Abstract
Hydrodynamics on non-commutative space is studied based on a formulation of hydrodynamics by Y. Nambu in terms of Poisson and Nambu brackets. Replacing these brackets by Moyal brackets with a parameter $\theta$, a new hydrodynamics on non-commutative space is derived. It may be a step toward to find the hydrodynamics of granular materials whose minimum volume is given by $\theta$. To clarify this minimum volume, path integral quantization and uncertainty relation of Nambu dynamics are examined.
Highlights
IntroductionWe construct a new hydrodynamics on non-commutative space through the replacement of the Poisson and Nambu brackets by the Moyal ones
To clarify the minimum volume, we examine the quantization of the Nambu dynamics in the path integral formulation
In order to introduce the finite size of the space point or the finite size of the element of the fluid, Poisson and Nambu brackets are replaced by the corresponding Moyal brackets
Summary
We construct a new hydrodynamics on non-commutative space through the replacement of the Poisson and Nambu brackets by the Moyal ones. In three dimensional phase space, the quantum Nambu dynamics is a closed string theory In this way the uncertainty relation which gives the basis of minimum volume, is clarified. It is interesting to consider different hydrodynamics on different non-commutative spaces with different quantization methods, and compare the obtained results to the experimental data which seems to be compiled so far for various granular materials. This is, beyond the scope of this paper.
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