$$\left[J_i,J_j \right]=i\epsilon_{ijk}J_k$$
$$\left[J_i,M_j \right]=i\epsilon_{ijk}M_k$$
$$\left[M_i,M_j \right]=-i\epsilon_{ijk}J_k$$
where $J_i$ is the generator of rotation of Lorentz group, $M_i$ is the generator of boost of Lorentz group
In many textbook of QFT, they say that the second one implies that the generators of the boosts transform like a vector under rotations. But I can't see it explicitly. Can anyone give
me the explanation.
Best Answer
Vectors transform linearly, \begin{equation} x _i \rightarrow A _{ ij } x _j \end{equation} through some transformation matrix $A$.
Now consider the transformation of $ M _i $: \begin{align} e ^{ i \theta _j J _j } M _i e ^{ - i J _j \theta _j } & = \left( 1 + i \theta _j J _j - ... \right) M _i \left( 1 - i \theta _j J _j - ... \right) \\ & = M _i + i \theta _j \left[ J _j , M _i \right] + ... \end{align} Now if $ \left[ J _j , M _i \right] $ is only proportional to $ M _j $ as above then infinitesimally, \begin{align} e ^{ i \theta _j J _j } M _i e ^{ - i J _j \theta _j } & = M _i + i \theta _j \epsilon _{ jik} M _k \\ &= (\delta_{ik}+i\theta_j\epsilon_{jik})M_k\\ &= A _{i,j}M _j \end{align} for some transformation matrix $A$ as required.