I am trying to derive the algebra and I get a factor of 2 wrong...(adsbygoogle = window.adsbygoogle || []).push({});

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]

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# Lorentz commutator

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