- #1
spookyfish
- 53
- 0
The Lorentz group generators, in any representation, satisfy the commutation relation
[tex]
[S^{\mu \nu}, S^{\rho \sigma}] = i \left( g^{\nu \rho}S^{\mu \sigma} -g^{\mu \rho} S^{\nu \sigma} -g^{\nu \sigma}S^{\mu \rho} +g^{\mu \sigma} S^{\nu \rho} \right)
[/tex]
and the Lorentz transformation is
[tex]
\Lambda=\exp(-i \omega_{\mu \nu} S^{\mu \nu}/2)
[/tex]
My question is: is it possible to prove the formula for the generators (the first formula I wrote), from the definition of the Lorentz group
[tex]
\Lambda^T g \Lambda =g
[/tex]
[tex]
[S^{\mu \nu}, S^{\rho \sigma}] = i \left( g^{\nu \rho}S^{\mu \sigma} -g^{\mu \rho} S^{\nu \sigma} -g^{\nu \sigma}S^{\mu \rho} +g^{\mu \sigma} S^{\nu \rho} \right)
[/tex]
and the Lorentz transformation is
[tex]
\Lambda=\exp(-i \omega_{\mu \nu} S^{\mu \nu}/2)
[/tex]
My question is: is it possible to prove the formula for the generators (the first formula I wrote), from the definition of the Lorentz group
[tex]
\Lambda^T g \Lambda =g
[/tex]