# Spontaneous symmetry breaking: How can 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 $Q$ is spontaneously broken, that would means $Q|0\rangle\ne0$, and hence the symmetry transformation transforms continuously $|0\rangle$ 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 $Q$ be a well defined operator which acting on a state (the vacuum) actually gives a state (another "vacuum") out of the space considered?

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It is also helpful to note that the charge operator is in fact not well-defined for infinite volumes and $Q|0\rangle\neq 0$, and in this infinite volume limit all vacua are orthogonal to each other, and hence would be part of an uncountable orthonormal basis, which prohibits them to be part of a signle separable hilbert space.