advanced_tools:group_theory:casimir_operators

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+ | ====== 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> | ||

+ | |||

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