N-electron probability densities and current densities
For an \(N\)-electron system, the associated probability density operator (in coordinate representation) can be taken as the sum of all the contributions for every electron: \[\begin{equation} \require{physics} \rho^c(\vb{r}) = \sum_i^N \delta(\vb{r} - \vb{r}_i) \thinspace . \end{equation}\] Its expectation value indeed yields the \(N\)-electron probability density: \[\begin{align} & \int \cdots \int \dd{\vb{r}_1} \cdots \dd{\vb{r}_N} \Psi^\dagger(\vb{r}_1, \ldots, \vb{r}_N) \thinspace \rho^c(\vb{r}) \thinspace \Psi(\vb{r}_1, \ldots, \vb{r}_N) \\ & \hspace{24pt} = N \int \cdots \int \dd{\vb{r}_2} \cdots \dd{\vb{r}_N} \Psi^\dagger(\vb{r}, \vb{r}_2, \ldots, \vb{r}_N) \Psi(\vb{r}, \vb{r}_2, \ldots, \vb{r}_N) \\ & \hspace{24pt} = \rho(\vb{r}) \thinspace , \end{align}\] which, following McWeeny’s conventions (McWeeny 1960), integrates to the total number of electrons \(N\): \[\begin{equation} \int \dd{\vb{r}} \rho(\vb{r}) = N \thinspace . \end{equation}\] This means that the \(N\)-electron probability density is not a probability-theoretic probability density function because it does not integrate to \(1\), but rather to \(N\).
Analogously, the \(N\)-electron probability current density operator has contributions from each electron: \[\begin{equation} \vb{j}^c(\vb{r}) = - \frac{i}{2} \sum_i^N \Bigg( \delta(\vb{r} - \vb{r}_i) \thinspace \grad + \grad \thinspace \delta(\vb{r} - \vb{r}_i) + 2 i \thinspace \vb{A}(\vb{r}) \thinspace \delta(\vb{r} - \vb{r}_i) \Bigg) \thinspace , \end{equation}\] whose expectation value must lead to the \(N\)-electron probability current density: \[\begin{align} \vb{j}(\vb{r}) &= \int \cdots \int \dd{\vb{r}_1} \cdots \dd{\vb{r}_N} \Psi^\dagger(\vb{r}_1, \ldots, \vb{r}_N) \thinspace \vb{j}^c(\vb{r}) \thinspace \Psi(\vb{r}_1, \ldots, \vb{r}_N) \\ & = - \frac{N i}{2} \int \cdots \int \dd{\vb{r}_2} \cdots \dd{\vb{r}_N} \Bigg( \Psi^\dagger(\vb{r}, \vb{r}_2, \ldots, \vb{r}_N) \thinspace [ \grad{ \Psi(\vb{r}, \vb{r}_2, \ldots, \vb{r}_N) } ] \\ & \hspace{60pt} - [ \grad{ \Psi^\dagger(\vb{r}, \vb{r}_2, \ldots, \vb{r}_N) } ] \thinspace \Psi(\vb{r}, \vb{r}_2, \ldots, \vb{r}_N) \\ & \hspace{60pt} + 2i \thinspace \Psi^\dagger(\vb{r}, \vb{r}_2, \ldots, \vb{r}_N) \Psi(\vb{r}, \vb{r}_2, \ldots, \vb{r}_N) \thinspace \vb{A}(\vb{r}, t) \Bigg) \thinspace . \end{align}\] This formula can also be found in (Jusélius 2004), equation (4.8) by realizing that Juselius writes the associated current density.
Employing a spinor basis or a spin-orbital basis, we may analogously quantize the \(N\)-electron density and current density operators.