The single-particle propagator re-visited, Chapter 9, appendix B-G

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. (Heisenberg picture) , is the exact normalized wave function for the interacting N-particle system, and T{.} is the time order operator defined as Single-particle Green’s function propagator Mathematical expression for single-particle Green’s function propagator: T {A(t1)B(t2) ...} = (-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. k2 k1 Systematic method for drawing diagrams in the Goldstone and Feynman prescriptions • Non-interacting fermions hole line particle line = + + + + + + + + ... + ≡ ≡ ≡ …for translation see table 4.2

4. …Goldstone and Feynman prescriptions,Non-interacting fermions Time order has no significance: ≡ k2 No hole or particle lines, arrows indicate direction of momentum flow t’2 = + + + +… q t’1 k1 Use Feynman propagator: iG0(q,t’2-t’1)

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

6. Diagram rules for single-particle propagator, table 9.1 = k, ω k, ω = k, ω or = or k, ω k, ω l k Each intermediate frequency ω: = q, ε m n Each intermediate momentum, k: -1 = Fermion loop

7. Examples t2 k q t’ p k-q p+q t q k t1 k,ω q,ε k-q, ω-ε p,β p+q, β+ε q,ε k,ω

8. Examples k, ω l, ε k, ω 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)