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

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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 densit

9、y 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 that the Feynman prescription save much work in drawing n-th order graphs and in the evaluation process. We need to use G(k,)= G+(k,)+ G_(k,)