Consider the time-dependent Schroedinger equation for a single non-relativistic particle with mass $m = 1$. The kinetic energy of the system is [ref] $$ T := \tfrac{\hbar^2}{2} \int |\nabla \psi|^2 dx . $$
Setting $\psi =: \sqrt{\rho} e^{i S / \hbar}$ (see pilot-wave formalism) one finds the decomposition $$ T = A + B := \tfrac12 \int |\nabla S|^2 \rho dx + \tfrac{\hbar^2}{8} I(\rho) $$ where $$ I(\rho) = \int |\nabla \rho|^2 \rho^{-1} dx $$ is the Fisher information measure (whose derivative is proportional the Bohm potential).
The first term $A$ has the obvious classical interpretation of kinetic energy of the density $\rho$ propelled by the velocity field $\nabla S$.
My main question is:
- Could you please point to some references where this decomposition appears?
Additionally, I would like to know
- What is a possible meaning of the second term $B$?
- Are there similar decompositions for other quantities?