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 advanced_tools:group_theory:casimir_operators [2017/12/17 11:53] advanced_tools:group_theory:casimir_operators [2017/12/17 12:53] (current)jakobadmin [Student] Both sides previous revision Previous revision 2017/12/17 12:53 jakobadmin [Student] 2017/12/17 12:51 jakobadmin [Student] 2017/12/17 12:05 jakobadmin created Previous revision 2017/12/17 12:53 jakobadmin [Student] 2017/12/17 12:51 jakobadmin [Student] 2017/12/17 12:05 jakobadmin created Line 1: Line 1: + ====== Casimir Operators ====== + +  ​ + + Casimir operators are crucial to understand representations of groups and are often used as labels for [[advanced_notions:​elementary_particles|elementary particles]]. ​ + + + +  ​ + + + 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. + ​ + ​ +  ​ + + 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 + + + C_2(r) = T^A T^A \, , + + + 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 |}} + +  ​ + + + The motto in this section is: //the higher the level of abstraction,​ the better//. + ​ + + ​ +  ​ + + --> Example1# + + + <-- + + --> Example2:# + + + <-- + +  ​ + ​ +  ​ + + ​ +