pt. 1. Symmetry breaking in classical systems. Symmetries of a classical system ; Spontaneous symmetry breaking ; Symmetries in classical field theory ; General properties of solutions of classical field equations ; Stable structures, Hilbert sectors, phases ; Stability under space translations. Positive energy ; Noether theorem and symmetry breaking ; Examples ; The Goldstone theorem ; Appendixes: Properties of the free wave propagator; The Cauchy problem for small times; The global Cauchy problem; The non-linear wave equation with driving team; Time independent solutions defining physical sectors
pt. 2. Symmetry breaking in quantum systems. Quantum mechanics. Algebraic structure and states ; Fock representation ; Non-fock representations ; Mathematical description of infinitely extended quantum systems ; Physically relevant representations ; Cluster property and pure phases ; Examples ; Symmetry breaking in quantum systems ; Examples ; Constructive symmetry breaking ; Symmetry breaking in the Ising model ; Thermal states ; Fermi and bose gas at non-zero temperature ; Breaking of continuous symmetries. Goldstone's theorem ; The Goldstone theorem at non-zero temperature ; The Goldstone theorem for relativistic local fields ; An extension of Goldstone theorem to non-symmetric Hamiltonians ; Symmetry breaking in gauge theories.