On energy-momentum transfer of quantum fields

TL;DR

This paper establishes bounds on quantum field operators related to energy-momentum transfer, analyzing their singularities and showing that outside the origin, contributions are limited to Dirac measures on hypersurfaces.

Contribution

It introduces new bounds and the concept of energy-momentum scaling degree in momentum space for quantum fields, advancing understanding of their singularity structure.

Findings

Bounded operators satisfy specific norm inequalities involving decay functions.

Singularities of operators are characterized by Dirac measures on hypersurfaces.

Outside the origin, only hypersurface-supported contributions are allowed under decay conditions.

Abstract

We prove the following theorem on bounded operators in quantum field theory: if $\|[B,B^*(x)]\|\leq \mathrm{const} D(x)$, then $\|B^k_\pm(\nu)G(P^0)\|^2\leq\mathrm{const}\int D(x-y)d|\nu|(x)d|\nu|(y)$, 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).