Doubt in Weinberg's book on Quantum Field Theory In page number 59 of his book on QFT, Weinberg mentions that for the operator $U$, defined for infinitesimal parameters $\omega$ and $\epsilon$ as:
\begin{equation}
U(1+\omega,\epsilon)=1+\dfrac{1}{2}i\omega_{\rho\sigma}J^{\rho\sigma}-\dfrac{1}{2}i\epsilon_{\rho}P^{\rho}+...\tag{2.4.3}
\end{equation}
(eq.(2.4.3)) to be unitary, the operators $J$ and $P$ must be Hermitian. But the Poincaré group is not compact and hence should have no non-trivial unitary representations of finite dimension. Doesn't Weinberg's statement violate this as he is assuming U to be unitary, even though $1+\omega$ and $\epsilon$ both belong to the Poincaré group?
 A: You are right that non compact groups should not have finite dimensional unitary representations. But the generators $J^{\rho\sigma}$ and $P^\rho$ do not act on a finite dimensional vector space here. Using Weinberg notation they act on states $|p,\sigma, n, \ldots\rangle$ and, in particular, $p$ does not take values in a bounded set, the energies are arbitrarily high (indeed $J^{\rho\sigma}$ can boost). Equivalently you can think of these operators in the differential form $P^\rho = -\mathrm{i}\partial^\rho$ and $J^{\rho\sigma} = -\mathrm{i}(x^\rho \partial^\sigma - x^\sigma\partial^\rho)$. In any case they would act on $C^\infty$ functions which live in an infinite dimensional vector space.
Your point might be raised later on when he discusses massless representations. In fact the little group for a massless representation is $\mathrm{ISO}(2)$ (like Poincaré but in two dimensions). Since it is non compact we expect unitary representations to be infinite dimensional, nevertheless it is not the case: the photon and the graviton have two helicities! This is correct because such representations trivialize the action of the translation generators in $\mathrm{ISO}(2)$ (which have nothing to do with actual four dimensional translations by the way), thus effectively reducing it to $\mathrm{SO}(2)$. There is also the possibility of keeping such two dimensional translations and, sure enough, we end up with infinite dimensional representations, which are called continuous spin representations.
