Why the generator operators of a compact Lie algebra are Hermitian?

In summary, the conversation discusses the topic of generator matrices of a compact Lie algebra and their relation to Hermitian matrices. The Peter-Weyl theorem is mentioned as a possible explanation, and further discussion delves into the concept of unitary representations and their connection to hermitian generators. Ultimately, the question of why generator matrices of a compact Lie algebra are Hermitian is posed, with the possibility of finding an answer in a cited paper. The conversation ends with a clarification on the terminology used and a suggestion to look into the Peter-Weyl theorem for further understanding.
  • #1
ndung200790
519
0
Why generator matrices of a compact Lie algebra are Hermitian?I know that generators of adjoint representation are Hermitian,but how about the general representaion of Lie groups?
 
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  • #3
ndung200790 said:
Why generator matrices of a compact Lie algebra are Hermitian?I know that generators of adjoint representation are Hermitian,but how about the general representaion of Lie groups?
Perhaps Theorem 3.11 in this paper.
 
  • #4
ndung200790 said:
Why generator matrices of a compact Lie algebra are Hermitian?I know that generators of adjoint representation are Hermitian,but how about the general representaion of Lie groups?
I think it is true only in case of su(n),where it follows follows from U+U=1.
 
  • #5
andrien said:
I think it is true only in case of su(n),where it follows follows from U+U=1.
Are you just guessing, or did you find an error in the paper that I cited.
 
  • #6
Well,the theorem given is talking about unitary representation of some operator representation of some compact group G.This unitary representation is different from the unitarity of operator.A unitary representaion simply concerns about finding an orthogonal set of basis in the group space,so I don't see a direct connection of that theorem with the unitarity which leads to hermitian generator.Probably you know how to make a connection.
 
  • #7
andrien said:
Well,the theorem given is talking about unitary representation of some operator representation of some compact group G.This unitary representation is different from the unitarity of operator.A unitary representaion simply concerns about finding an orthogonal set of basis in the group space,so I don't see a direct connection of that theorem with the unitarity which leads to hermitian generator.Probably you know how to make a connection.
I have no idea what you're talking about, andrien. He says the operator is unitary.

Thus the operator O′= SOS−1 is unitary on S, and its matrix representations will also be unitary.
It's a unitary operator with a unitary matrix. Unitary means unitary. I see nothing subtle going on.
 
  • #8
ndung200790 said:
Why generator matrices of a compact Lie algebra are Hermitian?I know that generators of adjoint representation are Hermitian,but how about the general representaion of Lie groups?

You must first distinguish between the matrices forming a group, forming a Lie algebra, forming the matrix elements of a linear operator representing a group/Lie algebra on some (topological) vector space. With this being said, let's try to see what can be understood from your first question: so you've got a compact Lie algebra of matrices and I assume the 'generator matrices' of it are the basis vectors of the linear space underlying the algebra. Why would they necessarily be Hermitean ? I see no reason for it.
 
  • #9
dextercioby said:
I see no reason for it.
You might, if you took a look at the paper I cited in #3 above. The proof is trivial.
 
  • #10
His 1st question was not related to representations. Surely, the irreds of a compact Lie group are equivalent to unitary ones as a result of Peter-Weyl's theorem. We use this result in Quantum Mechanics. That's why I wrote the first 2 sentences. Because I was suspecting a wrong terminology.
 

1. Why are the generator operators of a compact Lie algebra Hermitian?

The generator operators of a compact Lie algebra are Hermitian because they are related to the rotation and symmetry operations of a physical system. Hermitian operators are required for these operations to preserve the inner product, which is essential for the consistency of physical laws.

2. What is the significance of compactness in relation to the Hermitian property of generator operators?

The compactness of a Lie algebra is related to the compactness of the Lie group it represents. Compact Lie groups have a finite size and are bounded, which ensures that their generator operators are also bounded and therefore Hermitian.

3. How are Hermitian generator operators related to the unitarity of a physical system?

The Hermitian property of generator operators is a necessary condition for the unitarity of a physical system. Unitarity is the requirement for the total probability of a system to remain constant over time, and Hermitian operators ensure that this is the case.

4. Are all generator operators of a compact Lie algebra Hermitian?

No, not all generator operators of a compact Lie algebra are Hermitian. This property is only true for those generator operators that correspond to the continuous symmetries of a physical system. There may be additional generator operators that are not Hermitian, but they are not relevant to the physical symmetries.

5. How does the Hermitian property of generator operators relate to the conservation of physical quantities?

The Hermitian property of generator operators is related to the conservation of physical quantities through Noether's theorem. This theorem states that for every continuous symmetry of a physical system, there exists a corresponding conserved quantity. The Hermitian property ensures that the generator operators associated with these symmetries are also conserved, leading to the conservation of physical quantities.

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