⚛️ Introduction to Quantum Field Theory

Learn quantization from classical field theory, canonical quantization of Klein-Gordon and Dirac fields, equal-time commutation and anticommutation relations, and construction of Fock space. Implement description of multi-particle states using creation and annihilation operators and normal ordering concept, and understand applications to excited states in materials (phonons, magnons).

Learn Fourier expansion and mode analysis of free scalar fields, Dirac fields, and electromagnetic fields, and derivation of Feynman propagators and Green functions. Implement causality and iε prescription, Wick rotation and path integrals, Källén-Lehmann representation, and understand applications to lattice vibrations and electron-phonon coupling.

Learn interaction picture and Dyson series, S-matrix definition and causality, Gell-Mann-Low theorem and LSZ reduction formula. Implement calculation of scattering amplitudes and differential cross sections, T-products and time-ordered products, Wick's theorem, and understand applications to electron-electron interactions and Coulomb scattering.

Learn Feynman diagram rules and topology, connected and disconnected diagrams, correspondence between vertices and propagators, and regularization of loop integrals. Implement φ⁴ theory, one-loop calculations in quantum electrodynamics (QED), vacuum polarization and vertex corrections, and understand applications to screening effects in solids and RPA approximation.

Learn the origin and classification of UV divergences, dimensional regularization and minimal subtraction, renormalization group equations and β functions, and construction of effective field theory. Implement Wilson's renormalization group, critical phenomena and phase transitions, and Landau-Ginzburg theory, and understand applications to phase transitions and critical exponents in materials.