====== Casimir Operators ======

<tabbox Why is it interesting?> 

Casimir operators are crucial to understand representations of groups and are often used as labels for [[advanced_notions:elementary_particles|elementary particles]]. 



<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> 

The Casimir operators are those operators that can be built from the generators of a given group that commute with all generators of the group. 
Therefore their value is invariant and can be used to characterize the irreducible representations.

This means in practice that the Casimir operators simply yield a fixed (=invariant) number for each representation that we use to label [[advanced_tools:group_theory:representation_theory|representations]].

There is always a quadratic Casimir operator

\begin{equation}
C_2(r) = T^A T^A \, ,
\end{equation}

where $T^A$ denotes the $d(r) \times d(r)$ matrices that represent the generators in the representation $r$.

The following tables list the quadratic Casimir operators, denotes $C_2(r)$ (and Dynkin indices, denoted d(r)) for the most important representations:

{{ :advanced_tools:group_theory:group-invariants.jpg?nolink |}}
{{ :advanced_tools:group_theory:group-invariants2.jpg?nolink |}}
{{ :advanced_tools:group_theory:group-invariants3.jpg?nolink |}}
 
<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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