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--- advanced_tools:group_theory:casimir_operators [2017/12/17 12:05]
jakobadmin created
+++ advanced_tools:group_theory:casimir_operators [2017/12/17 12:53]
jakobadmin [Student]
@@ Line 15: / Line 15: @@
 <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.
+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 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>

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