On Energy-Momentum Transfer of Quantum Fields

Abstract

Abstract. We prove the following theorem on bounded operators in quantum field theory:is a function weakly decaying in spacelike directions, B k ± are creation/annihilation parts of an appropriate time derivative of B, G is any positive, bounded, non-increasing function in L 2 (R), and ν is any finite complex Borel measure; creation/annihilation operators may be also replaced by B k t with B k t ( p) = | p| kB ( p). We also use the notion of energymomentum 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 ofB( 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).Mathematics Subject Classification (2010). 46L40, 81T05.

Massless Asymptotic Fields and Haag–Ruelle Theory

“…By Lemma 14 in Appendix A of [11] we have |φpsq| ď const pλ`|s|q´2. Thus φ is absolutely integrable.…”

“…with δ P p0, 1q, δ ă n´p3{2q. This function satisfies the assumptions of Lemma 14 in [11] with γ " δ, so hp xq is integrable. Consider relation (14) with the replacements B Ñ C " p1´λ 2 ∆q 2 B , k Ñ n´δ , f ps, lq Ñ l bps, lq .…”

“…Our main tool in this section will be a partial result of Proposition 9 in [11]. We rewrite the necessary result with the use of the norms }.}…”

“…Fields are assumed to satisfy some weak condition on the decay of their commutators in spacelike directions, but need not be local. We believe that the limits of strict locality may be to narrow in a constrained theory like QED; we refer the reader to our earlier papers [10], [11] for more comments on nonlocality and our specific choice of commutator laws. The smearing applied here to the fields still cuts the infrared-singular degrees of freedom, but no further assumptions beside relativistic positivity of energy spectrum in the Hilbert space are needed.…”

Lieb–Robinson Bounds, Arveson Spectrum and Haag–Ruelle Scattering Theory for Gapped Quantum Spin Systems

“…its division into pure-point, absolutely continuous and singular continuous parts) remains to a large extent unexplored. Some steps to close this gap, with physical applications in mind, have been taken in [16,25,38]. Finally, we note that partial derivatives of a function f ∈ C ∞ (∆), where ∆ is an open subset of Γ, are independent of a chosen parametrization of the torus, an observation which is implicitly used e.g.…”