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advanced_tools:group_theory:casimir_operators [2017/12/17 11:53] |
advanced_tools:group_theory:casimir_operators [2017/12/17 12:51] jakobadmin [Student] |
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+ | ====== Casimir Operators ====== | ||
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+ | <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]]. | ||
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+ | |||
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+ | <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> | ||
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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. | ||
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+ | 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]]. | ||
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+ | 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> | ||
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+ | <note tip> | ||
+ | The motto in this section is: //the higher the level of abstraction, the better//. | ||
+ | </note> | ||
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+ | | ||
+ | <tabbox Examples> | ||
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+ | --> Example1# | ||
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+ | |||
+ | <-- | ||
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+ | --> Example2:# | ||
+ | |||
+ | |||
+ | <-- | ||
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+ | <tabbox FAQ> | ||
+ | | ||
+ | <tabbox History> | ||
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+ | </tabbox> | ||
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