Many body theory 1

A.Y. 2020/2021
Overall hours
Learning objectives
The main objective of the course is to provide an accurate presentation of important techniques needed in the study of many-particle systems in condensed matter, statistical mechanics, nuclear physics. It is a course in non-relativisti, low-energy field theory, with many particles. The main topics are: second quantization, Hartree-Fock equations, the Gell-Mann and Low theorem, Wick's theorem, time-ordered and retarded Green functions, Feynman's diagrams, Dyson equations and resummation of Hartree and RPA terms, Hedin's equations, linear response theory, quasiparticles. The theory is illustrated by applications to
The homogeneous electron gas.
Expected learning outcomes
To write operators in second quantization.
The comprehension of the Hartree Fock approximation, and knowledge of the HF properties of the electron gas.
The meaning of T-ordered and retarded Green funcions, and their relation in frequency space. To evaluate a Lehmann expansion.
To pass from an equation of motion to a Dyson equation. The structure of poles and their meaning.
The meaning of normal ordering and contraction, and the conditions for the validity of Wick's theorem, evaluation of correlators.
Knowledge of the Feynman rules and their origin. To write the analytic expression of a Feynman diagram in x and k spaces.
Meaning of the generalized dielectric function, and of the RPA approximation.
To know and apply the linear response theory.
Evaluate the effective mass and the dispersion law of a quasiparticle.
Course syllabus and organization

Single session

Lesson period
First semester
Lessons will be delivered online, as scheduled, via ZOOM platform. The lesson is recorded and made available with lecture notes and useful material in ARIEL.
The exam is oral, online (ZOOM), of approximate duration 1H. Questions may include some calculations to be carried on paper or graphic tablet. Two written home-works are to be submitted before the oral session.
Course syllabus
Second quantization. Field operators. Hartree-Fock and Thomas-Fermi approximations. Green functions for fermions in ground state. Gell-Mann and Low theorem. Wick's theorem and Feynman's diagrammatic expansion. Self-energy, polarisation, effective potential vertex function. Hedin's equations. Lehmann's expansion and retarded functions. Linear response. Applications to the interacting electron gas (RPA, screening, plasma oscillations, total energy). Poles and quasiparticles.
Prerequisites for admission
- Basics of structure of matter (Fermi gas, dielectric function, specific heat)
- Basics of mathematical methods (complex integration and residues, Fourier transform, convolution, Euler's Gamma function, distributions)
- Basics of quantum mechanics (harmonic oscillator, H atom, Dirac's formalism, Heisenberg and interaction pictures, Dyson expansion of propagator, translations and rotations, spin and Pauli matrices, identical particles)
Teaching methods
Lesson with blackboard. (if the lesson is online the presentation will be based on written notes and use of graphic tablet)
Teaching Resources
Online notes:
Main textbook: Fetter e Walecka, Quantum theory of Many Particle Systems reprint Dover Ed.
During the course some specific books and readings are suggested.
Assessment methods and Criteria
Oral colloquium of about 1H duration, with discussion of 3-4 written exercises (prepared at home and freely chosen among proposed ones, or from the textbook), discussion of a free topic of the program, followed by questions to ascertain comprehension of main aspects of the course, orders of magnitude and links with other courses.
Lessons: 42 hours