✎ Stone-von Neumann Theorem [The Physics Travel Guide]

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====== Stone-von Neumann Theorem ======

<tabbox Why is it interesting?> 
The Stone-von Neumann theorem is important, for example, to understand why symmetry breaking is possible in quantum field theory. 
<tabbox Layman>

<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 Student>

<blockquote> In the algebraic
approach, the fundamental structure of a quantum theory is given by an abstract algebra of the canonical commutation relations, which can
be given various different representations in terms of subalgebras of the
bounded operators on a Hilbert space. Two such representations, each
consisting of a Hilbert space $H$ and the set of bounded operators $\hat O$ defined
on it, are unitarily equivalent if there is a (one-to-one, linear, norm preserving)
map $U:G\to H'$ such that $(\forall i)(U^{-1}\hat{O}'_iU=\hat{O}_i)$. The Stone–von 
Neumann theorem guarantees that in the finite-dimensional case all irreducible
representations of the abstract algebra are unitarily equivalent,
but in the infinite-dimensional case there are unitarily inequivalent representations
of the algebra.<cite>http://publish.uwo.ca/~csmeenk2/files/HiggsMechanism.pdf</cite></blockquote>
 
<tabbox Researcher>

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

--> Example1#

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--> Example2:#

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* A[[https://www.math.umd.edu/~jmr/StoneVNart.pdf| Selective History of the Stone-von Neumann Theorem]] by Jonathan Rosenberg

</tabbox>

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