Find Generators of Lorentz Group: Jackson's List & K Matrix

[eoghan]

Hi!
I'm trying to find the generators of the Lorentz group. Jackson lists them all, for example, the generator of a boost along x is:
$$K=\left( \begin{array}{c}<br /> 0\;1\;0\;0 \\<br /> 1\;0\;0\;0 \\<br /> 0\;0\;0\;0 \\<br /> 0\;0\;0\;0 <br /> \end{array} \right)$$

Now, what I don't understand is: this matrix is a covariant, contravariant, or mixed tensor? I mean, should I write
$$K_{\mu\nu}\:\:,\:\:K_{\mu}\;^{\nu}\:\:,\:\:K^{\mu \nu}...$$ or what else?

 

[Dick]

A Lorentz transformation takes a vector to a vector. A generator is the derivative of that with respect to a parameter in the transformation. Wouldn't that make it a mixed tensor? Like $x'^\mu = T_\nu^\mu x^\nu$.