Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch

Adopt as a modelling postulate of this framework the following constitutive branches for the material described by the macroscopic fields of Polarization, magnetization, bound and free sources, and the fields D and H.

(a) [Material rest frame] The relations below are stated in a material rest frame: an inertial frame $(o;e_0,e_1,e_2,e_3)$ of the Minkowski model of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields (Inertial frames and inertial charts of the affine Minkowski model) in whose coordinates each modelled material element has fixed spatial position $x$, hence history $t\mapsto (ct,x)$ on the declared time interval. When this is the laboratory frame, the components below are laboratory components. Boosting the material changes both the fields and the rest observer; this postulate does not assert that the relations below hold with the same constants in every frame.

(b) [Elementary inertial-rest branch] In the rest frame, with constant symmetric positive-definite matrices $\epsilon ,\mu \in {\mathbb{R}}^{3\times 3}$ (the isotropic scalar cases $\epsilon =\epsilon \mathbf{1}$, $\mu =\mu \mathbf{1}$ with $\epsilon ,\mu >0$ included), the model takes

$D=\epsilon E,B=\mu H,J_c=\sigma E,$

where $\sigma \in {\mathbb{R}}^{3\times 3}$ is symmetric positive semidefinite and $J_c$ is the conduction part of the free current, $J_f=J_c+J_{\mathrm{e}\mathrm{x}\mathrm{t}}$ for a declared external free current $J_{\mathrm{e}\mathrm{x}\mathrm{t}}:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ in $\mathrm{A}{\mathrm{m}}^{-2}$. For a real matrix $T$, symmetric means $T^{\mathsf{T}}=T$, positive definite means $v^{\mathsf{T}}Tv>0$ for every nonzero $v\in {\mathbb{R}}^3$, and positive semidefinite means $v^{\mathsf{T}}Tv\ge 0$ for every $v$. Entries of $\epsilon ,\mu ,\sigma$ carry respectively $\mathrm{F}{\mathrm{m}}^{-1}$, $\mathrm{H}{\mathrm{m}}^{-1}$ and $\mathrm{S}{\mathrm{m}}^{-1}$, where $\mathrm{F}=\mathrm{C}\mathrm{/}\mathrm{V}$, $\mathrm{H}=\mathrm{T}{\mathrm{m}}^2\mathrm{/}\mathrm{A}$ and $\mathrm{S}=\mathrm{A}\mathrm{/}\mathrm{V}$; these make the three displayed relations dimensionally consistent. Local, linear and nondispersive (instantaneous) response is part of this restricted assumption. In the isotropic case an exact broadband interpretation of the constant-coefficient law additionally requires $\epsilon \mu \ge c^{-2}$, so that the characteristic speed $1\mathrm{/}\sqrt{\epsilon \mu }$ does not exceed the vacuum speed $c$; without that condition, and in the anisotropic branch, the constant-coefficient law is a declared finite-band effective phase-response model and carries no universal front-speed claim.

(c) [Causal memory branch] Alternatively, at each fixed material position $x$, declare a bounded continuous full history $E_x:\mathbb{R}\to {\mathbb{R}}^3$ in $\mathrm{V}{\mathrm{m}}^{-1}$ and its response $P_x:\mathbb{R}\to {\mathbb{R}}^3$ in $\mathrm{C}{\mathrm{m}}^{-2}$. Write $E(t)=E_x(t)$ and $P(t)=P_x(t)$ in the following formula; any field on a smaller time interval uses a declared full-history extension, not unspecified past values. The polarisation response is the causal convolution

$P(t)={\epsilon }_0{\int }_0^{\mathrm{\infty }}K(s)E(t-s)ds$

for a declared real matrix kernel $K:[0,\mathrm{\infty })\to {\mathbb{R}}^{3\times 3}$ with entries in $L^1(0,\mathrm{\infty })$ (The class $L^1(\mu )$ of integrable functions) and units ${\mathrm{s}}^{-1}$; the integral then converges absolutely because $\mathrm{\mid }{K}_{jk}(s)E_k(t-s)\mathrm{\mid }\le (\sup \mathrm{\mid }{E}_k\mathrm{\mid })\mathrm{\mid }{K}_{jk}(s)\mathrm{\mid }$. The convolution uses only the past history; its continuity and causality are proved in pthm-em-causal-dispersive-response, while passivity is an additional model condition. Spatial or temporal differentiability of the resulting fields requires separate hypotheses when used. A finitely-subtracted analytic response may instead be declared by a dimensionless scalar transfer function $\chi$ that is holomorphic on the open upper half-plane, continuous on its closure, of class $C^1$ on the real boundary away from a real subtraction point ${\omega }_0$, holomorphic on a neighbourhood of ${\omega }_0$ in $\mathbb{C}$, and satisfies $\mathrm{\mid }\chi (z)\mathrm{\mid }\le C(1+\mathrm{\mid }z{\mathrm{\mid }}^{n-1})$ on $\{\mathrm{Im}z\ge 0\}$ outside a compact set for some integer $n\ge 1$ and constant $C>0$ (Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions). Here $z,{\omega }_0$ are numerical angular frequencies in ${\mathrm{s}}^{-1}$ and the growth bound uses those SI numerical coordinates. Require also $\chi (-\omega )=\overline{\chi (\omega )}$ for every real $\omega$, hence $\chi (0)\in \mathbb{R}$, so that the response is defined on real harmonic histories independently of their representation. For $\omega \mathrm{\ne }0$ the amplitude at that frequency is unique and reversing frequency conjugates it; the symmetry gives the same real output. At $\omega =0$ only $\mathrm{Re}e$ determines the input, and real $\chi (0)$ makes the output depend only on that input. For $e\in {\mathbb{C}}^3$, real $\omega$, and componentwise real part, declaring this transfer means adopting the harmonic relation $E(t)=\mathrm{Re}(e{e}^{-i\omega t})\mapsto P(t)=\mathrm{Re}({\epsilon }_0\chi (\omega )e{e}^{-i\omega t})$; upper-half-plane holomorphy alone does not construct a time-domain response for arbitrary histories.

(d) [Status] These are adopted model assumptions, one branch at a time; the framework does not derive them from the vacuum equations or from coarse graining, and no microscopic mechanism, material data or empirical validity is claimed. Ferroelectric and ferromagnetic response, hysteresis, nonlinear response and general nonlocal response lie outside the branches declared here.

The equations of The macroscopic Maxwell equations in terms of D, H and the free sources are formulated for arbitrary declared $C^2$ maps $P,M$; the branches above are the additional modelling assumptions by which a given medium is closed. Clause (a) fixes the frame in which the constant coefficients and kernels are declared, clause (b) is a restricted instantaneous branch with explicit positivity and the separate isotropic broadband condition, and clause (c) retains a causal memory response whose analysis is the separate result pthm-em-causal-dispersive-response. No branch is deduced from the vacuum equations, from the averaging convention of the item post-em-coarse-graining or from material data, and none is asserted to hold for any actual substance; the exclusions in (d) delimit the model.