Prove the following commutation relations

[Hibarikyoya]

Homework Statement

I have to prove that:

$$[J^k, J^i] = i \epsilon^{kij} J^j$$

.

Relevant Equations

where $$J^k = \frac{1}{M} (L^{-1})^k_{\mu} W^{\mu},$$ in which M is a real number (with the meaning of a mass), $L(\vec{P},M)$ is a Lorentz transformation (in particular a boost in the direction identified by the momentum $\vec{P}$). I can also provide the explicit form of this transformation, but I think is not needed for this exercise. $W^{\mu}$ is the Pauli-Lubanski four vector. Moreover k, i and j run from 1 to 3 and they are spatial indices, while the greek indices run from 0 to 3

I tried in this way:
$$[J^k, J^i] = \frac{1}{M^2}\ (L^{-1})^k_{\mu}\ (L^{-1})^i_{\nu}\ [W^{\mu}, W^{\nu}] $$
$$ = \frac{1}{M^2}\ (L^{-1})^k_{\mu}\ (L^{-1})^i_{\nu}\ (-i) \epsilon^{\mu \nu \rho \sigma} W_{\rho} P_{\sigma}.$$
At this point I had no idea how to going on with the calculation. Can anyone help me?

 

[anuttarasammyak]

Dealing with
$$\mathbf{J}=\mathbf{r}\times\mathbf{p}$$
is a primitive way. Do you like to solve the problem in an advanced way ?

 

[vanhees71]

Hint: What you are supposed to prove is that the Pauli-Lubanski vector provides the generators (Lie-algebra basis elements) of the little group for the massive representations, where indeed the little group is the SO(3) (represented by representations of the SU(2), i.e., its covering group in QT).

Obviously the idea is to consider the Pauli-Lubanski-vector operators on the subspace of fixed three-momentum $\vec{P}$. The boost transforms to the rest frame of the particle, where $P^{\mu}/M=(1,0,0,0)$, and where $W^{\mu}=(0,\vec{W})$.

In other words: For your prove you can simply work in the restframe of the particle and the three spatial components of $\vec{W}$ wrt. this frame.

 

[Hibarikyoya]

Hint: What you are supposed to prove is that the Pauli-Lubanski vector provides the generators (Lie-algebra basis elements) of the little group for the massive representations, where indeed the little group is the SO(3) (represented by representations of the SU(2), i.e., its covering group in QT).

Obviously the idea is to consider the Pauli-Lubanski-vector operators on the subspace of fixed three-momentum $\vec{P}$. The boost transforms to the rest frame of the particle, where $P^{\mu}/M=(1,0,0,0)$, and where $W^{\mu}=(0,\vec{W})$.

In other words: For your prove you can simply work in the restframe of the particle and the three spatial components of $\vec{W}$ wrt. this frame.

Very nice idea. The professor showed us his solution and it was a mess