Anonymous
11/9/2025, 1:50:20 PM
No.16841392
>>16840973
calling a 0-form a "scalar superpotential" with unit Weber is a unit fallacy: the electromagnetic field is a 2-form [math]F=dA\in\Omega^2(M)[/math], current is a closed 3-form [math]J\in\Omega^3(M)[/math] with [math]dJ=0[/math], and the equations [math]dF=0[/math], [math]d{*}F=J[/math] fix what can carry flux; flux is the pairing [math]\langle F,\mathcal{S}\rangle=\int_{\mathcal{S}}F\in\mathrm{Wb}[/math] for any oriented 2-current [math]\mathcal{S}[/math], i.e. Weber is attached to the 2-form-2-current pairing, not to an arbitrary 0-form; a 0-form [math]\phi\in\Omega^0(M)[/math] pairs with 0-currents (weighted points), so declaring [math][\phi]=\mathrm{Wb}[/math] merely ensures that [math]\sum_i \phi(x_i)[/math] has units Weber, which is irrelevant to magnetic flux and cannot replace [math]\int_{\mathcal{S}}F[/math]; the only scalar with units of Weber that naturally appears is the gauge function [math]\chi\in\Omega^0(M)[/math] in [math]A\mapsto A+d\chi[/math], [math]\varphi\mapsto \varphi-\partial_t\chi[/math], with [math][A]=\mathrm{Wb}/\mathrm{m}[/math], [math][\chi]=\mathrm{Wb}[/math], and [math]F=dA[/math] unchanged, i.e. pure redundancy, not a physical superpotential; if you try to elevate that scalar to a potential of a potential by setting [math]A=d\phi[/math], then [math]F=dA=d^2\phi=0[/math] and all fluxes vanish (by Stokes, [math]\oint_{\partial \mathcal{S}}A=\int_{\mathcal{S}}F=0[/math]), contradicting any nontrivial [math]\langle F,\mathcal{S}\rangle[/math].
calling a 0-form a "scalar superpotential" with unit Weber is a unit fallacy: the electromagnetic field is a 2-form [math]F=dA\in\Omega^2(M)[/math], current is a closed 3-form [math]J\in\Omega^3(M)[/math] with [math]dJ=0[/math], and the equations [math]dF=0[/math], [math]d{*}F=J[/math] fix what can carry flux; flux is the pairing [math]\langle F,\mathcal{S}\rangle=\int_{\mathcal{S}}F\in\mathrm{Wb}[/math] for any oriented 2-current [math]\mathcal{S}[/math], i.e. Weber is attached to the 2-form-2-current pairing, not to an arbitrary 0-form; a 0-form [math]\phi\in\Omega^0(M)[/math] pairs with 0-currents (weighted points), so declaring [math][\phi]=\mathrm{Wb}[/math] merely ensures that [math]\sum_i \phi(x_i)[/math] has units Weber, which is irrelevant to magnetic flux and cannot replace [math]\int_{\mathcal{S}}F[/math]; the only scalar with units of Weber that naturally appears is the gauge function [math]\chi\in\Omega^0(M)[/math] in [math]A\mapsto A+d\chi[/math], [math]\varphi\mapsto \varphi-\partial_t\chi[/math], with [math][A]=\mathrm{Wb}/\mathrm{m}[/math], [math][\chi]=\mathrm{Wb}[/math], and [math]F=dA[/math] unchanged, i.e. pure redundancy, not a physical superpotential; if you try to elevate that scalar to a potential of a potential by setting [math]A=d\phi[/math], then [math]F=dA=d^2\phi=0[/math] and all fluxes vanish (by Stokes, [math]\oint_{\partial \mathcal{S}}A=\int_{\mathcal{S}}F=0[/math]), contradicting any nontrivial [math]\langle F,\mathcal{S}\rangle[/math].