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

Abstract

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. *

Plane waves in noncommutative fluids

“…This type of commutative fluids has been discussed in [37] in the Kähler parametrization, while in [43] it was shown that they can be quantized using canonical methods. Then the equation (35) takes the following simpler form…”

“…The choice of the commutative fluid potentials is not unique. When it is made in terms of real functions θ(x), α(x) and β(x) it is called the Clebsch parametrization [35,36] while the fluid potentials given in terms of one real θ(x) and two complex functions z(x) andz(x), respectively, define the so called Kähler parametrization [37,38,39,40,41,42,43,44].…”

Noncommutative fluid dynamics in the Snyder space-time

“…As shown in [16], the description of the fluid degrees of freedom in terms of fluid potentials allows one to lift the obstruction to inverting the symplectic form in the canonical phase space of the fluid variables. (For other applications of the Kähler parametrization of the fluid potentials see [18,20,21,22,23,24,25]. )…”

Quantum fluids in the Kähler parametrization

“…The consequences of the constraints (9) to the classical theory were analyzed in [8]. In [14] the above results were derived by a different method and the system was quantized for the particular choice K(z,z) = zz and f (ρ) ∼ ρ 2 by applying the canonical quantization method.…”

“…Therefore, it is certainly interesting to understand the quantum systems that correspond to these models. In [14] we have quantized a simple fluid characterized by the Kähler potential of the complex plane K(z,z) = zz and by the function f (ρ) ∼ ρ 2 by applying the canonical quantization methods of the Quantum Field Theory.…”