1. The single-particle propagator re-visited, Chapter 9, appendix B-G Single-particle Green’s function propagator Systematic method for drawing diagrams in the Goldstone and Feynman prescriptions Examples Spectral density function Conclusions

2. Single-particle Green’s function propagator Mathematical expression for single-particle Green’s function propagator: is the exact normalized wave function for the interacting N-particle system, and T{.} is the time order operator defined as, (Heisenberg picture) T {A(t 1 )B(t 2 )...} = (-1) P x operators 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 c’s for the case of equal times. Examples: The minus sign in G ¯ agrees with (4.29)

3. Systematic method for drawing diagrams in the Goldstone and Feynman prescriptions Non-interacting fermions k2k2 k1k1 +... ≡ ≡ ≡ p article line hole line = + +++++++ …for translation see table 4.2

4. …Goldstone and Feynman prescriptions, Non-interacting fermions k2k2 k1k1 =++++… No hole or p article lines, arrows indicate direction of momentum flow Use Feynman propagator: iG 0 (q,t’ 2 -t’ 1 ) t’ 2 t’ 1 q Time order has no significance: ≡

5. … Goldstone and Feynman prescriptions, Interacting fermions (second order) k k k … Feynman prescription Goldstone prescription … equivalent See figs. 9.33-35

6. Diagram rules for single-particle propagator, table 9.1 k, ω or = = k, ω or = k, ω k l mn q, ε = Fermion loop = Each intermediate frequency ω: Each intermediate momentum, k:

7. Examples t1t1 t t’ t2t2 q k-q q p p+q k k k,ω q,ε p+q, β+ε p,β q,ε k-q, ω-ε

8. Examples k, ω l, ε but, …same as (4.62)

9. Spectral density function For large systems (electron gas or nuclear matter, not for atoms of finite nuclei) For large systems where the energy levels are closely spaced, Spectral density function A + :

10. Conclusions We have defined the single-particle Green’ function using the occupation number formalism We have discussed method to draw systematically n-th order graphs We 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,ω)= G + (k,ω)+ G _ (k,ω)