QUANTUM-MECHANICS FOR THE SWIMMING OF MICROORGANISM IN 2-DIMENSIONS

In a two dimensional fluid, there are only two classes of swimming way s of micro-organisms, i.e., ciliated and flagellated motions. Towards understanding of this fact, we analyze the swimming problem by using w (1+infinity) and/or W-1+infinity algebras. In the study of the relatio nship between these two algebras, there appear wave functions expressi ng the shape of microorganisms. In order to construct well-defined qua ntum mechanics based on W-1+infinity algebra and wave functions, essen tially only two different kinds of definitions are allowed for the her mitian conjugate and the inner products of the wave functions. These t wo definitions are related with the shapes of ciliates and flagellates . The formulation proposed in this paper using W-1+infinity algebra an d wave functions is quantum mechanics of fluid dynamics where the stre am function plays the role of the Hamiltonian. We also consider the ar ea-preserving algebras which arise in the swimming problem of micro-or ganisms in a two dimensional fluid. These algebras are larger than the usual w(1+infinity) and W-1+infinity algebras. We give a free field r epresentation of this extended W-1+infinity algebra.