A guide to. Feynman diagrams in the many-body problem

Transcription

1 A guide to. Feynman diagrams in the many-body problem Richard D. Mattuck SECOND EDITION

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3 PAGE Preface to second edition v Preface to first edition. vi i 0. The Many-Body Problem for Everybody What the many-body problem is about Simple example of non-interacting fictitious bodies Quasi particles and quasi horses Collective excitations Feynman Diagrams, or how to Solve the Many-Body Problem by means o f Pictures Propagators-the heroes of the many-body problem Calculating propagators by Feynman diagrams : the drunken man propagator Propagator for single electron moving through a metal Single-particle propagator for system of many interacting particles The two-particle propagator and the particle-hole propagator The no-particle propagator (` vacuum amplitude') Classical Quasi Particles and the Pinball Propagator Physical picture of quasi particle The classical quasi particle propagator Calculation of the propagator by means of diagrams Quantum Quasi Particles and the Quantum Pinball Propagator The quantum mechanical propagator The quantum pinball game Disappearance of disagreeable divergences Where the diagram expansion of the propagator really comes from Energy and lifetime of an electron in an impure metal Quasi Particles in Fermi Systems Propagator method in many-body systems Non-interacting Fermi system in external potential ; particle-hole picture A primer of occupation number formalism (second quantization) Propagator for non-interacting Fermi system in external perturbin g potential Interacting Fermi system The `quasi-physical' nature of Feynman diagrams Hartree and Hartree-Fock quasi particles Hartree-Fock quasi particles in nuclear matter Quasi particles in the electron gas, and the random phase approxima - tion 94

4 YAVt: 5. Ground State Energy and the Vacuum Amplitude or `No-particle Propagator' Meaning of the vacuum amplitude The pinball machine vacuum amplitude Quantum vacuum amplitude for one-particle system Linked cluster theorem for one-particle system Finding the ground state energy in one-particle system The many-body case Bird's-Eye View of Diagram Methods in the Many-Body Problem Occupation Number Formalism (Second Quantization) The advantages of occupation number formalism Many-body wave function in occupation number formalism Operators in occupation number formalism Hamiltonian and Schrödinger equation in occupation number formalism Particle-hole formalism Occupation number formalism based on single-particle positio n eigenstates Bosons More about Quasi Particles Introduction A soluble fermion system : the pure Hartree model Crude calculation of quasi particle lifetime General form of quasi particle propagator The Single-Particle Propagator Re-visited Second quantization and the propagator Mathematical expression for the single-particle Green's functio n propagator Spectral density function Derivation of the propagator expansion in the many-body case Topology of diagrams Diagram rules for single-particle propagator Modified propagator formalism using chemical potential, Beyond Hartree-Fock : the single pair-bubble approximation Dyson's Equation, Renormalization, RPA and Ladder Approximations General types of partial sums Dyson's equation Quasi particles in low-density Fermi system (ladder approximation) Quasi particles in high-density electron gas (random phase approxi - mation) The general `dressed' or `effective' interaction The scattering amplitude Evaluation of the pair bubble ; Friedel oscillations. 197

5 PAGE 11. Self-Consistent Renormalization and the Existence of the Fermi Surface Dressed particle and hole lines, or `clothed skeletons' Existence of quasi particles when the perturbation expansion is valid Existence of the Fermi surface in an interacting system Dressed vertices Ground State Energy of Electron Gas and Nuclear Matter Review Diagrams for the ground state energy Ground state energy of high-density electron gas : theory of Gell-Mann and Brueckner Brief view of Brueckner theory of nuclear matter Collective Excitations and the Two-Particle Propagator Introduction The two-particle Green's function propagator Polarization (` density fluctuation ') propagator Retarded polarization propagator and linear response The collective excitation propagator Plasmons and quasi plasmons Expressing the two-particle propagator in terms of the scattering amplitude Fermi Systems at Finite Temperature Generalization of the T=0 case Statistical mechanics in occupation number formalism The finite temperature propagator The finite temperature vacuum amplitude The pair-bubble at finite temperature Diagram Methods in Superconductivity Introduction Hamiltonian for coupled electron-phonon system Short review of BCS theory Breakdown of the perturbation expansion in a superconductor A brief look at Nambu formalism Treatment of retardation effects by Nambu formalism Transition temperature of a superconductor Phonons From a Many-Body Viewpoint (Reprint) 275

6 17. Quantum Field Theory of Phase Transitions in Fermi Systems Introduction Qualitative theory of phase transitions Anomalous propagators and the breakdown of the perturbation serie s in the condensed phase The generalized matrix propagator Application to ferromagnetic phase in system with 8-function inter - action Divergence of the two-particle propagator and scattering amplitud e at the transition point Feynman Diagrams in the Kondo Problem Introduction Second-order (Born) approximation Parquet approximation with bare propagators Self-consistently renormalized s-electrons Strong-coupling approximation with self-consistently renormalized pseudofermions and vertices The Renormalization Group Introduction Review of effective interaction in the high-density electron gas Renormalization group for interaction propagators in the high-densit y electron gas Transforming from one transformed quantity to another : the functional equation of the renormalization group Lie equation for the renormalization group Solution of the Lie equation Appendices d. Finding fictitious particles with the canonical transformation A. Dirac formalism. 345 B. The time development operator, U(t) C. Finding the ground state energy from the vacuum amplitude D. The 17(t) operator and its expansion E. Expansion of the single-particle propagator and vacuum amplitude F. Evaluating matrix elements by Wick's theorem G. Derivation of the graphical expansion for propagator and vacuu m amplitude H. The spectral density function. 372 I. How the i8 factor is used J. Electron propagator in normal electron-phonon system K. Spin wave functions L. Summary of different kinds of propagators and their spectral representations and analytic properties M. The decoupled equations of motion for the Green's function expresse d as a partial sum of Feyman diagrams. 391 N. The reduced graph expansion. 395

7 Answers to Exercises. 402 References 414 Index 422