How to be causal: time, spacetime, and spectra

TL;DR

This paper clarifies a simple causality definition, links it to the Kramers-Kronig relations, and extends these concepts to spacetime causality and Maxwell's equations, enhancing understanding of dynamical models and wave signals.

Contribution

It introduces a reformulation of the Kramers-Kronig relations applicable to spacetime causality and demonstrates their use in analyzing Maxwell's equations.

Findings

Reformulated Kramers-Kronig relations for spacetime causality

Clarified the causal structure of Maxwell's equations

Provided a framework linking causality to wave signals and light-cone structures

Abstract

I explain a simple definition of causality in widespread use, and indicate how it links to the Kramers Kronig relations. The specification of causality in terms of temporal differential eqations then shows us the way to write down dynamical models so that their causal nature /in the sense used here/ should be obvious to all. To extend existing treatments of causality that work only in the frequency domain, I derive a reformulation of the long-standing Kramers Kronig relations applicable not only to just temporal causality, but also to spacetime "light-cone" causality based on signals carried by waves. I also apply this causal reasoning to Maxwell's equations, which is an instructive example since their casual properties are sometimes debated.