The goal of the course Field theory 2 is extension of the knowledge achieved through the course Field theory 1 in sense of ovewhelming the techniques needed for higher orders in pertubation theory. The course Field theory 2 assures deep understanding of the mathematical structure of the field theory with emphasis on evaluation and effects of higher order contributions in field theory, and elimination of divergencies in it. These methods cover an additional set of large group of observables and techniques needed in the courses Particle physics and Physics beyond Standard model, as well as in the postgraduate courses.
LEARNING OUTCOMES AT THE LEVEL OF THE PROGRAMME:
Upon completing the degree, students will be able to:
1. KNOWLEDGE AND UNDERSTANDING
1.1 formulate, discuss and explain the basic laws of physics including mechanics, electromagnetism and thermodynamics
1.2 demonstrate a thorough knowledge of advanced methods of theoretical physics including classical mechanics, classical electrodynamics, statistical physics and quantum physics
1.3 demonstrate a thorough knowledge of the most important physics theories (logical and mathematical structure, experimental support, described physical phenomena)
2. APPLYING KNOWLEDGE AND UNDERSTANDING
2.1 identify the essentials of a process/situation and set up a working model of the same or recognize and use the existing models
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 approximative expression of the observable corrections (e.g. a cross section) due to soft virtual and real photons for simpler diagrams
- evaluate vertex correction for quantum electrodynamics as well as for vertices in any other theory, for example Yukawa type theories
- evaluate observables as anomalous magnetic moment of a fermion at first nontrivial order in pertubation theory
- analyze ultraviolet divergencies for one-loop diagrams
- find imaginary part of the amplitudes at one-loop level
- evaluate the dependence of the coupling constants as a function of the momentum transfer of the exchanged particles, e.g. charge as a function of photon momentum and Yukawa coupling as a function of the scalar particle momentum
- renormalize simpler theories at the one-loop level
In the first part of the course Field theory 2 one gets acknowledged with nonpertubative methods in field theory: Kaellen-Lehmann spectral representation of scalar (and any other) propagator and LSZ formalism. Using them the proof of the formula for the S-matrix element, introduced in the course Field theory 1, is given.
In the second part the pertubative corrections on the tree level diagrams are analyzed. First, the infrered divergencies at the first order, and then at any order of pertubation theory appearing in soft bremsstrahlung processes and exchages of soft virtual photons. The proof of finiteness of these corrections at any order of pertubation theory is given. Second, the structure of the fermion photon-fermion vertex is provided on basis of symmetries. Through pertubative evaluation of the vertex function and fermion self-energy diagram part of basic techniques for evaluating loop diagrams is introduced, together with notions of regulatization of amplitudes an their renormalization, renormalization constants for the vertex function (Z1) and fermion line (Z2). These two renormalization constants are shown to be equal in quntum electrodynamics, leading to non-renormalization of the fermionic current. Third optical theorem is introduced through S-matrix and then for Feynman diagrams. Using it the expression for the decay rate, introduced by analogy with the expression for cross section in course Field theory 1, is prooved. Fourth, the Ward-Takahashi identity is prooved in any order of pertubation theory, and from it, the proof of the Ward identity and equality of renormalization constants Z1 and Z2 at any order of pertubation theory. Fifth, through analysis of the photon self energy the dimensional regularization is introduced, renormalization of the photon line is assigned to the charge renormalization, while the finite part of the renormalization leads to the vaccum correction of the Coulomb law, and photon momentum-square dependendence of the fermion charge.
In the third part the notion of renormalization is systematically introduced. Superficial degree of divergence, and methods of finding superfitial degrees of divergence are introduced, then the divergent diagrams for quantum electrodynamics and phy^4 theory are found and notion of Dyson renormalizable theories is introduced. Finally, the renormalized pertubation theory is introduced and applied to renormalize phy^4 theory, Yukawa theory and quantum electrodynamics.
1. week: Deriving potentials from the S-matrix elements.
2. week: Field-strength renormalization.
3. week: LSZ reduction formula. Nonpertubative proof of the formula for S-matrix element(s).
4. week: Soft bremsstrahlung at the first order in pertubation theory.
5. Structure of the electron vertex function.
6. week: Evaluation of the electron vertex function and its ultraviolet behaviour: vertex renormalization, anomalous magnetic moment.
7. week: Infrared divergencies of electron vertex function. Summation and interpretation of the infrared divergencies at all orders in pertubation theory.
8. week: Fermion self energy: position of the pole and cut, renormalization constant Z2, connection between renormalization of the fermion line and vertex, renormalization constant Z1 and its connection to Z2.
9. week: Optical theorem: S-matrix description, description in terms of Feynman rules, proof of the decay width formula.
10 week: Ward-Takahashi identity.
11. and 12. week: Renormalization of the electric charge.
13. week: Counting ultraviolet divergencies. Renormalization pertubation theory.
14. week: Structure of the phi^4 theory and Yukawa theory at one loop level.
15. week Renormalization of quantum electrodynamics.
The exercises cover through examples the topics passed on lectures.
REQUIREMENTS FOR STUDENTS:
Students have to attend the lectures and excercises 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.