Spinor Fields

Abstract

A spinor representation of the generalized energy-momentum density 4-vector is proposed, and examples of such representations for various particles and fields are given. This representation corresponds to the classical representation of the particle’s own rotation, which is described by the diagonal matrix of the moment of inertia. The concept of self-angular rotation of a particle is defined as a spatial characteristic of the field, at each point of which there is a local vortex rotation with an angular velocity Ω – a spinor field. The matrix representation of the vortex rotation Ω (spinor) and the values of the components of such a representation are derived from the matrix representation of the Lorentz transformation. The traditional concept of spin-orbit interaction, as the interaction of the magnetic moment of a particle with the magnetic field of orbital motion, is presented as the interaction of a charged particle with a spinor field. Solutions to the problems of particle motion in an external spinor field in the case of a hydrogen-like atom and planetary motion, splitting of the electron energy levels of an atom in an external magnetic field, deflection of a photon by the gravitational field, and representations in metric spaces are presented.

1. Introduction

When I meet God, I am going to ask Him two

questions: Why relativity? And why turbulence?

I really believe He will have an answer for the first.

Attributed to Werner Heisenberg (1901–1976).

The material presented in this chapter is based on the new approaches of relativistic and quantum mechanics developed in the works [1, 2, 3, 4]. Equations, which are obtained by applying the invariance principle for the total four-dimensional momentum of the system “field + particle,” have some significant advantages as compared with its analog equations such as the Klein-Fock-Gordon and Dirac equations. For instance, the problem of a hydrogen-like atom has solutions for an arbitrary value of the interaction constant not restricted to whatever the atomic number of the nucleus (we recall that for the Dirac equation the atomic number is restricted to Z < 137).

In contrast to the well-known equations of relativistic and quantum mechanics, the energy levels of the ground state of the particle for the considered equation prove to be limited by the size of the spatial characteristic. This property directly reflects the uncertainty principle in that, irrespective of the well depth value, the particle can be localized in a bound state only if the well width is larger than the half-wavelength of the particle.

For the problem of the passage of a particle through a potential barrier, if the energy of the particle does not exceed the height of the potential barrier, then the transmission coefficient is equal to zero regardless of the height of the barrier. In this case, there is no contradiction like Klein’s paradox.

The equations are applicable for different types of particles and interactions. The analysis of the solutions shows full compliance with the principles of relativistic and quantum mechanics, and the solutions are devoid of any restrictions on the nature and magnitude of the interactions.

However, this new theory was presented without an explicit representation of the spinor properties of fields and systems (Table 1). In this chapter, based on the representation of local relativistic rotation by the Lorentz transformation matrix, the representations of the four-dimensional energy-momentum vector for various spinor fields and systems, the results of corresponding solutions of the new equations of relativistic and quantum mechanics are given.

Representation of the generalized momentum
$\begin{array}{l}\mathbf{P}=\left(\begin{array}{l}\frac{\mathit{mc}}{\sqrt{1-{\beta }^2}}+\frac{q}{c}\phi ,\\ \frac{\mathit{mc}}{\sqrt{1-{\beta }^2}}\mathbf{\beta }+\frac{q}{c}\mathbf{A}\end{array}\right)\\ {\mathbf{P}}^2={\epsilon }^2-{\mathbf{p}}^2=I^2\ne \mathit{\text{invariant}}\end{array}$	$\begin{array}{l}\mathbf{P}=\left(\begin{array}{l}\frac{\mathit{mc}}{\sqrt{1-{\beta }^2}}+\frac{q}{c}\frac{\phi +\mathbf{\beta }\cdot \mathbf{A}}{\sqrt{1-{\beta }^2}},\\ \frac{\mathit{mc}}{\sqrt{1-{\beta }^2}}\mathbf{\beta }+\frac{q}{c}\frac{\phi \mathbf{\beta }+{\mathbf{A}}_{\parallel }}{\sqrt{1-{\beta }^2}}+\frac{q}{c}{\mathbf{A}}_{\perp }\end{array}\right)\\ {\mathbf{P}}^2={\epsilon }^2-{\mathbf{p}}^2=I^2=\mathit{\text{invariant}}\end{array}$
Invariant of the generalized momentum
${\mathbf{P}}^2={\left(\epsilon –\frac{q}{c}\phi \right)}^2-{\left(\mathbf{p}–\frac{q}{c}\mathbf{A}\right)}^2={\left(\mathit{mc}\right)}^2$	${\mathbf{P}}^2={\epsilon }^2-{\mathbf{p}}^2={\left(\mathit{mc}+\frac{q}{c}\phi \right)}^2-{\left(\frac{q}{c}\mathbf{A}\right)}^2$
Space-time interval
$\begin{array}{l}{\mathit{ds}}^2=I^2\left(d{\tau }^2-{\mathit{dx}}^2-{\mathit{dy}}^2-{\mathit{dz}}^2\right)=g_{\mathit{ik}}{\mathit{dx}}^i{\mathit{dx}}^k\\ I^2={\left(\mathit{mc}\right)}^2\end{array}$	$\begin{array}{l}{\mathit{ds}}^2=I^2\left(d{\tau }^2-{\mathit{dx}}^2-{\mathit{dy}}^2-{\mathit{dz}}^2\right)=g_{\mathit{ik}}{\mathit{dx}}^i{\mathit{dx}}^k\\ I^2={\left(\mathit{mc}+\frac{q}{c}\phi \right)}^2-{\left(\frac{q}{c}\mathbf{A}\right)}^2\end{array}$
Hamilton-Jacobi relativistic equation
${\left(\frac{\partial S}{\partial \tau }-\frac{q}{c}\phi \right)}^2-{\left(-\frac{\partial S}{\partial \mathbf{r}}-\frac{q}{c}\mathbf{A}\right)}^2={\left(\mathit{mc}\right)}^2$	${\left(\frac{\partial S}{\partial \tau }\right)}^2-{\left(\frac{\partial S}{\partial \mathbf{r}}\right)}^2={\left(\mathit{mc}+\frac{q}{c}\phi \right)}^2-{\left(\frac{q}{c}\mathbf{A}\right)}^2$
Hamilton-Jacobi-Einstein equation
$g_{\mathit{ik}}\sqrt{-g}\left(\frac{\partial S}{\partial x^i}-A_i\right)\left(\frac{\partial S}{\partial x^k}-A_k\right)={\left(\mathit{mc}\right)}^2$	$g_{\mathit{ik}}\sqrt{-g}\frac{\partial S}{\partial x^i}\frac{\partial S}{\partial x^k}={\left(\mathit{mc}+\frac{q}{c}\phi \right)}^2-{\left(\frac{q}{c}\mathbf{A}\right)}^2$
Klein-Fock-Gordon equation
$\begin{array}{l}{\left(i\mathrm{\hslash }\frac{\partial }{\partial \tau }-\frac{q}{c}\phi \right)}^2\mathrm{\Psi }-{\left(-i\mathrm{\hslash }\frac{\partial }{\partial \mathbf{r}}-\frac{q}{c}\mathbf{A}\right)}^2\psi =\\ -{\left(\mathit{mc}\right)}^2\psi \end{array}$	$\begin{array}{l}\frac{{\partial }^2\psi }{\partial {\tau }^2}-\frac{{\partial }^2\psi }{\partial {\mathbf{r}}^2}=\\ -\frac{{\left({\mathit{mc}}^2+\mathit{q\phi }\right)}^2-{\left(q\mathbf{A}\right)}^2}{{\mathrm{\hslash }}^2{c}^2}\psi \end{array}$
Klein-Fock-Gordon equation in metric spaces
$\begin{array}{l}\frac{1}{\sqrt{-g}}\left(i\mathrm{\hslash }\frac{\partial }{\partial x^i}-\frac{q}{c}{A}_i\right)\times \\ \left(g_{\mathit{ik}}\sqrt{-g}\left(-i\mathrm{\hslash }\frac{\partial }{\partial x^k}-\frac{q}{c}{A}_k\right)\psi \right)=-{\left(\mathit{mc}\right)}^2\psi \end{array}$	$\begin{array}{l}\frac{1}{\sqrt{-g}}\frac{\partial }{\partial x^i}\left(g_{\mathit{ik}}\sqrt{-g}\frac{\partial \psi }{\partial x^k}\right)=\\ -\frac{{\left({\mathit{mc}}^2+\mathit{q\phi }\right)}^2-{\left(q\mathbf{A}\right)}^2}{{\mathrm{\hslash }}^2{c}^2}\psi \end{array}$
Dirac equation in 1/2 spin spaces
$\begin{array}{l}\left(\widehat{\epsilon }-\frac{q}{c}\phi \right)\varphi -\mathbf{\sigma }\cdot \left(\widehat{\mathbf{p}}-\frac{q}{c}\mathbf{A}\right)\mathrm{\chi }=\mathit{mc\varphi },\\ \left(\widehat{\epsilon }-\frac{q}{c}\phi \right)\mathrm{\chi }+\mathbf{\sigma }\cdot \left(\widehat{\mathbf{p}}-\frac{q}{c}\mathbf{A}\right)\varphi =-\mathit{mc}\mathrm{\chi }.\end{array}$	$\begin{array}{l}\widehat{\epsilon }\varphi -\mathbf{\sigma }\cdot \widehat{\mathbf{p}}\mathrm{\chi }=\left(\mathit{mc}+\frac{q}{c}\phi \right)\varphi -\frac{q}{c}\mathbf{\sigma }\cdot \mathbf{A}\mathrm{\chi },\\ \widehat{\epsilon }\mathrm{\chi }+\mathbf{\sigma }\cdot \widehat{\mathbf{p}}\varphi =-\left(\mathit{mc}+\frac{q}{c}\phi \right)\mathrm{\chi }-\frac{q}{c}\mathbf{\sigma }\cdot \mathbf{A}\varphi .\end{array}$

Table 1.

Old and new [1, 2, 3, 4] concepts and equations of relativistic and quantum mechanics.

Since within the framework of the new, generalized relativistic theory, there is an exact correspondence between the representations of relativistic, quantum mechanics, and general relativity, in the chapter spinor properties and equations are also presented in metric spaces.

2. The spin concept

Although in physics the concept of spin arose as a property of the proper rotation of a particle (electron, G. E. Uhlenbeck, S. Goudsmit), as a result of quantization of the self-angular momentum, further development by W. Pauli and P. Dirac led to the description of spin as a property of space itself, in which we describe particles. Their interactions corresponded to the data of the physical experiment, when they were presented in spinor spaces.

One of the first, the Dirac equation, which describes systems in spaces with spin 1/2, the solution in the case of the hydrogen atom gives a very good match with the real spectrum of the hydrogen atom. But the hydrogen atom problem (Figure 1),

Figure 1.

The problem of the motion of two bodies can be represented as a problem of the motion of one body.

which consists of a proton and an electron, is solved for a generalized particle with the reduced mass $m$ of an electron $m_1$ and a proton $m_2$ (the two-body problem), whose coordinates $\mathbf{r}$ do not coincide with either the coordinates of the electron ${\mathbf{r}}_1$ or the coordinates of the proton ${\mathbf{r}}_2$, and the spin of this generalized particle (the sum of the spins of the proton and electron) can only have values of 0 or 1.

Obviously, Dirac’s spin 1/2 refers to the properties of the space in which the hydrogen atom is described, not to an electron, a proton, or a hydrogen atom. Similarly, regardless of the spin of the nucleus and the spin of the electron shell of hydrogen-like atoms, the spectrum, and fine splitting is described by solving the Dirac equation with spin 1/2. The same is true for other problems, regardless of the properties of the components of the physical system themselves—solutions to the Dirac equations describe systems with only a spin of 1/2.

Naturally, the spinor properties of space do not in any way describe the physical properties of the self-angular momentum of a particle or a vortex (turbulent) field (the flow of a liquid or gas) in the classical sense, and therefore, the statement arose that spin has no classical analog (Figure 2). And the problem of describing the self-angular momentum of the particles and the vortex field remained unresolved.

Figure 2.

The concept of spin in quantum mechanics does not describe the physical properties of the vortex motion or proper rotation of bodies in the classical sense.

Note that the description of the motion of an asymmetric spinning top, represented by the tensor of the moment of inertia, has no analog in quantum mechanics since the modern concept of spin does not imply any representations of the concept of the moment of inertia (Figure 2). But in the case of nonspherical nuclei (Figure 3), the projection of the self-angular momentum must have different spin values relative to the main axes of inertia (rotation) of the nucleus.

Figure 3.

The projections of the self-angular momentum (spin) of nonspherical nuclei have different values in the directions of the main axes of inertia.

The main, fundamental physical variable, for which the variational principles and equations of relativistic and quantum mechanics are formulated, is the energy-momentum density 4-vector, so all the properties of the system, also spinor, must be initially reflected in the representation of the energy-momentum 4-vector $\mathbf{P}=\left(\epsilon ,\mathbf{p}\right)$.

3. Orbital and vortex motions of the continuous media

Imagine holding a bicycle wheel by the axle at a distance $l$ and rotating around its axis with an angular velocity of $\mathbf{\Omega }$ (Figure 4).

Figure 4.

Bicycle wheel at a distance $l$ rotating around axis with an angular velocity of $\mathbf{\Omega }$.

Since the wheel rotates freely around its axis, there is no moment of own rotation of the wheel itself, and it makes an only translational motion (movement without its own rotation). Energy $W$ is expressed by the usual formula of kinetic energy through the mass m and velocity $\mathbf{v}$ of the wheel movement $W=m{\mathbf{v}}^2/2={\mathit{ml}}^2{\mathbf{\Omega }}^2/2$.

If there is any friction of the axle, then the rotational motion of the axle will gradually be transmitted to the wheel and eventually, the wheel will rotate with the same angular speed. Note that in this case, we considered the option of a rigidly fixed axis (rigidly fixed center of mass) wheel. This corresponds to a rigid spin-orbit interaction when the angular velocities of rotation are the same.

In this case, the energy is equal to the sum of the kinetic energy of the wheel and the energy of the wheel’s own rotation with the moment of inertia I (a sample conclusion of the Huygens-Stern theorem)

$$W=m{\mathbf{v}}^2/2+I{\mathbf{\Omega }}^2/2=\left({\mathit{ml}}^2+I\right){\mathbf{\Omega }}^2/2.$$E1

In other cases, in terms of energy, there will be other ratios of translational energy and energy of its own rotation (spin). In particular, the motion of the Moon corresponds to the case of a rigidly fixed center of mass (the Moon is constantly facing the Earth on the same side), and the movement of the Earth around the Sun corresponds to the case of a freely fixed center of mass (the tilt of the axis, the period of its own rotation is in no way related to the movement around the Sun).

Note that regardless of the distance from the center of rotation, the angular velocity of its own rotation and energy is constant. If the system is represented as a medium with distributed local rotation, then it can be described by the energy density of vortex rotation with an angular velocity $\mathbf{\Omega }$. A good example of such a system is a permanent magnet, where at each point of the medium there are eddy currents (rotating electric field) and, accordingly, the magnetic field of these currents (rotating electric field).

To illustrate the vortex motion of the distributed systems, let us consider an example of a large, thin hard disk on which much identical small metal (heavy) spinners of mass m and moment of inertia I are attached perpendicularly and densely (Figure 5). The mass of the disk relative to the total mass of spinners $\mathrm{\Sigma }m$ can be neglected. If the spinners do not have axial friction, then when the disk rotates, the spinners will rotate around the axis of the disk, but will not rotate around their own axis—the arrows will always be turned toward the original direction (Figure 5). In this case, the rotation energy of the system W is determined only by the sum of the kinetic energies of the orbital rotation of the spinners $W=\mathrm{\Sigma }m{{\mathbf{v}}_i}^2/2=\mathrm{\Sigma }{{\mathit{mr}}_i}^2{\mathbf{\Omega }}^2/2$ relative to the axis of the disk, and when the disk stops, the energy of the system becomes zero.

Figure 5.

Spinners of mass m and moment of inertia I are perpendicularly and densely fixed on the thin hard disk.

In the case when the spinners have some small axial friction (like the spin-orbit interaction of an electron in an atom or the connection of the moon’s rotation with the Earth’s rotation due to tides), then when the disk rotates, due to friction, after some time the spinners will begin to rotate around its own axis and with the angular speed of rotation of the disk $\mathbf{\Omega }$ (Figure 6). In this case, during rotation, the spinners are at any moment oriented toward the center of the disk axis (like a Moon), which corresponds to the rotation of the disk with the axes of the spinners fixed rigidly—at each point, the spinners rotate around their own axis with the angular speed of rotation of the disk. The energy of the system will be represented as the sum of the kinetic energy of the orbital rotation and the energy of the own rotation of the spinners in the form of $W=\mathrm{\Sigma }{{\mathit{mr}}_i}^2{\mathbf{\Omega }}^2/2+\mathrm{\Sigma }I{\mathbf{\Omega }}^2/2$.

Figure 6.

Spinners rotate around their axis with the angular velocity of the disk $\mathbf{\Omega }$.

If we stop the rotation of the disk in such an established stationary state, then the energy of the orbital motion of the spinners will be reset, but the energy of its own rotation $\mathrm{\Sigma }I{\mathbf{\Omega }}^2/2$ will be preserved (Figure 7).

Figure 7.

The spinners keep spinning after the disk has stopped.

The picture, obtained after stopping the disk, represents a distributed system with local vortex motion. Regardless of the distance from the center of the disk, the angular velocity of rotation and energy of the spinners are constant. If the system is represented as a medium with distributed local rotation, then it can be described by the energy density of vortex rotation with an angular velocity $\mathbf{\Omega }$.

Let us emphasize that although the described system does not have orbital momentum, the generalized angular momentum and the total energy of the system are not zero and have minimal internal angular momentum and energy. It should be noted that in the ground state, both the angular momentum and the energy of the system have corresponding minimum values due to the internal rotation—spin.

A good example of such a system is a permanent magnet, where eddy currents (rotating electric field) and, accordingly, the magnetic field of these currents (rotating electric field) exist at each point of the medium (Figure 8).

Figure 8.

Eddy currents and magnetic field exist at each point of the medium.

Vortex and circular fields should be distinguished: in vortex fields, the rotor is nonzero at any point in the field, and in circular fields, it is zero (Figure 9). Such is the electric field outside the alternating current solenoid, where the magnetic field is zero.

Figure 9.

Vortex and circular fields of the solenoid.

A time-varying magnetic field in the solenoid generates, induces an electric field $\mathbf{E}$, which is described by Maxwell’s equations. Such a field is described in cylindrical coordinates and is represented by the vector potential $\mathbf{A}=\mathbf{A}\left(\mathbf{r},t\right)$ in the form [5].

$$\mathbf{A}\left(\mathbf{r},t\right)=\left\{\begin{array}{l}\frac{1}{2}\left[\mathbf{B}\left(t\right)\times \mathbf{r}\right],r\le R_S;\\ \frac{{R_S}^2}{2\left(x^2+y^2\right)}\left[\mathbf{B}\left(t\right)\times \mathbf{r}\right],r>R_S;\end{array}\right.$$E2

where $R_S$ is a radius of the solenoid.

For the fields of the solenoid, we have

$$\mathbf{B}=\mathrm{rot}\mathbf{A}=\left\{\begin{array}{l}\mathbf{B},r\le R_S,\\ 0,r>R_S.\end{array}\right.\mathbf{E}=-\frac{1}{c}\frac{\partial \mathbf{A}}{\partial t}=\left\{\begin{array}{l}\frac{1}{2c}\left[\mathbf{r}\times \frac{\partial \mathbf{B}}{\partial t}\right],r\le R_S,\\ \frac{{R_S}^2}{2c\left(x^2+y^2\right)}\left[\mathbf{r}\times \frac{\partial \mathbf{B}}{\partial t}\right],r>R_S.\end{array}\right.$$E3

Inside the solenoid, the electric field is vortex – $\mathrm{rot}\mathbf{E}\ne 0$, and outside the solenoid it is circular and the magnetic field is zero – $\mathrm{rot}\mathbf{E}=0$.

4. Representation of the spinor fields in Minkowski spaces

When representing continuously distributed systems, one should average the energy of the spinners over the occupied volume and describe the continuous medium with the energy-momentum distribution density.

If we want to describe the spatial properties of the vortex fields, then rotation angular velocity $\mathbf{\Omega }$ at a given point in space can serve as such a kinematic variable. The occurrence of vortex rotation distributed in space in the framework of the relativistic theory is described by the Lorentz transformation, calculating the energy-momentum distribution density of local rotations.

For fields, we have [6]

$$\begin{array}{l}\left({\phi }^{\prime },{\mathbf{A}}^{\prime }\right)=\left(\frac{\phi +\mathbf{\beta }\cdot \mathbf{A}}{\sqrt{1-{\beta }^2}},\frac{\phi \mathbf{\beta }+q{\mathbf{A}}_{\parallel }}{\sqrt{1-{\beta }^2}}+q{\mathbf{A}}_{\perp }\right)=\\ \left(\phi ,\mathbf{A}\right)+\left(\left(\gamma -1\right)\phi +\gamma \mathbf{\beta }\cdot \mathbf{A},\mathit{\gamma \phi }\mathbf{\beta }+\left(\gamma -1\right)\left(\mathbf{A}\cdot \mathbf{\beta }\right)\mathbf{\beta }/{\beta }^2\right).\end{array}$$E4

where $\gamma =1/\sqrt{1-{\beta }^2}$ is a Lorentz factor. This transformation can be presented in matrices form

$$\left({\phi }^{\prime },{\mathbf{A}}^{\prime }\right)=\left(\phi ,\mathbf{A}\right)+\widehat{\mathrm{T}}\left(\phi ,\mathbf{A}\right).$$E5

where a Lorentz transformation has a form [1].

$$\widehat{\mathrm{T}}=\gamma ‖\begin{array}{cccc}0& {\beta }_1& {\beta }_2& {\beta }_3\\ {\beta }_1& 0& 0& 0\\ {\beta }_2& 0& 0& 0\\ {\beta }_3& 0& 0& 0\end{array}‖+\left(\gamma -1\right)‖\begin{array}{cccc}1& 0& 0& 0\\ 0& \frac{{\beta }_1{\beta }_1}{{\beta }^2}& \frac{{\beta }_1{\beta }_2}{{\beta }^2}& \frac{{\beta }_1{\beta }_3}{{\beta }^2}\\ 0& \frac{{\beta }_2{\beta }_1}{{\beta }^2}& \frac{{\beta }_2{\beta }_2}{{\beta }^2}& \frac{{\beta }_2{\beta }_3}{{\beta }^2}\\ 0& \frac{{\beta }_3{\beta }_1}{{\beta }^2}& \frac{{\beta }_3{\beta }_2}{{\beta }^2}& \frac{{\beta }_3{\beta }_3}{{\beta }^2}\end{array}‖.$$E6

The matrices of the invariant representation of a four-dimensional vector, which preserves the vector module in four-dimensional space, form the Poincare group (inhomogeneous Lorentz group). In addition to displacements and rotations, the group contains space-time reflection representations $\widehat{\mathrm{P}},\widehat{\mathrm{T}}$ and inversion$\widehat{\mathrm{P}}\widehat{\mathrm{T}}=\widehat{\mathrm{I}}$.

In the case of a vortex field, each point can be attributed to a local rotation with an angular velocity $\mathbf{\Omega }$ and velocity $\mathbf{v}=\left[\mathbf{\Omega }\times \mathbf{r}\right]$ at a distance $r$ from the chosen point of rotation. First, we consider rotation only around one axis of the coordinate system $\widehat{\mathbf{z}}$. Then, when choosing a cylindrical coordinate system, we have

$$\gamma =1/\sqrt{1-{\left(v/c\right)}^2}=1/\sqrt{1-{\left[\mathbf{\Omega }\times \mathbf{r}\right]}^2/c^2}=1/\sqrt{1-{\left(r/r_h\right)}^2},$$E7

$$\begin{array}{l}\widehat{\mathrm{T}}=\mathit{\gamma \beta }‖\begin{array}{cccc}0& cos\left(\varphi \right)& sin\left(\varphi \right)& 0\\ cos\left(\varphi \right)& 0& 0& 0\\ sin\left(\varphi \right)& 0& 0& 0\\ 0& 0& 0& 0\end{array}‖+\\ \left(\gamma -1\right)‖\begin{array}{cccc}1& 0& 0& 0\\ 0& {cos}^2\left(\varphi \right)& sin\left(\varphi \right)cos\left(\varphi \right)& 0\\ 0& sin\left(\varphi \right)cos\left(\varphi \right)& {sin}^2\left(\varphi \right)& 0\\ 0& 0& 0& 0\end{array}‖,\end{array}$$E8

were $r_h=c/\mathrm{\Omega }$ the event horizon radius. At each point in space, we must average the value of the energy-momentum and assign it to the selected point. From (7), (8) it follows that when averaging in volume $\pi {r_h}^{2}d$($d$is the thickness of the disk), only the average values of the diagonal elements of the matrix are nonzero:

$$\begin{array}{l}\widehat{\overline{\mathrm{T}}}=\frac{1}{V}\underset{V}{\int }\widehat{\mathrm{T}}\mathit{dv}=\\ \frac{1}{\pi {r_h}^{2}d}\underset{0}{\overset{d}{\int }}\mathit{dz}\cdot \underset{0}{\overset{r_h}{\int }}r\left(\frac{1}{\sqrt{1-r^2/{r_h}^2}}-1\right)\mathit{dr}\cdot \underset{0}{\overset{2\pi }{\int }}‖\begin{array}{ccc}1& 0& 0\\ 0& {cos}^2\left(\varphi \right)& sin\left(\varphi \right)cos\left(\varphi \right)\\ sin\left(\varphi \right)cos\left(\varphi \right)& 0& {sin}^2\left(\varphi \right)\end{array}‖\mathit{d\varphi }=\\ ‖\begin{array}{ccc}1& 0& 0\\ 0& 1/2& 0\\ 0& 0& 1/2\end{array}‖,\end{array}$$E9

where

$$\frac{2\pi }{\pi {r_h}^2}{\int }_0^{r_h}r\left(\frac{1}{\sqrt{1-r^2/{r_h}^2}}-1\right)\mathit{dr}=\underset{0}{\overset{1}{\int }}\left(\frac{1}{\sqrt{1-x^2}}-1\right)2\mathit{xdx}={\left.-2\sqrt{1-x^2}-x^2\right|}_0^1=1.$$E10

Averaging over the entire volume to the event horizon $r_h$, we obtain a result independent of the speed of angular rotation $\mathbf{\Omega }$ This seemingly unexpected result is due to the inverse dependence of the event horizon $r_h=c/\mathrm{\Omega }$ on the angular velocity of rotation. The greater the speed of angular rotation, the greater the relativistic compression and, accordingly, the increase in energy density due to a decrease in the volume of integration, so the integral does not depend on the angular velocity $\mathbf{\Omega }$ This means that own rotation, spin is an invariant for any reference frames, in the approaches of general relativity.

Based on the principle of superposition and additivity, such elementary excitations of the generating field in a unit of volume can be any integer, so that in the general case $n$

$$\widehat{\mathbf{\Omega }}={‖\begin{array}{ccc}1/2& 0& 0\\ 0& 1/2& 0\\ 0& 0& 0\end{array}‖}_{\widehat{\mathbf{z}}}\to \pm \frac{1}{2}\times n=s_z\to {‖\begin{array}{ccc}{s}_z& 0& 0\\ 0& s_z& 0\\ 0& 0& 0\end{array}‖}_{\widehat{\mathbf{z}}},$$E11

where is a $\widehat{\mathbf{\Omega }}$ representation matrix of the rotation (spinor), and ${\mathrm{\Omega }}_x={\mathrm{\Omega }}_y=s_z$ – the value of components for the $\widehat{\mathbf{x}},\widehat{\mathbf{y}}$ axis’s when rotating around an axis $\widehat{\mathbf{z}}$. As we see, the rotation along the selected axis generates two equal perpendicular spatial components of the generalized momentum with half-integer coefficients (Figure 10).

Figure 10.

Rotation along the selected axis generates two equal perpendicular spatial components with half-integer coefficients.

Accordingly, taking into account the independent rotations in all axes, for the matrix of the spinor representation we have

$$\begin{array}{l}\widehat{\mathbf{\Omega }}={‖\begin{array}{ccc}0& 0& 0\\ 0& s_x& 0\\ 0& 0& s_x\end{array}‖}_{\widehat{\mathbf{x}}}+{‖\begin{array}{ccc}{s}_y& 0& 0\\ 0& 0& 0\\ 0& 0& s_y\end{array}‖}_{\widehat{\mathbf{y}}}+{‖\begin{array}{ccc}{s}_z& 0& 0\\ 0& s_z& 0\\ 0& 0& 0\end{array}‖}_{\widehat{\mathbf{z}}}=\\ ‖\begin{array}{ccc}{s}_y+s_z& 0& 0\\ 0& s_x+s_z& 0\\ 0& 0& s_x+s_y\end{array}‖.\end{array}$$E12

Thus, the spinor, the averaged Lorentz transformation for local rotations, is represented by a diagonal matrix or column as

$$\begin{array}{l}\widehat{\mathbf{\Omega }}=\left\{s_y+s_z,s_x+s_z,s_x+s_y\right\},s_x,s_y,s_z=0,\pm \frac{1}{2},\pm 1,\pm \frac{3}{2},\dots ,\pm \frac{n}{2},\dots \\ {\widehat{\mathbf{\Omega }}}^2=2\left({s_x}^2+{s_y}^2+{s_z}^2+s_x{s}_y+s_x{s}_z+s_y{s}_z\right).\end{array}$$E13

The index symbols of the components of the matrix are specifically selected to indicate from rotation around which axes these components originated:

$${\mathrm{\Omega }}_x=s_y+s_z,{\mathrm{\Omega }}_y=s_x+s_z,{\mathrm{\Omega }}_z=s_x+s_y.$$E14

Negative signs of the projections of the spinor matrix $\widehat{\mathbf{\Omega }}$ appeared during the transformation of reflections and inversions (antiparticles).

$$\begin{array}{l}\widehat{\mathbf{T}}=‖\begin{array}{cccc}\pm m& 0& 0& 0\\ 0& s_y+s_z& 0& 0\\ 0& 0& s_x+s_z& 0\\ 0& 0& 0& s_x+s_y\end{array}‖,\\ \mathbf{T}=\left(\pm m,s_y+s_z,s_x+s_z,s_x+s_y\right).\end{array}$$E15

where $m$ is integer.

If a scalar field $\phi$ is given in space, at each point of which the vortices are excited, then the vector potential of such a field can be represented as $\mathbf{A}=\phi \mathbf{\Omega }$, and the generalized energy-momentum $\mathbf{P}$ as (Table 2)

$\left(s_y+s_z,s_x+s_z,s_x+s_y\right)$	${\mathbf{\Omega }}^2$	$1-{\mathbf{\Omega }}^2$	$\left(s_y+s_z,s_x+s_z,s_x+s_y\right)$	${\mathbf{\Omega }}^2$	$1-{\mathbf{\Omega }}^2$
(1/2, 1/2, 0)	1/2	1/2	(2, 1, 0)	5	–4
(1, 0, 0)	1	0	(3/2, 3/2, 1)	11/2	−9/2
(1, 1/2, 1/2)	3/2	−1/2	(2, 1, 1)	6	−5
(1, 1, 0)	2	−1	(5/2, 1/2, 0)	13/2	−11/2
(3/2, 1/2, 0)	5/2	−3/2	(2, 3/2, 1/2)	13/2	−11/2
(1, 1, 1)	3	−2	(5/2, 1, 1/2)	15/2	−13/2
(3/2, 1, 1/2)	7/2	−5/2	(2, 2, 0)	8	−7
(2, 0, 0)	4	−3	(5/2, 3/2, 0)	17/2	−15/2
(3/2, 3/2, 0)	9/2	−7/2	(2, 3/2, 3/2)	17/2	−15/2
(2, 1/2, 1/2)	9/2	−7/2	(3, 0, 0)	9	−8
(2, 1, 0)	5	−4	(2, 2, 1)	9	−8

Table 2.

Possible structures and invariants for spinor fields (component signs are arbitrary).

$$\begin{array}{l}\mathbf{P}=\left(\phi ,\mathbf{A}\right)=\left(\phi ,\phi \mathbf{\Omega }\right),{\mathbf{P}}^2={\phi }^2\left(1-{\mathbf{\Omega }}^2\right),\\ 1-{\mathbf{\Omega }}^2=1-2\left({s_x}^2+{s_y}^2+{s_z}^2+s_x{s}_y+s_x{s}_z+s_y{s}_z\right).\end{array}$$E16

If for qualitative evaluation, we assume that the energy of such elementary excitation corresponds to the quantum of the rotator energy $E={\mathit{mc}}^2=\mathrm{\hslash \Omega }$, then we get

$$\mathtt{ƛ}=\mathrm{\hslash }/\mathit{mc}=c/\mathrm{\Omega }=r_h,$$E17

that is, the event horizon radius (“particle size”) is the wavelength of the particle $r_h=\mathtt{ƛ}$.

Since the spin of the field has been determined locally, its direction from point to point can change, provided that the internal structure is preserved. The spinor field, as in the general case of any vector potential $\mathbf{A}=\phi \mathbf{\Omega }$, must satisfy the basic equations for the fields [2].

Spin is invariant from the point of view of general relativity since it does not depend on the state of motion—the speed and rotation of the reference frame, in which we describe the spinor field.

The spinor properties of fields are unchanged and do not depend on the method and sources of their creation. For each case of motion of a particle in a given field, the spinor properties of the field are determined by the physical nature of the field itself. Spin is a fundamental, unchanging characteristic of the field. Whichever way it is created, it will be with the same back.

Spin in the expression of potential energy describes the system of particle + field, the spinor properties of which are described by the above expressions, regardless of which components are included in the system. The new field of interaction is also represented by one of the above configurations. Whether the spinor field of interaction is a simple sum of spinor fields or another configuration can be found by comparing with experimental data. For each system, a spin configuration should be selected so that the results of the calculation coincide with the experimental data or with the already known characteristics of the system.

The spin of the nucleus and the presence of other electrons with their spins do not determine the spin of the electromagnetic field created by it in atoms. Fine splitting, which directly depends on the spin of the interaction field, does not depend on the spin of the nuclei and electron shell of the atoms. For example, the value of fine splitting of isotope atoms Potassium ³⁹K, ⁴¹K, and ⁴⁰K with spines 3/2, 3/2, and 4; Rubidium ⁸⁷Rb and ⁸⁵Rb with spines 3/2 и 5/2; Hydrogen ¹H, ²H, ³H with spines 1/2, 1, 1/2 [7].

Also in the solar system, the spin of the gravitational field does not depend on the positions of celestial bodies and their rotation.

Saying that a particle has spin, within the framework of the foregoing, means that the particle has a spinor field, and this property manifests itself at any point in space during any interactions. Spin is a spatial characteristic and is not attributed to any point particle. Therefore, the spin of elementary particles is not determined by their internal structure, but is determined by the spin of the interaction fields created by these particles. The phrase “spin-orbit interaction” in this case means the interaction of the orbital moment with the spinor field.

5. Examples of spinor fields representation

5.1 Electromagnetic wave: photon

The spin of an electromagnetic wave with potential $\phi$ is represented as a rotating electric field with a vector potential $\mathbf{A}=\mathbf{\Omega }\phi$. Given that the magnetic field $\mathbf{B}$ of the wave is defined as the rotor of the vector potential

$$\mathbf{B}=\mathrm{rot}\left(\phi \mathbf{\Omega }\right)=-\left[\mathbf{\Omega }\times \frac{\partial \phi }{\partial \mathbf{r}}\right]=\left[\mathbf{\Omega }\times \mathbf{E}\right],$$E18

and must satisfy the condition $div\left(\phi \mathbf{\Omega }\right)=\mathbf{\Omega }\cdot \mathbf{E}=0,$ get ${\mathbf{B}}^2={\mathbf{\Omega }}^2{\mathbf{E}}^2$. So how for an electromagnetic wave ${\mathbf{B}}^2={\mathbf{E}}^2,{\mathbf{\Omega }}^2=1$ and the invariant is zero, then the only structure corresponding to these conditions is $\mathbf{\Omega }=\left(1,0,0\right)$ (Table 2) with a single projection on the axis of propagation (spin) 1. Thus, the four-dimensional vector of the electromagnetic wave is represented as

$$\mathbf{P}=\left(\phi ,\phi \mathbf{\Omega }\right)=\left(\phi ,\phi ,0,0\right)=\left(\phi ,\phi \mathbf{n}\right),{\mathbf{P}}^2=0,$$E19

where n is the unit vector in the direction of wave propagation.

For a monochromatic wave propagating in the direction of the axis $\widehat{\mathbf{z}}$ with a potential $\phi \left(x,y\right)$ and wavelength $\lambda$, we have

$$\mathbf{P}=\left(\epsilon ,\mathbf{p}\right)=\phi \left(x,y\right)exp\left(\frac{\tau -z}{\lambda }i\right)\left(1,\widehat{\mathbf{z}}\right),{\mathbf{P}}^2=0,\frac{\partial \epsilon }{\partial \tau }+div\mathbf{p}=0.$$E20

For transverse waves of other fields, you can select one-component structures with other integers in the form $\mathbf{\Omega }=\left(m,0,0\right),m=1,2,3,\dots$ (Table 2).

5.2 Stationary magnetic field

$$\begin{array}{l}\mathbf{P}=\left(0,\phi \mathbf{\Omega }\right);\mathbf{B}=\mathrm{rot}\left(\phi \mathbf{\Omega }\right)=\left[\mathbf{\Omega }\times \frac{\partial \phi }{\partial \mathbf{r}}\right]=\\ \left({\mathrm{\Omega }}_y\frac{\partial \phi }{\partial z}-{\mathrm{\Omega }}_z\frac{\partial \phi }{\partial y},-{\mathrm{\Omega }}_x\frac{\partial \phi }{\partial z}+{\mathrm{\Omega }}_z\frac{\partial \phi }{\partial x},{\mathrm{\Omega }}_x\frac{\partial \phi }{\partial y}-{\mathrm{\Omega }}_y\frac{\partial \phi }{\partial x}\right)=\left(0,0,B\right).\end{array}$$E21

For solenoidal magnetic fields having cylindrical symmetry, we have ${\mathrm{\Omega }}_z=0$ and ${\mathrm{\Omega }}_x={\mathrm{\Omega }}_y=1/2$. The solution for such a system is the vector potential in the form of a structure. $\mathbf{\Omega }=\left(1/2,1/2,0\right)$, ${\mathbf{\Omega }}^2=1/2$ (Table 2) with one of the linear functions as an electric field potential

$$\phi \mathbf{\Omega }=\phi \left(1/2,1/2,0\right);\phi ={\phi }_0\left(x-y\right);\mathbf{B}=\left[\mathbf{\Omega }\times \mathbf{E}\right]=\left(0,0,E\right).$$E22

5.3 Spinor field with potential $\phi =\alpha /r$

For the potential in the form of $\phi =\alpha /r$ we have

$$\mathbf{P}=\left(\frac{\alpha }{r},\frac{\alpha }{r}\mathbf{\Omega }\right)=\frac{\alpha }{r}\left(1,\mathbf{\Omega }\right),{\mathbf{P}}^2=\frac{{\alpha }^2}{r^2}\left(1-{\mathbf{\Omega }}^2\right)$$E23

The spinor field of this potential can have different structures and invariant values $1-{\mathbf{\Omega }}^2$. For Coulomb interaction $\mathbf{\Omega }=\left(1,1,0\right),1-{\mathbf{\Omega }}^2=-1$, and gravity – $\mathbf{\Omega }=\left(2,2,0\right)$, $1-{\mathbf{\Omega }}^2=-7$ (Table 2). Such structures and values of the spinor representation provide an accurate description of the spectrum of hydrogen-like atoms, the value of fine splitting, the secular shift of Mercury’s perihelion, and the angle of deflection of the light beam by the gravitational field of the Sun.

6. Equations of relativistic and quantum mechanics with spinor fields

Adding spinor fields in the equations of relativistic and quantum mechanics for a particle in an external field (Table 1) is not associated with any difficulties, since the spin properties of the field are clearly represented as the vector potential of interaction $\mathbf{A}=\phi \mathbf{\Omega }$. Equations can be represented as

$${\left(\frac{\partial S}{\partial \tau }\right)}^2-{\left(\frac{\partial S}{\partial \mathbf{r}}\right)}^2=\frac{{\left({\mathit{mc}}^2+\mathit{q\phi }\right)}^2-{\left(\mathit{q\phi }\right)}^2{\mathbf{\Omega }}^2}{c^2}.$$E24

$$\frac{{\partial }^2\psi }{\partial {\tau }^2}-\frac{{\partial }^2\psi }{\partial {\mathbf{r}}^2}=-\frac{{\left({\mathit{mc}}^2+\mathit{q\phi }\right)}^2-{\left(\mathit{q\phi }\right)}^2{\mathbf{\Omega }}^2}{{\mathrm{\hslash }}^2{c}^2}\psi .$$E25

Accordingly, the Hamiltonian of the system can be represented as an expression (4)

$$\mathrm{H}=\frac{E^2-m^2{c}^4}{2{\mathit{mc}}^2}=\frac{{\mathbf{p}}^2}{2m}+\mathit{q\phi }+\frac{q^2{\phi }^2}{2{\mathit{mc}}^2}\left(1-{\mathbf{\Omega }}^2\right).$$E26

By presenting momentum as $\mathbf{p}={\mathbf{p}}^{\prime }+\mathit{q\phi }\mathbf{\Omega }/c$, the Hamiltonian can be represented is as

$$\mathrm{H}=\frac{1}{2m}{{\mathbf{p}}^{\prime }}^2+\frac{\mathit{q\phi }}{\mathit{mc}}{\mathbf{p}}^{\prime }\cdot \mathbf{\Omega }+\mathit{q\phi }+\frac{q^2{\phi }^2}{2{\mathit{mc}}^2},$$E27

and we see that the angular momentum and energy of the particle in the ground state have additional angular momentum and spin state energy. And the expression $\mathit{q\phi }{\mathbf{p}}^{\prime }\cdot \mathbf{\Omega }/\mathit{mc}$is “spin-orbit interaction”—the interaction of a particle with a spinor field.

If other external fields are present, such as a field with a vector potential $\mathbf{A}$, then the equations are represented as

$${\left(\frac{\partial S}{\partial \tau }\right)}^2-{\left(\frac{\partial S}{\partial \mathbf{r}}\right)}^2=\frac{{\left({\mathit{mc}}^2+\mathit{q\phi }\right)}^2-q^2{\left(\mathbf{A}+\phi \mathbf{\Omega }\right)}^2}{c^2}$$E28

$$\frac{{\partial }^2\mathrm{\Psi }}{\partial {\tau }^2}-\frac{{\partial }^2\mathrm{\Psi }}{\partial {\mathbf{r}}^2}=-\frac{{\left({\mathit{mc}}^2+\mathit{q\phi }\right)}^2-q^2{\left(\mathbf{A}+\mathbf{\Omega }\phi \right)}^2}{{\mathrm{\hslash }}^2{c}^2}\mathrm{\Psi }.$$E29

The Hamiltonian of the system can be represented as

$$\mathrm{H}=\frac{E^2-m^2{c}^4}{2{\mathit{mc}}^2}=\frac{{\mathbf{p}}^2}{2m}-\mathit{q\phi }-\frac{q^2{\phi }^2}{2{\mathit{mc}}^2}\left(1-{\mathbf{\Omega }}^2\right)-\frac{\mathit{q\phi }}{{\mathit{mc}}^2}\mathbf{\Omega }\cdot \mathbf{A}-\frac{q^2}{2{\mathit{mc}}^2}{\mathbf{A}}^2$$E30

By presenting momentum as $\mathbf{p}={\mathbf{p}}^{\prime }+q\left(\phi \mathbf{\Omega }+q\mathbf{A}\right)/c$, the Hamiltonian can be represented is as

$$\mathrm{H}=\frac{1}{2m}{{\mathbf{p}}^{\prime }}^2+\frac{q}{\mathit{mc}}{\mathbf{p}}^{\prime }\cdot \left(\phi \mathbf{\Omega }+\mathbf{A}\right)+\mathit{q\phi }+\frac{q^2{\phi }^2}{2{\mathit{mc}}^2}.$$E31

7. Some results of solutions of equations of relativistic and quantum mechanics with spinor fields

7.1 Secular shift of Mercury’s perihelion

Let us consider the motion of a particle with the mass $m$ in the field of a point body of mass $m^{\prime }$. Then the problem reduces to an investigation of the motion of the particle in the centrally symmetric gravitational field with the potential $-G{m}^{\prime }/r$, where $G$ is the gravitational constant, $r_g=G{m}^{\prime }/c^2$.

Choosing the polar coordinates $\left(r,\phi \right)$ in the plane of motion, we obtain the Hamilton-Jacobi (24) equation in the form

$${\left(\frac{\partial S}{\partial \tau }\right)}^2-{\left(\frac{\partial S}{\partial r}\right)}^2-\frac{1}{r^2}{\left(\frac{\partial S}{\partial \phi }\right)}^2={\left(\mathit{mc}\right)}^2\left(1-\frac{2r_g}{r}-\left({\mathbf{\Omega }}^2-1\right)\frac{{r_g}^2}{r^2}\right).$$E32

Let us represent the action $S$ in the form

$$S=-\mathit{\epsilon \tau }+\mathit{M\phi }+\mathit{mcf}\left(r\right),$$E33

where $\epsilon$ and $M$ are the constant energy and angular momentum, respectively. As a result, we obtain

$${\left(\frac{\partial f\left(r\right)}{\partial r}\right)}^2={\left(\frac{\epsilon }{\mathit{mc}}\right)}^2-1-\frac{2r_g}{r}-\frac{{\left(M/\mathit{mc}\right)}^2-\left({\mathbf{\Omega }}^2-1\right){r_g}^2}{r^2}$$E34

and

$$f\left(r\right)=\int \sqrt{{\left(\frac{\epsilon }{\mathit{mc}}\right)}^2-1+\frac{2r_g}{r}-\frac{{\left(M/\mathit{mc}\right)}^2-\left({\mathbf{\Omega }}^2-1\right){r_g}^2}{r^2}}\mathit{dr}.$$E35

We find trajectories from the condition $\partial S/\partial M=0$, with the use of which we obtain,

$$\phi =-\int \frac{M/{\mathit{mcr}}_g}{\sqrt{{\left(\frac{\epsilon }{\mathit{mc}}\right)}^2-1+2\frac{r_g}{r}-\left({\left(M/{\mathit{mcr}}_g\right)}^2-{\left({\mathbf{\Omega }}^2-1\right)}^2\right)\frac{{r_g}^2}{r^2}}}d\frac{r_g}{r},$$E36

which results in the solution

$$\begin{array}{l}r=r_g\left({\left(\frac{M}{{\mathit{mcr}}_g}\right)}^2+1\right)\times \\ \frac{1}{1+\sqrt{{\left(\frac{\epsilon }{\mathit{mc}}\right)}^2\left(1+{\left(\frac{M}{{\mathit{mcr}}_g}\right)}^2\right)-{\left(\frac{M}{{\mathit{mcr}}_g}\right)}^2}cos\left(\phi \sqrt{1-\left({\mathbf{\Omega }}^2-1\right){\left(\frac{{\mathit{mcr}}_g}{M}\right)}^2}\right)}.\end{array}$$E37

The secular shift for the gravitational field (23) with $\mathbf{\Omega }=\left(2,2,0\right),1-{\mathbf{\Omega }}^2=-7$ is calculated by the formula

$$\mathrm{\Delta }\phi =2\pi -\frac{2\pi }{\sqrt{1-\left({\mathbf{\Omega }}^2-1\right){\left(G\frac{m{m}^{\prime }}{\mathit{cM}}\right)}^2}}\approx -7\pi {\left(G\frac{m{m}^{\prime }}{\mathit{cM}}\right)}^2.$$E38

For the Schwarzschild metric, the formula is: $-6\pi {\left(\mathit{Gm}{m}^{\prime }/\mathit{cJ}\right)}^2$ [6].

7.2 Photon deflection by the gravitational field of the sun

For a photon whit energy $E$ and equivalent mass$m=E/c^2$, from (20) the Eq. (24) represented in the form

$$\begin{array}{l}{\left(\frac{\partial S}{\partial \tau }\right)}^2-{\left(\frac{\partial S}{\partial \mathbf{r}}\right)}^2={\left(\mathit{mc}\right)}^2\left({\left(1-\frac{r_g}{r}\right)}^2-{\left({\mathbf{\Omega }}_p+\frac{r_g}{r}{\mathbf{\Omega }}_g\right)}^2\right)=\\ {\left(\mathit{mc}\right)}^2\left(-\frac{2r_g}{r}+{\mathbf{\Omega }}_p\cdot {\mathbf{\Omega }}_g\frac{2r_g}{r}+\left(1-{{\mathbf{\Omega }}_g}^2\right)\frac{{r_g}^2}{r^2}\right).\end{array}$$E39

Choosing the polar coordinates $\left(r,\phi \right)$ in the plane of motion, we obtain the Hamilton-Jacobi (24) equation in the form

$${\left(\frac{\partial S}{\partial \tau }\right)}^2-{\left(\frac{\partial S}{\partial r}\right)}^2-\frac{1}{r^2}{\left(\frac{\partial S}{\partial \phi }\right)}^2={\left(\mathit{mc}\right)}^2\left(-\frac{2r_g}{r}-\left({{\mathbf{\Omega }}_g}^2-1\right)\frac{{r_g}^2}{r^2}\right)$$E40

In this case, we used the choice of special directions of the spins of the particle and the field when ${\mathbf{\Omega }}_p\cdot {\mathbf{\Omega }}_g=\left(0,0,1\right)\cdot \left(2,2,0\right)=0$(spin-spin interactions are absent).

Let us represent the action $S$ in the form

$$S=-\mathit{mc\tau }+\mathit{M\phi }+f\left(r\right)=-\mathit{mc\tau }+\mathit{\text{mcRφ}}+f\left(r\right),M=\mathit{mcR},$$E41

where $M$ and $R$ are the constant angular momentum and impact parameter, respectively. As a result, we obtain

$$f\left(r\right)=\mathit{mc}\int \sqrt{1+\frac{2r_g}{r}-\frac{R^2-\left({{\mathbf{\Omega }}_g}^2-1\right){r_g}^2}{r^2}}\mathit{dr}.$$E42

We find trajectories from the condition $\partial S/\partial M=0$, with the use of which we obtain,

$$\phi =-\int \frac{R}{\sqrt{1+\frac{2r_g}{r}-\frac{R^2-\left({{\mathbf{\Omega }}_g}^2-1\right){r_g}^2}{r^2}}}d\frac{1}{r}.$$E43

The photon deflection is calculated by the formula

$$\mathrm{\Delta }\phi =2arcsin\frac{1}{\sqrt{1-\left({{\mathbf{\Omega }}_g}^2-1\right){\left(\frac{r_g}{R}\right)}^2}}\approx -2\sqrt{7}\frac{r_g}{R}=-2\sqrt{7}G{m}^{\prime }/c^{2}R=-5.3G{m}^{\prime }/c^{2}R.$$E44

For the Schwarzschild metric, the formula is: $\mathrm{\Delta }\phi =-4G{m}^{\prime }/c^{2}R$ [6].

7.3 The hydrogen atom problem

For the problem of a hydrogen-like atom without external fields, we have [4].

$$\begin{array}{l}\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial \psi }{\partial r}\right)+\frac{1}{r^{2}sin\theta }\frac{\partial }{\partial \theta }\left(sin\theta \frac{\partial \psi }{\partial \theta }\right)+\\ \frac{1}{r^2{sin}^2\theta }\frac{{\partial }^2\psi }{\partial {\phi }^2}+\frac{1}{{\mathrm{\hslash }}^2{c}^2}\left(E^2-{\left({\mathit{mc}}^2-\frac{{\mathit{Ze}}^2}{r}\right)}^2+\frac{Z^2{e}^4}{r^2}{\mathbf{\Omega }}^2\right)\psi =0\end{array}$$E45

For the radial part of the solution get

$$\frac{d^{2}R}{d{\rho }^2}+\frac{2}{\rho }\frac{\mathit{dR}}{\mathit{d\rho }}-\frac{l\left(l+1\right)+\left(1-{\mathbf{\Omega }}^2\right)Z^2{\alpha }^2}{{\rho }^2}R=-\left(\frac{N}{\rho }-\frac{1}{4}\right)R,$$E46

Energy levels of the hydrogen atom are (Figure 11a)

Figure 11.

Fine splitting structures.

$${Е}_{n,l}={\mathit{mc}}^2\sqrt{1-\frac{Z^2{\alpha }^2}{{\left(n_{\mathrm{r}}+1/2+\sqrt{{\left(l+1/2\right)}^2-Z^2{\alpha }^2}\right)}^2}}.$$E47

The fine splitting value

$${Е}_{2,1}-{Е}_{1,0}\simeq -\frac{Z^2{\alpha }^2}{12}.$$E48

The ground states are

$${Е}_{n,j}={\mathit{mc}}^2\sqrt{1-\frac{Z^2{\alpha }^2}{{\left(1/2+\sqrt{{\left(l+1/2\right)}^2-Z^2{\alpha }^2}\right)}^2}},l+1/2>\sqrt{Z^2{\alpha }^2+\sqrt{\mathit{Z\alpha }-1/2}}.$$E49

Ground state with $Z>68$ has a $l$ more than $0$, and the greater the atomic number, the discretely it increases in increments of 1.

From the Dirac equation with $\mathbf{\Omega }=\left(1,1,0\right),{\mathbf{\Omega }}^2=2$ we get [4].

$${Е}_{n,j}={\mathit{mc}}^2\sqrt{1-\frac{Z^2{\alpha }^2}{{\left(n_r+\sqrt{{\left(j+1/2\right)}^2-Z^2{\alpha }^2}\right)}^2}}.$$E50

The fine splitting value

$${Е}_{2,3/2}-{Е}_{1,1/2}\simeq -\frac{Z^2{\alpha }^2}{32}.$$E51

The ground states are (Figure 11a)

$${Е}_{n,j}={\mathit{mc}}^2\sqrt{1-\frac{Z^2{\alpha }^2}{{\left(j+1/2\right)}^2-Z^2{\alpha }^2}},j+1/2>\sqrt{2}\mathit{Z\alpha }.$$E52

Ground state with $Z>1/\sqrt{2}\alpha \simeq 97$ has a $j$ more than $1/2$, and the greater the atomic number, the discretely it increases in increments of 1.

As we can see, the magnitude of the spin-orbit interaction is obtained with a reverse sign and is twice as large as the relativistic splitting of the electron energy levels, which leads to a shift and a change in the order of the splitting levels (Figure 11a).

Dirac’s solution prompt that if 1/2 is added to the orbital momentum $l\Rightarrow l+1/2=j$ and 1 to $n_r\Rightarrow n_r-1/2$ in solutions without spatial spin (47), then the results of the solutions will be the same. That is, it should be taken into account that in the expression of the orbital moment and energy there are initially present (26–27), independent of the state and fields of the system, a spatial spin with the corresponding value $j=l+1/2$and energy $n=n_r+l+1\Rightarrow n+j+1/2$.

7.4 Hydrogen atom in a magnetic field

For the hydrogen atom problem in a constant homogeneous magnetic field B we have (30)

$$\frac{q^2}{2m^2{c}^4}{\left(\frac{Q}{r}\mathbf{\Omega }-\frac{1}{2}\left[\mathbf{B}\times \mathbf{r}\right]\right)}^2={\alpha }^2\left(2Z^2\frac{{\mathtt{ƛ}}^2}{r^2}+\frac{1}{2}{\left(\frac{r}{r_h}\right)}^2{sin}^2\theta +Z\frac{\mathtt{ƛ}}{r_h}sin\theta \right),$$E53

where

$$\mathtt{ƛ}=\frac{\mathrm{\hslash }}{\mathit{mc}},\alpha =\frac{e^2}{\mathrm{\hslash }c},r_h=\frac{c}{\omega };\omega =\frac{\mathit{eB}}{2\mathit{mc}}.$$E54

As we can see, the splitting of levels in a constant magnetic field due to the linear dependence on the field $Z{\alpha }^2\left(\mathtt{ƛ}/r_h\right)sin\theta$does not depend on the state of the atom.

7.5 About hyperfine splitting

On hyperfine splitting: why when an electron interacts with nuclear spin, there are only two levels, and a complex structure manifests itself only in the external magnetic field (Figure 12).

Figure 12.

Theoretical magnetic field dependence of F_g = 2, 3 ground hyperfine levels of ⁸⁵Rb. Red lines: Calculations by the coupled basis theory; black lines: Calculations as given by Eq. (12) (HPB regime). Ground levels for the transitions 4–9 are indicated as (4)g − (9)g [8].

As has been shown in solutions to the hydrogen-like atom problem for spinor fields, the magnitude of the spin-orbit interaction $\mathbf{L}\cdot \mathbf{S}$is obtained with a reverse sign and is twice as large as the relativistic splitting of the electron energy levels, which leads to a shift and a change in the order of the splitting levels (Figure 11a). In the ground state, when the nuclear spin $\mathbf{I}$ interacts with the spinor field (Figure 11b), the spin-spin interaction of the nucleus $\mathbf{I}\cdot \mathbf{S}$_, there will be two times less orbital interaction and relativistic splitting (convergence points of black lines in Figure 12) will be accurately compensated by the magnitude of the spin-spin interaction of the nucleus (red lines).

8. Representation of the relativistic and quantum mechanics equations in metric spaces

The representation of the equations of relativistic and quantum mechanics in metric spaces means that coordinate transformations have been proposed that bring the equations from Table 1 to equations with a constant, unit invariant on the right part

$$\begin{array}{l}{\left(\frac{\partial S}{\partial \tau }\right)}^2-{\left(\frac{\partial S}{\partial \mathbf{r}}\right)}^2=I^2\Rightarrow g_{\mathit{ik}}\sqrt{-g}\frac{\partial S}{\partial x^i}\frac{\partial S}{\partial x^k}=1,\\ \frac{{\partial }^2\psi }{\partial {\tau }^2}-\frac{{\partial }^2\psi }{\partial {\mathbf{r}}^2}=-\frac{I^2}{{\mathrm{\hslash }}^2}\psi \Rightarrow \frac{1}{\sqrt{-g}}\frac{\partial }{\partial x^i}\left(g_{\mathit{ik}}\sqrt{-g}\frac{\partial \psi }{\partial x^k}\right)=-\psi .\end{array}$$E55

For clarity, consider the problems with spherical symmetry. For the Hamilton-Jacobi equation in spherical coordinates we have

$${\left(\frac{\partial S}{\partial \tau }\right)}^2-{\left(\frac{\partial S}{\partial r}\right)}^2-\frac{1}{r^2}{\left(\frac{\partial S}{\partial \phi }\right)}^2-\frac{1}{r^2{sin}^2\theta }{\left(\frac{\partial S}{\partial \theta }\right)}^2=I{\left(r\right)}^2.$$E56

and after the coordinates are transformed $\tau ,\mathbf{r}\to {\tau }^{\prime },{\mathbf{r}}^{\prime }$, they must be represented in metric spaces as

$${\left(\frac{\partial S}{\partial {\tau }^{\prime }}\right)}^2-g^2\left(r^{\prime }\right){\left(\frac{\partial S}{\partial r^{\prime }}\right)}^2-\frac{1}{{r^{\prime }}^2}{\left(\frac{\partial S}{\partial \phi }\right)}^2-\frac{1}{{r^{\prime }}^2{sin}^2\theta }{\left(\frac{\partial S}{\partial \theta }\right)}^2=1.$$E57

To do this, divide the Eq. (56) by $I{\left(r\right)}^2$

$$\frac{1}{I^2}{\left(\frac{\partial S}{\partial \tau }\right)}^2-\frac{1}{I^2}{\left(\frac{\partial S}{\partial r}\right)}^2-\frac{1}{I^2{r}^2}{\left(\frac{\partial S}{\partial \phi }\right)}^2-\frac{1}{I^2{r}^2{sin}^2\theta }{\left(\frac{\partial S}{\partial \theta }\right)}^2=1.$$E58

and we immediately find an implicit transformation for$r$

$$I\left(r\right)r=r^{\prime }.$$E59

and

$$\frac{r^2}{{r^{\prime }}^2}{\left(\frac{\partial S}{\partial \tau }\right)}^2-\frac{r^2}{{r^{\prime }}^2}{\left(\frac{\partial S}{\partial r}\right)}^2-\frac{1}{{r^{\prime }}^2}{\left(\frac{\partial S}{\partial \phi }\right)}^2-\frac{1}{{r^{\prime }}^2{sin}^2\theta }{\left(\frac{\partial S}{\partial \theta }\right)}^2=1.$$E60

It is convenient to choose a new coordinate system associated with a particle, where the velocity is zero $d{r}^{\prime }/d{\tau }^{\prime }\equiv 0$. Then,

$$\mathit{d\tau }=\frac{\partial \tau }{\partial {\tau }^{\prime }}d{\tau }^{\prime }+\frac{\partial \tau }{\partial r^{\prime }}d{r}^{\prime }=\left(\frac{\partial \tau }{\partial {\tau }^{\prime }}+\frac{\partial \tau }{\partial r^{\prime }}\frac{d{r}^{\prime }}{d{\tau }^{\prime }}\right)d{\tau }^{\prime }=\frac{\partial \tau }{\partial {\tau }^{\prime }}d{\tau }^{\prime }.$$

$$\frac{\partial S}{\partial \tau }=\frac{\partial S}{\partial {\tau }^{\prime }}\frac{d{\tau }^{\prime }}{\mathit{d\tau }}+\frac{\partial S}{\partial r^{\prime }}\frac{d{r}^{\prime }}{d{\tau }^{\prime }}=\frac{\partial S}{\partial {\tau }^{\prime }}\frac{\partial {\tau }^{\prime }}{\partial \tau }.$$E61

and we get

$$\frac{r^2}{{r^{\prime }}^2{\left(\frac{\partial \tau }{\partial {\tau }^{\prime }}\right)}^2}{\left(\frac{\partial S}{\partial {\tau }^{\prime }}\right)}^2-\frac{r^2}{{r^{\prime }}^2{\left(\frac{\partial r}{\partial r^{\prime }}\right)}^2}{\left(\frac{\partial S}{\partial r^{\prime }}\right)}^2-\frac{1}{{r^{\prime }}^2}{\left(\frac{\partial S}{\partial \phi }\right)}^2-\frac{1}{{r^{\prime }}^2{sin}^2\left(\theta \right)}{\left(\frac{\partial S}{\partial \theta }\right)}^2=1.$$E62

$$g\left(r^{\prime }\right)=\frac{r}{r^{\prime }}/\frac{\partial r}{\partial r^{\prime }}$$E63

Note that if the $I\left(r\right)=\mathit{\text{const}}$, then $g\left(r^{\prime }\right)=1$ – only the linear scale changes.

8.1 Field with scalar potential $\phi \left(r\right)=-r_g/r$

For an invariant in the form $I\left(r\right)=1-r_g/r$ (Coulomb, gravitational field) we find

$$r^{\prime }=\mathit{rI}\left(r\right)=r-r_g,r=r^{\prime }+r_g,\frac{\partial r}{\partial r^{\prime }}=1,$$E64

$$\tau =\frac{r}{r^{\prime }}{\tau }^{\prime }=\left(1+\frac{r_g}{r^{\prime }}\right){\tau }^{\prime },\frac{\partial \tau }{\partial {\tau }^{\prime }}=\frac{r}{r^{\prime }}=1+\frac{r_g}{r^{\prime }},$$E65

$$g\left(r^{\prime }\right)=\frac{r}{r^{\prime }}/\frac{\partial r}{\partial r^{\prime }}=1+\frac{r_g}{r^{\prime }}.$$E66

From (43), (67), we get

$${\left(\frac{\partial S}{\partial {\tau }^{\prime }}\right)}^2-{\left(1+\frac{r_g}{r^{\prime }}\right)}^2{\left(\frac{\partial S}{\partial r^{\prime }}\right)}^2-\frac{1}{{r^{\prime }}^2}{\left(\frac{\partial S}{\partial \phi }\right)}^2-\frac{1}{{r^{\prime }}^2{sin}^2\left(\theta \right)}{\left(\frac{\partial S}{\partial \theta }\right)}^2=1.$$E67

If the field is spinor with invariant $I^2={\left(1-r_g/r\right)}^2-{\left(\mathbf{\Omega }{r}_g/r\right)}^2$, then from getting

$$r=r_g+\sqrt{{r^{\prime }}^2+{\mathbf{\Omega }}^2{r_g}^2},\tau =\frac{r}{r^{\prime }}{\tau }^{\prime }=\left(\frac{r_g}{r^{\prime }}+\sqrt{1+{\mathbf{\Omega }}^2{r_g}^2/{r^{\prime }}^2}\right){\tau }^{\prime }.$$E68

and

$$g^2\left(r^{\prime }\right)=\left(1+{\mathbf{\Omega }}^2\frac{{r_g}^2}{{r^{\prime }}^2}\right){\left(1+\frac{r_e}{r^{\prime }}+\frac{{\mathbf{\Omega }}^2{r_g}^2/{r^{\prime }}^2}{1+\sqrt{1+{\mathbf{\Omega }}^2{r_g}^2/{r^{\prime }}^2}}\right)}^2.$$E69

8.2 Stationary magnetic field

For a magnetic field with invariant $I^2=1-r^2/{r_B}^2$, where $r_B$ the magnetic event horizon (54), is in cylindrical coordinates we have

$${\left(\frac{\partial S}{\partial \tau }\right)}^2-{\left(\frac{\partial S}{\partial r}\right)}^2-\frac{1}{r^2}{\left(\frac{\partial S}{\partial \phi }\right)}^2-{\left(\frac{\partial S}{\partial z}\right)}^2=I^2=1-{\left(\frac{r}{r_B}\right)}^2,$$E70

and we get ($\tau$ and $z$ are transforming similarly)

$${\left(\frac{\partial S}{\partial {\tau }^{\prime }}\right)}^2-2\frac{1-{\left(2r^{\prime }/r_B\right)}^2}{1+\sqrt{1-{\left(2r^{\prime }/r_B\right)}^2}}{\left(\frac{\partial S}{\partial r^{\prime }}\right)}^2-\frac{1}{{r^{\prime }}^2}{\left(\frac{\partial S}{\partial \phi }\right)}^2-{\left(\frac{\partial S}{\partial z^{\prime }}\right)}^2=1.$$E71

$$g^2\left(r^{\prime }\right)=2\frac{1-{\left(2r^{\prime }/r_B\right)}^2}{1+\sqrt{1-{\left(2r^{\prime }/r_B\right)}^2}}$$E72

8.3 Representations of the space-time interval and the wave equation

The results of the metric representations are easily portable to represent the space-time interval (it should be noted that for the space-time interval, the metric tensor is inverse). For example, for a gravitational field with a potential $r_g/r^{\prime }$, we have (68)

$$\begin{array}{l}{\mathit{ds}}^2=I^2\left(d{\tau }^2-{\mathit{dr}}^2-r^{2}d{\phi }^2-r^2{sin}^2\left(\theta \right)d{\theta }^2\right)=\\ =d{{\tau }^{\prime }}^2-\frac{d{r^{\prime }}^2}{g^2\left(r^{\prime }\right)}-{r^{\prime }}^{2}d{\phi }^2-{r^{\prime }}^2{sin}^2\left(\theta \right)d{\theta }^2,\\ {\mathit{ds}}^2=d{s^{\prime }}^2=d{{\tau }^{\prime }}^2-\frac{1}{{\left(1+r_g/r^{\prime }\right)}^2}d{r^{\prime }}^2-{r^{\prime }}^{2}d{\phi }^2-{r^{\prime }}^2{sin}^2\left(\theta \right)d{\theta }^2.\end{array}$$E73

and the wave equation in cylindrical and spherical coordinates ($\tau$ and $z$ are transforming similarly)

$$\frac{{\partial }^2\psi }{\partial {{\tau }^{\prime }}^2}-{\left(1+\frac{r_g}{{\rho }^{\prime }}\right)}^2\frac{{\partial }^2\psi }{\partial {{\rho }^{\prime }}^2}-\frac{1}{{\rho }^{\prime }}\left(1+\frac{r_g}{{\rho }^{\prime }}\right)\frac{\partial \psi }{\partial {\rho }^{\prime }}-\frac{1}{{{\rho }^{\prime }}^2}\frac{{\partial }^2\psi }{\partial {\phi }^2}-\frac{{\partial }^2\psi }{\partial {z^{\prime }}^2}=\psi ,$$E74

$$\frac{{\partial }^2\psi }{\partial {{\tau }^{\prime }}^2}-{\left(1+\frac{r_g}{r^{\prime }}\right)}^2\frac{{\partial }^2\psi }{\partial {r^{\prime }}^2}-\frac{2}{r^{\prime }}\left(1+\frac{r_g}{r^{\prime }}\right)\frac{\partial \psi }{\partial r^{\prime }}-\frac{1}{{r^{\prime }}^2}\frac{{\partial }^2\psi }{\partial {\phi }^2}-\frac{1}{{r^{\prime }}^{2}sin\left(\theta \right)}\frac{\partial }{\partial \theta }\left(sin\left(\theta \right)\frac{\mathrm{\partial \Psi }}{\partial \theta }\right)=\psi$$E75

For more complex cases, we will only point out that first, you need to convert the coordinates to bring the expression of the invariant to a diagonal (spherically or cylindrically symmetric) form, and then transform the resulting equation and invariant in the above way.

8.4 Inverse problem

If an equation is given in metric space that has spherical symmetry, it is easy to perform inverse transformations and find the corresponding invariant (field) from (59), (64)

$$g\left(r^{\prime }\right)=\frac{r}{r^{\prime }}\frac{d{r}^{\prime }}{\mathit{dr}};\frac{\mathit{dr}}{r}=\frac{d{r}^{\prime }}{r^{\prime }g\left(r^{\prime }\right)};r=e^{\int \frac{d{r}^{\prime }}{r^{\prime }g\left(r^{\prime }\right)}},I\left(r\right)=\frac{r^{\prime }}{r}.$$E76

Within the framework of developed approaches, there is an exact correspondence of representations of equations in metric spaces. This is important because it is possible to unambiguously find out which fields correspond to given metric spaces and vice versa.

Naturally, all solutions of equations in metric spaces correspond to solutions of the original equations of relativistic and quantum mechanics having corresponding trajectories of motion and describing the quantum properties of systems.

8.5 Equations in the gravitation field

From the point of view of general relativity, the gravitational field is universal, and all physical processes are described already in metric space. In general, the equations can be represented as follows:

$$g_{\mathit{ik}}\sqrt{-g}\frac{\partial S}{\partial x^i}\frac{\partial S}{\partial x^k}=I^2,$$E77

$$\frac{1}{\sqrt{-g}}\frac{\partial }{\partial x^i}\left(g_{\mathit{ik}}\sqrt{-g}\frac{\partial \psi }{\partial x^k}\right)=-\frac{I^2}{{\mathrm{\hslash }}^2}\psi .$$E78

where $g_{\mathit{ik}}$ is the metric tensor determined by a given gravitational field.

After the appropriate conversion $x\to x^{\prime }$

$$\begin{array}{l}{\left(\frac{\partial S}{\partial {\tau }^{\prime }}\right)}^2-{\left(\frac{\partial S}{\partial {\mathbf{r}}^{\prime }}\right)}^2={I_g}^2\left(x^{\prime }\right)I^2\left(x\left(x^{\prime }\right)\right)\\ \frac{{\partial }^2\psi }{\partial {{\tau }^{\prime }}^2}-\frac{{\partial }^2\psi }{\partial {{\mathbf{r}}^{\prime }}^2}=-\frac{{I_g}^2\left(x^{\prime }\right)I^2\left(x\left(x^{\prime }\right)\right)}{{\mathrm{\hslash }}^2}\psi .\end{array}.$$E79

For example, if a hydrogen-like atom is in a constant homogeneous gravitational spinor field $\left({\phi }_0,{\phi }_0\mathbf{\Omega }\right)$, in the field in which the particle has an invariant $I_g=\left(1-2{\phi }_0-{{\phi }_0}^2\left({\mathbf{\Omega }}^2-1\right)\right)$ is considered, we get

$$\begin{array}{l}{\left(\frac{\partial S}{\partial {\tau }^{\prime }}\right)}^2-{\left(\frac{\partial S}{\partial r^{\prime }}\right)}^2-\frac{1}{{r^{\prime }}^2}{\left(\frac{\partial S}{\partial \phi }\right)}^2-\frac{1}{{r^{\prime }}^2{sin}^2\left(\theta \right)}{\left(\frac{\partial S}{\partial \theta }\right)}^2={I_g}^2{I_e}^2=\\ \left(1-2{\phi }_0-{{\phi }_0}^2\left({{\mathbf{\Omega }}_g}^2-1\right)\right)\left(1-\frac{2r_e}{r\left(r^{\prime }\right)}-\left({{\mathbf{\Omega }}_g}^2-1\right)\frac{{r_e}^2}{r^2\left(r^{\prime }\right)}\right).\end{array}$$E80

A constant homogeneous gravitational field changes the scale of the coordinates by a constant coefficient.

9. Conclusion

Spinor representation of the generalized energy-momentum density 4-vector is proposed, based on the representation of local relativistic rotation by the Lorentz transformation matrix. This representation corresponds to the classical representation of the particle’s own rotation, which is described by the diagonal matrix of the moment of inertia.

The spin of the field is invariant from the point of view of general relativity, it does not depend on the state of the system and the sources of its creation.

Spin is a fundamental, unchanging characteristic of the field. Whichever way it is created, it will be with the same back. The spin of particles is not determined by their internal structure, but is determined by the spin of the interaction fields created by these particles.

Spin is a spatial characteristic and is not attributed to any point particle. The phrase “spin-orbit interaction” in this case means the interaction of the orbital moment with the spinor field.

Solutions to the problems of the motion of particles in various external spinor fields are presented. The results of solving these problems show that the developed approach correctly describes the physical properties of the interaction of particles and fields.

The proposed representation of spinor fields applies to the equations of relativistic and quantum mechanics and their representation in metric spaces.

On the other hand, these proposed approaches are still new and need more detailed theoretical and experimental studies and a more developed and rigorous formulation of the mathematical foundations of the new theory.

All aspects and a more complete presentation of the new theory will be presented shortly in the author’s book “Relativistic and Quantum Mechanics – with new formulations of principles and theory”.

Conflict of interest

The author declares no conflict of interest.