In this paper, we construct for the first time the noncommutative fluid with the deformed Poincaré invariance. To this end, the realization formalism of the noncommutative spaces is employed and the results are particularized to the Snyder space. The noncommutative fluid generalizes the fluid model in the action functional formulation to the noncommutative space. The fluid equations of motion and the conserved energy-momentum tensor are obtained. *

In this paper, we propose a first order action functional for a large class of systems that generalize the relativistic perfect fluids in the Kähler parametrization to noncommutative spacetimes. The noncommutative action is parametrized by two arbitrary functions K(z,z) and f ( √ −j 2 ) that depend on the fluid potentials and represent the generalization of the Kähler potential of the complex surface parametrized by z andz, respectively, and the characteristic function of each model. We calculate the equations of motion for the fluid potentials and the energy-momentum tensor in the first order in the noncommutative parameter. The density current does not receive any noncommutative corrections and it is conserved under the action of the commutative generators P µ but the energy-momentum tensor is not. Therefore, we determine the set of constraints under which the energy-momentum tensor is divergenceless. Another set of constraints on the fluid potentials is obtained from the requirement of the invariance of the action under the generalization of the volume preserving transformations of the noncommutative spacetime. We show that the proposed action describes noncommutative fluid models by casting the energy-momentum tensor in the familiar fluid form and identifying the corresponding energy and momentum densities. In the commutative limit, they are identical to the corresponding quantities of the relativistic perfect fluids. The energy-momentum tensor contains a dissipative term that is due to the noncommutative spacetime and vanishes in the commutative limit. Finally, we particularize the theory to the case when the complex fluid potentials are characterized by a function K(z,z) that is a deformation of the complex plane and show that this model has important common features with the commutative fluid such as infinitely many conserved currents and a conserved axial current that in the commutative case is associated to the topologically conserved linking number. *

We discuss the quantization of a class of relativistic fluid models defined in terms of one real and two complex conjugate potentials with values on a Kähler manifold, and parametrized by the Kähler potential K(z, z) and a real number λ. In the hamiltonian formulation, the canonical conjugate momenta of the potentials are subjected to second class constraints which allow us to apply the symplectic projector method in order to find the physical degrees of freedom and the physical hamiltonian. We construct the quantum theory for that class of models by employing the canonical quantization methods. We also show that a semiclassical theory in which the Kähler and the complex potential are not quantized has a highly degenerate vacuum. Also, we define and compute the quantum topological number (quantum linking number) operator which has non-vanishing contributions from the Kähler and complex potentials only. Finally, we show that the vacuum and the states formed by tensoring the number operators eigenstates have zero linking number and show that linear combinations of the tensored number operators eigenstates which have the form of entangled states have non-zero linking number. *

In this paper we address the problem of the quantization of the perfect relativistic fluids formulated in terms of the Kähler parametrization. This fluid model describes a large set of interesting systems such as the power law energy density fluids, Chaplygin gas, etc. In order to maintain the generality of the model, we apply the BRST method in the reduced phase space in which the fluid degrees of freedom are just the fluid potentials and the fluid current is classically resolved in terms of them. We determine the physical states in this setting, the time evolution and the path integral formulation. *

We study the dynamics of the noncommutative fluid in the Snyder space perturbatively at the first order in powers of the noncommutative parameter. The linearized noncommutative fluid dynamics is described by a system of coupled linear partial differential equations in which the variables are the fluid density and the fluid potentials. We show that these equations admit a set of solutions that are monocromatic plane waves for the fluid density and two of the potentials and a linear function for the third potential. The energy-momentum tensor of the plane waves is calculated. *