A Quest for Systematic Constitutive Formulations for General Field and Wave Systems Based on the Volterra Differential Operators

−A systematic formulation of constitutive relations for general field systems is explored, which takes into account the main material features. These are dispersion, inhomogeneity, which can be present in linear and nonlinear systems. There are two main difficulties associated with existing representations: Dispersion is usually referred to in the spectral domain, while inhomogeneity is obviously a spatiotemporal phenomenon; moreover, the existing representations involve spatiotemporally dependent integrals, and those suggest non-local interaction which raises relativistic causality issues. The present approach introduces new representations in terms of Volterra differential operators, which obviate these difficulties within the domain of their validity, allowing for spatiotemporal representation of both dispersion and inhomogeneity, in linear and nonlinear systems. Minkowski’s methodology for representation of material properties in the presence of moving media is re-examined in view of the new constitutive relations, especially as regards Maxwell’s equations of Electrodynamics.