Energies and collapse times of symmetric and symmetry-breaking states of finite systems with a U(1) symmetry

We study quantum systems of volume V, which will exhibit the breaking of a U(1) symmetry in the limit of V \to \infty, when V is large but finite. We estimate the energy difference between the `symmetric ground state' (SGS), which is the lowest-energy state that does not breaks the symmetry, and a `pure phase vacuum' (PPV), which approaches a symmetry-breaking vacuum as V \to \infty. Under some natural postulates on the energy of the SGS, it is shown that PPVs always have a higher energy than the SGS, and we derive a lower bound of the excess energy. We argue that the lower bound is O(V^0), which becomes much larger than the excitation energies of low-lying excited states for a large V. We also discuss the collapse time of PPVs for interacting many bosons. It is shown that the wave function collapses in a microscopic time scale, because PPVs are not energy eigenstates. We show, however, that for PPVs the expectation value of any observable, which is a finite polynomial of boson operators and their derivatives, does not collapse for a macroscopic time scale. In this sense, the collapse time of PPVs is macroscopically long.