The goal of the course Field theory 1 is extension and supplementation the knowledge achieved through the courses Quantum mechanics, Classical electrodynamics and Relativistic quantum mechanics in sense of their role in quantum field theories.
LEARNING OUTCOMES AT THE LEVEL OF THE PROGRAMME:
Upon completing the degree, students will be able to:
1. KNOWLEDGE AND UNDERSTANDING
1.2 demonstrate a thorough knowledge of advanced methods of theoretical physics including classical mechanics, classical electrodynamics, statistical physics and quantum physics
2. APPLYING KNOWLEDGE AND UNDERSTANDING
2.2 evaluate clearly the orders of magnitude in situations which are physically different, but show analogies, thus allowing the use of known solutions in new problems;
2.3 apply standard methods of mathematical physics, in particular mathematical analysis and linear algebra and corresponding numerical methods
3. MAKING JUDGEMENTS
3.2 develop a personal sense of responsibility, given the free choice of elective/optional courses
4. COMMUNICATION SKILLS
4.3 develop the written and oral English language communication skills that are essential for pursuing a career in physics
5. LEARNING SKILLS
5.1 search for and use physical and other technical literature, as well as any other sources of information relevant to research work and technical project development (good knowledge of technical English is required)
5.3 carry out research by undertaking a PhD
LEARNING OUTCOMES SPECIFIC FOR THE COURSE:
Upon completing the degree, students will be able to:
- find Noether currents for any field theory
- explain the content and structure of a general quantum field; explain the Lorentz transformation of any specific quantum field as well as the creation and annihilation
operatirs contained in them
- apply the Dyson's formula and the Wick's theorem for evaluation of correlation functions and S-matrix elements
- analize the quantum field theories defined by a Lagrangian and write the corresponding Feynman rules in x- and p- space
- evaluate cross sections and decay rates in the leading order of pertubation theory (tree diagrams)
In the first part of the course the students meet with notions of action, Lagrangian and momentum density, Euler-Lagrange equations of motion, Wigner's theorem on symmetry transformations, quantization of harmonic oscillator, causality problem in relativistic quantum mechanics, Lorentz transformations of coordinates, momenta, states in a Hilbert space and fields, and notion of propagator for the Klein-Gordon field. Using the knowledge students achieved through the course Relativistic quantum mechanics, same notions are introduced for the Dirac field. In the second part of the course the interactions are introduced for scalar field theory, the Dyson formula and Wick's theorem are derived, notion of Dyson's renormalizability is introduced, and Feynman rules in x- and p- space are derived. Using the Yukawa theory the Feynman rules for fermions are derived, and than by analogy the Feynman rules for quantum electrodynamics are derived, stressing the role of the Ward identity. Finally the photon propagator is derived using the covariant quantization and Gupta-Bleuler's mechanism.
1. week: Causality and necessity of the field theory. Basics of field theory: action, Lagrangian density, Euler-Lagrange equations of motion, momentum density, Hamiltonian density. Noether's theorem and examples for it.
2. week: Quantization of the Klein-Gordon (KG) field, quantization of the KG field, Poincare transformation of the states and creation operators.
3. week: The KG field in space-time: space-time dependence of the KG field, causality, Green functions for the KG field, propagator for the KG field.
4. week: Poincare transformations of coordinates, momenta and fields: generaly and for the vector field as an example.
5. week: Various representations of the Poincare symmetry.
6. week: Dirac and Weyl equations. Noether charges for Dirac field.
7. week: Free-particle solutions in the chiral representation, general solution of the Dirac equation.
8. week: Quantization of the Dirac field. Lorentz transformation of the Dirac field and corresponding one-particle states.
9. week: Spin and spin-projectors for the Dirac field. Dirac propagator and Dirac Green functions.
10. week: Interacting fields: examples of phi^4 theory, Yukawa theory and quantum electrodynamics
11. week: Derivation of the Dyson's formula for correlation function of two scalar fields and for any number of scalar fields. Derivation of the Wick's theorem.
12. Feynman rules for phi^4 theory in x- and p- space. Cross section and decay width from the experimental point of view.
13. week S-matrix and asymptotic states. Cross section and decay width from the theoretical point of view.
14. week: Formula for evaluation of S-matrix elements in terms of Feynman diagrams. Feynman rules for phi^4 theories and for Yukawa theory.
15. week: Feynman rules for quantum electrodynamics. Example of a cross-section evaluation.
The exercises cover through examples the topics passed on lectures.
REQUIREMENTS FOR STUDENTS:
The students have to attend the lectures and exercises regularly
GRADING AND ASSESSING THE WORK OF STUDENTS:
The exam has three parts: solving homework problems, written examination and oral examination. Part of written examination points may be acheved through homeworks. The problems given on written examination are similar in logic and content as those passed through lectures and exercises. Through written examination the calculation techniques of the students are examined. Through oral examination the knowledge of the logical structures and notions are examined.
- M. E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory, Addison Wesley, 1995
- S. Weinberg, The Quantum Theory of Fields, I, Cambridge, 1995