Clarification of requirements for the oral part of the state final examination in the field of Theoretical Physics

The green headings should always form one question in the state final exam and should correspond to what is stated in Karolinka valid from the academic year 2020/2021.

Please note that the Master studies are in Czech and therefore this list is purely informative.

Common requirements

Relativistic Physics

Foundational principles of special and general relativity: Newton's law, the concept of an inertial frame; the principle of special relativity and tensor notation of physical laws; the finiteness and invariance of the speed of light, basic properties of Lorentz transformations; the universality of gravitational interaction, the equivalence principle and its experimental verification, the concept of a local inertial frame; the principle of general covariance, the form of physical laws
Spacetime, four-dimensional formalism, coordinate transformations: space-time symmetry of the Lorentz transformation and the four-dimensional paradigm, Minkowski spacetime, real four-dimensional formalism; tensor quantities and invariants, the metric tensor, behavior of quantities under coordinate transformations; spacetime diagrams, the light cone, behavior of coordinate axes under (Lorentz) transformation; description of curved spacetimes
Parallel transport and the geodesic equation, metric and affine connection, covariant derivative: the problem of transition between tangent spaces at different points of spacetime, the geometric concept of parallel transport and its realization based on the equivalence principle; geodesics as the straightest curves, as worldlines of free test particles, and as extremal connections, the Newtonian limit of the geodesic equation; the general metric tensor, the affine connection and Christoffel symbols; covariant and absolute derivatives
Frequency shift in a gravitational field: time dilation and the problem of clock synchronization in special and general relativity; the frequency shift of radiation between two static observers in a stationary spacetime, the Newtonian limit of this relation, examples and derivation from the equivalence principle; the frequency shift between a freely orbiting satellite on a circular orbit and a static observer, time dilation in satellite navigation (GPS)
Spacetime curvature: Ricci identities and the introduction of the Riemann tensor, basic properties of the Riemann tensor, the complete set of its symmetries and the number of independent components; the geometric and physical significance of the Riemann tensor — non-integrability of parallel transport, the equation of geodesic deviation; the Ricci tensor and the Ricci scalar; the second Bianchi identities
Energy-momentum tensor, conservation laws and equations of motion: the energy-momentum tensor for charged incoherent dust and for the electromagnetic field, the meaning of its components, conservation laws (for a system subject to external forces and for a closed system); perfect fluid — derivation of the continuity equation and Euler equations from the conservation laws
Einstein's gravitational field equations: Newton's gravitational field equation and its criticism from the viewpoint of theory of general relativity; the right-hand side of Einstein's equations: the energy-momentum tensor, the left-hand side of Einstein's equations: the distinguished role of the Riemann tensor, determination of constants based on the second Bianchi identities, the requirement of energy-momentum tensor conservation, and the Newtonian limit of the equations; Einstein's equations as a mathematical problem, their nonlinearity and the self-interaction of the source, comparison with the content of Maxwell's equations; interpretation of the cosmological term
Schwarzschild solution of Einstein's equations: the Schwarzschild metric as an exact spherically symmetric solution of the vacuum Einstein equations without a cosmological term — outline of derivation (introduction of coordinates, the metric of a general spherically symmetric spacetime, solving the Einstein equations — Birkhoff's theorem), basic properties (symmetries, asymptotic flatness, light cones, the horizon as a null hypersurface, causal boundary, the static limit and the surface of infinite frequency shift, the black hole as a dynamic region inside the horizon, the central singularity); timelike and null geodesics in the Schwarzschild field (constants of motion, discussion of radial motion by the effective potential method), circular geodesics
Homogeneous and isotropic cosmological models: basic observational findings about the large-scale universe, cosmological expansion, the 'big bang' and the age of the universe; homogeneity and isotropy geometrically, 'cosmic time' and the synchronous coordinate system, metrics of spaces with constant curvature; 'cosmic fluid', the role of matter and radiation during the evolution of the universe; the Friedmann equation and the role of its individual terms, discussion of possible evolutions by the effective potential method; rewriting the equation in terms of Ω-factors, observational data and the idea of the relative magnitudes of terms, accelerated expansion and the problem of so-called dark energy

Quantum Physics

Description of states and observables in quantum theory: Hilbert space, density matrix, spectral decomposition, commuting operators, various representations, change of basis, examples of systems — state spaces and operators, description of measurement
Unitary time evolution: Schrödinger, Heisenberg, and Dirac representations, equations of motion in these representations, Ehrenfest's theorem, stationary states and integrals of motion
Quantum theory of angular momentum: commutation relations, relation to rotations, examples of angular momentum operators — spin and orbital angular momentum
Fundamentals of angular momentum addition: Clebsch-Gordan coefficients, examples of addition for two and three particles with spin, construction of spin-orbitals, irreducible representations of the rotation group and Wigner D-matrices, vector and tensor operators in quantum mechanics, the Wigner-Eckart theorem
Systems of several indistinguishable particles: Bose and Fermi statistics, second quantization, examples — helium atom, several particles in a potential well
Stationary perturbation theory: energy corrections to 1st and 2nd order, corrections for stationary states, degenerate spectrum
Ritz variational principle: the principle and examples of applications, excited states — the Hylleraas-Undheim theorem
Time-dependent perturbation theory: Dyson expansion, transition probability in first and second order, the case of a continuous spectrum. Particle in a spherically symmetric field: radial Schrödinger equation, stationary states, scattering states, phase shifts and differential cross section
Equations of relativistic quantum mechanics for particles with spin 0, 1/2, and 1: Klein-Gordon equation for a free particle — solutions with positive and negative energy, continuity equation; Dirac equation for a free particle — relativistic invariance, discrete symmetries C, P, T, plane waves and description of spin states, helicity and chirality; Weyl equation and its symmetry properties; Proca equation for a free particle — two equivalent forms of the equation, plane waves, description of spin states
Dirac equation for a particle in an electromagnetic field: transition to the Pauli equation and the spin magnetic moment, hydrogen-like atom and the fine structure of energy levels
Quantization of free fields and their particle interpretation: canonical quantization of the Klein-Gordon field, equal-time commutators, energy and momentum of the quantized field, creation and annihilation operators, Fock space, particles and antiparticles; quantization of the Dirac field and anticommutation relations; canonical quantization of the Proca field; quantization of the electromagnetic field
Field interactions: examples of interaction Lagrangians: Yukawa-type interaction, interaction of a vector field with fermions, quantum electrodynamics, direct four-fermion interaction, the principle of local internal symmetry
S-matrix and simple Feynman diagrams: Dyson expansion for the evolution operator in the interaction representation, S-matrix and the invariant transition amplitude, tree-level Feynman diagrams
Calculation of decay probability and reaction cross section: differential decay probability per unit time and differential cross section for a two-particle collision: general formulas and their possible integration for a two-particle final state, example of scattering in an external electromagnetic field

Statistical Physics

Statistical description of thermodynamics: microscopic and macroscopic states, statistical description and ergodicity, the thermodynamic limit, thermodynamic and statistical entropy, the partition function and thermodynamic potentials in the statistical description
Fundamental statistical ensembles: the microcanonical, canonical, and grand canonical statistical ensemble, their equivalence
Fluctuations of thermodynamic quantities: the Gibbs and Einstein methods, stability conditions for thermodynamic potentials
Quantum statistical mechanics: postulates of quantum statistical mechanics, the density matrix and the representation space of Fermi and Bose systems, the equation of state and the classical limit
Ideal Bose-Einstein gas of massive particles: chemical potential and Bose-Einstein condensation
Gas of massless bosons: photons and phonons, heat capacity of solids, blackbody radiation
Degenerate electron gas: the Fermi level, Sommerfeld expansion, heat capacity of metals, Pauli paramagnetism
Fundamentals of non-ideal gas theory: molecular forces, the van der Waals equation, the cluster and virial expansions
Fundamentals of non-equilibrium statistical physics: time evolution of the distribution function, hierarchical evolution equations (BBGKY), binary collisions and the Boltzmann kinetic equation

Plasma Physics and Solid State Physics

Basic concepts of plasma theory: definition of plasma, Debye radius, plasma frequency, plasma parameter
Plasma drifts in electric and magnetic fields, adiabatic invariants
Kinetic theory of plasma, Landau damping: kinetic equation, phase space, Liouville's theorem, Boltzmann equation, Vlasov equation, BBGKY, Landau damping, kinetic beam instability
Collision term and relaxation: Coulomb collisions, time scales in plasma, entropy, the H-theorem
Magnetohydrodynamic description of plasma: equations of motion in hydrodynamics, basic magnetohydrodynamic effects (diffusion, flux freezing), linear and nonlinear waves in plasma
Solid state as a quantum mechanical many-body problem: the adiabatic approximation, the Hellmann-Feynman theorem
Harmonic approximation of atomic motion: phonon dispersion bands, polarization vectors, the Brillouin zone, Born-von Kármán boundary conditions
Diffraction from a lattice: Fermi's golden rule, averaging over lattice vibrations, Bragg scattering, the Debye-Waller factor
Electronic band structure: Bloch's theorem, relativistic corrections to the Schrödinger equation, the Brillouin zone, Born-von Kármán boundary conditions
Thermodynamic properties of crystals: specific heat of phonons, specific heat of electrons

Computational Physics

Representation of real numbers on a computer, rounding error: floating-point representation, machine epsilon, (relative) rounding error, the effect of rounding error on the result of arithmetic operations
Algorithm stability and problem conditioning: stable and unstable algorithms, relation to rounding error, forward and backward stability, the condition number of a problem, ill-conditioned and well-conditioned problems, illustration of these concepts with examples
Approximation and interpolation of functions: representation of continuous quantities on a computer, function approximation and approximation error, polynomial interpolation and its accuracy: Lagrange and Newton interpolation polynomials, Hermite interpolation, cubic splines
Numerical differentiation of functions, finite differences: numerical differentiation using interpolation polynomials, finite differences: forward, backward, and centered, Richardson extrapolation, discretization error, optimal choice of discretization step with respect to rounding error
Numerical integration of functions: quadrature formulas and their order, Newton-Cotes formulas and their error, the Euler-Maclaurin summation formula, the Romberg algorithm, Gaussian quadrature, relation to orthogonal polynomials
Solving nonlinear equations: the fixed-point theorem of a contraction mapping, iterative methods, convergence rate and its improvement, the bisection method, Newton's method
Solving systems of linear equations: Gaussian elimination, LU decomposition and its stability, pivoting, Cholesky decomposition, computational complexity of methods, tridiagonal matrices
Basic methods for integrating ordinary differential equations: initial value and boundary value problems, numerical solution of initial value problems: single-step Runge-Kutta methods, multi-step linear methods, local and global discretization error, order of the method

Specialization

Students choose two areas from the following list.

Mathematical Methods

Fundamentals of measure theory: σ-algebra, Borel sets, outer measure, complete measure, Lebesgue measure, measurable functions, simple functions; signed measure (charge), Hahn decomposition, total variation of a measure
Banach and Hilbert spaces, linear operators and functionals: Banach fixed-point theorem; operators and functionals; representation of a linear functional on a Hilbert space; spectrum, resolvent set, point, continuous, and residual spectrum, spectral radius, compact operators, spectrum of a compact operator; dual operators and spaces, duality, adjoint operator, self-adjoint operator; basis composed of eigenvectors, orthonormal basis in a Hilbert space composed of polynomials and recurrence relation; the Hilbert-Schmidt theorem
Equations of mathematical physics and their basic properties, special functions: the transport equation, solution by the method of characteristics; the heat equation, existence and uniqueness of solutions; the wave equation, finding the elementary wave function in one spatial dimension, d'Alembert's formula, the wave cone and the finite speed of information propagation; the Laplace-Poisson equation, elementary solutions, solution on a sphere, Poisson's integral; the Gamma and Beta functions and their use in computations; Bessel functions, Legendre, Laguerre, and Hermite polynomials; hypergeometric series
Definition of distributions and basic operations with distributions
Fourier transform of functions and distributions: Fourier transform for functions from L¹(Rⁿ), S(Rⁿ) — the space of rapidly decreasing functions, L²(Rⁿ); tempered (Schwartz) distributions S'(Rⁿ) and their Fourier transform; inversion theorems for functions from the spaces L¹(Rⁿ), S(Rⁿ), L²(Rⁿ), S'(Rⁿ); theorems on the relation between the Fourier transform and differentiation, the Fourier transform and convolution, also for distributions; Fourier transform of a distribution with compact support, specifically the Dirac distribution; the use of distributions for finding fundamental solutions of ordinary and partial differential equations
Differentiable manifolds and their tangent spaces, exterior calculus: topological and differentiable manifolds, tangent spaces, vector and tensor fields, maps between manifolds, submanifolds, induced maps, Lie bracket of vector fields, Lie derivative, antisymmetric forms, exterior product, exterior derivative, exact and closed forms, integration of antisymmetric forms, Stokes' theorem
Riemannian geometry and covariant derivative: covariant derivative and parallel transport along curves, torsion, curvature and their geometric meaning, the space of connections, the connection associated with coordinates and a normalized basis, Riemannian and pseudo-Riemannian metric, the Levi-Civita tensor, the Hodge dual, the metric volume element, the Riemannian (metric) connection, isometries and Killing vectors, extremal properties of geodesics
Vector bundles: vector fiber bundles, vector fields — sections, covariant derivatives on vector bundles, the vector potential and the curvature tensor, electromagnetic and gauge fields in the language of vector bundles, gauge transformations and the gauge group
Lie groups and Lie algebras: Lie group as a smooth manifold, left-invariant vector fields, the Lie algebra of a Lie group, structure constants and the Killing metric, one-parameter subgroups, the exponential map, the universal covering group, the action of a group on a manifold and its generators
Fundamentals of group representation theory: reducible, irreducible, and completely reducible representations, character of a representation, Schur's lemmas and their consequences for finite groups — orthogonality relations, the number and dimensions of irreducible representations of finite groups. Representations of the groups SO(3) and SU(2): the relation between representations of a Lie group and its Lie algebra, representations of the Lie algebra su(2), representations of the group SU(2), single-valued and double-valued representations of the group SO(3)

Relativistic Theory of Gravitation

Lie derivative, symmetries, and Killing vectors: the flow of a vector field and induced maps of tangent spaces, pull-back and push-forward; derivation of index expressions for the Lie derivative of scalars, vectors, and covectors; spacetime symmetries in the language of Lie derivatives and Killing vector fields, corresponding coordinate statements; basic properties of Killing vectors and consequences of their existence (constants of geodesic motion)
Riemann and Weyl curvature tensors, geodesic deviation: Ricci identities and the Riemann tensor, the fully covariant version of the tensor, mathematical properties and the number of independent components, geometric interpretation of the Riemann tensor (non-integrability of parallel transport), physical interpretation of the Riemann tensor (the question of geodesic deviation); the second Bianchi identities; the Ricci tensor and scalar; the Weyl tensor and its significance, the number of independent components
Algebraic classification of spacetimes: the null tetrad and its Lorentz transformations; the Weyl tensor, its tetrad components and their behavior under Lorentz transformations; privileged null directions, their multiplicity and the corresponding algebraic types; important algebraically special spacetimes and their interpretation (types D and N)
Timelike and null congruences: kinematics of timelike and null congruences — expansion, vorticity, and shear of the congruence, corresponding scalars; the relation between the evolution of these properties along the congruence and the geometry of spacetime; Frobenius' theorem, hypersurface-orthogonal vector fields
Spaces of constant curvature (Minkowski, de Sitter, anti-de Sitter): maximally symmetric vacuum spacetimes and conformal flatness; curvature given solely by the cosmological constant; natural global coordinates of Minkowski, de Sitter, and anti-de Sitter spacetimes; their Penrose diagrams, the acceleration and cosmological horizons in Minkowski and de Sitter respectively; de Sitter as a contracting and then expanding 3-sphere; closed timelike curves in anti-de Sitter versus its covering space
Exact solutions of Einstein's equations describing stationary black holes, laws of dynamics: the Schwarzschild, Reissner-Nordström, Kerr, and Kerr-Newman solutions of Einstein's equations, their properties and analytic extensions; their spatial and spacetime structure; the question of inertial dragging in the vicinity of rotating sources; uniqueness theorems for stationary black holes, the laws of their dynamics (illustration on the Kerr-Newman solution) and the question of energy extraction from black holes
Linearized gravity theory and plane gravitational waves: the concept of a weak field and the derivation of linearized gravity theory, the linearized Einstein equations and the Lorenz gauge condition; a plane harmonic wave as a solution of the linearized field equations, the question of purely coordinate waves and fixing the remaining gauge freedom (TT gauge), decomposition of the solution into two polarization modes; the behavior of test particles during the passage of a wave
Exact spacetimes with gravitational waves: the exact type-N gravitational wave as a transverse effect of geodesic deviation encoded in Ψ₄; the pp-wave class in flat space, the Brinkmann and Rosen forms of plane waves; the Kundt class of non-expanding gravitational waves; the Robinson-Trautman class of expanding gravitational waves; sandwich and impulsive gravitational waves with an arbitrary cosmological constant
Lagrangian formalism in general relativity, conservation laws: variational derivation of Einstein's equations, the Hilbert action, the Lagrangian for the geometric and source sides of Einstein's equations, the question of the boundary term; the introduction of the energy-momentum tensor, invariance under diffeomorphisms and conservation laws; 'derivation' of the Levi-Civita connection based on independent variation with respect to the metric and its derivatives; the relation between conservation laws and the character of the symmetry group (Lie groups, infinite-dimensional pseudogroups of gauge transformations)
3+1 decomposition of spacetime, the initial value problem and Hamiltonian formalism in general relativity: 3+1 decomposition of spacetime — the metric of spacelike hypersurfaces, lapse and shift; the covariant derivative on the hypersurface, the extrinsic curvature tensor; projection of the Riemann tensor: the Gauss-Codazzi equations, decomposition of the Ricci tensor and scalar curvature; the 3+1 formulation of Einstein's equations, constraints and evolution equations; Hamiltonian formulation of field theory (illustrated on scalar and EM fields), Hamiltonian formulation of Einstein's equations — constraints and evolution equations (evolution equations without derivation), the question of boundary terms and integral quantities (ADM mass)

Theoretical Astrophysics and Cosmology

Classical and relativistic theory of stellar structure, radial oscillations and stability: separation of short-range and long-range forces, 16+3B quantities for the description of a static and spherically symmetric star with B baryon species, metric parametrization; Einstein's equations — the equation for mass and the equation for the gravitational potential, their interpretation; the TOV equation, its discussion and integration for uniform density; variational derivation of the TOV equation; the equation of thermal equilibrium, the energy transport equation (by convection, by radiation); the dynamical problem of convective stability — radial adiabatic oscillations, derivation of the equations for initial values and dynamical equations, the Sturm-Liouville problem for determining oscillation modes (the idea of a variational solution)
Stellar evolution and its final stages, gravitational collapse, supernovae, black holes: typical stellar parameters, the number of baryons, neutrons, and electrons (Y-factors) and their evolution during the life of stars; the general strategy for solving the system of equations for stellar equilibrium and the resulting (radius-mass) equilibrium diagram, stable branches of white dwarfs and neutron stars, typical densities, energy scales, and the concept of cold catalyzed matter; the Chandrasekhar limit; the Oppenheimer-Snyder exact spherically symmetric gravitational collapse, matching of the interior and exterior solutions; energy sources of supernova explosions
Equations of state for a degenerate gas, white dwarfs, neutron stars: degenerate gas, Fermi momentum and Fermi energy, the question of gas ideality; the pressure integral, its evaluation for a degenerate gas; equations of state of a degenerate ideal gas in the non-relativistic and ultra-relativistic limits; a rough derivation of the Chandrasekhar limit, its fundamental significance; the phenomenology of high densities: neutronization (inverse beta decay), neutron drip, nucleon breakup and the quark-gluon phase
Non-relativistic radiative processes in astrophysics: the electromagnetic field and radiation of non-relativistic particles; the dipole approximation, the multipole expansion of radiation; Thomson scattering; the braking (radiation reaction) force; radiation of a harmonically bound particle and scattering from it
Relativistic radiative processes in astrophysics: the electromagnetic field and radiation of relativistic particles; transformation of power and specific intensity; synchrotron radiation; bremsstrahlung; Compton and inverse Compton scattering
Homogeneous and isotropic cosmological models: the cosmological principle; the Friedmann-Lemaître-Robertson-Walker metric; kinematics of the FLRW metric: geodesics, cosmological redshift, cosmological horizons; matter and radiation and their cosmological evolution; the Friedmann equations; an overview of models of the evolution of the universe
Cosmological distances, light propagation, gravitational lensing: proper and comoving distance, distances determined from the decline in luminosity and from the decline in angular diameter; the appearance of cosmologically distant objects; gravitational lensing, the lens equation; a simple point lens, properties of images, the Einstein radius; weak lensing by continuously distributed matter
The early universe and its thermal history: thermal and chemical equilibrium, the Boltzmann transport equations and their solutions; freeze-out of interactions, decoupling of particles from the thermal bath; hot and cold relics and their residual densities, dark matter from the perspective of particle physics, WIMPs; the thermal history of the universe, the evolution of plasma during the 'first three minutes', electroweak and QCD phase transitions, neutrino decoupling, positron annihilation; primordial nucleosynthesis, the role of deuterium, the evolution of the relative abundance of neutrons and protons, deuterium and ⁴He, baryons and photons, the baryon number problem; the last scattering of cosmic microwave background photons, recombination, the Saha equation
Evolution of the cosmic plasma in linear perturbation theory: the cosmic plasma after nucleosynthesis — components and their interactions; linear perturbation theory with scalar perturbations on an FLRW background; the structure of the perturbed Boltzmann equations for photons, cold dark matter, and baryons; the evolution of temperature and temperature fluctuations of radiation; the evolution of densities, density fluctuations, and velocities of dark matter and baryons
Evolution of density perturbations, structure formation: perturbative description of density fluctuations; linear collapse and the Jeans criterion; cosmological evolution and the spectrum of density fluctuations; nonlinear collapse and structure formation; results of cosmological simulations of structure evolution
Cosmic microwave background radiation and its anisotropy: the cosmic microwave background radiation and its history, the significance of recombination; the multipole decomposition of temperature anisotropies; evolution on large scales, evolution of the monopole before recombination, the Sachs-Wolfe effect; evolution on smaller scales, tight coupling between baryons and photons, acoustic oscillations, the sound horizon, damping on small scales; the connection between the spatial structure of fluctuations at the time of recombination and the observed anisotropies; the structure of the observed anisotropy spectrum, the integrated Sachs-Wolfe effect, acoustic peaks, Silk damping

Advanced Quantum Mechanics

Fundamentals of quantum scattering theory of a particle on an external potential: time-dependent formulation of single-channel scattering theory: Møller operators, conservation laws, S-matrix and differential cross section; time-independent formulation, Green's functions, stationary scattering states, the Lippmann-Schwinger equation and its relation to the solution of the Schrödinger equation; the T-matrix
Scattering on a spherically symmetric potential and analytic properties of scattering quantities: partial wave decomposition of scattering quantities, the radial Schrödinger and Lippmann-Schwinger equations, phase shifts; the Jost function, poles of the S-matrix, Levinson's theorem, virtual states, resonances; the low-energy approximation, scattering length
Fundamentals of multichannel scattering theory: channels and channel spaces; Møller operators and S-matrix; time-independent formulation, the Lippmann-Schwinger equation and the T-matrix; the behavior of scattering quantities near thresholds, Wigner cusps; differential cross sections
Approximate methods for many-body systems: basic principles of the methods, the Hartree-Fock approximation, electron correlation, the configuration interaction (CI) method, the coupled clusters method, and density functional theory (DFT)
Structure of atoms and molecules: electronic states of atoms and molecules and their classification; vibrational states of molecules; Wigner d-matrices as stationary states for a symmetric top, molecular rotations
Approximate methods of scattering theory and their applications: the Born series; variational principles in quantum scattering theory; the R-matrix method; electron scattering on atoms and molecules in the static-exchange approximation; vibrational and rotational excitation of molecules by collisions
Decoherence and effective reduction: reduction (collapse) of the quantum state, composition of systems, restriction to a subsystem, averaging over the environment, effective reduction — equivalence of reduction and restriction, entropy of mixed states
Quantum mechanics and hidden variable theories: the EPR system, entangled states, locality of quantum measurements and 'nonlocality' of correlations, Bell inequalities, the inequivalence of quantum mechanics and local hidden variable theories
Feynman formulation of quantum mechanics: quantum histories, quantum indistinguishability, amplitudes and probabilities, rules for amplitudes, the path integral for a non-relativistic particle, the free Green's function, inclusion of interactions, the equivalence of the expansion of the amplitude in the number of interactions and the perturbative solution of the Schrödinger equation, the WKB approximation and the classical limit
Interpretations of quantum mechanics: the collapse of the quantum state and unitary evolution, the ontological nature of collapse, the reality of unmeasured quantities, delayed measurements, collapse in relativistic theory, the many-worlds interpretation, hidden variable theories, the Feynman formulation, decoherence

Quantum Field Theory

Propagator of a quantized field: definition and computation of the propagator as a time-ordered contraction, examples of the scalar, Dirac, and Proca fields, the electromagnetic field in a physical non-covariant gauge, the propagator as a Green's function
Covariant quantization of the electromagnetic field: the Gupta-Bleuler method, the propagator in a general covariant gauge, gauge independence of the scattering amplitude at the tree-diagram level
Systematics of the Dyson expansion of the S-matrix in the interaction representation: Wick's theorems, the normal form of the S-matrix operator — expansion into normal products of field operators, normal ordering in the interaction in quantum electrodynamics
Second-order processes in quantum electrodynamics: Compton scattering, elastic scattering of electrons and positrons, muon pair production in electron-positron annihilation, two-photon annihilation of an electron-positron pair
Diagrams with closed internal-line loops: ultraviolet divergences and their regularization: dimensional regularization, the Pauli-Villars method
Divergence index of a one-particle irreducible diagram: derivation of the expression for the divergence index of a 1PI graph in a field theory model with a polynomial-type interaction, the relation to the dimension of the interaction Lagrangian, specifics of models with a massive vector field
Techniques for practical computation of one-loop Feynman diagrams: Feynman parametrization, symmetric integration, basic integral formulas for dimensional regularization
Examples of computable diagrams without ultraviolet divergences: photon-photon scattering and the relevant effective Lagrangian in the low-energy limit, the anomalous magnetic moment of the electron — Schwinger's correction
Basic renormalization techniques: identification of ultraviolet-divergent parts of one-loop graphs, bare and renormalized quantities, counterterms
Types of renormalization counterterms in quantum electrodynamics: vacuum polarization, electron self-energy and vertex correction, renormalization constants Z₁, Z₂, and Z₃, the Ward identity, the on-shell renormalization scheme

Advanced Statistical Physics

Phase transitions: classification of phase transitions, phase coexistence, spontaneous symmetry breaking, the order parameter, the Ising and Heisenberg lattice models, Landau mean-field theory
Critical phenomena and universality: the scaling hypothesis, critical exponents and relations between them, the Landau-Ginzburg-Wilson model, renormalization group theory, universality classes
Complex systems: self-organized critical phenomena, spin glasses, random graphs, the replica method. Diagrammatic methods for many-body quantum systems: time and temperature Green's functions, the Matsubara formalism, the connected cluster theorem, the Schwinger-Dyson equation, analytic properties of Green's functions
Systems of interacting fermions: quasiparticles and their properties, Landau's scattering function, Fermi liquid theory
Theory of superconductivity: Cooper instability, BCS theory, the Nambu formalism, thermodynamic and spectral properties of superconductors
Linear response theory: the Kubo formula, the fluctuation-dissipation theorem, the Kramers-Kronig relations, Onsager reciprocal relations
Many-body quantum systems out of equilibrium: perturbation expansion on a complex-time contour, the equation of motion for the correlation function, the Keldysh-Schwinger and Kadanoff-Baym formalisms
Kinetic equations: reduced densities and the Wigner distribution function, the BBGKY hierarchy of kinetic equations, the Vlasov-Landau mean-field approximation, the Boltzmann equation, the H-theorem
Stochastic processes: time correlation functions, the Markov approximation, detailed balance, the Pauli master equation, Brownian motion and diffusion, the Langevin equation, the Fokker-Planck equation

Plasma Theory and Radiation

High-temperature and thermonuclear plasma: definition, derivation of the Debye length, plasma frequency, polarization drift
Magnetohydrodynamic equilibrium: the MHD model for equilibrium, calculation of radial pressure equilibrium and toroidal force balance for basic plasma configurations (z-pinch, theta-pinch, screw pinch), introduction of the rotational transform
Magnetohydrodynamic stability: the concept of magnetic field line freezing into the plasma, classification of MHD instabilities, calculation of kink, interchange, and ballooning instabilities, the concept of linear stability, reconnection, beam instability
Principles of plasma confinement: inertial confinement, magnetic confinement (Tokamak, Stellarator, Reversed Field Pinch, Spheromak), magnetic structures in stellar atmospheres
Transport in plasma: classical and neoclassical particle transport, the transport barrier, the bootstrap current
Radiative processes: radiation of a moving charge, bremsstrahlung, synchrotron radiation, the plasma radiation mechanism, Compton scattering, scattering from bound particles
Radiation (magneto)hydrodynamics: the geometric optics limit, the kinetic formulation of radiative transfer — the transfer equation, radiation pressure, stellar winds and accretion, shock waves, the passage of radiation through plasma (dispersion, Faraday rotation)
General-relativistic kinetic theory: phase space, Liouville's theorem, equations of motion for charged particles and the electromagnetic field, equations of motion for a continuum, the passage of radiation through a gravitational field (gravitational lenses, cosmic microwave background radiation)
Numerical modeling of plasma: particle models, mesh methods, numerical instabilities

Computational Physics

Matrix factorizations and their use in numerical linear algebra: LU decomposition and its stability, QR decomposition and singular value decomposition of a matrix and their use in solving overdetermined systems and ill-conditioned problems, the use of QR decomposition in finding eigenvalues of a matrix
Iterative methods in numerical linear algebra: linear iterative methods: the Jacobi method, the Gauss-Seidel method, the successive overrelaxation method, their convergence; the multigrid method; the conjugate gradient method, preconditioning; the Jacobi method for finding eigenvalues of real symmetric and Hermitian matrices
Integration of ordinary differential equations: stability and convergence of single-step methods and multi-step linear methods, Dahlquist's theorem, absolute stability and the region of absolute stability, stability of systems of ordinary differential equations, stiff systems, A-stability
Finite difference method for partial differential equations: derivation and error of finite difference formulas, the general s-step scheme for a linear PDE, examples of schemes for basic parabolic and hyperbolic PDEs; the order of the method and consistency, convergence and stability, Lax's theorem; the CFL condition, von Neumann stability analysis
Finite element method for boundary value problems: the weak and variational formulation of problems described by elliptic PDEs, the Ritz and Galerkin methods: discretization using a suitable compact basis, error estimation of the finite element method; illustration on a specific problem
Discrete Fourier transform and its applications: basic properties of the discrete Fourier transform, the Nyquist critical frequency, aliasing; the fast Fourier transform; applications to the computation of convolutions and the solution of differential equations
Fundamentals of the Monte Carlo method: the central limit theorem, error estimation of the Monte Carlo method, applications to integration, generation of random variables with a given distribution function: the Metropolis-Hastings algorithm
Fundamentals of molecular dynamics: equations of motion of a classical many-body system and their numerical integration: the Verlet method, Gear integrators; comparison of methods using integrals of motion, computation of mean values of macroscopic quantities
Fundamentals of quantum simulations: the quantum Monte Carlo method and computation of the ground state of a system using the variational principle, simulations at finite temperature, ab initio numerical simulations