  We show that superconductors have a thin spectrum associated with spontaneous
symmetry breaking similar to that of antiferromagnets, while still being in
full agreement with Elitzur's theorem, which forbids the spontaneous breaking
of local (gauge) symmetries. This thin spectrum in the superconductors consists
of in-gap states that are associated with the spontaneous breaking of a global
phase symmetry. In qubits based on mesoscopic superconducting devices, the
presence of the thin spectrum implies a maximum coherence time which is
proportional to the number of Cooper pairs in the device. Here we present the
detailed calculations leading up to these results and discuss the relation
between spontaneous symmetry breaking in superconductors and the Meissner
effect, the Anderson-Higgs mechanism and the Josephson effect. Whereas for the
Meissner effect a symmetry breaking of the phase of the superconductor is not
required, it is essential for the Josephson effect.
