On energy-momentum transfer of quantum fields

We prove the following theorem on bounded operators in quantum field theory: if $\|[B,B^*(x)]\|\leq \mathrm{const} D(x)$, then $\|B^k_\pm(ν)G(P^0)\|^2\leq\mathrm{const}\int D(x-y)d|ν|(x)d|ν|(y)$, where $D(x)$ is a function weakly decaying in spacelike directions, $B^k_\pm$ 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 $ν$ is any finite complex Borel measure; creation/annihilation operators may be also replaced by $B^k_t$ with $\check{B^k_t}(p)=|p|^k\check{B}(p)$. 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).

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