- #1
Bobhawke
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Question in the title, ie why is [tex] Tr(T_{a_1}T_{a_2}...T_{a_n}) [/tex] independent of which representation we choose, where the Ts are a matrix representation of the group generators.
The Casimirs are important because they are operators that measure the total energy and momentum of a system. They are used to classify and characterize different representations of a group, which is essential in understanding the symmetries and properties of a physical system.
It means that the values of the Casimirs do not change when the basis of the representation is changed. This is a fundamental property of Casimir operators, and it allows for a consistent and universal way of characterizing the different representations of a group.
The Casimir operators are constructed from the generators of the Lie algebra of a group. They are quadratic combinations of these generators and are used to label the different representations of a group. In this way, the Casimirs are intimately connected to the Lie algebra and are crucial in the study of group representations.
Yes, the Casimir operators can be used to determine the irreducible representations of a group. Each irreducible representation has a unique set of Casimir values, which can be used to identify and classify the representations. This is a powerful tool in the study of group theory.
No, there are other operators that are also independent of the representation, such as the center of the group and the group's identity element. However, the Casimirs are the most commonly used and are essential in understanding the symmetries and properties of a physical system.