# A question about a commutation relation

Here $a$, $b$,$c$, $d$ $u$, $v$ range from 0 to 4 and the metric $g^{ab}=\text{diag}(-1,1,1,1)$.

16 $4 \times 4$ matrices $M^{ab}$ are defined as follows: \begin{equation*} (M^{ab})_{uv} = -i(\delta^a_u\delta^b_v-\delta^b_u\delta^a_v) \end{equation*}

Then equivalently, \begin{equation*} (M^{ab})^{u}\;_{v} = -i(g^{ua}\delta^b_v-g^{ub}\delta^a_v) \end{equation*}

What is the meaning of these matrices? And these matrices are said to have the following commutation relation: \begin{equation*} [M^{ab}, M^{cd}]=i(g^{ac}M^{bd}-g^{ad}M^{bc}-g^{bc}M^{ad}+g^{bd}M^{ac}) \end{equation*}

How can I efficiently prove this relation? All I can think of is to compare the both side by components. Could anyone please help me with proving this relation?

The matrices $M^{ab}$ span a 4 dimensional representation of the Lorentz algebra $so(1,3)$. The commutation relation you want to prove is the defining property of the Lorentz algebra. This means that you can use these matrices to generate Lorentz transformations via exponantiation.
• I think $(M^{ab})^u_v$ expression would be more convenient for this. Right? – Keith Mar 21 '18 at 8:31