Hydrodynamics of Superfluid Quantum Space: particle of spin-1/2 in a magnetic field

TL;DR

This paper extends the hydrodynamic model of superfluid quantum space to include spin-1/2 particles in magnetic fields, linking quaternion algebra and SU(2) symmetry to quantum spin behavior.

Contribution

It introduces a quaternion-based extension of the Navier-Stokes equation to describe spin-1/2 particles in magnetic fields within a superfluid quantum space framework.

Findings

Extended Navier-Stokes equation includes electromagnetic forces.

Spin behavior modeled as motion of a spin flag on a 2D sphere.

Quaternion group describes spin-1/2 dynamics and electromagnetic interactions.

Abstract

The modified Navier-Stokes equation describing the velocity field in the superfluid quantum space is loaded by the external Lorentz force introducing electromagnetic fields. In order to open the path for getting the \Schrodinger-Pauli equation describing the behavior of a particle with spin-1/2 in the magnetic field we need to extend the continuity equation to take into account conservation of spin flows on the 3D sphere. This extension includes conservation of the density distribution function in 6D space, that is a multiplication of the 3D Euclidean space by the 3D sphere of unit radius. The special unitary group SU(2) underlies the rotations of the spin on this sphere. This group is isomorphic to the group of quaternions containing the real 4x4 matrices of norm 1. Transition to the quaternion group opens up the way to the possibility of describing the spin-1/2 behavior in a magnetic field as a motion of a spin flag on the 2D sphere. Maxwell's electromagnetic field theory manifests itself in the quaternion group basis by the natural manner.