New metastable states in quantum field theory

It is shown that any scalar theory with spontaneous symmetry breaking of a U(1) or SU(2) internal symmetry possesses, besides the usual asymmetric vacuums characterized by a constant value of the vacuum expectation of the field, other classically stable coordinate-dependent periodic field configurations. The quantization of these configurations leads, as in the former case, to the existence of an infinite collection of new metastable ground states for which the internal symmetry is spontaneously broken. In the special case of a U(1) -symmetric theory, which is discussed in detail, it is found to lowest order in perturbation theory that the small oscillations about these configurations consist of a massless excitation (Goldstone boson) and a massive one. Moreover, it is shown that, in spite of the fact that these classical solutions are not displacement invariant, there is no spontaneous breakdown of momentum conservation. The reason for this is that for these solutions any space translation can always be compensated by a U(1) transformation. The generator of this combined transformation can then be identified as the momentum operator, and it is shown to commute with the Hamiltonian. Nevertheless, spontaneous breakdown of Lorentz symmetry does occur and is particularly manifested in the energy-momentum relationship of the one-particle states. This breakdown of Lorentz invariance invalidates the hypothesis of the Goldstone theorem, and therefore owing to renormalization effects the Goldstone boson may acquire a mass. It is also argued that, by a proper choice of an arbitrary parameter which appears in the classical solutions, Lorentz invariance can be restored approximately. It can be shown that these solutions do not exist for groups U(N) with N > 2. Another possible application of these solutions to the description of metastables phases in the Landau-Ginzburg theory of critical phenomena is suggested