Why is it interesting?

Layman

Student

Fabri and Picasso showed in 1966 that if the vacuum state $|0\rangle$ is invariant under translations, it follows that the vacuum state is invariant under the internal symmetry or there is no state corresponding to $\hat Q |0\rangle$ in the Hilbert space of the theory. ($\hat Q $ is the Noether charge that corresponds to the internal symmetry.)

Fabri and Picasso (1966) showed that if the vacuum state $|0\rangle$ is translationally invariant, then the vacuum is either invariant under the internal symmetry, $Q |0\rangle = 0$ , or there is no state corresponding to $Q |0\rangle$ in the Hilbert space. The second case corresponds to SSB. The symmetry is hidden in that there is no unitary operator to map a physical state to its symmetric counterparts; instead, the symmetry is (roughly speaking) a map from one Hilbert space of states to an entirely distinct space. This is usually described as ‘vacuum degeneracy’, although each distinct Hilbert space has a unique vacuum state. […]

http://publish.uwo.ca/~csmeenk2/files/HiggsMechanism.pdf

Researcher

Examples

Example1

Example2:

FAQ

Proof

See the footnote 4 in http://publish.uwo.ca/~csmeenk2/files/HiggsMechanism.pdf