Lecture 1 - Review of the basics of complex numbers, geometrical interpretation in terms of the Argand-Wessel plane. DeMoivre's theorem and applications. Branch points and branch cutsLecture 2 - Cauchy-Riemann equations. Holomorphic functions and harmonic functions. Contour integrationLecture 3 - Cauchy's integral theorem. integral formula. Taylor and Laurent series. Singularities. Residue theoremLecture 4 -Applications of the residue theorem. Semi-circular contours. Mouse hole contours. Keyhole integralsLecture 1 - Lie Algebra (vector space, dual basis, matrix representations), the groups Cn and Sn.Lecture 2 - The groups O(n), SO(n), U(n), SU(n); Lie groups and Lie algebras; structure constantsLecture 3 - Lie algebras: the Adjoint representation, compact, simple and semi-simple Lie Algebras; highest-weight representations of su(2).Lecture 4 - The Cartan subalgebra, roots and weights, rank-2 and higher algebrasLecture 1 - Distribuitions, Test functions, Derivatives of distributions, Multiplication of distributions with functionsLecture 2 - Composition of distributions with functions, Orthogonal functions, Sturm-Liouville theory, Parseval's Theorem, Orthogonal polynomialsLecture 3 - Orthogonal polynomials, gamma function, Zeta function, Hypergeometric functionsLecture 4 - Asymptotic Series, Stirling's approximation, Saddle point methodLecture 1 - One-dimensional and multi-dimensional Gaussian integrals. Averages with the gaussian weight, Wick's theoremLecture 2 - Imaginary Gaussian Integrals. Grassman variables, definitions. Gaussian Integrals with Grassman variablesLecture 3 - Functionals and functional derivatives. Euler-Lagrange equationsLecture 4 - Noether's theorem. Functionals and Euler-Lagrange equations for continuous systems; energy-momentum tensorLecture 1 - 1d Boundary value problem, Poisson's equation, Green's identities, Method of imageLecture 2 - Fourier transform, Cauchy problem and diffusion equation, FT in quantum mechanicsLecture 3 - Classical electrodynamics, wave equation, Retarded potentials, Feynman-Wheeler theoryLecture 4 -Scalar field, Functional Integral, Propagator, Degrees of freedom in gauge theoriesLecture 1 - The wavefunction, Momentum and the time-independent Schroedinger equationLecture 2 - Schroedinger equation in 3D and angular momentumLecture 3 - Angular momentum and mixed statesLecture 4 - Quantum state tomography and teleportationStudent Presentations
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Lecture 1 - Maxwell's theory in relativistic notationsLecture 2 - Lorentz transformations, Time dilatation, Length contraction, Spacetime diagrams, E=mc^2Lecture 3 - Relativistic mechanics, Newton's bucket vs. Einstein's elevator, Principle of equivalenceLecture 4 Lecture 4b - Geodesic equation, Newtonian limit, Gravitational redshift, Classical field theory, ManifoldsLecture 7 - De Sitter space, Connection, Torsion, MetricityLecture 8 - Particle in curved spacetime, Parallel transport, Geodesics, Noether symmetriesLecture 9 - Geodesic deviation, Riemann tensor and its symmetries, Weyl tensorLecture 10 - Bianchi identities, Perfect fluid, Newtonian limit, Einstein's equationsLecture 11 - Cosmological principle, Maximally symmetric spaces, FRW metricLecture 12 - Friedmann's equations and their solutions, Einstein's static Universe, Dark matter and dark energyLecture 14 - Light and Particle motion in Schwarzschild geometry, Perihelion precession, Light bendingLecture 15 - Journey into a black hole: from Schwarzschild to KruskalLecture 1 - Motivations and axioms of quantum theoryLecture 2 - Axioms continued. Density matrix. The Schroedinger and the Heisenberg PicturesLecture 3 - Composite and Entangled Systems. Dual SpaceLecture 4 - Subsystems and Partial Trace. Schmidt Decomposition.Lecture 5 - Partial criteria for Entanglement. Von Neumann EntropyLecture 7 - Purification theorem. Measurement. Generalized measurement. Von Neuman's model of indirect measurementLecture 8 - Generalized measurement. Preparations/Measurements/Transformations. Stinespring dilatation theorem.Lecture 9 - Generalized transformations. Classical state update rule. Quantum state update rule.Lecture 10 - Leuder's - von Neuman postulate. Measurement decoherence. Neumark's theorem.Lecture 11 - State update rule under generalized measurement. Lueder's rule. Fault-tolerant threshold theorem. Dirac's idea: Poisson bracket (QM CM)Lecture 12 - Bell's inequality and nonlocality. Einstein-Polosky-Rosen paradox.Lecture 13 - Bell's theorem and nonlocality. Nonlocal games.Lecture 14 - Infinite dimensions (rigged Hilbert space). Path integrals. Continuous-time open system dynamics.Lecture 15 - Concepts in quantum information. Complexity classes. Quantum simulation. Circuit model of quantum computation.Lecture 2 - Hamiltonian quatization of scalar field theory. Vacuum energy.Lecture 3 - Quantization of Klein-Gordon field continued. The Feynman propagator. Noether's theorem.Lecture 4 - Energy-momentum tensor. Lorentz transformation and conserved charges. Interacting field theories. Irrelevant, relevant and marginal couplings. Fundamental assumptions of Perturbation theory.Lecture 5 - Two point function in the interacting theory. Time ordering and the Dyson series. The vacuum.Lecture 6 - Review of week 1. Normal ordering and Wick's theorem. Feynman diagrams and symmetry factors. Feynman rules. Cancelation of vacuum bubbles.Lecture 7 - Review of the previous lecture. Momentum space two point function. Momentum space Feyman rules. One particle irreductible diagrams. Resummation of the perturbation series. Structure of the particles states in the exact theory. Base mass and physical mass.Lecture 8 - Cross sections. The LSZ reduction formula. Dimensional regularization.Lecture 9 - Covariant quantization of Maxwell's theory.Lecture 10 - Feynman rules for scalar nucleon-meson theory. Decay amplitudes. Scalar electrodynamics.Lecture 11 - Left handed and right handed spinor representations of the Lorentz group. The Weyl equations. The Dirac equation. Clifford algebra.Lecture 1 - Introduction to phase transitions, the Ising model, Mean Field TheoryLecture 2 - Critical exponents α, β, γ, δ out of MFT, Hubbard Stratonovich TransformationLecture 3 -Spin-spin correlation function; calculation in the functional integral formalism and in MFTLecture 4 - Calculation of the correlation function in the MFT through Fourier transform; FluctuationsLecture 5 - Corrections in Cv from fluctuations, Ginzburg criterion, Landau-Ginzburg theoryLecture 6 - Wilsonian RG: fast and slow modesLecture 7 - Calculation of the first cumulant; the Gaussian F.p.; Feynman diagramsLecture 8 - Calculation of the 2nd cumulant; Wilson Fisher F.P.; linearized flow around the Gaussian F.P.Lecture 9 - Linearized flow around fixed points; calculation of critical exponents from RGLecture 10 - Mermin-Wagner theorem, lower critical dimensionLecture 11 - Results of 2+ε expansion; Topological order in d=2Lecture 12 - Electrostatic analogy, Duality transformation of the XY modelLecture 1 - Euclidean time, Path integrals, Relation between Euclidean field theory and statistical mechanicsLecture 2 - Operators and correlation functions in the path integral formalism, Free scalar field, Functional integration using spacetime discretizationLecture 3 - Free scalar field propagator, Wick's theoremLecture 4 - Quantization of φ4 theory, COrrelation functionsLecture 5 - Structure of perturbative expansion, Effective actionLecture 6 - One-loop effective action, Kallen-Lehmann spectral representationLecture 7 - Renormalization of φ4 theory at one loop with D=4Lecture 8 - Perturbative renormalization, Beta functionLecture 10 - Wilsonian renormalization of scalar field theory at one loop in the local potential approximationLecture 11 - Grassman variables, Berezin calculus, Fermionic functional integralsLecture 12 - Non-abelian gauge theory, Gauge fixingLecture 13 - Quantization of non-abelian gauge theory, Gauge fixingLecture 15 - Feynman rules for non-abelian gauge theory, Renormalization of non-abelian gauge theoryLecture 1 - Introduction to Condensed Matter, Order and symmetry, Crystal Lattices, symmetries, reciprocal spaceLecture 2 - Reciprocal lattice vectors and first Brillouin zone, Bloch's theorem; Phonons in 1D chains, dispersion, acoustical and optical modesLecture 3 - Quantum mechanics of lattice vibrations, Einstein model, heat capacity at high and low temperaturesLecture 4 - Bose-Einstein condensation for non-interacting bosons, critical temperature. Magnons and spin waves: 1D ferromagnetic Heisenberg chain, quantum mechanical treatmentLecture 5 - Specific heat and thermal dependence of magnetization in ferromagnets. Fermi gas: Fermi energy, density of states and specific heatLecture 6 - Particle hopping on a lattice, multi-band casesLecture 7 - Review of Lagrangian and Hamiltonian formalisms; particle in electromagnetic field; quantum motionsLecture 8 - Classical equations of motion following from a hopping Hamiltonian on a lattice; multi-band caseLecture 9 - Conductivity as follows from quasi-classical approach. Quantum Boltzmann approach, distribution funtion, Hall conductivityLecture 10 - Quantized Hall conductance in insulators, Chern numbers. Introduction to topological classification of gapped phases of non-interacting fermions.Lecture 11 - Non-interacting Fermi gas, properties of metals. Interacting fermions, Fermi-liquid, Hartree-Fock approximationLecture 12 - Landau Fermi-liquid theory; quasiparticles and their lifetime.Lecture 13 - Excitations and specific heat in Landau Fermi-liquid. Classical charged particle in magnetic field; quantum dynamics of an electron in magnetic field.Lecture 14 - Integer Quantum Hall effect: role of disorder and semi-classical percollation picture; edge statesLecture 15 - Fractional Quantum Hall effect, Laughlin wave functions, properties of Quantum Hall states
Lecture 1 - Historical background on particle physicsLecture 4 - Global symmetries of strong interactions and the Eightfold way. Chiral symmetryLecture 5 - Spontaneous symmetry breaking and Goldstone's theoremLecture 6 - Nonabelian gauge theories and Feynman Rules. QCD.Lecture 9 - Semi-leptonic decays in Fermi TheoryLecture 10 - SU(2) X U(1) theory of weak interactionsLecture 11 - Spontaneous symmetry breaking of SU(2) X U(1)Lecture 13 - Unitarity triangle and Higgs productionLecture 2 - Differential Forms, Exterior and Lie DerivativesLecture 3 - Lie Derivative contd, Killing vectors, Connections and Curvature, Cartan's Equations of StructureLecture 4 - Applying Cartan: spherically symmetric, static solutionsLecture 5 - The casual structure of spacetimeLecture 6 - The Einstein-Hilbert action (and beyond)Lecture 7 - Gravity and non-perturbative field theoryLecture 9 - Applications of Gauss-Codazzi, Israer Equations, Gibbons-Hawking TermLecture 10 - Black Hole Thermodynamucs, Euclidean MagicLecture 12 - Higher Dimensional Black Holes: KK Black Holes, Magnetic Monopole, SUGRA SolutionsLecture 13 - Perturbation Theory, Gregory-Laflamme Instability of Black StringLecture 14 - Acceleration Gravity: C-Metric, Cosmic StringsLecture 1 - Basic elements of interferometry, Mach-Zehnder interferometer, Elitzur-Vaidman bomb tester, quantum errasure, Hardy's paradox.Lecture 2 - Axioms for pure state Quantum Theory; No-cloning theorem; Quantum Zeno effectLecture 3 - Quantum optical interferometry, singles modes and coherent states; Hong-Ou-Mandel effect.Lecture 4 - Zou-Wang-Mandel experiment; Einstein's comments at the 1927 Solvay conference; Einstein-Podolsky-Rosen paradoxLecture 5 - The Harrigen-Spekkens classification scheme of ontological models; interpretations flow-chartLecture 6 - The de Broglie-Bohm model; measurements and non-locality of the modelLecture 7 - The many-worlds interpretation of quantum mechanics: axioms, consequences and crtiticismLecture 8 - Collaps models of quantum mechanics, GRW modelLecture 9 - Psi-epistemic models; the ontological excess baggage theoremLecture 10 - Toy models based on the balanced model principle, ContextualityLecture 11 - Epistemic vs. ontic interpretations of the wavefunction and the Pussey-Barrett-Rudolph theorem proving the reality of the wavefunctionLecture 12 - Rob Spekken's approach to non-contextuality; Generalized probability theoriesLecture 13 - Quantum circuits and measurements, the property of convexityLecture 14 - Quantum circuits: tomographic locality and Wooters hierarchy. Reasonable postulates for quantum theoryLecture 15 - The shape of quantum state space and the Fuchs approachLecture 1 - The notion of a quantum phase transition, universality, dynamical critical exponent, types of phase transitions and level crossingLecture 2 - Quantum Ising Moel: spontaneous symmetry breking and dephasing. Transfer matrix method in one dimensionLecture 3 - Mapping between classical and quantum Ising models, scaling limitLecture 4 - The method of duality in the study of 1D quantum Ising model. Orthogonality catastrophe. Fidelity between the states.Lecture 5 - Quantum geometric tensor. quantum XY-model, Berry curvatureLecture 6 - Locality in quantum many-body physics; Lieb-Robinson boundsLecture 7 - Lieb-Robinson bounds: bounds on correlation functions and Lieb-Mattis-Schultz theoremLecture 8 - Quasi-adiabatic connectivity; Lattice gauge theoryLecture 9 - Elitzur's theorem. Quantum phase transitions without symmetry breakingLecture 10 - Quantum lattice Z_2 gauge theory: ground states, dualityLecture 11 - Discrete gauge theory; Kitaev's toric code: states classificationLecture 12 - Kitaev's toric code: string operators, emergent fermionsLecture 13 - Topological order, entanglement, memoryLecture 14 - Topological order, entanglement, memory (continued); EquilibrationLecture 1 - Introduction to perturbative String TheoryLecture 2 - Free scalar field on the world-sheetLecture 5 - The central charge and the Weyl anomalyLecture 6 - BRST quatization of the string, ghostsLecture 7 - BRST quantization contd; the state-operator correspondenceLecture 10 - Strings in background fields; T-dualityLecture 11 - D-branes as sources of closed stringsLecture 13 - Super-particle world line formalism; Introduction to the superstringLecture 3 - FRW Spacetime: Metric, Hubble's Law, Cosmological RedshiftLecture 4 - FRW Spacetime: Horizons, Dynamics, Standard Cosmological ModelLecture 5 - Thermodynamics in expanding spacetimeLecture 8 - Cosmological constant problem, modified gravityLecture 12 - Euclidean trick: temperature of the Sitter horizon, Primordial perturbations InflationLecture 3 - Abelian and Non-abelian anomaliesLecture 5 - Path integral Derivation of the AnomalyLecture 6 - Topological Aspects of Abelian AnomaliesLecture 7 - Topological Aspects of Non-Abelian AnomaliesLecture 8 - Index Theorems and Characteristic ClassesLecture 9 - Index Theorems and Characteristic Classes (continued), Introduction to InstantonsLecture 10 - Introduction to Instantons (continued)Lecture 1 - Introduction to Quantum Gravity, Einstein-Hilbert ActionLecture 2 - Triads, Spin Connection, 3D GravityLecture 3 - Platini Action and its Symmetries, B-F TheoryLecture 4 - Canonical Analysis and Gravity HamiltonianLecture 5 -Systems with first class constraints, Constraint algebraLecture 6 - Parametrized particle and its quantizationLecture 7 - Towards quantizing gravity, SU(2) gymnasticsLecture 8 - Holonomies, Fluxes and their Poisson AlgebraLecture 10 - SU(2) gymnastics: Haar measure Peter-Weyl theoremLecture 13 - Tetrahedra: Solving the flatness constraintLecture 1 - qubits, unitary operations and quantum protocols (superdense coding and teleportation)Lecture 2 - Circuits, reversible computation, and universalityLecture 3 - universality cont., DiVincenzo criteria, nonlinear optics, survey of implementationsLecture 4 - the Church-Turing thesis, efficiency, strong Church-Turing thesis, complexity classes, black boxesLecture 5 - introductory quantum algorithms: Deutch-Jozsa and Simon's problemLecture 6 - the quantum Fourier transform and phase estimationLecture 7 - factoring, RSA, Shor's algorithm and order finding
Lecture 1 - Basics of Neutron Interferometry (NI) without spinLecture 3 - Discussion of Tutorial 1, Incoherence and Decoherence in NILecture 1 - Fermionic systems with quadratic HamiltoniansLecture 2 - Overview of the course; Quantum spin chains; Tensor product and graphical notationLecture 3 - MATLAB session I -- Quantum spin chains, exact diagonalizationLecture 4 - MATLAB session II -- Quantum spin chain by power methodLecture 5 - MATLAB session III -- Diagonalizing quadratic HamiltoniansLecture 6 - Basics of Entanglement: Definition, Schmidt decomposition, measure of entanglement. Shanon, von-Neumann and Renyi entropiesLecture 7 - MATLAB session IV -- Entanglement in quantum spin chainLecture 8 - MATLAB session V -- Entanglement in systems with free fermionsLecture 9 - Boundary law -- scaling of entanglement in systems of free fermions and toy modelLecture 10 - Entanglement as a theoretical tool: universality and entanglement spectrumLecture 11 - Basics of tensor networks -- definitions, notations, computational cost.Lecture 12 - Matrix product states: efficient manipulations.Lecture 13 - Matrix product states: entanglement entropy and ground states of gapped systemsLecture 14 - Multi-scale entanglement renormalization ansatz (MERA) and tree tensor networks (TTN)Lecture 15 - Projected entangled-pair states (PEPS), branching MERA and applications of tensor networksLecture 5 - Elements of string theory: type IIB SUGRA eomLecture 6 - Elements of string theory contd: SUGRA action and black brane solnsLecture 7 - More on Dp-branes, N=4 SYM as the low energy effective theory for D3-branes.Lecture 8 - "Constructing" gauge-string dualityLecture 9 - "Constructing" gauge-string duality contd.Lecture 11 - Scalar field in AdS5 (Euclidean propagator)Lecture 12 - Scalar field in AdS5 (Lorentzian propagator)Lecture 13 - BF bound, normalizable modes, quasinormal spectrumLecture 15 - Holographic calculation of transport coefficientsLecture 1 - Dark Matter: Evidence, hypothesis and challenges.Lecture 2 - FRW universe and equilibrium thermodynamics.Lecture 3 - Departure from equilibrium and the Boltzmann equation.Lecture 4 - Thermal relics, solving the Boltzmann equation and the WIMP miracle.Lecture 5 - Thermal freeze-out, general implications of WIMP models, variations on thermal freeze-out.Lecture 6 - Resonant enhancement and coannihilation. Introductin to direct detection.Lecture 8 - Spin dependent scattering. Experimental status of direct detection.Lecture 11 - Vacuum misalignment. QCD axion as DM.Lecture 12 - Introduction to Baryogenesis. Sakharov conditions.Lecture 1 - Introduction, FRW metric, Perfect fluidsLecture 2 - FRW universe, Newtonian theory of perturbationsLecture 3 - Growth of structure in Newtonian theoryLecture 4 - Metric perturbations and coordinate transformationsLecture 5 - Einstein's equations in a perturbed universe, Boltzmann equationLecture 7 - Boltzmann equation, Baryon acoustic oscillations
