✎ Casimir Operators [The Physics Travel Guide]

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

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

<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.
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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 $r$.
 
<tabbox Researcher>

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