How to Calculate Structure Constants of SU(N) Using Kroenecker Deltas?

In summary, the conversation discusses the use of Kroenecker deltas and structure constants in SU(N) and how to exploit them. The conversation also mentions useful identities, such as the Jacobi identities, and asks for book references for these identities and the contraction formula for structure constants in SU(N).
  • #1
Lester
33
0
Hi there,

Does anybody know how to exploit the product of structure constants of SU(N) through Kroenecker deltas? I mean

[tex]\sum_a f_{abc}f_{ade}[/tex]

I know this for SU(2) as in this case I have the Levi-Civita symbol but in other cases I was not able to recover it in literature. Any help appreciated.

Jon
 
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  • #2
Lester said:
Hi there,

Does anybody know how to exploit the product of structure constants of SU(N) through Kroenecker deltas? I mean

[tex]\sum_a f_{abc}f_{ade}[/tex]

I know this for SU(2) as in this case I have the Levi-Civita symbol but in other cases I was not able to recover it in literature. Any help appreciated.

Jon

It is given in terms of the totally symmetric coefficients [itex]d_{abc}[/itex] which vanish in SU(2);

[tex]f_{abe}f_{cde} = \frac{2}{n} (\delta_{ac}\delta_{bd} - \delta_{ad}\delta_{bc}) + ( d_{ace}d_{bde} - d_{bce}d_{ade})[/tex]

Another useful identities are (Jacobi identities)

[tex]f_{abe}d_{ecd} + f_{cbe}d_{aed} + f_{dbe}d_{ace} = 0[/tex]

and the usual one for the structure constants [itex]f_{abc}[/itex]


regards

sam
 
  • #3
Thanks a lot Sam. This was the formula I was looking for.

Jon
 
  • #4
Dear Sam & Lester

Can you please tell me any book reference for these identities involving $d^{abc}$ in jacobi identity and the structure constants contraction.

Thanks in advance
 
  • #5
Dear samalkhaiat

Your formula for jacobi identity is wrong. Actually it should have all d^{abc} instead of the f^{abc} everywhere. The correct Jacobi identity is:

[tex]d^{ace}d^{bde}+d^{ade}d^{bce}+d^{bae}d^{cde}=0[/tex]

Kindly provide me any references for contraction formula for structure constants of SU(N).

Thanks.
 

FAQ: How to Calculate Structure Constants of SU(N) Using Kroenecker Deltas?

1. What are the structure constants of SU(N)?

The structure constants of SU(N) are numerical values that describe the algebraic structure of the special unitary group SU(N). They quantify the commutation relations between the generators of the group and are used to study the group's representation theory.

2. How are the structure constants of SU(N) calculated?

The structure constants of SU(N) can be calculated through the use of the Cartan-Weyl basis, which is a set of basis elements for the Lie algebra of the group. These basis elements are used to construct the structure constants through the use of certain mathematical formulas and techniques.

3. What is the relationship between the structure constants of SU(N) and Lie algebra?

The structure constants of SU(N) are closely related to the Lie algebra of the group. They are used to define the structure of the Lie algebra and are a fundamental part of its representation theory. The structure constants can also be used to calculate the commutators and other important algebraic properties of the Lie algebra.

4. How do the structure constants of SU(N) impact the study of physics?

The structure constants of SU(N) have a significant impact on the study of physics, particularly in the field of quantum mechanics. They are used in the study of gauge theories, which are fundamental to our understanding of the interactions between elementary particles. The structure constants also play a role in the formulation of grand unified theories.

5. Are the structure constants of SU(N) the same for all values of N?

No, the structure constants of SU(N) are different for each value of N. This is because the structure of the group changes as N increases, leading to different commutation relations and different structure constants. However, there are certain patterns and relationships between the structure constants of different N values that can be studied and utilized in the analysis of the group.

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