Spontaneous Symmetry Breaking, Off-diagonal Long-range Order, and Nucleation of Quantum State

Spontaneous symmetry breaking originats in quantum mechanical measurement of the relevant observable defining the physical situation, order parameter is the average of this observable. A modification is made on the random-phase postulate validating the ensemble description. Off-diagonal long-range order, macroscopic wavefunction and interference effects in many-particle systems present when there is a so-called nucleation of quantum state, which is proposed to be the origin of spontaneous gauge symmetry breaking, for which nonconservation of particle number N is not essential. The approach based on nonvanishing expectation of the field operator, $<\hat{\psi}(\vec{r})>$, is only a coherent-state approximation in thermodynamic limit. When $N \to \infty$, this approach is equivalent, but $<\hat{\psi}(\vec{r})>$ is not the macroscopic wavefunction.