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theorems:fabri-picasso

# Fabri-Picasso Theorem

## Intuitive

Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.

## Concrete

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. […]

## Abstract

The motto in this section is: the higher the level of abstraction, the better.

## Proof

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