L^{\mu\nu} contains the three generators of rotation J^i and the three generators of boosts K^i.
They form the Poincaré generators together with the four translation operators.
<br />
\begin{aligned}<br />
& P^\mu &=~& -i\Big( ~~~~~~~~~-\frac{\partial}{\partial x^\mu} &\Big)&<br />
~~~\mbox{4 translation generators} \\<br />
& J^i &=~& -i\Big(\, ~~~~x^j\frac{\partial}{\partial x^k}-x^k\frac{\partial}{\partial x^j} &\Big)&<br />
~~~\mbox{3 rotation generators}~~~~~~ \\<br />
& K^i &=~& -i\Big( ~ - x^i\frac{\partial}{\partial x^o}-x^o\frac{\partial}{\partial x^i} &\Big)&<br />
~~~\mbox{3 boost generators}~~~~~~<br />
\end{aligned}<br />
In the image below you can see how they work. The \delta here is an infinitesimal small parameter.
You can for instance translate an arbitrary function over an infinitesimal small distance by
subtracting \delta\partial f/\partial x (The red and blue delta functions)
If you repeatedly apply the (1-\delta\partial_x f) operator then this amounts to an exponential function like
the one in your book. To translate over a distance \ell_x to the left you do:
<br />
\exp\left(\,i\ell_x P^x\right)\,f(x) ~=~<br />
\left\{1 +<br />
\frac{\ell_x }{1!}\,\frac{\partial }{dx } +<br />
\frac{\ell_x^2}{2!}\,\frac{\partial^2}{dx^2} +<br />
\frac{\ell_x^3}{3!}\,\frac{\partial^3}{dx^3} +<br />
\cdots\right\}f(x)<br />
The right hand side is just the standard Taylor series. If we write (x-a) for the displacement \ell_x and
let the operators act on f(a) then we get the familiar expression for the Taylor series.
<br />
f(a)+\frac {f'(a)}{1!} (x-a)+ \frac{f''(a)}{2!} (x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+ \cdots<br />
The rotate and boost operators work in just the same way. The matrix \epsilon_{\mu\nu} contains the three angles
by which you want to rotate and the three rapidities by which you want to boost just like \ell_x is the
distance by which you want to translate.
The above rotates/boosts works on a scalar field, that is they handle the coordinate transformation.
If you want to transform a (four) vector field then have to operate on the (four) vector parameters
as well because the vector transforms under a general Lorentz transformation.
Hans