The singlepaticle propagator revisited, Chapter 9, appendix BG单粒子传播的重新访问9章附录B G

1、the single-particle propagator re-visited,chapter 9, appendix b-gnsingle-particle greens function propagatornsystematic method for drawing diagrams in the goldstone and feynman prescriptionsnexamplesnspectral density functionnconclusionssingle-particle greens function propagatormathematical expressi

2、on for single-particle greens function propagator:0is the exact normalized wave function for the interacting n-particle system, and t. is the time order operator defined as2121210210(,)( )( )kkig k k ttt ct ct ( ),( )kkihtihtkkihtihtc tec ec tec e,(heisenberg picture)t a(t1)b(t2) . = (-1)p x operato

3、rs rearranged so that time decreases from left to right (assuming no two times are equal), = (-1)p x operators rearranged so all c+s stand to the left of the cs for the case of equal times.examples:212121210210212121012021(,)( )( ),(,)( )( ),kkkkigigkk ttctctttigigkk ttct cttt the minus sign in g ag

4、rees with (4.29) 0000:,()ifhhthen ggfeynman propagator systematic method for drawing diagrams in the goldstone and feynman prescriptionsnnon-interacting fermionsk2k1+ .particle linehole line=+for translation see table 4.2goldstone and feynman prescriptions, non-interacting fermionsk2k1=+no hole or p

5、article lines, arrows indicate direction of momentum flowuse feynman propagator: ig0(q,t2-t1) t2t1q00,0( , ),0( ,)kkkffkkfititttkkeetig k ttiig ki time order has no significance: goldstone and feynman prescriptions,interacting fermions (second order)kkkfeynman prescriptiongoldstone prescriptionequiv

6、alentsee figs. 9.33-35diagram rules for single-particle propagator, table 9.1k, ( ,)ig kk, 0,( ,),fkfkkkkiig kkkior=k, or=0( ,)exp(0 )igkik, k, klmnq, =klmnqivorivfermion loop=-1each intermediate frequency :2deach intermediate momentum, k:33(2 )kd korexamplest1ttt2qk-qqpp+qkk(1)01,(1)0000( 1)( ,)(,

7、)( , )(, )( , 2).iqp qiqdt dtig k ttivig kq ttigp ttigpq ttivig k tt k,k,q,p+q, +p,q,k-q,-200,200( 1)( ,)(,)22( ,)(,)p qqddig kig kqigpigpqiv examplesk, k, l, 2000( ,)( , )2ilkklldig kivig le01,0,2iffkllkdielki but,20( ,)( 1)flkkll kig kivsame as (4.62)spectral density function1110002()0100111001110

8、0( , )(),(),nnnnni eeeettknnnnknnknnnnnnnnigk tcecceeee for large systems (electron gas or nuclear matter, not for atoms of finite nuclei)0110002()0200,( , )(),( ,)()()nnnnnnnnittknnknnnthenigk tceoriigkcifor large systems where the energy levels are closely spaced, 020( ,)()nkndakdc spectral density function a+:()00( , )( ,)( ,)( ,)()ittigk tdakeiigkdaki conclusionsnwe have defined the single-particle green function using the occupation number formalismnwe have discussed method to draw systematically n-th order graphs nwe have seen th