Anonymous
11/9/2025, 1:52:10 PM
No.16841397
>>16840973
>>16841392
attempting to salvage the claim via a differential operator [math]\mathcal{D}:\Omega^0\to\Omega^2[/math] with [math]F=\mathcal{D}(\phi)[/math] collapses under the constraints [math]dF=0[/math] and [math]d{*}F=J[/math] for arbitrary [math]J[/math]: you would need [math]d\mathcal{D}=0[/math] as an operator identity and, simultaneously, [math]\phi\mapsto { *}\mathcal{D}(\phi)[/math] to be surjective onto exact 3-forms, which in turn recreates the usual potential [math]A[/math] (up to gauge) rather than a new scalar; in vacuum one can introduce Hertz/Debye superpotentials [math]\Pi\in\Omega^2[/math] with [math]A=\delta\Pi[/math], [math]F=d\delta\Pi[/math], but these are 2-forms (not scalars), metric-dependent through [math]\delta=-{*}d{*}[/math], and their dimensions follow [math][\Pi]=[A]\cdot\mathrm{length}=\mathrm{Wb}\cdot\mathrm{m}[/math], not Weber; units are fixed unambiguously by the kinematics and the action [math]S=\int \tfrac{1}{2\mu_0}F\wedge{*}F + A\wedge J[/math], giving [math][F]=\mathrm{Wb}/\mathrm{m}^2[/math] (hence the spatial [math]\mathbf{B}[/math] has [math]\mathrm{Wb}/\mathrm{m}^2[/math]) and [math][A]=\mathrm{Wb}/\mathrm{m}[/math], while the electric scalar potential has [math][\varphi]=\mathrm{V}[/math] and the magnetostatic scalar potential (when it exists on [math]\mathbf{J}=0[/math] simply connected regions via [math]\mathbf{H}=-\nabla\psi_m[/math]) has [math][\psi_m]=\mathrm{A}[/math]; confusing [math]\mathrm{Wb}[/math] (the unit of the integrated flux [math]\int_{\mathcal{S}}F[/math]) with [math]\mathrm{Wb}/\mathrm{m}^2[/math] (the areal density represented by the 2-form’s spatial components) is precisely the “integrand vs integral” mistake. declaring a pointwise scalar to "have unit Weber" does not make it a flux density, and it cannot reproduce F without either trivializing the field ([math]d^2=0[/math]) or smuggling in a noncanonical metric-dependent operator that reduces to the standard potentials anyway
>>16841392
attempting to salvage the claim via a differential operator [math]\mathcal{D}:\Omega^0\to\Omega^2[/math] with [math]F=\mathcal{D}(\phi)[/math] collapses under the constraints [math]dF=0[/math] and [math]d{*}F=J[/math] for arbitrary [math]J[/math]: you would need [math]d\mathcal{D}=0[/math] as an operator identity and, simultaneously, [math]\phi\mapsto { *}\mathcal{D}(\phi)[/math] to be surjective onto exact 3-forms, which in turn recreates the usual potential [math]A[/math] (up to gauge) rather than a new scalar; in vacuum one can introduce Hertz/Debye superpotentials [math]\Pi\in\Omega^2[/math] with [math]A=\delta\Pi[/math], [math]F=d\delta\Pi[/math], but these are 2-forms (not scalars), metric-dependent through [math]\delta=-{*}d{*}[/math], and their dimensions follow [math][\Pi]=[A]\cdot\mathrm{length}=\mathrm{Wb}\cdot\mathrm{m}[/math], not Weber; units are fixed unambiguously by the kinematics and the action [math]S=\int \tfrac{1}{2\mu_0}F\wedge{*}F + A\wedge J[/math], giving [math][F]=\mathrm{Wb}/\mathrm{m}^2[/math] (hence the spatial [math]\mathbf{B}[/math] has [math]\mathrm{Wb}/\mathrm{m}^2[/math]) and [math][A]=\mathrm{Wb}/\mathrm{m}[/math], while the electric scalar potential has [math][\varphi]=\mathrm{V}[/math] and the magnetostatic scalar potential (when it exists on [math]\mathbf{J}=0[/math] simply connected regions via [math]\mathbf{H}=-\nabla\psi_m[/math]) has [math][\psi_m]=\mathrm{A}[/math]; confusing [math]\mathrm{Wb}[/math] (the unit of the integrated flux [math]\int_{\mathcal{S}}F[/math]) with [math]\mathrm{Wb}/\mathrm{m}^2[/math] (the areal density represented by the 2-form’s spatial components) is precisely the “integrand vs integral” mistake. declaring a pointwise scalar to "have unit Weber" does not make it a flux density, and it cannot reproduce F without either trivializing the field ([math]d^2=0[/math]) or smuggling in a noncanonical metric-dependent operator that reduces to the standard potentials anyway