Quantization of the Relativistic Fluid in Physical Phase Space on K\"{a}hler Manifolds

TL;DR

This paper develops a quantum theory for a class of relativistic fluid models on Kähler manifolds, revealing a highly degenerate vacuum and a quantum linking number operator that distinguishes entangled states.

Contribution

It introduces a novel quantization approach for relativistic fluids on Kähler manifolds, including the construction of a quantum linking number operator and analysis of vacuum degeneracy.

Findings

Quantum theory constructed using canonical quantization methods.

Vacuum state exhibits high degeneracy when Kähler and complex potentials are classical.

Entangled states have non-zero quantum linking number, unlike vacuum and tensor product states.

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 K\"{a}hler manifold, and parametrized by the K\"{a}hler potential $K(z,\bar{z})$ and a real number $\lambda$. 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 K\"{a}hler 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 K\"{a}hler 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.