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Homework Statement
Derive the following commutation relations from the general commutation relation for the Lorentz generators:
[tex] [J_i,J_j]=i\hbar\epsilon_{ijk}J_k[/tex]
[tex][J_i,K_j]=i\hbar\epsilon_{ijk}K_k[/tex]
[tex][K_i,K_j]=-i\hbar\epsilon_{ijk}J_k[/tex]
Homework Equations
The commutator for the Lorentz generators:
[tex][M^{\mu\nu},M^{\rho\sigma}]= i\hbar ((g^{\mu\rho}M^{\nu\sigma} - (\mu \leftrightarrow \nu))-(\rho \leftrightarrow \sigma))[/tex][tex]J_i=\frac{1}{2}\epsilon_{ijk}M^{jk}[/tex]
[tex]K_i=M^{i0}[/tex]
The Attempt at a Solution
I've got the first one.
The second two I'm having slight problems and just need help finding my mistake.
For the second commutator, I have an extra factor of 1/2 on the RHS. I start from:
[tex][M^{jk},M^{j0}]= i\hbar ((g^{jj}M^{k0} - (\mu \doublearrow \nu))-(\rho \doublearrow \sigma))[/tex]
Only the first term on the right have side is non zero since all off diagonal g are 0.
Now this implies:
[tex][J_{i},K_j]= \frac{1}{2}\epsilon_{ijk}i\hbar M^{k0}[/tex]
How do I get rid of that pesky 1/2?
Similarly on the third commutator:
I start from the same place and get to the line:
[tex][K_i,K_j]=[M^{i0},M^{k0}]=-i\hbar M^{ij}[/tex]
I can't figure out how to put the RHS in terms of J_k without getting a factor of 2!
Any help will be appreciated. I'm sure its just stupid errors. Thanks!
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