- #1
spookyfish
- 53
- 0
I am trying to derive the algebra and I get a factor of 2 wrong...
Consider the Lorentz group elements near the identity
[tex]
\Lambda_1^\mu\,_\nu = \delta^\mu\,_\nu + \omega_1^\mu\,_\nu, \quad \Lambda_2^\mu\,_\nu = \delta^\mu\,_\nu + \omega_2^\mu\,_\nu
[/tex]
and write a representation as
[tex]
U(\Lambda)=U(1 +\omega)=1-\frac{1}{2}i\omega^{\mu \nu} M_{\mu \nu}
[/tex]
where [itex]M[/itex] is a generator. Now the term [itex]\Lambda \equiv (\Lambda_2^{-1}\Lambda_1^{-1}\Lambda_2 \Lambda_1) [/itex] belongs to the group and up to 2nd order is [itex] 1+[\omega_2,\omega_1] [/itex].
So its representation is
[tex]
U(\Lambda)=U(1+[\omega_2,\omega_1])=1-\frac{1}{2}i[\omega_2,\omega_1]^{\mu \nu} M_{\mu \nu}
[/tex]
I know this is wrong and I am supposed to get
[tex]
1-i[\omega_2,\omega_1]^{\mu \nu} M_{\mu \nu}
[/tex]
Consider the Lorentz group elements near the identity
[tex]
\Lambda_1^\mu\,_\nu = \delta^\mu\,_\nu + \omega_1^\mu\,_\nu, \quad \Lambda_2^\mu\,_\nu = \delta^\mu\,_\nu + \omega_2^\mu\,_\nu
[/tex]
and write a representation as
[tex]
U(\Lambda)=U(1 +\omega)=1-\frac{1}{2}i\omega^{\mu \nu} M_{\mu \nu}
[/tex]
where [itex]M[/itex] is a generator. Now the term [itex]\Lambda \equiv (\Lambda_2^{-1}\Lambda_1^{-1}\Lambda_2 \Lambda_1) [/itex] belongs to the group and up to 2nd order is [itex] 1+[\omega_2,\omega_1] [/itex].
So its representation is
[tex]
U(\Lambda)=U(1+[\omega_2,\omega_1])=1-\frac{1}{2}i[\omega_2,\omega_1]^{\mu \nu} M_{\mu \nu}
[/tex]
I know this is wrong and I am supposed to get
[tex]
1-i[\omega_2,\omega_1]^{\mu \nu} M_{\mu \nu}
[/tex]