User Tools

Site Tools


advanced_tools:group_theory:casimir_operators

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Previous revision
advanced_tools:group_theory:casimir_operators [2017/12/17 11:53]
advanced_tools:group_theory:casimir_operators [2017/12/17 12:53] (current)
jakobadmin [Student]
Line 1: Line 1:
 +====== 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> ​
 +
 +--> Example1#
 +
 + 
 +<--
 +
 +--> Example2:#
 +
 + 
 +<--
 +
 +<tabbox FAQ> ​
 +  ​
 +<tabbox History> ​
 +
 +</​tabbox>​
 +