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Statistical and Quantum Field Theory

The Statistical Physics program provides an overview of the theory of phase transition, be they continuous or discontinuous. Mean-field approaches of the Landau family will be introduced, together with renormalization group techniques.
The aim of the second part of the course on Quantum Physics is to introduce the Quantum Field Theory for scalar fields - only coupled via a simple self-interaction - up to the calculation of basic reaction probability amplitudes.

Syllabus : " Statistical and Quantum Field Theory "

- Lectures 40 hours, Tutorials 30 hours (2nd Semester) -

(Grégory Moreau, Emmanuel Trizac)

Chapter 1: Phase transitions and critical phenomena : qualitative approaches
Phase transitions: problems raised and classification
Order parameter and symmetry breaking
Magnetic models: Ising, Heisenberg, Potts and the like
Local order and correlation functions

Chapter 2: Going quantitative
From Weis molecular field to Landau approaches
Ginsburg Landau functionals

Chapter 3: Analytical Mechanics
Principle of least action, Hamiltonian
Euler-Lagrange equations
Classical field theory
Noether’s theorem

Chapter 4: Relativistic Quantum Framework
Klein-Gordon equation
Second quantization of a spin-0 field
Green function for a free-field
Canonical commutation relations

Chapter 5: Introduction to Quantum Field Theory
Harmonic Oscillators
Multi-particle states
Evolution operator
Simple scalar theory :
— perturbation theory
— scattering amplitudes

Recommended textbooks:

  • From Microphysics to Macrophysics, R. Balian
  • Basic Concepts for Simple and Complex Liquids, J.-L. Barrat and J.-P. Hansen
  • Le Bellac, Peres, Landau-Lifshitz, Parisi, Messiah, Feynman, Pitaevski and Stringari
  • Introduction to Gauge Field Theory, D. Bailin and A. Love
  • Student Friendly Quantum Field Theory, R. D. Klauber
  • A First Book of Quantum Field Theory, A. Lahiri and P. B. Pal
  • An Introduction To Quantum Field Theory, M. E. Peskin and D. V. Schroeder

Course prerequisites and corequisites

This course requires basic knowledge of probability theory (elementary laws generating functions, central limit theorem etc), Statistical Physics (see e.g. D. Chandler, Introduction to Modern Statistical Mechanics, or Diu et al, Physique Statistique), Quantum Mechanics (typically the content of the main chapters of the book « Quantum Mechanics » - Volume 1 & 2, by F. Laloë, B. Diu, C. Cohen-Tannoudji) and basic notions in Special Relativity (like the covariant formalism).

This Major course can be complemented by the Major course « Particles, Nuclei and the Universe » (1st semester) and of the Minor courses « Soft matter », « Complex Systems and Information theory » , « Experiments and Applications in Sub-atomic Physics » (2nd semester).

Course concrete goals

On completion of the course students should be able to:

— be familiar with first order or continuous phase transitions
— be able to perform Ginzburg-Landau analysis
— understand the rationale behind the renormalization group technique

— apply the Noether’s theorem to any case
— use path integral techniques
— master advanced relativistic quantum mechanics
— quantize spin-0 fields
— ultimately calculate reaction amplitudes in Quantum Field Theory.

See also the E.Trizac web page.