Spontaneous symmetry breaking: the vacuum be infinitly degenerate? In classical field theories, it is with no difficulty to imagine a system to have a continuum of ground states, but how can this be in the quantum case? Suppose a continuous symmetry with charge [itex]Q[/itex] is spontaneously broken, that would means [itex]Q|0\rangle\ne 0[/itex], and hence the symmetry transformation transforms continuously [itex]|0\rangle[/itex] into anther vacuum, but how can a separable Hilbert space have a continuum of vacuums deferent from each other? I saw somewhere that says the quantum states are built upon one vacuum, and others simply doesn't belong to it, what does this mean? and then how could [itex]Q[/itex] be a well defined operator which acting on a state (the vacuum) actually gives a state (another "vacuum") out of the space considered?