# MANY-BODY THEORY

**Academic Year 2022/2023**- Teacher:

**Giuseppe Gioacchino Neil ANGILELLA**

## Expected Learning Outcomes

The course aims at providing basic concepts and techniques in the theory of many particle physics, at low energies, i.e. in the non-relativistic regime. Selected examples from condensed matter theory will be treated in some detail.

The course requires basic knowledge of standard Quantum mechanics, Complex calculus, as well as Thermodynamics, Statistical mechanics, and Structure of matter.

*Knowledge and understanding.*

Critical understanding of the most advanced developments of Modern Physics, both theoretical and experimental, and their interrelations, also across different subjects.. Adequate knowledge of advanced mathematical and numerical tools, currently used in both basic and applied research. Remarkable acquaintance with the scientific method, understanding of nature, and of the research in Physics. During the course we will present both experimental facts and theoretical models concerning the properties of many-body systems, with reference to modern experiments and novel theoretical interpretations.

*Applying knowledge and understanding*

Ability to identify the essential elements in a phenomenon, in terms of orders of magnitude and approximation level, and being able to perform the required approximations. Ability to use analogy as a tool to apply known solutions to new problems (problem solving). In presenting a may-body effect, emphasis will be given to the most important magnitudes, introducing all other magnitudes as successive approximations.

*Making judgements*

Ability to convey own interpretations of physical phenomena, when discussing within a research team. Developing one's own sense of responsibility, through the choice of optional courses and of the final project. In presenting the different topics, both during the course and during the final exam, links will be given with other courses (mainly, but not only, belonging to the same curriculum), some of which optional, and with possible topics for a research final project, both experimental and theoretical.

*Communication skills*

Ability to discuss about advanced physical concepts, both in Italian and in English.

*Learning skills.*

Ability to acquire adequate tools for the continuous update of one's knowledge. Ability to access to specialized literature both in the specific field of one's expertise, and in closely related fields. Ability to exploit databases and bibliographical and scientific resources to extract information and suggestions to better frame and develop one's study and research activity. Ability to acquire, through individual study, knowledge in new scientific fields. We will often make reference to scientific papers, both reviews and research articles.

## Course Structure

Frontal lectures (in person).

Should the circumstances require online or blended teaching, appropriate modifications to what is hereby stated may be introduced, in order to achieve the main objectives of the course.

## Required Prerequisites

Fundamentals of quantum mechanics, complex analysis, thermodynamics, statistical mechanics, condensed matter.## Attendance of Lessons

Lectures attendance is usually mandatory (see the Regulations of the MSc Course in Physics).## Detailed Course Content

**Second quantization.** Identical particles. Bosons and fermions. Fock space. Creation and annihilation operators. Field operators. Examples: kinetic energy, spin, density, current, Coulomb interaction. Harmonic oscillator and electromagnetic field in second quantization: photons. Degenerate electron gas. Phonons. Electron-phonon interaction.

**Zero-temperature Green's functions.** Time dependence: Schrödinger, Heisenberg, and interaction pictures. Time ordering. Gell-Mann--Low's theorem. Green's functions and their physical meaning. Green's functions for fermions at T=0. Particles and holes. Lehmann representation. Advanced and retarded Green's functions. Causality and dispersion relations. Wick's theorem.

**Perturbation theory.** Two-body interaction. Feynman diagrams. Goldstone's theorem. Self-energy. Dyson equation. Hartree-Fock approximation. Renormalization: quasiparticles. Microscopic foundations of Landau theory of Fermi liquids. Polarizability and density-density correlation function. Random Phase Approximation.

**Linear response theory.** Kubo formulas and correlations. Single impurity in a degenerate electron gas: screening. Dielectric function and Lindhard function. Friedel oscillations. Plasmons. Plasmons in low-dimensional systems.

## Textbook Information

A. L. Fetter, J. D. Walecka, *Quantum Theory of Many-Particle Systems*, Dover (2003).

H. Bruus, K. Flensberg, *Many-body quantum theory in condensed matter physics*, Oxford University Press (2004).

A. A. Abrikosov, L. P. Gorkov, I. E. Dzyaloshinski, *Methods of Quantum Field Theory in Statistical Physics*, Dover (1975).

Ch. P. Enz, *Many-Body Theory Applied to Solid-State Theory*, World Scientific (1998).

G. D. Mahan, *Many-Particle Physics*, Plenum Press (1990).

J. W. Negele, H. Orland, *Quantum Many-Particle Systems*, Addison-Wesley (1988).

N. H. March, W. H. Young, S. Sampanthar, *The Many-Body Problem in Quantum Mechanics*, Dover (1995).

## Course Planning

Subjects | Text References | |

1 | See course contents. |

## Learning Assessment Procedures

Oral discussion on topics of the course and of interest for the candidate (e.g. for her/his MSc final project). The topics should be agreed with the teacher in advance.Evaluation criteria are: relevance of answer wrt question, detail level, ability to stipulate links with other topics of this or other courses, ability to formulate relevant examples, language skills, clarity of presentation.

Evaluation may also take place online, should circumstances require, and current regulations allow that.