How to be causal: time, spacetime, and spectra

Paul Kinsler

    Blackett Laboratory
    Imperial College London,
    Prince Consort Road,
    London SW7 2AZ
    United Kingdom.

    Eur. J. Phys. 32, 1687 (2011)
    arXiv:1106.1792
    doi:10.1088/0143-0807/32/6/022
 

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.