Quantum mechanics for the swimming of micro-organism in two dimensions

In a two dimensional fluid, there are only two classes of swimming ways of micro-organisms, i.e., ciliated and flagellated motions. Towards understanding of this fact, we analyze the swimming problem by using w1+∞ and/or W1+∞ algebras. In the study of the relationship between these two algebras, there appear wave functions expressing the shape of micro-organisms. In order to construct well-defined quantum mechanics based on W1+∞ algebra and wave functions, essentially only two different kinds of definitions are allowed for 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 W1+∞ algebra and wave functions is quantum mechanics of 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 a two dimensional fluid. These algebras are larger than the usual w1+∞ and W1+∞ algebras. We give a free field representation of this extended W1+∞ algebra.