Causality Sum Rules in Conventional Scattering Matrices

2026-08-10

AI Summary

This work addresses the long-standing challenge that conventional scattering matrices struggle to directly encode causality constraints, often relying on auxiliary variables that disconnect theory from experimental measurements. By introducing a domain-delay matrix defined via the earliest arrival time of each channel, the authors restore the causal time origin while preserving real-frequency passivity. Building upon assumptions of analyticity, transparency, and regularity, they construct a Schur-function framework that, for the first time, establishes direct causality sum rules within standard scattering matrices. This bridges fundamental causal theory with measurable scattering data, yielding new constraints on coherent superposition and multi-channel loss. The approach synergistically combines domain-delay transforms, Schur theory, Cayley–Herglotz representations, and singular value analysis, not only recovering Rozanov’s absorber limit and spherical multipole sum rules but also generalizing them to universal causal bounds governing insertion loss, singular-value suppression, and delay–bandwidth trade-offs in conditionally lossless systems.

Abstract

Scattering matrices are the standard experimental and computational description of photonic and electromagnetic devices. Passivity is explicit in the conventional incoming-outgoing matrix, whereas causality sum rules are usually formulated only after transforming the response into auxiliary variables. Here we show that these rules can be written directly in the conventional scattering matrix by removing the time advance introduced by the reference domain. Using the earliest-arrival delay of each channel, we define a domain-delayed matrix that preserves real-frequency passivity while restoring the causal time origin. Under explicit analyticity, transparency, and regularity assumptions, this matrix becomes a Schur function, enabling a Cayley-Herglotz construction. The resulting projected and determinant bounds constrain coherent channel superpositions and aggregate multichannel loss. The framework recovers Rozanov's absorber limit and spherical-multipole sum rules, while extending causality bounds to measurable quantities including insertion loss, suppressed singular-value channels, and conditional lossless delay-bandwidth trade-offs. Our work directly connects fundamental causality theory with experimentally accessible scattering data. The initial theoretical route is autonomously explored by Qiushi Engine, an AI research system for open-ended scientific discovery, and subsequently verified, refined, and developed by the authors, demonstrating a hybrid AI-human discovery workflow.

Problem

• causality
• scattering matrix
• sum rules
• passivity
• analyticity

Innovation

• causality sum rules
• scattering matrix
• domain-delayed matrix
• Schur function
• delay-bandwidth trade-off