Fisher Information and Electromagnetic Interacting Dirac Spinors

Abstract

In earlier works, it was demonstrated that Schrödinger’s equation, which includes interactions with electromagnetic fields, can be derived from a fluid dynamic Lagrangian framework. This approach treats the system as a charged potential flow interacting with an electromagnetic field. The emergence of quantum behavior was attributed to the inclusion of Fisher information terms in the classical Lagrangian. This insight suggests that quantum mechanical systems are influenced not just by electromagnetic fields but also by information, which plays a fundamental role in driving quantum dynamics. This methodology was extended to Pauli’s equations by relaxing the constraint of potential flow and employing the Clebsch formalism. Although this approach yielded significant insights, certain terms remained unexplained. Some of these unresolved terms appear to be directly related to aspects of the relativistic Dirac theory. In a recent work, the analysis was revisited within the context of relativistic flows, introducing a novel perspective for deriving the relativistic quantum theory but neglecting the interaction with electromagnetic fields for simplicity. This is rectified in the current work, which shows the implications of the field in the current context.

1. Introduction

Quantum theory is often interpreted using the Copenhagen perspective, which treats the quantum mechanical wave as an instrument for evaluating measurement outcomes rather than representing physical reality. This approach, influenced by Kantian philosophy, suggests that humans cannot perceive the true nature of reality (“ontology”) [1]. On the other hand, an alternative viewpoint, supported by Bohm [2,3,4], regards the quantum wave as physical, comparable to physical electromagnetic fields. Interpretations like this were the origins of models like Madelung’s fluid analogy [5,6], where the quantum wave’s squared magnitude is the fluid’s density, and the quantum wave’s phase corresponds to the potential of the fluid’s velocity. Notice, however, that this framework is restricted in scope, as it applies only to spinless electrons and fails to fully describe their properties, even at low velocities.

In 1927, Wolfgang Pauli [7] introduced a quantum equation specifically for describing spinors in a non-relativistic context. This equation features a Hamiltonian expressed as a two-dimensional operator matrix. Later, it was demonstrated that the Pauli equation could be interpreted through a fluid dynamics framework [8]. This insight holds particular importance, as advocates of the Copenhagen interpretation of quantum mechanics often point to the concept of spin as inherently quantum, lacking any classical analogy or interpretation. The fluid dynamic perspective challenges this notion by offering a classical-like representation of quantum spin phenomena.

Holland [3] and others applied a Bohmian interpretation to the Pauli equation, exploring its implications for quantum mechanics. However, their analysis did not delve into the connection between the Pauli theory and fluid dynamics, nor did it examine the concept of spin-related vorticity. To address this gap, the framework of spin fluid dynamics was later developed [8] to describe the behavior of a single electron with spin, providing a novel perspective on the interplay between quantum spin and fluid-like systems.

Reinterpreting Pauli’s spinor through fluid density and velocity variables builds on the foundation of Clebsch’s 19th-century work, which is integral to the Eulerian variational framework for fluid dynamics. Clebsch [9,10] introduced a four-function variational principle to describe the behavior of barotropic fluids in an Eulerian framework, a methodology later reintroduced by Davidov [11], who sought to quantize fluid dynamics. However, Davidov’s contributions were less recognized internationally due to language barriers. Separately, Eckart [12] provided a variational principle for fluid dynamics in the Lagrangian formulation, which contrasts with Clebsch’s approach focused on Eulerian dynamics. These interpretations provide deeper insights into fluid mechanics and its parallels with quantum systems.

Early efforts to establish variational principles for Eulerian fluid dynamics in the literature written in English were attempted by Herivel [13], Serrin [14], and Lin [15]. These approaches were notably complex, using multiple multipliers (as suggested by Lagrange’s method), as well as additional potential functions. This complexity resulted in systems requiring between seven and eleven independent functions—far exceeding the four functions needed to describe the Euler and mass conservation equations for barotropic flow. Consequently, these approaches were deemed useless for straightforward application.

Seliger and Whitham [16,17] have reinvented Clebsch’s principle of variation, using a four variable description of a barotropic fluid. Later, Lynden-Bell and Katz [18] developed a Lagrangian based on dual variables: $\lambda$ load and $\rho$ density. However, the suggested method implicitly defined the velocity $\overrightarrow{v}$, requiring a partial differential equation to relate $\overrightarrow{v}$ to $\rho$, $\lambda$, and their variations. To overcome this, Yahalom and Lynden-Bell [19] introduced an additional variational variable, enabling fully unconstrained variations and providing an explicit formula for $\overrightarrow{v}$.

One of the main difficulties in explaining quantum mechanics using fluid dynamics is interpreting thermodynamic properties. In conventional fluids, quantities such as specific enthalpy, pressure, and temperature are linked to specific internal energy, which depends uniquely on entropy and density as determined by the equation of state. This internal energy can be understood in terms of the fluid’s microscopic structure, with statistical physics describing how atoms, ions, electrons, and molecules interact via electromagnetic forces.

Quantum fluids, however, do not have the same kind of microscopic structure as traditional fluids. Despite this, the equations governing the dynamics of both spinless [5,6] and spin [8] quantum fluids contain terms that resemble internal energies. This raises the question of the origin of these internal energies. The idea that quantum fluids might possess a microscopic substructure conflicts with experimental evidence, showing that electrons are point particles with no internal structure.

The key to understanding this lies in the measurement theory, specifically Fisher information [20,21,22,23,24], which quantifies the standard deviation of a random variable. It was shown that the information quantifier defined by Fisher substitutes the internal energy for an electron, provided that this electron is both non-relativistic and spinless. Alternatively, one can look at Fisher information as the target function. The other part of the action serves to constrain the probability such that it is conserved. Thus the total probability in each infinitesimal volume is equal to the total probability flux (positive or negative) entering it. In this context, it is worthwhile to mention the important contribution of Heifetz and Cohen [25], who have underlined that for a spinless electron, the internal energy which is identified with Fisher information is also identified with the mean of quantum potential of the Madelung equation. Moreover, Bloch and Cohen [26] have shown that the identification of the mean quantum potential with Fisher information leads to, through the Cramér–Rao bound, to an uncertainty principle which is stronger, in general, than both Heisenberg and Robertson–Schrödinger uncertainty relations, allowing one to experimentally test the validity of such an identification.

Using the Fisher information approach one can also partially justify the “internal energy” of an electron with spin (but non-relativistic) [8,27,28,29,30,31,32,33].

Frieden [34] attempted to obtain a theoretical description of many physical systems using Fisher’s quantifier of information. In his method, one must always use an additional J term in the action principle, which is specific for each physical system. This term is chosen arbitrarily, without any clear justification, in order to produce the desired Lagrangian for the system.

When Clebsch developed his variational principle, relativity had not yet been introduced, so the need had not risen for a Lagrangian for relativistic Eulerian fluids (symmetrical under Lorentz group). Later, a series of papers [35,36,37] addressed this issue, introducing relativistic Clebsch flows. We demonstrated that relativistic Clebsch flows lead to a relativistic quantum theory by incorporating a Lorentz symmetric Fisher quantifier of information. However, this theory does not describe the Stern Gerlach terms; nevertheless, for slow velocity fields with zero rotation, this Lagrangian reduces to Schrödinger’s Lagrangian.

It seems necessary to compare the flow approach to the relativistic quantum theory with Dirac’s theory, which is widely acknowledged as the correct relativistic theory of the electron. This was already carried out by Takabayasi [38,39] many years before the development of the relativistic Clebsch formalism. The Takabayasi formalism involves the definition of a scalar and a pseudoscalar, two four vectors and one pseudo four vector, with a total of fourteen degrees of freedom. This is obviously much more than the eight degrees of freedom of the original Dirac formulation. However, the degrees of freedom are restricted by algebraic and differential restrictions. It is quite difficult to see any correspondence between this “fluid” and any natural fluid. Thus, it is beneficial to see if Dirac’s theory can be cast in terms of the relativistic Clebsch form, which is the purpose of the current paper. This task was partitioned [40,41] into multiple stages. In the first stage, the Dirac equation is reformulated using four variables (which we later connect to the velocity vector and density) instead of double the sum of variables (a complex four spinor or eight real variables) used in the equation of Dirac. In a previous work, this was completed by ignoring the electromagnetic interaction for simplicity. However, here, we also discuss the implication of a field on the Fisher information approach. For pedagogical reasons, this is carried out only after repeating the discussion of the same without an electromagnetic interaction.

2. Fisher Information

Let there be a random variable X with probability density function (PDF) $f_X\left(x\right)$. Fisher information for a PDF which is translationaly invariant is given by the following form:

$$F_I=\int dx{\left({\displaystyle \frac{d{f}_X}{dx}}\right)}^2{\displaystyle \frac{1}{f_X}}$$

(1)

It was shown [20,34] that the standard deviation ${\sigma }_X$ of any random variable is bounded from below, such that

$${\sigma }_X\ge {\sigma }_{Xmin}={\displaystyle \frac{1}{F_I}}$$

(2)

Hence, the more Fisher information we have about the variable, the smaller standard deviation we may achieve. Thus, our knowledge about the value of this random variable is greater. This is known as the Cramér–Rao inequality. Fisher information is most elegantly introduced in terms of the probability amplitude:

$$f_X=a^2\Rightarrow F_I=4\int dx{\left({\displaystyle \frac{da}{dx}}\right)}^2.$$

(3)

This is the only reason one can justify using the probability density amplitude (the square root of the probability density) in quantum mechanics instead of using the more intuitive probability density. This is explained in terms of simplifying the Fisher information expression above. We will be interested in a three-dimensional (and higher dimensional) random variable designating the position of an electron, as follows:

$$F_I=\int d^{3}x{\left(\overrightarrow{\nabla }{f}_{\overrightarrow{X}}\right)}^2{\displaystyle \frac{1}{f_{\overrightarrow{X}}}}=4\int d^{3}x{\left(\overrightarrow{\nabla }a\right)}^2\equiv \int d^{3}x{\mathcal{F}}_I$$

(4)

In the above equation, ${\mathcal{F}}_I\equiv 4{\left(\overrightarrow{\nabla }a\right)}^2$ is the Fisher information density.

3. Dirac Equation of a Free Field

Dirac’s equation is formulated using the following equation, where electromagnetic interactions are initially disregarded:

$$\left(i\hslash {\gamma }^{\mu }{\mathsf{\partial }}_{\mu }-mc\right)\Psi =0.$$

(5)

The Greek letter $\Psi$ denotes a 4-dimensional column vector of complex functions, known as a spinor. The matrices ${\gamma }^{\mu }$ are $4\times 4$-dimensional matrices of complex constants that satisfy specific anticommutation relations:

$$\left\{{\gamma }^{\mu },{\gamma }^{\nu }\right\}=2{\eta }^{\mu \nu }{I}_4,{\eta }^{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$$

(6)

$I_4$ is the identity matrix in four dimensions. In the following, indices taken from the Greek alphabet, e.g., $\mu ,\nu$, belong to the following set of natural numbers: $\{0,1,2,3\}$. Indices taken from the Latin alphabet, e.g., $i,j,k$, are taken from the following set of natural numbers: $\{1,2,3\}$. There are several possible representations of ${\gamma }^{\nu }$. We will use the following specific representation:

$${\gamma }^0=\left(\begin{array}{cc}{I}_2& 0\\ 0& -I_2\end{array}\right){\gamma }^i=\left(\begin{array}{cc}0& {\sigma }^i\\ -{\sigma }^i& 0\end{array}\right)$$

(7)

$${\sigma }^1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }^2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }^3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right).$$

(8)

$I_2$ is a $2\times 2$ identity matrix, m is the mass of the particle and c is the velocity of light in a vacuum. Equation (5) will be solved uniquely if initial values are provided, as follows:

$$\Psi (0,\overrightarrow{x})={\Psi }_0\left(\overrightarrow{x}\right)$$

(9)

At first glance, the equation appears unrelated to fluid dynamics because $\Psi$ depends on eight scalar quantities, while the barotropic fluid theory only requires four variables (half as many). However, the theory can be reformulated using fewer variables. To carry this out, we begin by expressing the four-dimensional spinor in terms of two-dimensional spinors:

$$\Psi =\left(\begin{array}{c}{\psi }_1\\ {\psi }_2\end{array}\right).$$

(10)

This form leads to initial conditions for both ${\psi }_1$ and ${\psi }_2$, as specified by Equation (9):

$${\psi }_1(0,\overrightarrow{x})={\psi }_{10}\left(\overrightarrow{x}\right),{\psi }_2(0,\overrightarrow{x})={\psi }_{20}\left(\overrightarrow{x}\right),{\Psi }_0=\left(\begin{array}{c}{\psi }_{10}\\ {\psi }_{20}\end{array}\right).$$

(11)

By inserting Equation (10) into Equation (5), we derive

$$(i\hslash {\mathsf{\partial }}_0-mc){\psi }_1+i\hslash {\sigma }^i{\mathsf{\partial }}_i{\psi }_2=0,(i\hslash {\mathsf{\partial }}_0+mc){\psi }_2+i\hslash {\sigma }^i{\mathsf{\partial }}_i{\psi }_1=0$$

(12)

We introduce hatted variables as follows:

$${\widehat{\psi }}_1\equiv e^{-{\displaystyle \frac{imc}{\hslash }}{x}_0}{\psi }_1=e^{-{\displaystyle \frac{im{c}^2}{\hslash }}t}{\psi }_1,{\widehat{\psi }}_2\equiv e^{-{\displaystyle \frac{imc}{\hslash }}{x}_0}{\psi }_2=e^{-{\displaystyle \frac{im{c}^2}{\hslash }}t}{\psi }_2,$$

(13)

in which the temporal coordinate $x_0$ (measured in meters) is related to t (measured in seconds) by the unit conversion factor c, such that $x_0=ct$. We can substitute

$${\psi }_1=e^{+{\displaystyle \frac{imc}{\hslash }}{x}_0}{\widehat{\psi }}_1,{\psi }_2=e^{+{\displaystyle \frac{imc}{\hslash }}{x}_0}{\widehat{\psi }}_2.$$

(14)

into Equation (12). This results in a simplified set of equations, as follows:

$$(i\hslash {\mathsf{\partial }}_0-2mc){\widehat{\psi }}_1+i\hslash {\sigma }^i{\mathsf{\partial }}_i{\widehat{\psi }}_2=0,i\hslash {\mathsf{\partial }}_0{\widehat{\psi }}_2+i\hslash {\sigma }^i{\mathsf{\partial }}_i{\widehat{\psi }}_1=0$$

(15)

The initial values at $x_0=0$ remain identical, since $\psi$ & $\widehat{\psi }$ are the same at the initial time.

$${\widehat{\psi }}_1(0,\overrightarrow{x})={\psi }_{10}\left(\overrightarrow{x}\right),{\widehat{\psi }}_2(0,\overrightarrow{x})={\psi }_{20}\left(\overrightarrow{x}\right).$$

(16)

${\widehat{\psi }}_2$ is easily obtained once ${\widehat{\psi }}_1$ is given.

$${\widehat{\psi }}_2(x_0,\overrightarrow{x})\left[{\widehat{\psi }}_1\right]={\widehat{\psi }}_2(0,\overrightarrow{x})-{\sigma }^i{\mathsf{\partial }}_i{\int }_0^{x_0}{\widehat{\psi }}_1(x_0^{\prime },\overrightarrow{x})d{x}_0^{\prime }$$

(17)

We define an additional function, which is shown below.

$$int{\widehat{\psi }}_1\equiv {\int }_0^{x_0}{\widehat{\psi }}_1(x_0^{\prime },\overrightarrow{x})d{x}_0^{\prime }\Rightarrow {\mathsf{\partial }}_{0}int{\widehat{\psi }}_1={\widehat{\psi }}_1,{\mathsf{\partial }}_0^{2}int{\widehat{\psi }}_1={\mathsf{\partial }}_0{\widehat{\psi }}_1.$$

(18)

We also define the time independent spinor as follows:

$$W\left(\overrightarrow{x}\right)\equiv -{\sigma }^k{\mathsf{\partial }}_k{\widehat{\psi }}_2(0,\overrightarrow{x}).$$

(19)

We can now express the left-hand side of Equation (15) using the definition of $int{\widehat{\psi }}_1$ and W, which are given in Equation (18) and Equation (19), respectively, as well as by using the Pauli matrix identity:

$${\sigma }^k{\sigma }^i={\delta }^{ki}{I}_2+i{ϵ}^{kil}{\sigma }^l,$$

(20)

in which ${\delta }^{ki}$ is Kronecker’s delta and ${ϵ}^{kil}$ is the antisymmetric tensor. We notice that the ${ϵ}^{kil}$ term does not contribute to a free Dirac spinor, but it does have profound consequences for a Dirac spinor influenced by a magnetic field, which will be discussed later. Consequently, the Dirac theory takes the following form:

$$({\mathsf{\partial }}^{\mu }{\mathsf{\partial }}_{\mu }+2{\displaystyle \frac{imc}{\hslash }}{\mathsf{\partial }}_0)int{\widehat{\psi }}_1=W\left(\overrightarrow{x}\right),{\widehat{\psi }}_2(x_0,\overrightarrow{x})={\widehat{\psi }}_2(0,\overrightarrow{x})-{\sigma }^i{\mathsf{\partial }}_{i}int{\widehat{\psi }}_1$$

(21)

Therefore, the aim in Dirac equation is to find a solution for the left-hand side of Equation (21), since the equation on the right simply provides an explicit relation for ${\widehat{\psi }}_2$ in terms of ${\widehat{\psi }}_1$. Furthermore, once we have a solution for ${\widehat{\psi }}_1$, the integral of ${\widehat{\psi }}_1$ can be immediately determined using Equation (18). By calculating the time differential of the equation on the left in (21), we see that ${\widehat{\psi }}_1$ is a solution of the following equation:

$$({\mathsf{\partial }}^{\mu }{\mathsf{\partial }}_{\mu }+2{\displaystyle \frac{imc}{\hslash }}{\mathsf{\partial }}_0){\widehat{\psi }}_1=0$$

(22)

The initial values for this differential equation of order 2 are determined by the initial values of Equation (15), as those values also specify the time differentials at $x_0=0$.

$$(i\hslash {\mathsf{\partial }}_0-2mc){\widehat{\psi }}_1{{|}_{x_0=0}+i\hslash {\sigma }^i{\mathsf{\partial }}_i{\psi }_{20}=0,{\mathsf{\partial }}_0{\widehat{\psi }}_2|}_{x_0=0}+{\sigma }^i{\mathsf{\partial }}_i{\psi }_{10}=0.$$

(23)

Or, more simply, we have

$${\mathsf{\partial }}_0{\widehat{\psi }}_1{|}_{x_0=0}={\displaystyle \frac{2mc}{i\hslash }}{\psi }_{10}-{\sigma }^i{\mathsf{\partial }}_i{\psi }_{20}$$

(24)

As the initial value of the function and the initial value of its first differential are given, we solve differential Equation (22) of order two uniquely. Thus, the Dirac equation is solved. We are reminded that one can use the original function ${\psi }_1$ through Equation (13), which will yield a Klein–Gordon-type equation:

$$({\mathsf{\partial }}^{\mu }{\mathsf{\partial }}_{\mu }+{\displaystyle \frac{m^2{c}^2}{{\hslash }^2}}){\psi }_1=0$$

(25)

with the following initial conditions:

$${\psi }_1{{|}_{x_0=0}={\psi }_{10},{\mathsf{\partial }}_0{\psi }_1|}_{x_0=0}={\displaystyle \frac{mc}{i\hslash }}{\psi }_{10}-{\sigma }^i{\mathsf{\partial }}_i{\psi }_{20}.$$

(26)

This is also equivalent to Dirac’s theory. However, it is important to note two things: first, in this case, the Klein–Gordon equation describes the evolution of a spinor with two dimensions, not a complex scalar evolution. Moreover, the interpretation suggested by Dirac differs significantly from the original Klein–Gordon theory. Specifically, the conserved probability four-current is

$$J^{\mu }\equiv \overline{\Psi }{\gamma }^{\mu }\Psi ,\overline{\Psi }\equiv {\Psi }^{†}{\gamma }^0$$

(27)

As a result, we obtain the probability density as follows:

$$J^0=\overline{\Psi }{\gamma }^0\Psi ={\Psi }^{†}{\left({\gamma }^0\right)}^2\Psi ={\Psi }^{†}\Psi ={\psi }_1^{†}{\psi }_1+{\psi }_2^{†}{\psi }_2\ge 0.$$

(28)

$J^0$ differs significantly from the probability density in Klein–Gordon’s theory. This probability may become unpositive and therefore unphysical. However, it is demonstrated from a mathematical point of view that the two theoretical frameworks involve equations of an identical form, though they differ in the mathematical variables used. For the Klein–Gordon framework, we consider complex scalars. On the other hand, in the Dirac framework, spinors are considered. At this point, we are able to highlight the correspondence to relativistic fluids, as both physical systems rely on the same quantity of four real functions.

4. Lagrangian Formalism

The Lagrangian density leads to Equation (25), as follows:

$${\mathcal{L}}_{KG}\equiv m\left({\displaystyle \frac{{\hslash }^2}{m^2}}{\mathsf{\partial }}^{\mu }{\psi }_1^{†}{\mathsf{\partial }}_{\mu }{\psi }_1-c^2{\psi }_1^{†}{\psi }_1\right),{\mathcal{A}}_{KG}\equiv \int d^{4}x{\mathcal{L}}_{KG}.$$

(29)

This equation holds as long as the variations are appropriately constrained at the boundaries, which are both spatial and temporal. Although it differs from the standard Lagrangian density of Dirac equation, it is demonstrated in the previous section that mathematical information in both is identical. Now, we express ${\psi }_1$ in the following form:

$${\psi }_1=\left(\begin{array}{c}{\psi }_{\uparrow }\\ {\psi }_{\downarrow }\end{array}\right).$$

(30)

By substituting this into the Lagrangian density, it follows that

$${\mathcal{L}}_{KG}=m\left({\displaystyle \frac{{\hslash }^2}{m^2}}{\mathsf{\partial }}^{\mu }{\psi }_{\uparrow }^{*}{\mathsf{\partial }}_{\mu }{\psi }_{\uparrow }-c^2{\psi }_{\uparrow }^{*}{\psi }_{\uparrow }+{\displaystyle \frac{{\hslash }^2}{m^2}}{\mathsf{\partial }}^{\mu }{\psi }_{\downarrow }^{*}{\mathsf{\partial }}_{\mu }{\psi }_{\downarrow }-c^2{\psi }_{\downarrow }^{*}{\psi }_{\downarrow }\right).$$

(31)

Next, we express the upper and lower components as follows:

$${\psi }_{\uparrow }=R_{\uparrow }{e}^{i{\displaystyle \frac{m}{\hslash }}{\nu }_{\uparrow }},{\psi }_{\downarrow }=R_{\uparrow }{e}^{i{\displaystyle \frac{m}{\hslash }}{\nu }_{\downarrow }}$$

(32)

Using Equation (32) in Equation (31), we obtain the following form:

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{KG}& =& {\mathcal{L}}_{KGq}+{\mathcal{L}}_{KGc}\hfill \\ \hfill {\mathcal{L}}_{KGq}& \equiv & {\displaystyle \frac{{\hslash }^2}{m}}\left({\mathsf{\partial }}^{\mu }{R}_{\uparrow }{\mathsf{\partial }}_{\mu }{R}_{\uparrow }+{\mathsf{\partial }}^{\mu }{R}_{\downarrow }{\mathsf{\partial }}_{\mu }{R}_{\downarrow }\right)\hfill \\ \hfill {\mathcal{L}}_{KGc}& \equiv & m\left(R_{\uparrow }^2\left({\mathsf{\partial }}^{\mu }{\nu }_{\uparrow }{\mathsf{\partial }}_{\mu }{\nu }_{\uparrow }-c^2\right)+R_{\downarrow }^2\left({\mathsf{\partial }}^{\mu }{\nu }_{\downarrow }{\mathsf{\partial }}_{\mu }{\nu }_{\downarrow }-c^2\right)\right).\hfill \end{array}$$

(33)

in which we partition ${\mathcal{L}}_{KG}$ into a quantum part ${\mathcal{L}}_{KGq}$ along with a classical component, ${\mathcal{L}}_{KGc}$. In the classical limit, as $\hslash \to 0$ is

$$\underset{\hslash \to 0}{lim}{\mathcal{L}}_{KGq}=0\Rightarrow \underset{\hslash \to 0}{lim}{\mathcal{L}}_{KG}={\mathcal{L}}_{KGc},$$

(34)

this leads to the following natural definition of the density $\overline{\rho }$:

$$\overline{\rho }\equiv m\left(R_{\uparrow }^2+R_{\downarrow }^2\right),tan\theta \equiv {\displaystyle \frac{R_{\downarrow }}{R_{\uparrow }}},\Rightarrow R_{\uparrow }=\sqrt{{\displaystyle \frac{\overline{\rho }}{m}}}cos\theta ,R_{\downarrow }=\sqrt{{\displaystyle \frac{\overline{\rho }}{m}}}sin\theta$$

(35)

From this, it follows that

$${\mathcal{L}}_{KGc}=\overline{\rho }\left[{cos}^2\theta {\mathsf{\partial }}^{\mu }{\nu }_{\uparrow }{\mathsf{\partial }}_{\mu }{\nu }_{\uparrow }+{sin}^2\theta {\mathsf{\partial }}^{\mu }{\nu }_{\downarrow }{\mathsf{\partial }}_{\mu }{\nu }_{\downarrow }-c^2\right].$$

(36)

Now, let us define

$$\begin{array}{ccc}\hfill \nu & \equiv & {\nu }_{\uparrow },\beta \equiv {\nu }_{\downarrow }-{\nu }_{\uparrow },\hfill \\ \hfill \alpha & \equiv & {\displaystyle \frac{-{\mathsf{\partial }}_{\mu }\nu {\mathsf{\partial }}^{\mu }\beta \pm \sqrt{{\left({\mathsf{\partial }}_{\mu }\nu {\mathsf{\partial }}^{\mu }\beta \right)}^2+{sin}^2\theta \left({\mathsf{\partial }}_{\mu }\beta {\mathsf{\partial }}^{\mu }\beta \right)\left({\mathsf{\partial }}^{\mu }\beta (2{\mathsf{\partial }}_{\mu }\nu +{\mathsf{\partial }}_{\mu }\beta )\right)}}{{\mathsf{\partial }}_{\mu }\beta {\mathsf{\partial }}^{\mu }\beta }}\hfill \end{array}$$

(37)

Using these, we define a four-dimensional Clebsch field:

$${\tilde{v}}_C^{\mu }\equiv \alpha {\mathsf{\partial }}^{\mu }\beta +{\mathsf{\partial }}^{\mu }\nu$$

(38)

where $\tilde{v}$ notation signifies that since $\alpha ,\beta$ and $\nu$ are defined with respect to a two spinor, and thus are not true Lorentz scalars, ${\tilde{v}}_C^{\mu }$ is therefore not a true Lorentz four vector. By substituting Equation (38) and using the definitions from Equation (37), we arrive at the result after some straightforward, though somewhat tedious, calculations:

$${\mathcal{L}}_{KGc}=\overline{\rho }\left[{\tilde{v}}_C^{\mu }{\tilde{v}}_{C\mu }-c^2\right].$$

(39)

We define the mass density in the rest frame as

$${\rho }_0={\displaystyle \frac{\overline{\rho }}{c}}\left[\sqrt{{\tilde{v}}_C^{\mu }{\tilde{v}}_{C\mu }}+c\right].$$

(40)

It follows that

$${\mathcal{L}}_{KGc}=c{\rho }_0\left[\sqrt{{\tilde{v}}_C^{\mu }{\tilde{v}}_{C\mu }}-c\right]={\mathcal{L}}_{\mathrm{Relativistic}\mathrm{Flow}}.$$

(41)

The non-quantum component of ${\mathcal{L}}_{KG}$ is mapped into the Lagrangian density of a classical relativistic flow, lacking internal energy (observe Equation (103) of [35]). It is important to notice that, in comparison to the Pauli spin flow which has a Lagrangian density containing a component that cannot be justified, the flow framework (see Equation (63) in [33]) is

$$\underset{\hslash \to 0}{lim}{\epsilon }_{qs}={\displaystyle \frac{1}{2}}(1-{\alpha }^2){\left(\overrightarrow{\nabla }\beta \right)}^2$$

(42)

In the theory of Dirac, there is a direct correspondence between components that appear in the classical section of Dirac’s Lagrangian and the relativistic flow dynamics Lagrangian, with no extra terms or inconsistencies.

5. The Dirac Quantum Term

Now, let us compare the Dirac quantum term ${\mathcal{L}}_{KGq}$ from Formula (33) with the Fisher information quantifier, which one usually attributes quantum behavior to. That is Formula (113) of [35], which is given below.

$${\mathcal{L}}_{RFq}={\displaystyle \frac{{\hslash }^2}{2m}}{\mathsf{\partial }}^{\mu }{a}_0{\mathsf{\partial }}_{\mu }{a}_0,a_0\equiv \sqrt{{\displaystyle \frac{{\rho }_0}{m}}}.$$

(43)

At first glance, these terms appear quite similar; however, upon a closer inspection, significant differences become apparent. First, ${\mathcal{L}}_{KGq}$ depends on two “density amplitudes” (one for each spin), whereas ${\mathcal{L}}_{RFq}$ depends on just a single amplitude. This may be related to the fact that every eigenstate of the Dirac Hamiltonian can accept two particles with a unique spin. Also, ${\mathcal{L}}_{KGq}$ is missing a division by 2. We address these issues by examining Formula (40), where it is underlined that $R_{\uparrow }$ and $R_{\downarrow }$ do not correspond directly to the mass density since

$${\rho }_0=\overline{\rho }\left[\sqrt{{\displaystyle \frac{{\tilde{v}}_C^{\mu }{\tilde{v}}_{C\mu }}{c^2}}}+1\right].$$

(44)

However, as stated in Equation (104) of [35],

$$\sqrt{{\tilde{\tilde{v}}}_{C\mu }{\tilde{v}}_C^{\mu }}=\left|{\tilde{v}}_{C0}\right|\sqrt{1-{\displaystyle \frac{{\overrightarrow{\tilde{v}}}_C^2}{{\tilde{v}}_{C0}^2}}}=\left|{\tilde{v}}_{C0}\right|\sqrt{1-{\displaystyle \frac{{\overrightarrow{v}}^2}{c^2}}}={\displaystyle \frac{|{\tilde{v}}_{C0}|}{\gamma }}$$

(45)

Also, according to Equation (101) of [35],

$$|{\tilde{v}}_{C0}|=c\lambda$$

(46)

If a classical flow has no internal energy and follows the laws of motion (see Formula (58) in [35]), we have

$$\lambda =\gamma .$$

(47)

Therefore, with the exception of quantum corrections, we have

$$\sqrt{{\tilde{v}}_{C\mu }{\tilde{v}}_C^{\mu }}\approx c$$

(48)

By substituting Equation (48) into Equation (44), we obtain

$${\rho }_0\approx 2\overline{\rho }.$$

(49)

Thus,

$$a_0=\sqrt{{\displaystyle \frac{{\rho }_0}{m}}}\approx \sqrt{{\displaystyle \frac{2\overline{\rho }}{m}}}=\sqrt{2}R,R^2\equiv R_{\downarrow }^2+R_{\uparrow }^2$$

(50)

Using R and $\theta$, the quantum component of the Lagrangian density can be written as

$${\mathcal{L}}_{KGq}={\displaystyle \frac{{\hslash }^2}{m}}\left({\mathsf{\partial }}^{\mu }{R}_{\uparrow }{\mathsf{\partial }}_{\mu }{R}_{\uparrow }+{\mathsf{\partial }}^{\mu }{R}_{\downarrow }{\mathsf{\partial }}_{\mu }{R}_{\downarrow }\right)={\displaystyle \frac{{\hslash }^2}{m}}\left({\mathsf{\partial }}^{\mu }R{\mathsf{\partial }}_{\mu }R+R^2{\mathsf{\partial }}^{\mu }\theta {\mathsf{\partial }}_{\mu }\theta \right).$$

(51)

Thus,

$${\mathcal{L}}_{KGq}\approx {\displaystyle \frac{{\hslash }^2}{2m}}\left({\mathsf{\partial }}^{\mu }{a}_0{\mathsf{\partial }}_{\mu }{a}_0+a_0^2{\mathsf{\partial }}^{\mu }\theta {\mathsf{\partial }}_{\mu }\theta \right).$$

(52)

It is important to note that in Dirac’s theory, R is not a probability amplitude, as stated in Equation (28):

$$J^0={\psi }_1^{†}{\psi }_1+{\psi }_2^{†}{\psi }_2=R^2+{\psi }_2^{†}{\psi }_2\ge R^2.$$

(53)

Therefore, the second term in the quantum Lagrangian density is probably not unexpected. Naturally, a complete calculation would require accounting for quantum effects, which were ignored in Formula (48).

6. Dirac Equation with an Electromagnetic Field

We shall now address Dirac’s equation in which electromagnetic interactions are added, as follows:

$$\left(i\hslash {\gamma }^{\mu }{D}_{\mu }-mc\right)\Psi =0,D_{\mu }\equiv {\mathsf{\partial }}_{\mu }+i{\displaystyle \frac{e}{\hslash }}{A}_{\mu },$$

(54)

where $A^{\mu }=({\displaystyle \frac{\varphi }{c}},\overrightarrow{A})$ is the electromagnetic four-dimensional vector potential, $A_{\mu }={\eta }_{\mu \nu }{A}^{\nu }$, and e is the charge of the particle under consideration. It is well known that the four potential is defined up to a local gauge transformation:

$$A_{\nu }^{\prime }=A_{\nu }-{\displaystyle \frac{\hslash }{e}}{\mathsf{\partial }}_{\nu }\alpha$$

(55)

For the vector potential $A_{\nu }^{\prime }$, one can construct a solution for the Dirac equation ${\Psi }^{\prime }$ by multiplying the solution $\Psi$ by an exponent of the imaginary number i times a local phase $\alpha$, such that

$${\Psi }^{\prime }=e^{i\alpha }\Psi .$$

(56)

Thus, we can always choose a gauge, such that

$$A_0=0,D_0={\mathsf{\partial }}_0.$$

(57)

This solution can be transformed into a more general solution $A_0^{\prime }\ne 0$ using the gauge transformation of Equation (55), as follows:

$${\mathsf{\partial }}_0\alpha =-{\displaystyle \frac{e{A}_0^{\prime }}{\hslash }}\Rightarrow \alpha =-{\displaystyle \frac{ec}{\hslash }}\int dt{A}_0^{\prime }+\overline{\alpha }\left(\overrightarrow{x}\right),$$

(58)

where $\overline{\alpha }\left(\overrightarrow{x}\right)$ is an arbitrary function of the spatial coordinates. Again, Equation (54) will be solved uniquely if initial values are provided, as follows:

$$\Psi (0,\overrightarrow{x})={\Psi }_0\left(\overrightarrow{x}\right)$$

(59)

At first glance, the equation appears unrelated to fluid dynamics because $\Psi$ depends on eight scalar quantities, while the barotropic fluid theory only requires four variables (half as many). However, the theory can be reformulated using fewer variables. To do this, we begin by expressing the four-dimensional spinor in terms of two-dimensional spinors as is the case for a free spinor:

$$\Psi =\left(\begin{array}{c}{\psi }_1\\ {\psi }_2\end{array}\right).$$

(60)

This form leads to initial conditions for both ${\psi }_1$ and ${\psi }_2$, as specified by Equation (59):

$${\psi }_1(0,\overrightarrow{x})={\psi }_{10}\left(\overrightarrow{x}\right),{\psi }_2(0,\overrightarrow{x})={\psi }_{20}\left(\overrightarrow{x}\right),{\Psi }_0=\left(\begin{array}{c}{\psi }_{10}\\ {\psi }_{20}\end{array}\right).$$

(61)

By inserting Equation (60) into Equation (54), we derive

$$(i\hslash D_0-mc){\psi }_1+i\hslash {\sigma }^i{D}_i{\psi }_2=0,(i\hslash D_0+mc){\psi }_2+i\hslash {\sigma }^i{D}_i{\psi }_1=0$$

(62)

Then, we introduce hatted variables, as follows:

$${\widehat{\psi }}_1\equiv e^{-{\displaystyle \frac{imc}{\hslash }}{x}_0}{\psi }_1=e^{-{\displaystyle \frac{im{c}^2}{\hslash }}t}{\psi }_1,{\widehat{\psi }}_2\equiv e^{-{\displaystyle \frac{imc}{\hslash }}{x}_0}{\psi }_2=e^{-{\displaystyle \frac{im{c}^2}{\hslash }}t}{\psi }_2.$$

(63)

We can substitute

$${\psi }_1=e^{+{\displaystyle \frac{imc}{\hslash }}{x}_0}{\widehat{\psi }}_1,{\psi }_2=e^{+{\displaystyle \frac{imc}{\hslash }}{x}_0}{\widehat{\psi }}_2.$$

(64)

into Equation (62). This results in a simplified set of equations, as follows:

$$(i\hslash D_0-2mc){\widehat{\psi }}_1+i\hslash {\sigma }^i{D}_i{\widehat{\psi }}_2=0,i\hslash D_0{\widehat{\psi }}_2+i\hslash {\sigma }^i{D}_i{\widehat{\psi }}_1=0$$

(65)

The initial values at $x_0=0$ remain identical, since $\psi$ and $\widehat{\psi }$ are the same at the initial time.

$${\widehat{\psi }}_1(0,\overrightarrow{x})={\psi }_{10}\left(\overrightarrow{x}\right),{\widehat{\psi }}_2(0,\overrightarrow{x})={\psi }_{20}\left(\overrightarrow{x}\right).$$

(66)

${\widehat{\psi }}_2$ is easily obtained once ${\widehat{\psi }}_1$ is given, as follows:

$${\widehat{\psi }}_2(x_0,\overrightarrow{x})\left[{\widehat{\psi }}_1\right]={\widehat{\psi }}_2(0,\overrightarrow{x})-{\sigma }^i{D}_i{\int }_0^{x_0}{\widehat{\psi }}_1(x_0^{\prime },\overrightarrow{x})d{x}_0^{\prime },$$

(67)

in which the gauge choice of Equation (57) is taken into account from now on in the current section. We define the additional function as

$$int{\widehat{\psi }}_1\equiv {\int }_0^{x_0}{\widehat{\psi }}_1(x_0^{\prime },\overrightarrow{x})d{x}_0^{\prime }\Rightarrow {\mathsf{\partial }}_{0}int{\widehat{\psi }}_1={\widehat{\psi }}_1,{\mathsf{\partial }}_0^{2}int{\widehat{\psi }}_1={\mathsf{\partial }}_0{\widehat{\psi }}_1$$

(68)

and we define the time independent spinor as

$$W_{em}\left(\overrightarrow{x}\right)\equiv -{\sigma }^k{D}_k{\widehat{\psi }}_2(0,\overrightarrow{x}).$$

(69)

We can express Equation (65) and, consequently, the Dirac theory in the following form:

$$\begin{array}{ccc}& & (i\hslash D_0-2mc){\widehat{\psi }}_1+i\hslash {\sigma }^k{D}_k({\widehat{\psi }}_2(0,\overrightarrow{x})-{\sigma }^i{D}_{i}int{\widehat{\psi }}_1)=0\hfill \end{array}$$

(70)

$$\begin{array}{ccc}& & {\widehat{\psi }}_2(x_0,\overrightarrow{x})={\widehat{\psi }}_2(0,\overrightarrow{x})-{\sigma }^i{D}_{i}int{\widehat{\psi }}_1\hfill \end{array}$$

(71)

Equation (70) can be expressed more simply as

$$({\mathsf{\partial }}_0+{\displaystyle \frac{2imc}{\hslash }}){\widehat{\psi }}_1-{\sigma }^k{\sigma }^i{D}_k{D}_{i}int{\widehat{\psi }}_1=W_{em}.$$

(72)

The above equation can be formulated in terms of $int{\widehat{\psi }}_1$ alone by using Equation (68):

$$({\mathsf{\partial }}_0^2+{\displaystyle \frac{2imc}{\hslash }}{\mathsf{\partial }}_0)int{\widehat{\psi }}_1-{\sigma }^k{\sigma }^i{D}_k{D}_{i}int{\widehat{\psi }}_1=W_{em}.$$

(73)

Using the Pauli matrix identity given in Equation (20), it follows that

$$({\mathsf{\partial }}_0^2+{\displaystyle \frac{2imc}{\hslash }}{\mathsf{\partial }}_0)int{\widehat{\psi }}_1-({\delta }^{ki}{I}_2+i{ϵ}^{kil}{\sigma }^l)D_k{D}_{i}int{\widehat{\psi }}_1=W_{em},$$

(74)

or, more concisely, we have

$$(D^{\mu }{D}_{\mu }+{\displaystyle \frac{2imc}{\hslash }}{\mathsf{\partial }}_0)int{\widehat{\psi }}_1-i{ϵ}^{kil}{\sigma }^l{D}_k{D}_{i}int{\widehat{\psi }}_1=W_{em},$$

(75)

Taking into account that

$${ϵ}^{kil}{D}_k{D}_i={ϵ}^{kil}{\displaystyle \frac{ie}{\hslash }}{\mathsf{\partial }}_k{A}_i,$$

(76)

and that the magnetic field $\overrightarrow{B}$ is defined as

$$\overrightarrow{B}\equiv \overrightarrow{\nabla }\times \overrightarrow{A}\Rightarrow B^l={ϵ}^{lik}{\mathsf{\partial }}_i{A}^k={ϵ}^{lki}{\mathsf{\partial }}_i{A}_k$$

(77)

we have

$${ϵ}^{kil}{D}_k{D}_i=-{\displaystyle \frac{ie}{\hslash }}{B}^l.$$

(78)

We may now write Equation (75) as

$$\left(D^{\mu }{D}_{\mu }+{\displaystyle \frac{2imc}{\hslash }}{\mathsf{\partial }}_0-{\displaystyle \frac{e}{\hslash }}\overrightarrow{B}·\overrightarrow{\sigma }\right)int{\widehat{\psi }}_1=W_{em},\overrightarrow{\sigma }\equiv \left({\sigma }^1,{\sigma }^2,{\sigma }^3\right).$$

(79)

Thus, we see that for an electromagnetic coupled $int{\widehat{\psi }}_1$, the upper and lower components (spin up and spin down) can influence one another, provided we have a non-zero magnetic field. In contrast, for the free case, the upper and lower component equations are decoupled. The electromagnetic terms in the above equation do not allow us to proceed as in the free case by just taking a time derivative of Equation (79). As such, an equation will depend on both $int{\widehat{\psi }}_1$ and ${\widehat{\psi }}_1$ (and not just on ${\widehat{\psi }}_1$). Indeed, for the free case, we may take a time derivative of any order, which makes the choice of our fluid variable quite arbitrary. This is obviously fixed by the electromagnetic interaction. To simplify our further calculations, we shall make the following assumption: Suppose that at some time ${\widehat{\psi }}_2=0$ (or rather more physically at that time $\left|{\widehat{\psi }}_2\right|\ll \left|{\widehat{\psi }}_1\right|$), we shall choose this time to be the origin of our temporal axis, such that $t=0$. By assumption, it follows that

$${\widehat{\psi }}_2(0,\overrightarrow{x})=0,W_{em}=0,{\widehat{\psi }}_2(x_0,\overrightarrow{x})=-{\sigma }^i{D}_{i}int{\widehat{\psi }}_1.$$

(80)

We can now write Equation (79) in the following form:

$$\left(D^{\mu }{D}_{\mu }+{\displaystyle \frac{2imc}{\hslash }}{\mathsf{\partial }}_0-{\displaystyle \frac{e}{\hslash }}\overrightarrow{B}·\overrightarrow{\sigma }\right)int{\widehat{\psi }}_1=0.$$

(81)

The following new variable will make this equation look more familiar:

$$\tilde{\psi }\equiv e^{+{\displaystyle \frac{imc}{\hslash }}{x}_0}int{\widehat{\psi }}_1,int{\widehat{\psi }}_1=e^{-{\displaystyle \frac{imc}{\hslash }}{x}_0}\tilde{\psi }.$$

(82)

Thus,

$$\left(D^{\mu }{D}_{\mu }+{\displaystyle \frac{m^2{c}^2}{{\hslash }^2}}-{\displaystyle \frac{e}{\hslash }}\overrightarrow{B}·\overrightarrow{\sigma }\right)\tilde{\psi }=0.$$

(83)

Moreover, we have

$${\widehat{\psi }}_2(x_0,\overrightarrow{x})=-e^{-{\displaystyle \frac{imc}{\hslash }}{x}_0}{\sigma }^i{D}_i\tilde{\psi }.$$

(84)

These equations are equivalent to the Dirac equation, provided that $\tilde{\psi }$ satisfies the initial conditions as follows:

$$\tilde{\psi }(0,\overrightarrow{x})=0,{\mathsf{\partial }}_0\tilde{\psi }(0,\overrightarrow{x})={\psi }_1(0,\overrightarrow{x}),$$

(85)

Equation (81) is a Klein–Gordon equation with an additional spin coupling magnetic field term. Of course, this equation in not a (complex) scalar equation, as was originally suggested by Klein and Gordon, but a two-dimensional spinor equation. As in the free case, we have shown that Dirac’s theory, which is usually represented as a four spinor theory, can be reduced to a two spinor theory, in which the other two spinor can be explicitly evaluated in terms of the first without the need to solve any equation. However, there are some straightforward mathematical operations that need to be performed.

7. Pauli’s Theory

Equation (62) is also the standard starting point for deriving Pauli’s equation. This is carried out by introducing primed variables (instead of the hatted variables of Equation (63)), in which the phase correction is applied in an opposite direction:

$${\psi }_1^{\prime }\equiv e^{+{\displaystyle \frac{imc}{\hslash }}{x}_0}{\psi }_1,{\psi }_2^{\prime }\equiv e^{+{\displaystyle \frac{imc}{\hslash }}{x}_0}{\psi }_2.{\psi }_1=e^{-{\displaystyle \frac{imc}{\hslash }}{x}_0}{\psi }_1^{\prime },{\psi }_2=e^{-{\displaystyle \frac{imc}{\hslash }}{x}_0}{\psi }_2^{\prime }.$$

(86)

In terms of the primed variables, Equation (62) takes the following form:

$$\begin{array}{ccc}& & i\hslash D_0{\psi }_1^{\prime }+i\hslash {\sigma }^i{D}_i{\psi }_2^{\prime }=0,\hfill \end{array}$$

(87)

$$\begin{array}{ccc}& & (i\hslash D_0+2mc){\psi }_2^{\prime }+i\hslash {\sigma }^i{D}_i{\psi }_1^{\prime }=0\hfill \end{array}$$

(88)

The crucial assumption leading to Pauli’s theory is

$$\left|2mc{\psi }_2^{\prime }\right|\gg \left|i\hslash D_0{\psi }_2^{\prime }\right|,$$

(89)

which is equivalent to

$$\left|2m{c}^2{\psi }_2^{\prime }\right|\gg \left|i\hslash c{D}_0{\psi }_2^{\prime }\right|=\left|i\hslash {\mathsf{\partial }}_t{\psi }_2^{\prime }-e\varphi {\psi }_2\right|.$$

(90)

In the above equation, we use the standard relation between the electromagnetic scalar potential $\varphi$ and the zeroth component of the four vector $A_0$, such that $\varphi =c{A}_0$ (we do not use the special gauge, which we used in the previous section). Now, if we assume that ${\psi }_2^{\prime }$ is an energy E eigenstate, such that ${\psi }_2^{\prime }\sim e^{{\displaystyle \frac{-iEt}{\hslash }}}$, the inequality (90) takes the following form:

$$2m{c}^2\gg \left|E-e\varphi \right|.$$

(91)

Assuming that $E-e\varphi >0$, the above inequality can be written as

$$m{c}^2\gg (E-e\varphi -m{c}^2)\equiv E_k.$$

(92)

Therefore, the assumption (89) can be interpreted as demanding, that the rest mass of the electron is much larger than the its kinetic energy $E_k$, which is what we expect for a non-relativistic particle. Based on Equation (89), we may now write Equation (88) as

$${\psi }_2^{\prime }\simeq -{\displaystyle \frac{i\hslash }{2mc}}{\sigma }^i{D}_i{\psi }_1^{\prime }.$$

(93)

Inserting Equation (93) into Equation (87) will result in

$$i\hslash D_0{\psi }_1^{\prime }={\displaystyle \frac{-{\hslash }^2}{2mc}}{\sigma }^k{D}_k\left({\sigma }^i{D}_i{\psi }_1^{\prime }\right)$$

(94)

in which the equal sign should be understood as being correct only in the non-relativistic limit. Equation (94) can also be casted in the following form:

$$i\hslash {\mathsf{\partial }}_t{\psi }_1^{\prime }=\left(e\varphi -{\displaystyle \frac{{\hslash }^2}{2m}}{\sigma }^k{\sigma }^i{D}_k{D}_i\right){\psi }_1^{\prime }.$$

(95)

Taking into account Equations (20) and (78), we obtain

$$i\hslash {\mathsf{\partial }}_t{\psi }_1^{\prime }=\left(e\varphi -{\displaystyle \frac{{\hslash }^2}{2m}}{(\overrightarrow{\nabla }-{\displaystyle \frac{ie}{\hslash }}\overrightarrow{A})}^2-{\displaystyle \frac{e\hslash }{2m}}\overrightarrow{B}·\overrightarrow{\sigma }\right){\psi }_1^{\prime }.$$

(96)

Thus, ${\psi }_1^{\prime }$ is Pauli’s two-dimensional spinor. The above equation can be written in a more familiar form using the momentum operator, $\widehat{\overrightarrow{p}}=-i\hslash \overrightarrow{\nabla }$, and Bohr’s magneton, ${\mu }_B\equiv {\displaystyle \frac{\left|e\right|\hslash }{2m}}=-{\displaystyle \frac{e\hslash }{2m}}$ (where the charge for an electron is assumed to be negative).

$$i\hslash {\mathsf{\partial }}_t{\psi }_1^{\prime }=\left(e\varphi +{\displaystyle \frac{1}{2m}}{(\widehat{\overrightarrow{p}}-e\overrightarrow{A})}^2+{\mu }_B\overrightarrow{B}·\overrightarrow{\sigma }\right){\psi }_1^{\prime }.$$

(97)

Compare this to Equation (46) of [33] in which a fluid dynamical interpretation of Pauli’s theory is discussed and attention is given to the terms in the theory that do not suffer such an interpretation.

8. Lagrangian Formalism of a DIRAC Two Spinor with an Electromagnetic Interaction

To continue to a fluid interpretation of an electromagnetic two spinor, we shall write down a Lagrangian density and action that will lead to Equation (81)

$${\mathcal{L}}_{KGem}\equiv m\left({\displaystyle \frac{{\hslash }^2}{m^2}}{D}^{\mu }{\tilde{\psi }}^{†}{D}_{\mu }\tilde{\psi }-c^2{\tilde{\psi }}^{†}\tilde{\psi }\right)-2{\mu }_B\left({\tilde{\psi }}^{†}\overrightarrow{\sigma }\tilde{\psi }\right)·\overrightarrow{B},{\mathcal{A}}_{KGem}\equiv \int d^{4}x{\mathcal{L}}_{KGem}.$$

(98)

Equation (81) holds as long as the variations of the above action are appropriately constrained at the boundaries which are both spatial and temporal. Although it differs from the standard Lagrangian density of the Dirac equation, it is demonstrated in the previous section that the mathematical information in both is identical. The total Lagrangian density of the system can be obtained when adding a field Lagrangian density to ${\mathcal{L}}_{KGem}$, which takes the following form in MKS units:

$${\mathcal{L}}_F\equiv -{\displaystyle \frac{1}{4{\mu }_0}}{F}_{\alpha \beta }{F}^{\alpha \beta }={\displaystyle \frac{1}{2}}({ϵ}_0{\overrightarrow{E}}^2-{\displaystyle \frac{1}{{\mu }_0}}{\overrightarrow{B}}^2),F^{\alpha \beta }={\mathsf{\partial }}^{\alpha }{A}^{\beta }-{\mathsf{\partial }}^{\beta }{A}^{\alpha }.$$

(99)

In the above equation, ${\mu }_0$ is the vacuum permeability, ${ϵ}_0$ is the vacuum susceptibility and $\overrightarrow{E}=-{\mathsf{\partial }}_t\overrightarrow{A}-\overrightarrow{\nabla }\varphi$ is the electric field. Thus, the total Lagrangian density is

$${\mathcal{L}}_T\equiv {\mathcal{L}}_F+{\mathcal{L}}_{KGem}.$$

(100)

Now, for standard macroscopic matter, we can write the Lagrangian density as

$${\mathcal{L}}_{Ts}\equiv {\mathcal{L}}_{Matter}+{\mathcal{L}}_I+{\mathcal{L}}_F,$$

(101)

where ${\mathcal{L}}_{Matter}$ is the Lagrangian density of matter and ${\mathcal{L}}_I$ is the interaction Lagrangian between matter and a field. ${\mathcal{L}}_I$ can be written for standard macroscopic matter in the following form:

$${\mathcal{L}}_I\equiv -A_{\alpha }{J}_{free}^{\alpha }+{\displaystyle \frac{1}{2}}{F}_{\alpha \beta }{M}^{\alpha \beta }=-\varphi {\rho }_{free}+\overrightarrow{A}·{\overrightarrow{J}}_{free}+\overrightarrow{E}·\overrightarrow{P}+\overrightarrow{B}·\overrightarrow{M}.$$

(102)

In the above equation, $J_{free}^{\alpha }=(c{\rho }_{free},{\overrightarrow{J}}_{free})$ is the free four current, $M^{\alpha \beta }$ is the magnetization polarization tensor, ${\rho }_{free}$ is the free charge density, ${\overrightarrow{J}}_{free}$ is the free current density, $\overrightarrow{P}$ is the matter polarization, and $\overrightarrow{M}$ is the matter’s polarization. It can easily be seen that ${\mathcal{L}}_{KGem}$ can also be partitioned to free and interaction components, such that

$${\mathcal{L}}_T={\mathcal{L}}_{KG}+{\mathcal{L}}_{KGI}+{\mathcal{L}}_F,$$

(103)

${\mathcal{L}}_{KG}$ is defined in Equation (29) (in which ${\psi }_1$ is replaced by $\tilde{\psi }$), and the interaction lagrangian density is given by

$${\mathcal{L}}_{KGI}\equiv {\displaystyle \frac{ie\hslash }{m}}{A}^{\mu }({\mathsf{\partial }}_{\mu }{\tilde{\psi }}^{†}\tilde{\psi }-{\tilde{\psi }}^{†}{\mathsf{\partial }}_{\mu }\tilde{\psi })+{\displaystyle \frac{e^2}{m}}{A}_{\mu }{A}^{\mu }{\tilde{\psi }}^{†}\tilde{\psi }-2{\mu }_B\left({\tilde{\psi }}^{†}\overrightarrow{\sigma }\tilde{\psi }\right)·\overrightarrow{B}$$

(104)

Comparing Equation (104) with Equation (102), one notices that the Dirac Lagrangian contains a magnetization term of the following form:

$$\overrightarrow{M}=-2{\mu }_B\left({\tilde{\psi }}^{†}\overrightarrow{\sigma }\tilde{\psi }\right),$$

(105)

where the polarization $\overrightarrow{P}$ is null. The definition of the “free” current is less straightforward. We already noticed in [42] that when the current depends on the electromagnetic potential, the linear form depicted in Equation (102) is not appropriate. This is the case in quantum mechanics and charged Eulerian flows. However, one can use a variation with respect to the four potential to derive a total current, as follows:

$$\delta {\mathcal{L}}_{KGI}=-J^{\mu }\delta A_{\mu }.$$

(106)

Taking into account Equation (104), this turns out to be

$$\begin{array}{ccc}\hfill J^{\mu }& =& J^{\mu free}+J^{\mu M}\hfill \\ \hfill J^{\mu free}& =& {\displaystyle \frac{ie\hslash }{m}}{A}^{\mu }({\tilde{\psi }}^{†}{\mathsf{\partial }}^{\mu }\tilde{\psi }-{\mathsf{\partial }}^{\mu }{\tilde{\psi }}^{†}\tilde{\psi })-{\displaystyle \frac{2e^2}{m}}{A}^{\mu }{\tilde{\psi }}^{†}\tilde{\psi }\hfill \\ \hfill J^{\mu M}& =& (0,\overrightarrow{\nabla }\times \overrightarrow{M}).\hfill \end{array}$$

(107)

This electric current seems to differ from the probability current defined in Equation (27). In particular, the charge density does not need to be positive; however, the probability density must be positive.

9. Fluid Interpretation in the Electromagnetic Interacting Case

We shall now attend to the task of formulating the Lagrangian density in terms of fluid dynamical Clebsch variables. This will be carried out along the same line as in the free case elaborated in Section 4. For this, we express $\tilde{\psi }$ in the following form:

$$\tilde{\psi }=\left(\begin{array}{c}{\psi }_{\uparrow }\\ {\psi }_{\downarrow }\end{array}\right).$$

(108)

By substituting this into the Lagrangian density, it follows that

$${\mathcal{L}}_{KGem}=m\left({\displaystyle \frac{{\hslash }^2}{m^2}}{D}^{\mu }{\psi }_{\uparrow }^{*}{D}_{\mu }{\psi }_{\uparrow }-c^2{\psi }_{\uparrow }^{*}{\psi }_{\uparrow }+{\displaystyle \frac{{\hslash }^2}{m^2}}{D}^{\mu }{\psi }_{\downarrow }^{*}{D}_{\mu }{\psi }_{\downarrow }-c^2{\psi }_{\downarrow }^{*}{\psi }_{\downarrow }\right)+\overrightarrow{M}\left[\tilde{\psi }\right]·\overrightarrow{B}.$$

(109)

Next, we express the upper and lower components as follows:

$${\psi }_{\uparrow }=R_{\uparrow }{e}^{i{\displaystyle \frac{m}{\hslash }}{\nu }_{\uparrow }},{\psi }_{\downarrow }=R_{\uparrow }{e}^{i{\displaystyle \frac{m}{\hslash }}{\nu }_{\downarrow }}$$

(110)

Using the definition of $D_{\mu }$ (see Equation (54)), we obtain

$$\begin{array}{ccc}\hfill D_{\mu }{\psi }_{\uparrow }& =& e^{i{\displaystyle \frac{m}{\hslash }}{\nu }_{\uparrow }}\left[{\mathsf{\partial }}_{\mu }{R}_{\uparrow }+{\displaystyle \frac{im}{\hslash }}{R}_{\uparrow }{\overline{\mathsf{\partial }}}_{\mu }{\nu }_{\uparrow }\right],D_{\mu }{\psi }_{\downarrow }=e^{i{\displaystyle \frac{m}{\hslash }}{\nu }_{\downarrow }}\left[{\mathsf{\partial }}_{\mu }{R}_{\downarrow }+{\displaystyle \frac{im}{\hslash }}{R}_{\downarrow }{\overline{\mathsf{\partial }}}_{\mu }{\nu }_{\downarrow }\right],\hfill \\ \hfill {\overline{\mathsf{\partial }}}_{\mu }& \equiv & {\mathsf{\partial }}_{\mu }+{\displaystyle \frac{e}{m}}{A}_{\mu }\hfill \end{array}$$

(111)

By inserting Equation (110) and Equation (111) into Equation (109), we obtain the following form:

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{KGem}& =& {\mathcal{L}}_{KGemq}+{\mathcal{L}}_{KGemc}\hfill \\ \hfill {\mathcal{L}}_{KGqem}& \equiv & {\displaystyle \frac{{\hslash }^2}{m}}\left({\mathsf{\partial }}^{\mu }{R}_{\uparrow }{\mathsf{\partial }}_{\mu }{R}_{\uparrow }+{\mathsf{\partial }}^{\mu }{R}_{\downarrow }{\mathsf{\partial }}_{\mu }{R}_{\downarrow }\right)+\overrightarrow{M}\left[\tilde{\psi }\right]·\overrightarrow{B}\hfill \\ \hfill {\mathcal{L}}_{KGcem}& \equiv & m\left(R_{\uparrow }^2\left({\overline{\mathsf{\partial }}}^{\mu }{\nu }_{\uparrow }{\overline{\mathsf{\partial }}}_{\mu }{\nu }_{\uparrow }-c^2\right)+R_{\downarrow }^2\left({\overline{\mathsf{\partial }}}^{\mu }{\nu }_{\downarrow }{\overline{\mathsf{\partial }}}_{\mu }{\nu }_{\downarrow }-c^2\right)\right).\hfill \end{array}$$

(112)

We have partitioned ${\mathcal{L}}_{KGem}$ into a quantum part ${\mathcal{L}}_{KGqem}$ along with a classical component, ${\mathcal{L}}_{KGcem}$. In the classical limit, $\hslash \to 0$ is

$$\underset{\hslash \to 0}{lim}{\mathcal{L}}_{KGqem}=0\Rightarrow \underset{\hslash \to 0}{lim}{\mathcal{L}}_{KGem}={\mathcal{L}}_{KGcem}$$

(113)

in which we recall that Bohr’s magneton is linear in the Planck constant ℏ. This leads to the following natural definition of the density $\overline{\rho }$:

$$\overline{\rho }\equiv m\left(R_{\uparrow }^2+R_{\downarrow }^2\right),tan\theta \equiv {\displaystyle \frac{R_{\downarrow }}{R_{\uparrow }}},\Rightarrow R_{\uparrow }=\sqrt{{\displaystyle \frac{\overline{\rho }}{m}}}cos\theta ,R_{\downarrow }=\sqrt{{\displaystyle \frac{\overline{\rho }}{m}}}sin\theta$$

(114)

From this, it follows that

$${\mathcal{L}}_{KGcem}=\overline{\rho }\left[{cos}^2\theta {\overline{\mathsf{\partial }}}^{\mu }{\nu }_{\uparrow }{\overline{\mathsf{\partial }}}_{\mu }{\nu }_{\uparrow }+{sin}^2\theta {\overline{\mathsf{\partial }}}^{\mu }{\nu }_{\downarrow }{\overline{\mathsf{\partial }}}_{\mu }{\nu }_{\downarrow }-c^2\right].$$

(115)

Now, let us define

$$\begin{array}{ccc}\hfill \nu & \equiv & {\nu }_{\uparrow },\beta \equiv {\nu }_{\downarrow }-{\nu }_{\uparrow },\hfill \\ \hfill \alpha & \equiv & {\displaystyle \frac{-{\overline{\mathsf{\partial }}}_{\mu }\nu {\overline{\mathsf{\partial }}}^{\mu }\beta \pm \sqrt{{\left({\overline{\mathsf{\partial }}}_{\mu }\nu {\overline{\mathsf{\partial }}}^{\mu }\beta \right)}^2+{sin}^2\theta \left({\overline{\mathsf{\partial }}}_{\mu }\beta {\overline{\mathsf{\partial }}}^{\mu }\beta \right)\left({\overline{\mathsf{\partial }}}^{\mu }\beta (2{\overline{\mathsf{\partial }}}_{\mu }\nu +{\overline{\mathsf{\partial }}}_{\mu }\beta )\right)}}{{\overline{\mathsf{\partial }}}_{\mu }\beta {\overline{\mathsf{\partial }}}^{\mu }\beta }}.\hfill \end{array}$$

(116)

We notice that $\alpha$ now depends on the electromagnetic four potential. Using this information, we define a four-dimensional electromagnetic modified Clebsch field as follows:

$${\overline{v}}_E^{\mu }\equiv \alpha {\overline{\mathsf{\partial }}}^{\mu }\beta +{\overline{\mathsf{\partial }}}^{\mu }\nu$$

(117)

It is easy to see that the modified electromagnetic modified Clebsch field is related to the fluid electromagnetic Clebsch field (see Equation (100) of [35]) as follows:

$${\overline{v}}_E^{\mu }=v_E^{\mu }+{\displaystyle \frac{e}{m}}\alpha A^{\mu }$$

(118)

By substituting in Equation (117) and using the definitions from Equation (116), we arrive at the following result after some straightforward, though somewhat tedious, calculations:

$${\mathcal{L}}_{KGc}=\overline{\rho }\left[{\overline{v}}_E^{\mu }{\overline{v}}_{E\mu }-c^2\right].$$

(119)

We define the mass density in the rest frame as

$${\rho }_0={\displaystyle \frac{\overline{\rho }}{c}}{\displaystyle \frac{\left[{\overline{v}}_E^{\mu }{\overline{v}}_{E\mu }-c^2\right]}{\sqrt{\left[v_E^{\mu }{v}_{E\mu }-c^2\right]}}}.$$

(120)

The above equation will become identical to the definition given in Equation (40) for a null four vector potential. It follows that

$${\mathcal{L}}_{KGcem}=c{\rho }_0\left[\sqrt{v_E^{\mu }{v}_{E\mu }}-c\right]={\mathcal{L}}_{\mathrm{Relativistic}\mathrm{Flow}}.$$

(121)

The non-quantum component of ${\mathcal{L}}_{KGcem}$ is mapped into the Lagrangian density of a classical relativistic flow, lacking internal energy (observe Equation (103) of [35]). It is important to notice that, in comparison to the Pauli spin flow which has a Lagrangian density containing an unjustified component that cannot be explained in the flow framework (see Equation (63) in [33]), we have

$$\underset{\hslash \to 0}{lim}{\epsilon }_{qs}={\displaystyle \frac{1}{2}}(1-{\alpha }^2){\left(\overrightarrow{\nabla }\beta \right)}^2$$

(122)

In the theory of Dirac, there is a direct correspondence between components that appear in the classical section of Dirac’s Lagrangian and the relativistic flow dynamics Lagrangian, with no extra terms or inconsistencies.

10. The Dirac Quantum Term in the Presence of an Electromagnetic Field

Now, let us compare the Dirac quantum term ${\mathcal{L}}_{KGqem}$ from Formula (112) with the Fisher information quantifier that one usually attributes quantum behavior to. That is Formula (113) of [35], which is given below as

$${\mathcal{L}}_{RFq}={\displaystyle \frac{{\hslash }^2}{2m}}{\mathsf{\partial }}^{\mu }{a}_0{\mathsf{\partial }}_{\mu }{a}_0,a_0\equiv \sqrt{{\displaystyle \frac{{\rho }_0}{m}}}.$$

(123)

At first glance, these terms appear quite similar; however, upon a closer inspection, significant differences become apparent. First, ${\mathcal{L}}_{KGqem}$ depends on two “density amplitudes” (one for each spin), whereas ${\mathcal{L}}_{RFq}$ depends on just a single amplitude. This may be related to the fact that every eigenstate of the Dirac Hamiltonian can accept two particles with a unique spin. Also, ${\mathcal{L}}_{KGq}$ is missing a division by 2. We address these issues by examining Formula (120), where it is underlined that $R_{\uparrow }$ and $R_{\downarrow }$ do not correspond directly to the mass density. However, as stated in Equation (104) of [35],

$$\sqrt{v_{E\mu }{v}_E^{\mu }}=\left|v_{E0}\right|\sqrt{1-{\displaystyle \frac{{\overrightarrow{v}}_E^2}{v_{E0}^2}}}=\left|v_{E0}\right|\sqrt{1-{\displaystyle \frac{{\overrightarrow{v}}^2}{c^2}}}={\displaystyle \frac{|v_{E0}|}{\gamma }}$$

(124)

Also, according to Equation (101) of [35],

$$|v_{E0}|=c\lambda$$

(125)

If a classical flow has no internal energy and follows the laws of motion (see Formula (58) in [35]), we have

$$\lambda =\gamma .$$

(126)

Therefore, with the exception of quantum corrections, we have

$$\sqrt{v_{E\mu }{v}_E^{\mu }}\approx c$$

(127)

By substituting Equation (127) into Equation (120), we obtain a free Dirac result as follows:

$${\rho }_0\approx 2\overline{\rho },$$

(128)

but only for a vanishing small ${\displaystyle \frac{e}{m}}\alpha A^{\mu }$, meaning that the quantum correction cannot be neglected and will have a significant effect on the Dirac fluid density.

11. Conclusions

By demonstrating mapping of the classical part of Dirac’s theory to relativistic fluid dynamics, we resolved the issue of certain unusual terms in the flow description of Pauli’s equation. Notice, however, that the quantum sector of Dirac’s formalism includes a redundant term (which also appears in the flow mapping of Pauli’s formalism [33]), being unexpected when only considering Fisher information. Therefore, a more in-depth study is needed: one that incorporates both quantum contributions to the $\lambda$ term—an aspect of relativistic fluid dynamics—and the distinct definition of the probability density in Dirac’s theory, accounting for all four spinor amplitudes. We will address this crucial task in a future study.

Takabayasi [38,39] claimed that his theory is equivalent to Dirac’s theory. We have shown that the fluid formalism presented in this paper is also equivalent to Dirac’s theory. Logically, it follows that the current fluid formalism is equivalent to Takabayasi’s theory. Notice, however, that the explicit relations between the quantities defined in Takabayasi’s theory and in the current fluid formalism are presumably complex and uninteresting.

Finally, we note that the nature of the quantum relativistic flow remains largely mysterious. An obvious question arises: “A flow of what?” This fundamental question has implications for the issues discussed earlier, such as the peculiar addition of a redundant Fisher term and the existence of electromagnetic fields. Riemann [43] has suggested that all basic physical entities originate from geometry, suggesting that the flow is a geometric representation of an elongated defect of spacetime (spatially thin but temporally extended). The location of this defect in the near future manifold is not deterministically determined, explaining the appearance of the Fisher information term.

We are reminded that Riemann’s ideas inspired Einstein to develop a highly successful theory of gravity, known as general relativity [44], which accounts for very fine effects of gravity through a spacetime metric. However, Weyl’s attempt [43] to geometrize the electromagnetic field using affine geometry was less successful. Similarly, Schrödinger’s [45] attempt to geometrize matter using a non-symmetric affine connection is also considered unsuccessful. Despite these setbacks, we remain hopeful that the current mapping of relativistic flow to Dirac’s theory might provide new insights into these early attempts and lead to some progress.