Functional Integration in Quantum Field Theory


Description: This course is intended as an introduction to the quantum field theory. It aims to explain those features of quantum and statistical field systems which result from their field-theoretic nature and, therefore, are common to different physical contexts. Among these features are renormalisation, effective interactions, running coupling constants and anomalous scaling dimensions. The main object for study and construction in this course is the effective action. The basic technical tool applied throughout the course is the path integral calculated via the loop expansion.

Quantum field theory can be considered as a mathematical tool for many physical disciplines, including elementary particle physics, solid state physics and physics of condensed matter. The technique of functional integration the most appropriate formalism which allows us to study field theory in a con\-text-in\-de\-pen\-dent way prior to making use of it in specific physical disciplines.

This course aims at giving a self-contained introduction in the formalism of functional integration in modern quantum field theory, it provides a set of simple examples of the physical concepts and mathematical ideas which where developed in that context.

Syllabus:

  1. Feynman path integral and its Euclidean counterpart in quantum mechanics. Green function.
  2. Quantum mechanical instantons, energy splitting in the double-well potential.
  3. Quantization of the fields, Casimir effect.
  4. Effective potential of the f4 model. Running coupling constant. Regularization and remormalization.
  5. Generating functional in quantim field theory. Effective action of the f4 model.
  6. The loop expansion of the effective action. Vertices.
  7. Evaluation of the divergent diagrams.
  8. Quantum mechanics with anticommuting variables. Fermionic oscillator.
  9. Functional integral over fermions.
  10. Supersymmetric oscillator.
  11. Fermionic corrections to the boson propagator in the model with Yukawa interaction.

Lectures