to proceed even further, to obtain the explicit generators. By considering an infinitesimal transformation with $U(\Lambda) =
{\ensuremath{\mathbb I}}+ {\ensuremath{\frac{i}{2}}}\omega_{\mu\nu}{{\ensuremath{\mathbb M}}}^{\mu\nu}$ we may express Eq. (\[e:arule\]) as $$\begin{aligned}
 &&U(\Lambda)a({\ensuremath{\textbf p}},s)U^\dag(\Lambda)\\
 &=& a({\ensuremath{\textbf p}}, s) + {\ensuremath{\frac{i}{2}}}\omega_{\mu\nu}[{{\ensuremath{\mathbb M}}}^{\mu\nu}, a({\ensuremath{\textbf p}}, s)]\\
&=& ({\ensuremath{\mathbb I}}+ \omega^0\,_i\frac{p^i}{2p^0})\biggl({\ensuremath{\mathbb I}}-i\bigl[\theta^i +
\omega_{0}\,^k\frac{\epsilon_{ijk}{p^j}}{p^0 + m}\bigr]J^i\biggr)_{s s'}({\ensuremath{\mathbb I}}+
\omega^i_\mu p^\mu\frac{\partial}{\partial p^i})a({\ensuremath{\textbf p}}, s') \label{e:44}\\
   &=& \biggl[ {\ensuremath{\mathbb I}}-i\omega_{0i} \biggl( \underbrace{
     \frac{\epsilon_{ijk}J^j p^k}{p^0+m} + p^0\frac{\partial}{i\partial p^i}
     + \frac{p^i}{2ip^0} }_{{\ensuremath{\mathbb K}}^i}  \biggl)
     - {\ensuremath{\frac{i}{2}}}\omega_{ij} \epsilon^{ijk} \biggr( \underbrace{ J^k
     + \epsilon^{klm}p^l \frac{\partial}{i\partial p^m} }_{{\ensuremath{\mathbb J}}^k} \biggr)
       \biggr]_{s s'}a({\ensuremath{\textbf p}},s') \\
  \label{e:ainvlo}
  &=&  a({\ensuremath{\textbf p}},s) - {\ensuremath{\frac{i}{2}}}\omega_{\mu\nu}\left(\int d^3\,{\ensuremath{\textbf p}}'\,{\ensuremath{\mathbb M}}^{\mu\nu}_{({\ensuremath{\textbf p}},s)({\ensuremath{\textbf p}}',s')}
  a({\ensuremath{\textbf p}}', s')\right)\end{aligned}$$ In terms of creation and annihilation operators, the explicit [*infinite-dimensional Hermitian*]{} generators of [*unitary*]{} transformations $U(\Lambda) = \exp({\ensuremath{\frac{i}{2}}}\omega_{\mu\nu}{{\ensuremath{\mathbb M}}}^{\mu\nu})$ for the [*non-compact*]{} Lorentz group are $$\begin{aligned}
\epsilon^{ijk}{{\ensuremath{\mathbb M}}}_{jk} &=& \int d^3{\ensuremath{\textbf p}}\,a^\dag({\ensuremath{\textbf p}},s)\biggr( J^i
+\epsilon_{ijk}p^j\frac{\partial}{i\partial{p^k}}\biggr)_{ss'}a({\ensuremath{\textbf p}},s'),\\
{{\ensuremath{\mathbb M}}}^{0i} &=& \int d^3{\ensuremath{\textbf p}}\,
a^\dag({\ensuremath{\textbf p}},s)\biggr(\frac{\epsilon^{ijk}{J_j}p_k}{p^0+m} +
p^0\frac{\partial}{i\partial {p^i}} +
  \frac{p^i}{2ip^0}\biggr)_{ss'}a({\ensuremath{\textbf p}},s').\end{aligned}$$ In general, we may also introduce the particle species label $n_i$ for the creation $a^\dagger({\ensuremath{\textbf p}},s,n_i)$ and annihilation $a({\ensuremath{\textbf p}},s,n_i)$ operators; and the expression of the generators will then include summing over all $n_i$.

It can be verified that the explicit generators $$\begin{aligned}
  \vec{{\ensuremath{\mathbb J}}} &=& 