Abstract
We prove the following theorem on bounded operators in quantum field theory: if \({\|[B,B^*(x)]\|\leqslant{\rm const}D(x)}\), then \({\|B^k_\pm(\nu)G(P^0)\|^2\leqslant{\rm const}\int D(x - y){\rm d}|\nu|(x){\rm d}|\nu|(y)}\), 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 \({\nu}\) 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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Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
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Herdegen, A. On Energy-Momentum Transfer of Quantum Fields. Lett Math Phys 104, 1263–1280 (2014). https://doi.org/10.1007/s11005-014-0710-5
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DOI: https://doi.org/10.1007/s11005-014-0710-5