Evolution of Correlation Properties and Appearance of Broken Symmetry in the Process of Bose-Einstein Condensation

We consider the self-evolution of strongly non-equilibrium interacting Bose gas. Due to the mere fact of large (as compared to unity) occupation numbers in the initial state the problem is directly reduced to the question of temporal evolution of the statistical matrix diagonal in the coherent-state representation. Strictly speaking, gauge symmetry is not destroyed even when the long-range ordering is completed, owing to the ineviatble averaging over the ensemble. Actual symmetry breaking can occur only as a result of introducing a small term of the Hamiltonian violating conservation of particles, or quantum-mechanical measurement, also implying non-conservation of particles.