=ΩΩ<---∞→T T T T i T H H x d i x d i x x x x T L D L D 4421)1(21exp exp )()(|)()(|limφφφφφφεT-T01x 02x整理于2011-3-6Functional Derivatives and Generating FunctionalThe functional derivative obeys the basic axiom (In four dimensions)：i j j j i ij j ik k x x x x =∂∂=∑δδδ⇒ )()()()()()()(4)4(x y y yJ d x J y x y J x J φφδδδδδ=-=⎰Example ：[][])()()()()()(exp )()()(exp )(444x V y V y J y d x J y y yJ d i x i y y yJ d i x J μμμμδδφφφδδ-∂=∂=⎰⎰⎰ （表面项相当于变分常数）Generationg Functional of correlation ：Def ：()[]⎰⎰+=)()(exp ][4x x J x d J Z φφL D So that ：()[]⎰⎰+=-φφφφδδδδJ x d i x x J Z x J x J i L D 421212ex p )()(][)()()(Therefore the two-point function is ：21021][)()(10|)()(|0=⎪⎪⎭⎫ ⎝⎛-⎪⎪⎭⎫ ⎝⎛->=<J J Z x J i x J i Z x x T δδδδφφFor free scalar field ：222121φφφμμm -∂∂=L⇒ ⎰⎰⎪⎭⎫ ⎝⎛--∂==φφ)(212244m x d x d S L Therefore ：()()⎰⎰⎰+⎭⎬⎫⎩⎨⎧⎥⎦⎤⎢⎣⎡-+-∂-=⎥⎦⎤⎢⎣⎡++-∂-)()()()()(2121)4(2244224x x J y x x i m x y d x d J i m x d x φδφεφφφεφWith the Gaussian intergration formulae ：)2exp(2)exp(22aJ a Jx axdx π=+-⎰∞∞- ⇒ ⎪⎭⎫⎝⎛⎪⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛⋅+⋅⋅--∞∞-∞∞-⎰⎰J JA A x J x A x dx dx dx nT n 1212121exp det )2(21exp ......πWhere x A J are matrixes. Therefore the two-point function ][J Z should be ：⎰⎰⎥⎦⎤⎢⎣⎡⎪⎭⎫ ⎝⎛+-=)()()()()(21exp ][4x x J y y x K x x d J Z φφφφDWhere ：def ：())()()4(22y x i m y x K -+-∂-=-δεWe can check these ：)()()()()()()()()())(()()()()()4(22)4()4(224)4()4(224)4(4y x i y x D i m y x i z y y x D i m y d z x i z y i m y x yD d z x z y K y x D i y d F x F xx F F -=-+--∂⇒-=--+--∂⇒-=-+--∂-⇒-=---⎰⎰⎰δεδδεδδεδ)(y x D F - 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 ：)()()()('4y J y x D y d i x x F --≡⎰φφUsing these we have ：()()()[]⎰⎰⎰⎰⎰⎰⎰⎰⎥⎦⎤⎢⎣⎡--⨯=⎥⎦⎤⎢⎣⎡⎪⎭⎫ ⎝⎛---+-∂-=⎥⎦⎤⎢⎣⎡⎪⎭⎫ ⎝⎛++-∂-=)()()(21exp )(exp ' )()()(21''21exp ' )()()()(21exp ][440444224224y J y x D x yJ xd d x d i y J y x iD x J y xd d i i m x d x x J x i m x x d J Z F F φφφεφφφφεφφL D D DFree Field ：)( )()()(21exp )()(21)()(21)( )()()(21exp )()(0|)()(|0214424241442121x x D y J y x D x yJ xd d x x D x yJ d y J y x yD d x J y J y x D x yJ xd d x J x J x x T F F F F F -=⎥⎦⎤⎢⎣⎡--⨯⎥⎦⎤⎢⎣⎡-----=⎥⎦⎤⎢⎣⎡--->=<⎰⎰⎰⎰δδδδδδφφFor 4φ theory ：44φλ！-=I L⎰⎰⎰⎰⎰⎰⎰⎰⎰⎰⎰⎰⎰⎥⎦⎤⎢⎣⎡⎪⎭⎫ ⎝⎛+-∂∂⨯⎥⎦⎤⎢⎣⎡+⎪⎪⎭⎫ ⎝⎛-+=⎥⎦⎤⎢⎣⎡⎪⎭⎫ ⎝⎛+-∂∂⨯⎥⎦⎤⎢⎣⎡⎪⎪⎭⎫ ⎝⎛-=⎥⎦⎤⎢⎣⎡⎪⎭⎫ ⎝⎛+-∂∂⨯⎥⎦⎤⎢⎣⎡⎪⎭⎫ ⎝⎛-=⎥⎦⎤⎢⎣⎡⎪⎭⎫ ⎝⎛+--∂∂=φφφφφδδλφφφφφδδλφφφφφφφλφφφλφφφφμμμμμμμμJ m x d i x J i x d J m x d i x J i x d i J m x d i x d i J m x d i J Z 2244442244442244442242121exp ......)(!41 2121exp )(!4exp 2121exp !4exp !42121exp ][~D D D D D D Where we make ：[]⎥⎥⎦⎤⎢⎢⎣⎡⎪⎪⎭⎫ ⎝⎛⎪⎪⎭⎫ ⎝⎛--=⎥⎦⎤⎢⎣⎡⎪⎪⎭⎫ ⎝⎛-→⎰⎰⎰4444)(!4exp )(exp )(exp x J i x d i x J i x d i x d i I I δδλδδφL L For the vertex ：()()()()()()⎰⎰⎰⎰⎰⎰⎰-=-----=----⨯----⎪⎭⎫ ⎝⎛-===xd i x x D x x D x x D x x xD d i y J y x yD dy J y x yD d y J y x yD d y J y x yD d x d i J Z x J x J x J x J F F F F F FF F J 4432144444404321)()()()()()()()( )()()()(!4!4][~)()()()(λλλδδδδδδδδ整理于2011-3-71x 2x 3x 4xQuantization of the Electromagnetic Field ﹡The difficutlies of questing gauge fieldTransformation of the gauge field μA ：)(1x eA A αμμμ∂+→Lagrangian of electromagnetic ：μνμνF F 41-=LTherefore the conjugate momentum ：μμπA ∂∂=L However ：00=π so we cannot write down the commutation relations like ：)y x ()]y (),x ([)3(-=δπμννμig APath intergral formula ：()[]⎰⎰∂∂-∂=⎥⎦⎤⎢⎣⎡-=)()(2141244x A g x A x d F F x d S ννμμνμμνμνFourier Transformation ： ⇓()()[]⎰-+-=)(~)(~221244k A k k g k k A k d S ννμμνμπ However if we define ：2kk k gνμμνμν-=∆⇒ μνλνμν∆=∆∆ That means ：3=∆μνμνg Therefore ：μν∆ 不可逆。