1、第四章 正則量子化與路徑積分LagraianL L（,rr）向量場變量rrxLagrangian 密度xd3)(xLL,rr（）LagrangianxdS4)(L（,rr）)(xLdx作用量（action)3213dxdxdxdxxddxfour-dimensional space-time正則量子化之一般原理Hamilton原理0S場方程（Euler方程）)()(xxrrrurrrrxx,)()()()(xxxrrr0)(xrOn Surface)(ururrrxdS,4LL0,4,4rrrrrxxdxxdLLLrurd,)(L0rrrrx,2,1,0,LL之場方程1x2x)(x)(x)(H
2、amitonian),()()(dH,3rrrrxxx L)()(L)(xxxrrrLr之共軛動量場)(3xxdHHamitonian 密度正則量子化 （Canonical Quantization）)(),(,),(xxitxtxrssr0),(),(),(,),(txtxtxtxsrsr相對論規範下的不變性 Lorentz 轉換x：zxyxxxctx321,逆變（contravariant）x：zxyxxxctx3210,協變（convariant）度規張量1000010000100001)g()(ggvvvxxvgvxxvg，)(gggvvv1000010000100001dAlembe
3、rt 算符312222c1ivvigxt相對論規範意味之不變性 座標系轉換axxvv xxxxgggvvvvggvv 0a非均勻 Lorentz 轉換（轉換）Poincare0a均勻 Lorentz 轉換Lorentz 群之分類1gg2或13122v1)()(g1gkkvsgndetProper orthochronousL11improper orthochronous*L1-1time-reflection type*L-1-1Space-time inversion type*L-11*spatial reflection*time reflection*space-time inver
4、sion1000010000100001P1000010000100001T1000010000100001PTLLorentz group(L.G.)Lrestricted L.G.(is an invariant subgroup)_LLLorthochronous L.G.LLLproper L.G._0LLLOrthochronous L.G.子群LL_PLL_TLLTP子集合LLLLLLLPTTPTNoether 定理)()()(xxxrrr變分)()()(xxxrrr全變分)()()()(xxxxrrrr)(0)(2xxxrr)(0)(2xxxrr)(),()(),(,r,rxxx
5、xrr L LsystemIin LsystemIin L),(),(),(),(xxxxrrrrLL L L)(02xxLL=0,rrrr,LLL)()(rrrxxr,r,rLLLxxxxrrrr,r,rLLL)()()(vvxgxxxxLLxL0vvvrrxgxxLLLr,r,vrvT能量-動能張量0j0j,x當中vvrxTr,Lj),(jvrx依不同之守恆量而定ClassicQuantum函數 算符 若0j則稱其為流異常無窮小 Lorentz 轉換Noethe 定理之應用局部連續轉換移動轉動規範守恆定律動量角動量 電荷vwvvwgggvv帶入g)(0ggg2wwwvvg0ww（局部連續轉
6、換）xxxxvvvvxwxgvvxvvxwx6個獨立變量 波函數之轉換關係)()()(1axFxFx)()()()(1axSxSNaxSx11)()()(SSSSSS1)det()det(SS1det SS 為正算符vvvwg)()g()(xwSxvv)()()g(xwSxSvvv)()(21)(21)()(xwSSSSxxvvvvvvS反稱對稱0vvwSx21)()(xv反稱 純移動線動量守恆vvxwx00vrwvvxTx,j)(,xTTvvvv任意量0=00,vT當中vvrvTg,LLr,jd0jd4xx廣義 Gauss 發散定理取)(11tx)(22txxdd120303jdjd021x
7、x),(jd),(jd203103txxtxx)(1tJ)(2tJ0dtdJ當中),(jd)(J3txxt0ddtd3vxTvP當中vvrvTg,LLr,00vHxxPrrr),()(d,r3 LHamitonian 算符H3,2,1ivirrixxxxP)()(d3線動量算符 轉動不變性角動量守恆vvvxwx0svrsvrSw21vvvvxwTxwT2121xwTSwvvsvrsvr,L21jvwwTxTxSvvsvrsr)(21,LvvwM2100j,vMGauss 廣義散度定理取0vvxMM03dvvsvrsrTxTxxSxx003)()(d0)(dtdvM空間分量取3,2,1,vji,
8、ijjisijrsrijTxTxSxM003d自旋空間角動量),()I,I,(II123123321MMM時空分量（oi)ooioixMM3d),(03,0201MMMK)()(),(d003xSxtxxTxtpsoirsrboost 向量，規範不變性電荷守恆),(,rrrr LL)()1()()()(xixexxrrirr)()1()()()(xixexxrrirr全域相位變換若)(x則為局域相位變換)()()(xxixrrr)()()(xxixrrrrrii)()(jr,r,LL)(j(x)rrir,r,LL當中)()()()()(jxxxxixrrrr)()()()(dQ3xxxxxiq
9、rrrr0HQ,0dtdQ電荷守恆微小常數已知)()()(),(d)(Q,3rxqxxxxiqxrsrs)(xxirs若QQQQQQQ)Q(rrrqQQQQQrrrqQq)-Q(QQrrQq)Q(QQrreigenvalueeigenstate路徑積分的一般原理Heisenberg 矩陣力學Schrdinger 波動力學Feynman 路徑積分代數形式局域微分形式全域積分形式正則經典力學Hamiton-Jacobi 方程Lagrange 力學Hamilton力學傳播子（propagator）)(tHtih)(|)(exp)(tttiHt 座標表象)(/)(expd)(3trrttiHrxtr
10、xtr 3d),(),(trtrK ),(tr傳播子),(),(d3trtrtrKx ),(tr),(tr),(tr ),(tr K的能量表象nEnnHrnnttiHnnrtrtrKnn /)(exp),()(/)(exp)(rttiErnnnnnnn /exp)(/exp)(tiErtiErnnnnn),(),(trtrnnn 當 nnnrrrrtrtrKttt)()()(),(輸入輸出傳播子的組合規則1t1tt r1rr ),(),(d),(111113trtrtrkxtr),(),(d113trtrtrkx ),(),(),(dd1111133trtttrktrtrkxx),(),(dt
11、rrrtrkx ),(),(d),(111113trtrktrtrkxtrtrk2ttttttn,210rrrrrrn,210),()(),(ddd),(1122,1111132313trtrktrtrktrtrkxxxtrtrknnnnnnn )(GK滿足的微分方程定義ikG ttxxttxxGitx),(),(d),(3t1)(Q)(),(d)()(3xxxxGixtt)()()()()H(xtttixttxti0)()(xxHti)()(dd)(tti)()(xtti)()()(),()(d3xttixxxGxHtixi)()()(d33xxxttxi)()()(),()(H43xxxx
12、ttxxGxti（Green 函數）位形空間中的路徑積分)(2mpH2xv一維勢場)(xv中粒子運動的 Hamiltonxttixtxtx /)H(exp),(kNie)(H/jijxxxdINj,2,1)(2(2xvmpie)()2(.2xvimpiee)(2 jNjNxxxd111Nx1Nx2Nx2Nxxx0v和2p互易2,BABABAeeeeBAeet txtxtxN )()()(itxtt Ntxxtxtx)()(1)(2.2NxvipmiNxeex)(exp1122NNmpiNxvixex)(expdp1122NNpmiNxvixeppx)(exp)(dp11p2NNpmiNxvix
13、pex)(exp),(),(pd111pNNNpNNxvitxtx)(exp)(2)(expdp)2(11121NNNNNxvittmpxxpi)(exp2)(exp212121NNNxvixximim/)2(exp)2(1),(223tmprpitrptmiprp2exp)(2)()(2exp212121NNNxvxxmiimNnNXxx10lim)(exp2)(21(exp2,211221NNNNxxiimmxvxmiimL ),(dt1expdx)2(2),(j211210limxximimtxtxkttNjNL),(exp)(D(xxitxttL1,1dIjxNjxxxjjNjpppj
14、jj,2,1dIp當中jjjxxxxjjjpppp 相空間中的路徑積分xk pIxIpIx2)(jkkxpijpkjexPx3N 323121232expdddddppmixxxpppk222232exp)(expppmixxxvip121122exp)(expppmixxxvip以為例來推導 )()()()(2expddddd01221222312123xvxvxvipppmixxppp011112222333xppxxppxxppx3321xpie2321xpieL 121236ddddd21xxpppexp)(2212223pppmi)(233xxp)()(011122xxpxxp)()()(123xvxvxvi NjjjjjjtxtxxvixxpipmitxDtpDtxtxk1112),(),()()(2exp)()(),(kjNKNJNdxdp111221)(2(1211jjjjjNjxvmpxxpi)(jxHtttNj d1 dtpxHxpixDpDtt),(exp dtxxLixDpDtt),(exp
