Quantum Mechanics for the Swimming of Micro-Organism in Two Dimensions

TL;DR

This paper develops a quantum mechanical framework using $W_{1+ abla}infty$ algebras to analyze the swimming behaviors of micro-organisms in two-dimensional fluids, linking fluid dynamics with algebraic structures.

Contribution

It introduces a novel quantum mechanics formulation based on $W_{1+ abla}infty$ algebra for micro-organism swimming, connecting shape functions with algebraic definitions.

Findings

Wave functions express microorganism shapes.

Two definitions of wave function conjugates relate to organism types.

Extended area-preserving algebras are constructed and represented.

Abstract

In two dimensional fluid, there are only two classes of swimming ways of micro-organisms, {\it i.e.}, ciliated and flagellated motions. Towards understanding of this fact, we analyze the swimming problem by using $w_{1+\infty}$ and/or $W_{1+\infty}$ algebras. In the study of the relationship between these two algebras, there appear the wave functions expressing the shape of micro-organisms. In order to construct the well-defined quantum mechanics based on $W_{1+\infty}$ algebra and the wave functions, essentially only two different kinds of the definitions are allowed on the hermitian conjugate and the inner products of the wave functions. These two definitions are related with the shapes of ciliates and flagellates. The formulation proposed in this paper using $W_{1+\infty}$ algebra and the wave functions is the quantum mechanics of the fluid dynamics where the stream function plays the role of the Hamiltonian. We also consider the area-preserving algebras which arise in the swimming problem of micro-organisms in the two dimensional fluid. These algebras are larger than the usual $w_{1+\infty}$ and $W_{1+\infty}$ algebras. We give a free field representation of this extended $W_{1+\infty}$ algebra.