Quantization of the relativistic fluid in physical phase space on Kaehler manifolds

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 Kaehler manifold, and parametrized by the Kaehler 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 Kaehler and the complex potentials are not quantized has a highly degenerate vacuum. We define and compute the quantum topological number (quantum linking number) operator which has nonvanishing contributions from the Kaehler and complex potentials only. Also, 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 tensor product of number operators eigenstates which have the form of entangled states have nonzero linking number