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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 12:51] jakobadmin [Student] |
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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. | + | 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 $r$. | ||
<tabbox Researcher> | <tabbox Researcher> |