====== Unitary Representation ======

<tabbox Why is it interesting?> 

To make sense of the probabilistic interpretation of quantum theory, we need to use unitary representations (see e.g. page 69 in [[http://isites.harvard.edu/fs/docs/icb.topic521209.files/QFT-Schwartz.pdf|Schwartz's book]]). Expressed differently, we are interested in representations of given groups on the Hilbert space in a quantum field theory.

Many important groups are non-compact (e.g. the Poincare group and the conformal group) and there is a theorem that tells us that all unitary representations of a non-compact group are infinite-dimensional.

"//The infinite-dimensional representations are
considered unphysical because we never see particle states in nature labelled by extra continuous
parameters.//" http://homepages.uc.edu/~argyrepc/cu661-gr-SUSY/susy2001.pdf

Thus it is often a non-trivial problem to find make sense of the unitary representations of such groups.


<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> 
<WRAP tip> A unitary representation is defined as a representation where all generators $G$ act as hermitian matrices: $$ G^\dagger G= 1$$ </WRAP>
 
<tabbox Researcher> 

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

  
<tabbox Examples> 

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<tabbox FAQ> 
  
<tabbox History> 

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