(adsbygoogle = window.adsbygoogle || []).push({}); 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, buthow 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? andthen how could [itex]Q[/itex] be a well defined operatorwhich acting on a state (the vacuum) actually gives a state (another "vacuum") out of the space considered?

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# Spontaneous symmetry breaking: How can the vacuum be infinitly degener

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