On energy-momentum transfer of quantum fields

We prove the following theorem on bounded operators in quantum field theory: if $\mathrm{\parallel }[B,B^{\ast }(x)]\mathrm{\parallel }\le \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}D(x)$, then $\mathrm{\parallel }{B}_{\pm }^k(\nu )G(P^0){\mathrm{\parallel }}^2\le \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\int D(x-y)d\mathrm{\mid }\nu \mathrm{\mid }(x)d\mathrm{\mid }\nu \mathrm{\mid }(y)$, where $D(x)$ is a function weakly decaying in spacelike directions, $B_{\pm }^k$ are creation/annihilation parts of an appropriate time derivative of $B$, $G$ is any positive, bounded, non-increasing function in $L^2(\mathbb{R})$, and $\nu$ is any finite complex Borel measure; creation/annihilation operators may be also replaced by $B_t^k$ with $\check{B_t^k}(p)=\mathrm{\mid }p{\mathrm{\mid }}^k\check{B}(p)$. We also use the notion of energy-momentum scaling degree of $B$ with respect to a submanifold (Steinmann-type, but in momentum space, and applied to the norm of an operator). These two tools are applied to the analysis of singularities of $\check{B}(p)G(P^0)$. We prove, among others, the following statement (modulo some more specific assumptions): outside $p=0$ the only allowed contributions to this functional which are concentrated on a submanifold (including the trivial one – a single point) are Dirac measures on hypersurfaces (if the decay of $D$ is not to slow).