✎ Fabri-Picasso Theorem [The Physics Travel Guide]

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====== Fabri-Picasso Theorem======

<note tip>
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.
</note>
  
<tabbox 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.)

<blockquote>
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. [...]

<cite>http://publish.uwo.ca/~csmeenk2/files/HiggsMechanism.pdf</cite></blockquote>
 
<tabbox Abstract>

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

<tabbox FAQ> 
  
<tabbox Proof> 
See the footnote 4 in http://publish.uwo.ca/~csmeenk2/files/HiggsMechanism.pdf
</tabbox>

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