## Proceedings of the International Conference on Geometry, Integrability and Quantization

### Geometric Flow Appearing in Conservation Law in Classical and Quantum Mechanics

Naohisa Ogawa

#### 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.

#### Article information

Dates
First available in Project Euclid: 21 December 2018

https://projecteuclid.org/ euclid.pgiq/1545361495

Digital Object Identifier
doi:10.7546/giq-20-2019-215-226

Mathematical Reviews number (MathSciNet)
MR3887752

Zentralblatt MATH identifier
07060440

#### Citation

Ogawa, Naohisa. Geometric Flow Appearing in Conservation Law in Classical and Quantum Mechanics. Proceedings of the Twentieth International Conference on Geometry, Integrability and Quantization, 215--226, Avangard Prima, Sofia, Bulgaria, 2019. doi:10.7546/giq-20-2019-215-226. https://projecteuclid.org/euclid.pgiq/1545361495