The general equations of a gauge invariant, classical theory of the electrodynamics of material media are obtained. The gauge invariance is insured by taking the equations $\nabla ·\mathrm{D}=\rho , \nabla \times \mathrm{H}-{\mathrm{D}}^{\prime }=\mathrm{J}$ as conditions auxiliary to the variation principle $\delta \int \int \{L+\Sigma \stackrel{}{n}{\theta }_n[{N_n}^{\prime }+\nabla ·(N_n{\mathrm{V}}_n\left)\right]dvdt\}=0.\mathrm{}$ The Lagrangian function, $L$, depends on D, H, $N_n$, ${\mathrm{V}}_n$, ${\theta }_n$ and possibly their derivatives; here $N_n$ is the numerical density of atoms in the state $n$, ${\mathrm{V}}_n$ their macroscopic or average velocity, and ${\theta }_n$ is a variable that functions as the velocity potential in some cases and has the dimensions of action. The electromagnetic potentials enter the theory as Lagrangian multipliers only.

It is shown that if there is only one state and $L=\frac{1}{2}\mathrm{Nm}{\mathrm{V}}^2-\frac{\left(\frac{{\hslash }^2}{8m}\right){\left(\nabla N\right)}^2}{N}+\frac{1}{2}({\mathrm{H}}^2-{\mathrm{D}}^2),$ then the Schrödinger wave equation is obtained on making the substitution $\psi =N^{\frac{1}{2}}\exp (-\frac{i\theta }{\hslash }).\mathrm{}$

If the atoms are stationary (so that terms in ${\mathrm{V}}_n$ may be neglected), and $L=\Sigma \stackrel{}{n}{N}_n{W}_n+\mathrm{D}·\mathrm{P}+\frac{1}{2}({\mathrm{H}}^2-{\mathrm{D}}^2),$ where $W_n$ is the energy of the $n\mathrm{th}$ state, and $\mathrm{P}=\Sigma {\Sigma }_{\mathrm{}(m,n)\mathrm{}}{\left(N_m{N}_n\right)}^{\frac{1}{2}}{\mathrm{P}}_{\mathrm{mn}}cos[\frac{({\theta }_m-{\theta }_n)}{\hslash }+{\alpha }_{\mathrm{nm}}]$ is the polarization of the medium, an adequate theory of dispersion results. However, the spontaneous transitions are not correctly accounted for by the equations.

If the electromagnetic fields are neglected and $L=\frac{1}{2}\mathrm{Nm}{\mathrm{V}}^2-U\left(N\right),$ the equations are those for the irrotational motion of a gas, with $\frac{\theta }{m}$ as the velocity potential.

• Received 9 September 1938

DOI:https://doi.org/10.1103/PhysRev.54.920