A non relativistic electron with a spin is described by Pauli's equation. It has been shown that this system can be interpreted as a vortical fluid which has both similarities and differences with classical ideal flows. Moreover, it was demonstrated that the internal energy of the spin fluid can partially be interpreted in terms of Fisher Information. While previous work on the subject has mainly ignored electromagnetic fields which are represented by a vector potential here we remove this limitation and study the system under general electromagnetic interaction.
An electron moving at velocities much lower that the speed of light with a spin, is described by a wave function which is a solution of Pauli's equation. It has been demonstrated that this system can be viewed as a vortical fluid which has remarkable similarities but also differences with classical ideal flows. In this respect, it was shown that the internal energy of the Pauli fluid can be interpreted, to some degree, as Fisher Information. In previous work on this subject, electromagnetic fields which are represented by a vector potential were ignored, here we remove this limitation and study the system under general electromagnetic interaction.
We develop a quantum representation for Newtonian viscous fluid flows by establishing a mapping between the Navier-Stokes equation (NSE) and the Schrödinger-Pauli equation (SPE). The proposed nonlinear SPE incorporates the two-component wave function and the imaginary diffusion. Consequently, classical fluid flow can be interpreted as a non-Hermitian quantum spin system. Using the SPE-based numerical simulation of viscous flows, we demonstrate the quantum/wavelike behavior in flow dynamics. Furthermore, the SPE equivalent to the NSE can facilitate the quantum simulation of fluid dynamics. Published by the American Physical Society 2024
Classical particle mechanics on curved spaces is related to the flow of ideal fluids, by a dual interpretation of the Hamilton-Jacobi equation. As in second quantization, the procedure relates the description of a system with a finite number of degrees of freedom to one with infinitely many degrees of freedom. In some two-dimensional fluid mechanics models a duality transformation between the velocity potential and the stream function can be performed relating sources and sinks in one model to vortices in the other. The particle mechanics counterpart of the dual theory is reconstructed. In the quantum theory the strength of sources and sinks, as well as vorticity are quantized; for the duality between theories to be preserved these quantization conditions must be related.
In the first part (Secs. I and II) of this paper, starting from the Pauli current, we obtain the decomposition of the nonrelativistic field velocity into two orthogonal parts: (i) the ``classical'' part, that is, the velocity
w=p/m in the center of mass (c.m.), and (ii) the ``quantum'' part, that is, the velocity V of the motion of the c.m. frame (namely, the internal ``spin motion'' or Zitterbewegung). By inserting such a complete, composite expression of the velocity into the kinetic-energy term of the nonrelativistic classical (i.e., Newtonian) Lagrangian, we straightforwardly get the appearance of the so-called quantum potential associated, as it is known, with the Madelung fluid. This result provides further evidence of the possibility that the quantum behavior of microsystems is a direct consequence of the fundamental existence of spin. In the second part (Secs. III and IV), we fix our attention on the total velocity
v=w+V, now necessarily considering relativistic (classical) physics. We show that the proper time entering the definition of the four-velocity ${v}^{\ensuremath{\mu}}$ for spinning particles has to be the proper time \ensuremath{\tau} of the c.m. frame. Inserting the correct Lorentz factor into the definition of ${v}^{\ensuremath{\mu}}$ leads to completely different kinematical properties for
v2. The important constraint ${p}_{\ensuremath{\mu}}{v}^{\ensuremath{\mu}}=m,$ identically true for scalar particles but just assumed a priori in all previous spinning-particle theories, is herein derived in a self-consistent way.