Geometric Flow Appearing in Conservation Law in Classical and Quantum Mechanics

Abstract

The appearance of a geometric flow in the conservation law of particle number in classical particle diffusion and in the conservation law of probability in quantum mechanics is discussed in the geometrical environment of a two-dimensional curved surface with thickness $\epsilon$ embedded in $\mathbb E^3$. In such a system, two-dimensional conservation law needs an additional term just like an anomaly. The additional term can be obtained by the $\epsilon$ expansion. This term has a Gaussian and a mean curvature dependence and can be written as the total divergence of geometric flow $J^i_{G}$. This fact holds in both classical and quantum mechanics.