The quantum hydrodynamic like equations as a function of two real sets of variables, the 4x4 action matrix and the 4 dimensional wave function modulus vector of the Dirac equation, are derived in the present work. The paper shows that in the low velocity limit the equations lead to the hydrodynamic representation of the Pauli equation for charged particle with spin given by Janossy and by Bialynicki.The Lorentz invariance of the relativistic quantum potential that generates the non-local behavior of the quantum mechanics is discussed.
The authors explore certain interconnections between density-functional theory and quantum fluid dynamics of many-electron systems, in the relativistic domain following the hydrodynamical approach adopted by Takabayashi for the one-particle Dirac equation. In order to build a 'classical' hydrodynamical interpretation, the spinor formulation is transformed into a tensor formulation by defining a number of density functions (local observables). These lead to six 'classical' fluid dynamic equations, together with two subsidiary conditions, for a complete specification of the system. The various density functions and the hydrodynamical equations are physically interpreted. The relativistic hydrodynamics discussed here correspond to a 'spinning' fluid. The net many-electron fluid consists of components each of which is characterised by fluid dynamic quantities corresponding to each spinor. The net hydrodynamical quantities are obtained by summing over the occupied spinors. Thus, the earlier nonrelativistic 'classical' picture of the many-electron fluid as a collection of individual fluid components is also valid in the relativistic domain.
The Dirac equation for an electron subjected to a potential A mu is shown to be equivalent to the following classical equations for a beam of non-interacting electrons subjected to A mu in addition to 'quantum' scalar and four-vector potentials together with an anomalous electromagnetic field affecting the spin: the Hamilton-Jacobi equation for the momentum, the continuity equation for the charge distribution, Frenkel's relativistic equation for the spin and a new equation giving a phase theta as an integral along the electron's path. These equations are coupled through the 'quantum' potentials and fields. In the classical limit theta is constant along the electron's path and the equations decouple. Two approximation procedures based on these coupled hydrodynamic equations are given: (1) the semiclassical to lowest order in powers of h(cross), (2) the distorted wave Glauber approximation assumes the solution of the Dirac equation and hence of the hydrodynamic equations for an unperturbed A mu and that the electrons travel along the same four-dimensional paths in the perturbed potential, but with different 'speed'.
Abstract In previous papers we have shown how Schrödinger equations which include an electromagnetic field interaction can be deduced from a fluid dynamical Lagrangian of a charged potential flow that interacts with an electromagnetic field. The quantum behaviour was derived from Fisher information terms which were added to the classical Lagrangian. It was thus shown that a quantum mechanical system is drived by information and not only electromagnetic fields. This program was applied also to Pauli’s equations by removing the restriction of potential flow and using the Clebsch formalism. Although the analysis was quite successful there were still terms that did not admit interpretation, some of them can be easily traced to the relativistic Dirac theory. It is thus suggested to repeat the analysis for a relativistic flow, relating it to the Dirac theory by adding invariant four dimensional Fisher information terms. It is shown that while the classical parts of a classical fluid and a Dirac fluid can be mapped, the Fisher information term of Dirac theory is non-trivial. L Quantum = L Classical Fluid + L Fisher Information
The distributed system
SD described by the Dirac equation is investigated simply as a dynamic system, i.e. without usage of quantum principles. The Dirac equation is described in terms of hydrodynamic variables: 4-flux
ji, pseudo-vector of the spin
Si, an action
ℏϕ and a pseudo-scalar
κ. In the quasi-uniform approximation, when all transversal derivatives (orthogonal to the flux vector
ji) are small, the system
SD turns to a statistical ensemble of classical concentrated systems
Sdc. Under some conditions the classical system
Sdc describes a classical pointlike particle moving in a given electromagnetic field. In general, the world line of the particle is a helix, even if the electromagnetic field is absent. Both dynamic systems
SD and
Sdc appear to be non-relativistic in the sense that the dynamic equations written in terms of hydrodynamic variables are not relativistically covariant with respect to them, although all dynamic variables are tensors or pseudo-tensors. They becomes relativistically covariant only after addition of a constant unit timelike vector
fi which should be considered as a dynamic variable describing a space-time property. This "constant" variable arises instead of
γ-matrices which are removed by means of zero divizors in the course of the transformation to hydrodynamic variables. It is possible to separate out dynamic variables
κ,
κi responsible for quantum effects. It means that, setting
κ,κi≡0, the dynamic system
SD described by the Dirac equation turns to a statistical ensemble
EDqu of classical dynamic systems
Sdc.