Green’s Functions and Feynman Diagrams — Adding an Electron to the Electron Sea and Removing It Again

Add one electron to a non-interacting Fermi sea (the sea of electrons), and remove it after a while. The complex amplitude that records “how much of the added excitation is still alive” is the one-particle Green’s function $G (x, y)$ . In ① you pick the six first-order Feynman diagrams (Fig. 9.1), and in ② the ten second-order diagrams (Fig. 9.8), one at a time, and read each in three layers: “facts on the diagram” → “animation of a representative story” → “factors of the formula”. In ③ you can move the controls to see how the real and imaginary parts of the self-energy show up in the spectral function and the lifetime. A Feynman diagram is not a classical trajectory of an electron but a ledger showing how the lines of an integral expression are connected — this is the theme running through the whole simulator. In ④ the lead role changes from the electron to the phonon: taking the harmonic oscillator as G⁰ and the cubic and quartic anharmonic terms as vertices, you read three diagrams (loop, tadpole, and bubble) and check the Dyson equation against the Morse-type potential, which has an exact solution.

This is “the Green’s function told with schematic diagrams.” The diagrams are redrawn to match the topologies of Figs. 9.1, 9.7, 9.8, and 9.16 of Fetter–Walecka, Quantum Theory of Many-Particle Systems (Japanese translation); the notation (iG = ⟨T[ψψ^†]⟩, time on the vertical axis, both ends of a wavy line at the same time, a factor of −1 per closed loop) also follows that book. The “story” in each animation is one representative time ordering, chosen for intuition; in reality the times and momenta of all internal vertices are integrated over. The linewidth in ③ comes from a model with the Fermi-liquid form −Im Σ ∝ ξ² + (πk_BT)², not from a calculation for any particular material. For the detailed derivation and a complete reading of all 16 diagrams, see Why Green’s Functions Can Be Calculated with Diagrams (in Japanese). For a review of second quantization, see also the harmonic oscillator simulator.
(In Appendix C of the Mochizuki Research Group’s Introduction to Linear Response Theory (in Japanese), the same figures are called Fig. C.2, Fig. C.7, Fig. C.11, and Fig. C.12, respectively. This simulator uses the original numbers Fig. 9.1, Fig. 9.7, Fig. 9.8, and Fig. 9.16 throughout.)