User:Wing gundam/sandbox

Gauss's law	$\nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}$	$\scriptstyle \partial V$ $\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {Q(V)}{\varepsilon _{0}}}$
Gauss's law for magnetism	$\nabla \cdot \mathbf {B} =0$	$\scriptstyle \partial V$ $\mathbf {B} \cdot \mathrm {d} \mathbf {A} =0$
Maxwell–Faraday equation (Faraday's law of induction)	$\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$	$\oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =-\iint _{S}{\frac {\partial \mathbf {B} }{\partial t}}\cdot \mathrm {d} \mathbf {A}$
Ampère's circuital law (with Maxwell's correction)	$\nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}$	$\oint _{\partial S}\mathbf {B} \cdot \mathrm {d} \mathbf {l} =\mu _{0}I_{S}+\mu _{0}\varepsilon _{0}\iint _{S}{\frac {\partial \mathbf {E} }{\partial t}}\cdot \mathrm {d} \mathbf {A}$

$\color {Salmon}{\begin{array}{lcl}\nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}\\\nabla \cdot \mathbf {B} =0\\\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}\\\nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\\\\\partial _{\mu }A^{\mu }=0\\\partial ^{\nu }\partial _{\nu }A^{\mu }=\mu _{0}J^{\mu }\end{array}}$

You make a number of even stranger claims, like stating that thought ID is "begging the question," as if testing the predictions of a model is equivalent to assuming it is true to prove itself. Making predictions and testing them against reality is precisely how scientific models are falsified. And the suggestion that because philosophy incorporates more modes of thought than science, science cannot impact philosophical conclusions is clear sophistry.

In fact, neuroscience and related fields are the only direct, falsifiable investigations into the mind-body problem, so excluding them is beyond incomplete, it is blatantly dishonest. If you think there is a better way to structure their inclusion, then you should work toward that. But don't make this a power struggle over what evidence is allowed inclusion and what is not. That in itself is the heart of POV.

I'm not going to go through every point, and to be honest I don't even have a position on them all, but I do think you have to change your attitude here. Dualism is not a purely speculative question, and has not been for decades. Scientific facts will bear on this article if it is to ever make GA criteria again.

— Eebster the Great (talk), Talk:Dualism (philosophy of mind)#, Wikipedia

[1], [2], [3]

physics

$U(r)=\hbar c{\frac {\ {g_{e}}^{2}}{4\pi r}}={\frac {\hbar c\alpha }{r}}$

$F(r)=\hbar c{\frac {\ {g_{e}}^{2}}{4\pi r^{2}}}={\frac {\hbar c\alpha }{r^{2}}}$

where:

• $g_{e}$ is the electromagnetic gauge coupling constant, related to the fine structure constant by $\alpha ={\frac {{g_{e}}^{2}}{4\pi }}$. It's the value of an elementary charge when electric charge is nondimensionalized, and more fundamental than $\alpha$.

For compound charges,

$U(r)=\hbar c{\frac {\ q_{1}q_{2}}{4\pi r}}=\hbar c\alpha {\frac {n_{1}n_{2}}{r}}$

$F(r)=\hbar c{\frac {\ q_{1}q_{2}}{4\pi r^{2}}}=\hbar c\alpha {\frac {n_{1}n_{2}}{r^{2}}}$

where:

• $q_{i}=g_{e}n_{i}$
• and $n_{i}$ is the integer number of charges

$0=T^{\mu \nu }{}_{;\nu }=\nabla _{\nu }T^{\mu \nu }=T^{\mu \nu }{}_{,\nu }+T^{\sigma \nu }\Gamma ^{\mu }{}_{\sigma \nu }+T^{\mu \sigma }\Gamma ^{\nu }{}_{\sigma \nu }$

physics 2

$z=\sum _{k=0}^{\infty }\sum _{\{\mathbf {x} _{1},\cdots ,\mathbf {x} _{k}\}\,\subset \,\mathbb {R} ^{3}}{\frac {1}{k!}}c_{\mathbf {x} _{1}\cdots \mathbf {x} _{k}}\theta _{\mathbf {x} _{1}}\cdots \theta _{\mathbf {x} _{k}}$

derivative:

$i\hbar c\gamma ^{\mu }\partial _{\mu }\psi =mc^{2}\psi$

$i\hbar c\gamma ^{\mu }(\partial _{\mu }+ieA_{\mu })\psi =mc^{2}\psi$

$-(\hbar c)^{2}(\partial _{0}+ieA_{0})^{2}\psi =-(\hbar c)^{2}(\partial _{i}+ieA_{i})^{2}\psi +(mc^{2})^{2}\psi$

$-({\frac {1}{c}}\partial _{t}+ieA_{0})^{2}\psi =-\nabla ^{2}\psi +{\lambda \!\!\!\!-}_{c}^{2}\psi$

free:

$-{\frac {1}{c^{2}}}\partial _{t}^{2}\phi =-\nabla ^{2}\phi +{\lambda \!\!\!\!-}_{c}^{2}\phi$

$-{\frac {1}{c^{2}}}\partial _{t}^{2}(e^{-i\omega _{c}t}\psi )=-\nabla ^{2}(e^{-i\omega _{c}t}\psi )+{\lambda \!\!\!\!-}_{c}^{2}(e^{-i\omega _{c}t}\psi )$

$-{\frac {1}{c^{2}}}{\cancel {e^{-i\omega _{c}t}}}(-i\omega _{c}+\partial _{t})^{2}\psi =-{\cancel {e^{-i\omega _{c}t}}}\nabla ^{2}\psi +{\cancel {e^{-i\omega _{c}t}}}{\lambda \!\!\!\!-}_{c}^{2}\psi$

$-{\frac {1}{c^{2}}}(-i\omega _{c}+\partial _{t})^{2}\psi =-\nabla ^{2}\psi +{\lambda \!\!\!\!-}_{c}^{2}\psi$

$-{\frac {1}{c^{2}}}(-\omega _{c}^{2}-2i\omega _{c}\partial _{t}+\partial _{t}^{2})\psi =-\nabla ^{2}\psi +{\lambda \!\!\!\!-}_{c}^{2}\psi$

$-{\frac {1}{c^{2}}}({\cancel {-\omega _{c}^{2}}}-2i\omega _{c}\partial _{t}+\partial _{t}^{2})\psi =-\nabla ^{2}\psi +{\cancel {{\lambda \!\!\!\!-}_{c}^{2}\psi }}$

$-{\frac {1}{c^{2}}}(-2i\omega _{c}\partial _{t}+\partial _{t}^{2})\psi =-\nabla ^{2}\psi$

${\frac {2i{\lambda \!\!\!\!-}_{c}}{c}}\partial _{t}\psi -{\cancel {{\frac {1}{c^{2}}}\partial _{t}^{2}\psi }}=-\nabla ^{2}\psi$

• $(\hbar c)^{2}{\frac {2i\omega _{c}}{c^{2}}}\partial _{t}\psi =-(\hbar c)^{2}\nabla ^{2}\psi$

${\frac {1}{2\hbar \omega _{c}}}{\frac {2i{\lambda \!\!\!\!-}_{c}}{c}}\partial _{t}\psi '=-{\frac {1}{2\hbar \omega _{c}}}\nabla ^{2}\psi '$

${\frac {i}{\hbar c^{2}}}\partial _{t}\psi '=-{\frac {1}{2\hbar \omega _{c}}}\nabla ^{2}\psi '$

$i\hbar \partial _{t}\psi '=-{\frac {\hbar ^{2}c^{2}}{2\hbar \omega _{c}}}\nabla ^{2}\psi '$

$i\hbar \partial _{t}\psi '=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi '$

again:

$-(\hbar c)^{2}({\frac {1}{c}}\partial _{t}+ieA_{0})^{2}\phi =-(\hbar c)^{2}\nabla ^{2}\phi +E_{0}^{2}\phi ,\quad A_{0,0}=0$

$-(\hbar c)^{2}({\frac {1}{c^{2}}}\partial _{t}^{2}+2ieA_{0}{\frac {1}{c}}\partial _{t}-e^{2}A_{0}^{2})\phi =-(\hbar c)^{2}\nabla ^{2}\phi +E_{0}^{2}\phi$

subtitute, factor, divide,

$\left\{-(\hbar c)^{2}({\frac {1}{c^{2}}}(-\omega _{c}^{2}-2i\omega _{c}\partial _{t}+\partial _{t}^{2})+2ieA_{0}{\frac {1}{c}}(-i\omega _{c}+\partial _{t})-e^{2}A_{0}^{2})\psi =-(\hbar c)^{2}\nabla ^{2}\psi +E_{0}^{2}\psi \right\}{\frac {1}{2E_{0}}}$

cancel,

$\left\{-(\hbar c)^{2}({\frac {1}{c^{2}}}({\cancel {-\omega _{c}^{2}}}-2i\omega _{c}\partial _{t}+\partial _{t}^{2})+2ieA_{0}{\frac {1}{c}}(-i\omega _{c}+\partial _{t})-e^{2}A_{0}^{2})\psi =-(\hbar c)^{2}\nabla ^{2}\psi +{\cancel {E_{0}^{2}}}\psi \right\}{\frac {1}{2E_{0}}}$

$-{\frac {(\hbar c)^{2}}{2E_{0}}}({\frac {1}{c^{2}}}(-2i\omega _{c}\partial _{t}+\partial _{t}^{2})+2ieA_{0}{\frac {1}{c}}(-i\omega _{c}+\partial _{t})-e^{2}A_{0}^{2})\psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi$

$i\hbar \partial _{t}\psi -{\frac {\hbar ^{2}}{2E_{0}}}\partial _{t}^{2}\psi -{\frac {(\hbar c)^{2}}{2E_{0}}}(2ieA_{0}{\frac {1}{c}}(-i\omega _{c}+\partial _{t})-e^{2}A_{0}^{2})\psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi$

$i\hbar \partial _{t}\psi -{\frac {\hbar ^{2}}{2E_{0}}}\partial _{t}^{2}\psi -{\frac {(\hbar c)^{2}}{E_{0}}}ieA_{0}{\frac {1}{c}}(-i\omega _{c}+\partial _{t})\psi +{\frac {(\hbar c)^{2}}{2E_{0}}}e^{2}A_{0}^{2}\psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi$

$i\hbar \partial _{t}\psi -{\cancel {{\frac {\hbar ^{2}}{2E_{0}}}\partial _{t}^{2}\psi }}-\hbar ceA_{0}\psi -{\cancel {{\frac {\hbar ^{2}}{mc}}ieA_{0}\partial _{t}\psi }}+{\cancel {{\frac {\hbar ^{2}}{2m}}e^{2}A_{0}^{2}\psi }}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi$

$i\hbar \partial _{t}\psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +\hbar ceA_{0}\psi \quad$ as expected. e is dimensionless coupling, $A_{0}\propto {\frac {1}{L}}$