How Does the Casimir Operator Function in SU(2) Algebra?

In summary, the conversation discusses the definition of Tij, a second order Casimir operator for SU(2), and an error in the calculation for C2. The participants also inquire about the expressions of three generators of SU(2) in terms of a_i and a^\dagger_i.
  • #1
hjlim
4
0
Hi all,

If I define Tij = a+i aj, then

C2 = T11T11 + T12T21 + T21T12 + T22T22 is a second order casimir operator.

For SU(2), it's [tex]\frac{N}{2}[/tex] ([tex]\frac{N}{2}[/tex] + 1)

But as I calculate it directly,
C2 = a+1 a1a+1 a1 + a+1 a2a+2 a1 + a+2 a1a+1 a1 + a+2 a2a+2 a2 =
a+1 a1a+1 a1 + a+1 (a+2a2 + 1)a1 + a+2 (a+1a1 + 1)a2 + a+2 a2a+2 a2 =
N1N1 + N1(N2 + 1) + N2(N1 + 1) + N2N2 = (N1 + N2)2 + N1 + N2 = N(N + 1)

which is different from above. Can you let me know what is wrong with my argument? Thank you very much!
 
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  • #2
How are the three generators of SU(2) expressed in terms of the [tex]a_i[/tex] and [tex]a^\dagger_i[/tex]?

Btw, you have a typo in the next to last term on the first line of the big calculation. The subscript 1 should be a 2. It doesn't influence your calculation.

Torquil
 

What is the Casimir operator in su(2)?

The Casimir operator in su(2) is a mathematical tool used in the study of the special unitary group SU(2). It is a mathematical operator that commutes with all other operators in the group and is used to determine the irreducible representations of the group.

How is the Casimir operator calculated in su(2)?

The Casimir operator in su(2) is calculated using the Lie algebra of the group, which consists of the generators of the group and their commutation relations. The specific formula for calculating the Casimir operator varies depending on the representation of the group being studied.

What is the significance of the Casimir operator in su(2)?

The Casimir operator in su(2) is significant because it allows us to classify the irreducible representations of the group, which are fundamental to understanding the group's structure and properties. It also plays a crucial role in the study of symmetry and group theory in physics.

Can the Casimir operator be applied to other Lie groups?

Yes, the concept of the Casimir operator can be extended to other Lie groups, and it is an essential tool in the study of these groups. However, the specific formula for calculating the Casimir operator will differ for each group.

What are some real-world applications of the Casimir operator in su(2)?

The Casimir operator in su(2) has various applications in physics, particularly in quantum mechanics and particle physics. It is used in the study of atomic and molecular systems, nuclear structure, and the behavior of subatomic particles. It also has applications in other fields, such as crystallography and differential geometry.

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