Perhaps Theorem 3.11 in this paper.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.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?
Are you just guessing, or did you find an error in the paper that I cited.andrien said:I think it is true only in case of su(n),where it follows follows from U+U=1.
I have no idea what you're talking about, andrien. He says the operator is unitary.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.
It's a unitary operator with a unitary matrix. Unitary means unitary. I see nothing subtle going on.Thus the operator O′= SOS−1 is unitary on S, and its matrix representations will also be unitary.
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 might, if you took a look at the paper I cited in #3 above. The proof is trivial.dextercioby said:I see no reason for it.
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.
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.
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.
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.
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.