Infrared Safety from ZX-Diagrams: A Categorical Proof of Soft-QED as Open Quantum System

2026-06-29

AI Summary

This work addresses the infrared divergence problem in quantum electrodynamics by treating soft photons as an environment and hard processes as a system, establishing a non-perturbative, categorical framework for open quantum systems based on discard ZX-calculus. It provides the first diagrammatic, rigorous proof of the consistency between the Bloch–Nordsieck cancellation mechanism and the soft photon theorem. The study derives the corresponding Schur channel and an exact Lindblad generator, thereby elucidating the decoherence mechanism acting on off-diagonal elements of hard states. Furthermore, it introduces a local CPTP certification protocol that enables constant-time verification of trace preservation. By integrating categorical quantum semantics with nonequilibrium field theory, this approach offers a novel paradigm for infrared safety.

Abstract

The discard ZX-calculus, a diagrammatic language for mixed-state quantum mechanics, is used to give a nonperturbative, categorical proof of the Bloch-Nordsieck cancellation of infrared divergences in QED. Soft photons are treated as an open quantum system: the resolved charged particles and hard photons form the system, while photons below a detector resolution form the environment. The reduced hard channel is a completely positive trace-preserving (CPTP) map, and the soft-photon theorem replaces the full S-matrix by a controlled displacement operator whose Feynman-Vernon influence functional satisfies the equal-history normalization ${\cal F}[J,J]=1 $. In the ZX-calculus, this normalization is a single diagrammatic identity: the doubled displacement diagram collapses to the bare wire under the unitarity, cyclicity, and discard rules. The proof therefore serves as a categorical consistency check on the open-system treatment of soft QED given in a companion paper; it confirms that the physical derivation is logically complete and free of hidden assumptions about the infrared limit. For off-diagonal hard-state elements, the same diagram yields the coherent-state overlap, giving a first-principles account of soft-cloud decoherence. The soft-shell coarse graining is then constructed as a CPTP Schur channel whose infinitesimal limit produces the exact Lindblad generator with jump operators determined by the eikonal emission amplitudes. Finally, a local CPTP-certification pipeline is developed for non-Markovian process tensors, enabling constant-time verification of trace preservation in open quantum simulations. The framework bridges categorical quantum semantics, non-equilibrium field theory, and practical open-system compilation.

Problem

• infrared divergences
• soft QED
• open quantum system
• categorical proof
• decoherence

Innovation

• ZX-calculus
• infrared safety
• open quantum systems
• CPTP maps
• soft-photon theorem