Maxwell’s Hydrodynamic Equations

Hydrodynamic Equations of BEC Vacuum and Phonons

$\nabla \cdot E=\rho /{\epsilon }_0$

$i\hslash \frac{\partial \Psi }{\partial t}=\left(-\frac{{\hslash }^2}{2m}{\nabla }^2+V_{ext}+g{\left|\Psi \right|}^2\right)\Psi$

$\nabla \cdot B=0$

$\widehat{\varphi }\left(r\right)={\displaystyle {\sum }_k}\left(u_k\left(r\right){\widehat{a}}_k+v_k^{\mathrm{<mo>\; *\; </mo>}}\left(r\right){\widehat{a}}_k^{†}\right)$

$\nabla \times E=-\frac{\partial B}{\partial t}$

${\widehat{H}}_{\mathrm{int}}={\displaystyle {\sum }_{k\mathrm{,}k\mathrm{<mo>\; \text{'}\; </mo>}}{V}_{k\mathrm{,}k\mathrm{<mo>\; \text{'}\; </mo>}}{\widehat{a}}_k^{†}{\widehat{a}}_{k\mathrm{<mo>\; \text{'}\; </mo>}}^{†}{\widehat{a}}_{k\mathrm{<mo>\; \text{'}\; </mo>}}{\widehat{a}}_k}$

$\nabla \times B={\mu }_{0}J+{\mu }_0{\epsilon }_0\frac{\partial E}{\partial t}$

${\widehat{T}}_{\mu \nu }^{•}=\frac{2}{\sqrt{-g}}\frac{\delta S_{•}}{\delta g^{\mu \nu }}$