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Matching units does not determine the permittivity or the response law

Physical scope

A worked dimensional and model-identification example for the framework's SI conventions: the proposed law E=eps0 D is excluded by dimensional analysis, dimensionally admissible laws do not determine the permittivity or the response law, and a frequency-domain symbol epsilon(omega) is not an instantaneous time-domain coefficient. No measurement, no material data and no empirical claim.

Example

(a) [Dimensional exclusion of $E = ε_{0} D$ ] In the SI conventions of Framework-local SI units, the vacuum constants and the exact/measured status of their values: the electric constant $ε_{0} = (μ_{0} c^{2})^{- 1}$ carries $F m^{- 1} = C^{2} s^{2} {k g}^{- 1} m^{- 3}$ , the displacement $D$ of Polarization, magnetization, bound and free sources, and the fields D and H carries $C m^{- 2}$ , and the electric field $E$ of The electric and magnetic fields as fixed-time spatial vector fields carries $V m^{- 1} = k g m s^{- 2} C^{- 1}$ . The product $ε_{0} D$ therefore carries $F C m^{- 3} = C^{3} s^{2} {k g}^{- 1} m^{- 5}$ , which is not the dimension of $E$ : the proposed identification $E = ε_{0} D$ fails dimensionally in SI and cannot be a field relation of the framework.

(b) [Units do not determine the law] Solving the definition $D = ε_{0} E + P$ of Polarization, magnetization, bound and free sources, and the fields D and H gives $E = (D - P) / ε_{0}$ , which in vacuum ( $P = 0$ ) reads $E = ε_{0}^{- 1} D$ . Any proposed law $E = η D$ with a constant $η > 0$ of dimension $V m C^{- 1}$ is dimensionally admissible; dimensional consistency therefore does not identify $η$ , and it does not determine whether the response is local, linear or instantaneous. In the model of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch the scalar relation is $D = ε E$ , i.e. $η = ε^{- 1}$ , and $ε$ (or $χ_{e} = ε / ε_{0} - 1$ ) is fixed by the declared material model, not by units.

(c) [Frequency-domain symbols are not instantaneous coefficients] In the memory branch of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch(c) the response is a causal convolution in time; a scalar $ε (ω)$ introduced in the harmonic description is defined for a single harmonic history $E (t) = Re (e e^{- i ω t})$ by the response $D (t) = Re (ε (ω) e e^{- i ω t})$ ; for a scalar kernel this means $ε (ω) = ε_{0} (1 + \int_{0}^{\infty} K (s) e^{i ω s} d s)$ . This is the harmonic convention of Specified causal material response: transfer function, dispersion relations and the passivity condition, not an assumption that an arbitrary bounded continuous history has a Fourier transform. The framework therefore does not read $ε (ω)$ as an instantaneous multiplicative coefficient in the time domain: doing so silently replaces the convolution by an instantaneous law and is a model-identification error of the same kind as (a).

(d) [Scope] A dimensional and model-identification statement about the framework's own conventions; it asserts no measurement, no material property and no empirical claim.

Verification

technique · direct dimensional computation and immediate consequences of the definitions

Given: The SI unit conventions of Framework-local SI units, the vacuum constants and the exact/measured status of their values, the coefficient types of The electric and magnetic fields as fixed-time spatial vector fields and Polarization, magnetization, bound and free sources, and the fields D and H, and the declared material branches of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch, on the laboratory region of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields.

[L1] The electric constant is $ε_{0} = (μ_{0} c^{2})^{- 1}$ , carrying $F m^{- 1}$ ; the farad is $F = C V^{- 1}$ and the volt is $V = k g m^{2} s^{- 2} C^{- 1}$ , so $F m^{- 1} = C^{2} s^{2} {k g}^{- 1} m^{- 3}$ ; the electric field carries $V m^{- 1} = k g m s^{- 2} C^{- 1}$ (Framework-local SI units, the vacuum constants and the exact/measured status of their values, clauses (a)-(c); BIPM SI Brochure, Table 4, cited above).

[L2] The displacement is defined by $D = ε_{0} E + P$ and carries $C m^{- 2}$ (Polarization, magnetization, bound and free sources, and the fields D and H, clause (d)); the linear branch declares $D = ε E$ with constant positive matrix $ε$ , and the memory branch declares the causal convolution $P (t) = ε_{0} \int_{0}^{\infty} K (s) E (t - s) d s$ with $K \in L^{1} (0, \infty)$ (Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch, clauses (b)-(c)).

[L3] For a differentiable real primitive with an integrable derivative on a compact interval, its derivative integrates to its endpoint increment (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ).

1.1L1L2

Multiplying the dimension of $ε_{0}$ from [L1] by the dimension $C m^{- 2}$ of $D$ from [L2] gives $C^{3} s^{2} {k g}^{- 1} m^{- 5}$ , which differs from the dimension $k g m s^{- 2} C^{- 1}$ of $E$ in the exponent of $C$ and of $m$ ; the proposed law $E = ε_{0} D$ is therefore excluded.

1.2L2

Solving $D = ε_{0} E + P$ for $E$ gives $E = (D - P) / ε_{0}$ , so in vacuum $E = ε_{0}^{- 1} D$ ; a law $E = η D$ with constant $η > 0$ of dimension $V m C^{- 1}$ has the correct dimension for every such $η$ , so the unit check alone leaves $η$ free and says nothing about locality, linearity or instantaneity.

1.3L2L3

Give the witness physical units and the required vector type: fix $τ > 0$ in seconds and $E_{*} > 0$ in $V m^{- 1}$ , let $E : R \to R^{3}$ be $E (t) = E_{*} \min (1, \max (0, - t / τ)) e_{1}$ , and set $K (s) = τ^{- 1} 1_{[0, τ]} (s) I_{3}$ , where $I_{3}$ is the identity matrix. This real matrix kernel has units $s^{- 1}$ and integrable entries; the history is bounded and continuous, with $E (0) = 0$ and $E (- s) = E_{*} (s / τ) e_{1}$ for $0 \leq s \leq τ$ . The polynomial $s^{2} / (2 τ^{2})$ has derivative $s / τ^{2}$ by expanding its difference quotient, so [L3] evaluates the integral. Therefore $P (0) = ε_{0} E_{*} e_{1} \int_{0}^{τ} s / τ^{2} d s = ε_{0} E_{*} e_{1} / 2 \neq 0$ , while every instantaneous law $P = C E$ with a fixed matrix $C$ gives $P (0) = 0$ . Thus a memory response need not be an instantaneous coefficient law; replacing its harmonic symbol by a time-domain multiplier changes this declared model.

2.1givenstep 1.1step 1.2step 1.3∎

Steps 1.1-1.3 verify (a)-(c): the identification $E = ε_{0} D$ fails dimensionally, dimensionally admissible laws do not determine $η$ or the response type, and the frequency-domain symbol is defined by its action on harmonic histories rather than as an instantaneous coefficient, as the explicit history shows. The scope (d) holds: the example uses only the framework's unit and definition conventions, asserts no measurement, no material property and no empirical claim, and the material constant $ε$ or susceptibility $χ_{e}$ remains fixed by the declared model.

Matching units does not determine the permittivity or the response law

Example

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