- #1
Einj
- 470
- 59
Hello everyone,
I was wondering if anyone knows what the completeness relation for the fundamental representation of SO(N) is.
For example, in the SU(N) we know that, if [itex]T^a_{ij}[/itex] are the generators of the fundamental representation then we have the following relation
$$
T^a_{ij}T^a_{km}=\frac{1}{2}\left(\delta_{im}\delta_{jk}-\frac{1}{N}\delta_{ij}\delta_{km}\right)
$$
This follows from the fact that the [itex]T^a[/itex], together with the identity form a complete basis for the [itex]N\times N[/itex] complex matrices.
Does anyone know how to find the analogous for SO(N) (if any)?
Thanks a lot!
I was wondering if anyone knows what the completeness relation for the fundamental representation of SO(N) is.
For example, in the SU(N) we know that, if [itex]T^a_{ij}[/itex] are the generators of the fundamental representation then we have the following relation
$$
T^a_{ij}T^a_{km}=\frac{1}{2}\left(\delta_{im}\delta_{jk}-\frac{1}{N}\delta_{ij}\delta_{km}\right)
$$
This follows from the fact that the [itex]T^a[/itex], together with the identity form a complete basis for the [itex]N\times N[/itex] complex matrices.
Does anyone know how to find the analogous for SO(N) (if any)?
Thanks a lot!