Interacting representation of the Poincaré group In his QFT book, Weinberg claims what follows (Vol I, pag. 144-145): given a (free) representation of the Poincaré group with generators $\bf P$ (spatial translations), $H_0$ (time translations), $\bf J$ (rotations), and ${\bf K}_0$ (boosts), one obtains another (interacting) reprentation by replacing $H_0$ and ${\bf K}_0$ with 
$$
H = H_0 + \int \mathcal{H}({\bf x}, 0) d^3x \; \; \text{ and } \; \; {\bf K} =  {\bf K}_0 - \int {\bf x} \mathcal{H}({\bf x}, 0) d^3x,
$$ 
respectively, where $\mathcal{H}(x)= \mathcal{H}({\bf x}, t)$ is a self-adjoint operator density which is covariant and causal, that is:
$$
U_0(\Lambda, a) \mathcal{H}(x) U_0(\Lambda, a)^{-1}= \mathcal{H}(\Lambda x + a)
\; \;  \text{ and } \; \; [\mathcal{H}(x), \mathcal{H}(y )] = 0 \text{ for } (x - y)^2 \geq 0,
$$
where $U_0(\Lambda, a)$ is the free representation.
Weinberg provides only a partial proof of this claim. Does somebody know if this is a rigorous result, and where one can possibly find a complete/rigorous proof?
 A: There is no rigorous proof for this, as there is no known way to make sense of the interaction part of a local Hamiltonian at fixed time in such a way that $H$ becomes self-adjoint. The reason is that
quantum fields are distributions only, and the product of distributions is ill-defined in general. There are nonrigorous recipes for renormalizing such Poincare representations, but these recipes are defined perturbatively only.
Indeed, the construction of interacting local unitary reps of the Poincare group is completely unsettled in 4 dimensions. No example is known, neither are there nogo theorems.
A: Using the notation:
$ H = H_0 + V$ and  $ \mathbf{K} =  \mathbf{K_0}  + \mathbf{Z}$
The requirement that both free and interacting generators satisfy the
Poincaré algebra commutaion relations, lead to the following requirements:
(Please see the following book by Eugene Stefanovich (published in the arXiv) equations 6.22-6.26 (page 179):
$[\mathbf{J} , V] = [\mathbf{H_0} , V] = 0$
$[P_i , Z_j] = i \delta_{ij} V,  [J_i, Z_j] = i \epsilon_{ijk} Z_k$
$[K_{0[i}, Z_{j]}] +[Z_i,Z_j]=0 $, $[\mathbf{Z}, H_0] + [\mathbf{K_0}  , V] + [ \mathbf{Z}, V] = 0$;
Now, to verify that these relations are satisfied in the present case, please observe that due to the first given realtion expressing the transformation properties of the interaction Hamiltonian density that the action of the free Poincaré generators on the Hamiltonian density is by means of the well known differential operator realization:
[$H_0, \mathcal{H}(\mathbf{x}, 0)] = (\frac{\partial}{\partial t} \mathcal{H})(\mathbf{x}, 0)$
$[P_i , \mathcal{H}(\mathbf{x}, 0)] = i \frac{\partial}{\partial x_i} \mathcal{H}(\mathbf{x}, 0)$
$[J_i , \mathcal{H}(\mathbf{x}, 0)] = i\epsilon_{ijk} x_i \frac{\partial}{\partial x_j} \mathcal{H}(\mathbf{x}, 0)$
$[K_{0i}, \mathcal{H}(\mathbf{x}, 0)] = ((t \frac{\partial}{\partial x_i} - x_i\frac{\partial}{\partial t})\mathcal{H})(\mathbf{x}, 0) = -x_i(\frac{\partial}{\partial t} \mathcal{H})(\mathbf{x}, 0)$
What is left is to perform the substitutions. But in the addition of the given requirements, one must assume that the interaction Hamiltonian density vanishes sufficiently rapidly at infinity, and surface terms in integration by parts can be ignored. (Of course these are operators and one must specify the strength of convergence).
Here is a sample calculation of one of the required commutation
relations:
$[J_i , Z_j] = \int x_j i \epsilon_{ilm} x_l \frac{\partial}{\partial x_m} (\mathcal{H}(\mathbf{x}, 0) )d^3x$
Please observe that the Poincaré  generators do not act on the free
$x_i$'s in the integrand, because they are only dummy integration
variables. Thus after integration by parts we get:
$[J_i , Z_j] = -\int i \epsilon_{ilm}  (\delta_{jm} x_l + \delta_{lm} x_j) \mathcal{H}(\mathbf{x}, 0) d^3x = + i \epsilon_{ijl} x_l \mathcal{H}(\mathbf{x}, 0) d^3x = i \epsilon_{ijk} Z_k$
Finally, please let me remark that finding an exact representation of
 the interacting Poincaré algebra would require knowing the exact
 solution (Hilbert space and operator eigenvalues) of the interacting
 quantum field model, which is not known outside perturbation theory,
 however, the interacting Poincaré  generators can be deduced form the
 Lagrangian by means of Noether's theorem + canonical quantization. The  forms of interacting Poincare algebra were already studied by Dirac in  1949.
