Rotation and Boost Commutating on the Same Axis

[JordanGo]

I want to prove that:
$$[J_1,G_1] = 0$$
Where J is the rotation operator and G is the boost operator (subscript refers to the axis).

I am using the Jacobi identity:
$$[[J_1,J_2],G_3] = [[G_3,J_2],J_1] +[[J_1,G_3],J_2]$$

Using other identities, I got:
$$[J_3,G_3] = [G_2,J_2] - [G_1,J_1]$$

Now I'm not sure what to do, can someone help?

 

[Bill_K]

You will need to use information specifically about the Lorentz group and these two generators. For example you could write their action as 4 x 4 matrices on (x, y, z, t), and show that these matrices commute.

 

[haael]

I would start from the Poincare algebra:
$$\frac{ 1 }{ i }[M_{\mu\nu}, M_{\rho\sigma}] = \eta_{\mu\rho} M_{\nu\sigma} - \eta_{\mu\sigma} M_{\nu\rho} - \eta_{\nu\rho} M_{\mu\sigma} + \eta_{\nu\sigma} M_{\mu\rho}$$

Then the boosts are: (μ = 1, 2, 3)
$$G_{\mu} = M_{0\mu}$$

And the rotations: (μ, ρ, σ = 1, 2, 3; ρ ≠ σ; ρ ≠ μ; σ ≠ μ)
$$J_{\mu} = M_{\rho\sigma}$$

You will prove the commutativity knowing that certain components of the metric tensor are zero and the generator of the Lorentz transformations is antisymmetric.

 

[andrien]

I want to prove that:
$$[J_1,G_1] = 0$$
Where J is the rotation operator and G is the boost operator (subscript refers to the axis).

That is pretty simple,if you are allowed to use the commutation relation [J_i,G_j]=iε_ijkG_k,for i=j=1.you have ε_11k term,which is of course zero.(you can get this commutation using above post)