## Scope of questions for State exams of Master's degree program

Branch: Quantum Technologies

Subject: Quantum Field Theory

Subjects regarding the questions:

- 02KTP1 Quantum Field Theory 1
- 02KTP2 Quantum Field Theory 2

- Relativistic wave equation for a scalar particle, Klein-Gordon’s equation, Klein-Gordon’s equation in Feshbach-Villars representation, solution for a free particle, continuity equation and its problems, non-relativistic limit, Klein’s paradox
- Relativistic wave equation for a spin ½ particle, Dirac’s equation, Lorentz group and its representations, invariance of the Dirac‘s equation with respect to the proper Lorentz transformations and bilinear forms, solution of the Dirac’s equation for a free particle, Dirac’s, Weyl’s and Majorana’s representation, continuity equation and its problems, zitterbewegung, non-relativistic limit.
- Relativistic particle in an external electromagnetic field, principle of minimal coupling, relativistic particle in a spherically symmetric field, compatible observables, solution of the Driac’s equation for a hydrogen (and hydrogen-like) atom, fine structure of the spectrum.
- Canonical quantization of a scalar field, algebra of observables and particle interpretation, canonical quantization of the Dirac’s field, non-relativistic limit, Fock’s space and occupation numbers.
- Symmetries and conservations laws, Noether’s theorem and Ward’s identities, discrete P, T and C, symmetries, explicit forms of P, T and C operators for a Dirac’s particle, antiparticle
- Normal ordering, Feynman’s propagator for a scalar and Dirac’s field, interacting fields, Wick’s theorem and perturbation expansion, scattering processes, S and T matrices and Feynman’s rules, optical theorem and unitarity, cross-section and the decay of an unstable particle, renormalization for a φ
^{4}theory. - First quantization with the path integral, second quantization and the functional integral, partition sum and Wick’s theorem, Ward’s identities and anomalies, functional integral and non-relativistic limit, perturbation expansion of Green’s function with Feynman’s diagrams for a scalar field, dimensional regularization, generating functionals W and Г, connected and 1PI diagrams, evaluation of Feynman’s diagrams.
- Grassmann’s variables and Berezin functional integral for fermionic fields, perturbation expansion of Green’s function with Feynman’s diagrams for a fermionic field, evaluation of Feynman’s diagrams.
- Quantum electrodynamics, basic scattering processes in QED, evaluation of the S-matrix for Compton’s scattering, electron-positron annihilation, Møller‘s scattering and Bhabha’s scattering, S-matrix and LSZ formalism, Lehman’s representation of Green’s functions.
- Yang-Mills fields and their quantization, Faddeev-Popov ghost fields, calibration and ‘t Hooft’s trick, Goldstone’s theorem and Higgs mechanism, Callan-Symanzik equation of renormalization group and the β function, perturbation expansion of the β function for a scalar field, concept of asymptotic freedom, effective theory.