- #1

Hibarikyoya

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- 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?

$$[J^k, J^i] = \frac{1}{M^2}\ (L^{-1})^k_{\mu}\ (L^{-1})^i_{\nu}\ [W^{\mu}, W^{\nu}] $$

$$ = \frac{1}{M^2}\ (L^{-1})^k_{\mu}\ (L^{-1})^i_{\nu}\ (-i) \epsilon^{\mu \nu \rho \sigma} W_{\rho} P_{\sigma}.$$

At this point I had no idea how to going on with the calculation. Can anyone help me?