Fisher Information and Electromagnetic Interacting Dirac Spinors

1. Introduction

Quantum theory is often interpreted using the Copenhagen perspective, that 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 [2–4], 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 [14], Serrin [15], and Lin [16]. These approaches notably complex, using multiple multipliers as suggested by Lagrange’s method and 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, those approaches were deemed useless for straightforward application.

Seliger & Whitham [17] have reinvented Clebsch’s principle of variation, using a four variable description of a barotropic fluid. Later, Lynden-Bell & Katz [18] developed a Lagrangian based on dual variables: $\lambda$ load & $\rho$ 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 measurement theory, specifically Fisher information [20,21], which quantifies the standard deviation of a random variable. It was shown [21] that the information quantifier defined by Fisher substitutes the internal energy for an electron provided this electron is both non-relativistic and spinless, it also partially justifies the "internal energy" of an electron with spin (but non-relativistic) [21].

Frieden [22] 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 has not arised for a Lagrangian for relativistic Eulerian fluids (symmetrical under Lorentz group). Later a series of papers [24–26] addressed this issue, and introduces relativistic Clebsch flows. We demonstrated that relativistic Clebsch flows leads to relativistic quantum theory by incorporating a Lorentz symmetric Fisher quantifier of information. However, this is only correct for situations with slow velocity fields with zero rotational, this Lagrangian reduces to Schrödinger’s Lagrangian.

It seems a necessity to evaluate the flow approach to relativistic quantum theory with Dirac theory which is widely acknowledged. This was partitioned [27,28] into multiple stages, in the first stage the Dirac equation is reformulated using four fluid 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. While in a previous work this was done ignoring the electromagnetic interaction for simplicity here we discuss also the implication of a field on the Fisher information approach. For pedagogical reasons this is done only after a repetition of the discussion of the same without electromagnetic interaction.

2. Dirac Equation of a Free Field

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

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

(1)

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:

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

(2)

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

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

(3)

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

(4)

$I_2$ is a $2\times 2$ identity matrix, m is the mass of the particle and c is the velocity of light in vacuum. Equation (1) will be solved uniquely if initial values are provided, specifically:

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

(5)

At first glance, the equation appears unrelated to fluid dynamics because $\Psi$ depends on eight scalar quantities, while 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:

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

(6)

This form leads to initial conditions for both ${\psi }_1$ and ${\psi }_2$ as specified by equation equation (5):

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

(7)

Inserting equation (6) into equation (1) we derive:

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

(8)

Introducing the hatted variables:

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

(9)

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:

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

(10)

into equation (8) this results in a simplified set of equations:

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

(11)

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

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

(12)

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

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

(13)

Defining the additional function:

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

(14)

and the time independent spinor:

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

(15)

We can now express the left hand side of equation (11) using the definition of $int{\widehat{\psi }}_1$ and W given in equation (14) and equation (15) respectively, and using the Pauli matrix identity:

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

(16)

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 for a free Dirac spinor, but have profound consequences for a Dirac spinor influenced by a magnetic field to be discussed later. Consequently, the Dirac theory takes the following form:

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

(17)

Therefore, the chore in Dirac equation is to find a solution for the left hand side of equation (17), since the right equation 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 (14). By calculating the time differential of the left equation in (17), we see that ${\widehat{\psi }}_1$ is a solution of the equation:

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

(18)

The initial values for this differential equation of order 2 are determined by the initial values of equation (11), as those values also specify the time differentials at $x_0=0$.

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

(19)

Or more simply as:

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

(20)

As the initial value of the function and the initial value of its first differential are given, we solve the differential equation (18) of order two uniquely, and thus we have solved Dirac equation. We are reminded that one can use the original function ${\psi }_1$ through equation (9) which will yield a Klein Gordon type equation:

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

(21)

with the initial conditions:

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

(22)

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:

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

(23)

As a result, we obtain the probability density:

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

(24)

$J^0$ 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 identical form, though they differ in the mathematical variables used. For the Klein-Gordon framework, we consider complex scalars, whereas 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.

3. Lagrangian Formalism

The Lagrangian density leads to equation (21) :

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

(25)

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

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

(26)

By substituting this into the Lagrangian density it follows that:

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

(27)

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

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

(28)

Using equation (28) in equation (27), we obtain the following form:

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

(29)

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

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

(30)

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

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

(31)

From this, it follows that:

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

(32)

Now, let us define:

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

(33)

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

$$v_C^{\mu }\equiv \alpha {\partial }^{\mu }\beta +{\partial }^{\mu }\nu$$

(34)

By substituting equation (34) and using the definitions from equation equation (33), we arrive at the result after some straightforward, though somewhat tedious, calculations:

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

(35)

We define the mass density in the rest frame as:

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

(36)

It follows that:

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

(37)

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 [24]). It is important to notice that, in comparison to the Pauli spin flow which has a Lagrangian density containing a unjustified component that cannot be justified the flow framework (see Equation (63) in [23]):

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

(38)

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

4. The Dirac Quantum Term

Now, let us compare the Dirac quantum term ${\mathcal{L}}_{KGq}$ from formula (29) with the Fisher information quantifier that one usually attributes quantum behaviour to. That is formula (113) of [24], given below:

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

(39)

At first glance, these terms appear quite similar, but upon 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. Addressing these issues by examining formula (36), where it is underlined that $R_{\uparrow }$ & $R_{\downarrow }$ do not correspond directly to the mass density since:

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

(40)

However, as stated in Equation (104) of [24]:

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

(41)

Also according to Equation (101) of [24]:

$$|v_{C0}|=c\lambda$$

(42)

If a classical flow that has no internal energy and follows the laws of motion (see formula (58) in [24]):

$$\lambda =\gamma .$$

(43)

Therefore, with the exception of quantum corrections:

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

(44)

Substituting equation (44) into equation (40):

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

(45)

Thus:

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

(46)

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

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

(47)

Thus:

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

(48)

It is important to note that in Dirac’s theory, R is not a probability amplitude, as stated by equation (24):

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

(49)

Therefore, the second term in the quantum Lagrangian density is probably not unexpected. Naturally, a complete calculation would require accounting for quantum effects that were ignored in formula (44).

5. Dirac Equation with an Electromagnetic Field

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

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

(50)

in the above $A^{\mu }=({\displaystyle \frac{\varphi }{c}},\overrightarrow{A})$ is the electromagnetic four dimensional vector potential, and $A_{\mu }={\eta }_{\mu \nu }{A}^{\nu }$, 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:

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

(51)

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:

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

(52)

Thus we can always choose a gauge such that:

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

(53)

This solution can be transformed into a more general solution $A_0^{\prime }\ne 0$ using the gauge transformation of equation (51), that is:

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

(54)

$\overline{\alpha }\left(\overrightarrow{x}\right)$ is an arbitrary function of the spatial coordinates. Again equation (50) will be solved uniquely if initial values are provided, specifically:

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

(55)

At first glance, the equation appears unrelated to fluid dynamics because $\Psi$ depends on eight scalar quantities, while 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:

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

(56)

This form leads to initial conditions for both ${\psi }_1$ and ${\psi }_2$ as specified by equation equation (55):

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

(57)

Inserting equation (56) into equation (50) we derive:

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

(58)

Introducing the hatted variables:

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

(59)

We can substitute:

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

(60)

into equation (58) this results in a simplified set of equations:

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

(61)

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

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

(62)

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

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

(63)

in which the gauge choice of equation (53) is taken into account from now on in the current section. Defining the additional function:

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

(64)

and the time independent spinor:

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

(65)

We can express equation (61) and, consequently, the Dirac theory in the following form:

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

(66)

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

(67)

Equation (66) can be expressed more simply as:

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

(68)

The above equation can be formulated in terms of $int{\widehat{\psi }}_1$ alone by using equation (64):

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

(69)

Using the Pauli matrix identity given in equation (16) it follows that

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

(70)

or more concisely as:

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

(71)

Taking into account that:

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

(72)

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

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

(73)

Thus:

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

(74)

And we may now write equation (71) as:

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

(75)

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 contrary 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 (75) 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 $|{\widehat{\psi }}_2|\ll |{\widehat{\psi }}_1|$), we shall choose this time to be the origin of our temporal axis, that is $t=0$. By assumption it follows that:

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

(76)

We can now write equation (75) in the form:

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

(77)

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

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

(78)

Thus:

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

(79)

And also:

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

(80)

Those equations are equivalent to the Dirac equation provided that $\tilde{\psi }$ satisfied the initial conditions:

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

(81)

Equation (77) 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 the Dirac theory 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 (but there are some straightforward mathematical operations to perform).

6. Pauli’s Theory

Equation (58) is also the standard starting point for deriving Pauli’s equation. This is done by introducing primed variables (instead of the hatted variables of equation (59)) in which the phase correction is applied in an opposite direction:

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

(82)

In terms of the primed variables equation (58) takes the form:

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

(83)

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

(84)

The crucial assumption leading to Pauli’s theory is:

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

(85)

which is equivalent to:

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

(86)

in the above we use the standard relation between the electromagnetic scalar potential $\varphi$ and the zeroth component of the four vector $A_0$, that is: $\varphi =c{A}_0$ (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 eigen state such that: ${\psi }_2^{\prime }\sim e^{{\displaystyle \frac{-iEt}{\hslash }}}$ the inequality (86) takes the form:

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

(87)

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

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

(88)

so the assumption (85) 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 (85) we may now write equation () as:

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

(89)

Inserting equation (89) into equation (83) will result in:

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

(90)

in which the equal sign should be understood as being correct only in the non-relativistic limit. Equation (90) can also be casted in form:

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

(91)

Taking into account equation (16) and equation (74) we obtain:

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

(92)

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 ${\mu }_B\equiv {\displaystyle \frac{\left|e\right|\hslash }{2m}}=-{\displaystyle \frac{e\hslash }{2m}}$ (the charge for an electron is assumed negative).

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

(93)

Compare this to Equation (46) of [23] 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 interpretation.

7. Lagrangian Formalism of a Dirac Two Spinor with Electromagnetic Interaction

To continue to a fluid interpretation of an electromagnetic two spinor we shall write down a Lagrangian density and action that lead to equation (77) :

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

(94)

Equation (77) holds as long as the variations of the above action are appropriately constrained at the boundaries which are both spatial and temporal. While it differs from the standard Lagrangian density of 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 to ${\mathcal{L}}_{KGem}$ a field Lagrangian density which in MKS units takes the form:

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

(95)

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

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

(96)

Now for standard macroscopic matter we can write a Lagrangian density:

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

(97)

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

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

(98)

in the above $J_{free}^{\alpha }=(c{\rho }_{free},{\overrightarrow{J}}_{free})$ 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 a free and interaction components such that:

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

(99)

${\mathcal{L}}_{KG}$ is defined in equation (25) (in which ${\psi }_1$ is replaced by $\tilde{\psi }$), and the interaction lagrangian density is given by:

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

(100)

Comparing equation (100) with equation (98) one notices that the Dirac Lagrangian contains a magnetization term of the form:

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

(101)

the polarization $\overrightarrow{P}$ is null. The definition of the "free" current is less straightforward. We already noticed in [29] that when the current depends on the electromagnetic potential the linear form depicted in equation (98) 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:

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

(102)

Taking into account equation (100) this turns out to be:

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

(103)

This electric current seem to differ from the probability current defined in equation (23) in particular the charge density need not be positive while the probability density must be positive.

8. 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 done along the same line as in the free case elaborated in Section 3. For this we express $\tilde{\psi }$ in the form:

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

(104)

By substituting this into the Lagrangian density it follows that:

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

(105)

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

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

(106)

Using the definition of $D_{\mu }$ (see equation (50)) we obtain that:

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

(107)

Inserting equation (106) and equation (107) into equation (105), we obtain the following form:

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

(108)

in which 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, as $\hslash \to 0$:

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

(109)

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

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

(110)

From this, it follows that:

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

(111)

Now, let us define:

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

(112)

In which we notice that $\alpha$ now depends on the electromagnetic four potential. Using these, we define a four-dimensional electromagnetic modified Clebsch field:

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

(113)

It is easy to see that the modified electromagnetic modified Clebsch field is related to the fluid electromagnetic Clebsch field (see Equation (100) of [24]) by:

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

(114)

By substituting equation (113) and using the definitions from equation equation (112), we arrive at the result after some straightforward, though somewhat tedious, calculations:

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

(115)

We define the mass density in the rest frame as:

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

(116)

The above equation will become identical to the definition given in equation (36) for a null four vector potential. It follows that:

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

(117)

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 [24]). It is important to notice that, in comparison to the Pauli spin flow which has a Lagrangian density containing a unjustified component that cannot be explained in the flow framework (see Equation (63) in [23]):

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

(118)

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

9. The Dirac Quantum Term in the Presence of Electromagnetic Field

Now, let us compare the Dirac quantum term ${\mathcal{L}}_{KGqem}$ from formula (108) with the Fisher information quantifier that one usually attributes quantum behaviour to. That is formula (113) of [24], given below:

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

(119)

At first glance, these terms appear quite similar, but upon 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. Addressing these issues by examining formula (116), where it is underlined that $R_{\uparrow }$ & $R_{\downarrow }$ do not correspond directly to the mass density. However, as stated in Equation (104) of [24]:

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

(120)

Also according to Equation (101) of [24]:

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

(121)

If a classical flow that has no internal energy and follows the laws of motion (see formula (58) in [24]):

$$\lambda =\gamma .$$

(122)

Therefore, with the exception of quantum corrections:

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

(123)

Substituting equation (123) into equation (116) we obtain the free Dirac result:

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

(124)

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

10. Conclusion

Demonstrating mapping of the classical part of Dirac’s theory to relativistic fluid dynamics, we resolve the issue of certain unusual terms in the flow description of Pauli’s equation. Notice, however, the quantum sector of Dirac’s formalism includes a redundant term (that also appears in the flow mapping of Pauli’s formalism [23]), that is unexpected when considering only 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. This crucial task will be addressed in future work.

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 [33], has suggested that all basic physical entities originate from geometry, suggesting that the flow is a geometric representation of a elongated defect of space-time (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 [34], which accounts for very fine effects of gravity through a space-time metric. However, Weyl’s attempt [33] to geometrize the electromagnetic field using affine geometry was less successful. Similarly, Schrödinger’s [35] 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 theory might provide new insights into these early attempts and lead to some progress.