Basic Variables in the Presence of a Magnetostatic Field

We present our recent understanding of the issue of what properties constitute the basic variables in quantum mechanics for electrons in the presence of external electrostatic ${\cal{E}} ({\bf{r}}) = -$ {\boldmath $\nabla$} $v ({\bf{r}})$ and magnetostatic ${\bf{B}} ({\bf{r}}) =$ {\boldmath $\nabla$} $\times {\bf{A}} ({\bf{r}})$ fields. In this case, the relationship between the potentials $\{v, {\bf{A}} \}$ and the ground state wave function $\Psi$ can be many-to-one. We discuss our prior work\footnote{Pan and Sahni, Int. J. Quantum Chem. 110, 2833 (2010); J. Phys. Chem. Solids. 73, 630 (2012).} in which we claimed that the basic variables are the ground state density $\rho ({\bf{r}})$ and physical current density ${\bf{j}} ({\bf{r}})$. We prove here more fully this to be the case for the nondegenerate ground state for which $\Psi$ is real. The proof explicitly accounts for the many-to-one relationship between $\{v, {\bf{A}} \}$ and $\Psi$. We also draw parallels between our work on the density and physical current density functional theory and those of the Hohenberg-Kohn and Percus-Levy-Lieb definitions of density functional theory.