A simple real-space model for the free-electron wavefunction with spin is proposed, based on coherent vortices on the scale of h/mc, rotating at mc^2/h. This reproduces the proper values for electron spin and magnetic moment. Transformation to a moving reference frame turns this into a wave with the de Broglie wavelength. The mapping of the real two-dimensional vector phasor to the complex plane satisfies the Schrodinger equation. This suggests a fundamental role for spin in quantum mechanics.
A simple real-space model for the free-electron wavefunction with spin is proposed, based on coherent vortices on the scale of h/mc, rotating at mc^2/h. This reproduces the proper values for electron spin and magnetic moment. Transformation to a moving reference frame turns this into a wave with the de Broglie wavelength. The mapping of the real two-dimensional vector phasor to the complex plane satisfies the Schrodinger equation. This suggests a fundamental role for spin in quantum mechanics.
The nature of electron spin has presented an enigma right from the beginning of quantum mechanics. We suggest that a simple realistic picture of a real coherently rotating vector field can account for both the Schrodinger equation and electron spin in a consistent manner. Such a rotating field carries distributed angular momentum and energy in the same way as a circularly polarized electromagnetic wave. We derive the Schrodinger equation from the relativistic Klein-Gordon Equation, where the complex wave function maps onto a fixed-axis real rotating vector. Such a realistic picture can also explain the Stern-Gerlach experiment which first identified electron spin. Remarkably, the predictions of a two-stage Stern-Gerlach experiment within this realistic picture differ from those of the orthodox quantum superposition approach. This two-stage experiment has not actually been done, and could provide insights into the limits of realistic models. This realistic picture also avoids quantum paradoxes and enables realistic explanations for a variety of quantum phenomena.
A semiclassical model of the electron is presented based on the Principle of Helical Motion (“A free electron always moves at the speed of light following a helical motion, with a constant radius, and with the direction of movement perpendicular to the rotation plane”). This model interprets the Zitterbewegung as a real motion that causes rotation of the electron spin and its magnetic moment. Based on this model, the quantum magnetic flux and quantum Hall resistance are obtained as parameters of the electron and special relativity theory is derived from the helical motion of the electron. Finally, a fix is proposed for the De Broglie’s wavelength that questions the very validity of the Dirac equation.
The quantum relationship may be regarded as the equivalence between two expressions for the rest energy of the particle, if is considered as the spin angular velocity of the particle in its rest frame. The invariance of the relativistic space-time interval to such a spin motion (space isotropy) leads to the spin momentum for all structureless particles irrespective of their mass values. The inertia is an intrinsic property due to the spin motion of the particles. The signs of the mass values occurring in the solutions of the Dirac equation might be related to the orientation of the spin motion, as suggested by the fundamental relationship . Besides it deals with the electron, and more specifically with two key properties: its complex wavefunction, and its intrinsic spin. In the standard interpretation, there is no clear real-space picture of what is oscillating in the wave, or what is rotating in the spin. Indeed, it is generally believed that no simple model of rotation can account for the spin of the electron. On the contrary, the present paper shows that a crude mechanical model of coherently rotating vortices can account quantitatively not only for spin, but also for the wavefunction itself. The implications of this are discussed in this paper. La relación cuántica puede ser considerada como la equivalencia entre dos expresiones para la energía en reposo de la partícula, si se considera la velocidad angular de giro de partículas en su marco en reposo. La invariancia del intervalo relativista espacio- tiempo para tal movimiento de espín (isotropía espacial) conduce al impulso de espín para todas las partículas sin estructura, independientemente de sus valores de masa. La inercia es una propiedad intrínseca debido al movimiento de spin de las partículas. Los signos de los valores de masa que se producen en las soluciones de la ecuación de Dirac podrían estar relacionados con la orientación del espín, según lo sugerido por la relación fundamental . Además se refiere al electrón, y más concretamente con dos de los principales propiedades: su función de onda compleja, y su giro intrínseco. En su interpretación estándar, no hay una clara imagen del espacio real de lo que es oscilante en la onda, o lo que está girando en el espín. De hecho, es la creencia generalizada de que ningún modelo sencillo puede dar cuenta de la rotación de espín de los electrones. Por el contrario, en el presente trabajo se muestra que un crudo modelo mecánico de rotación de vórtices coherentes explica cuantitativamente no sólo el espín, sino también la propia función de onda. Las consecuencias de esto son examinadas en este trabajo.
Physical processes are usually described using four-dimensional vector quantities - coordinate vector, momentum vector, current vector. But at the fundamental level they are characterized by coordinate and momentum spinors. The propagation of fields and their interaction takes place at the spinor level, and since each spinor uniquely corresponds to a certain vector, the results of physical processes appear before us in vector form. For example, the relativistic Schrödinger equation and the Dirac equation are formulated by means of coordinate vectors, momentum vectors and quantum operators corresponding to them. In the Dirac equation the wave function is a spinor with complex components, but still coordinates and momentum are vectors. For a closed description of nature using only spinor quantities, it is necessary to have an equation similar to the Dirac equation in which momentum, coordinates and operators are spinors. It is such an equation that is presented in this paper. Quantum mechanics describes interactions using complex wave functions that cannot be observed directly; only upon measurement do they translate into probabilities expressed as the squares of the magnitudes of the wave functions. In the same way spinor quantities are not observable, but they completely determine observable vectors. In Section 2 of the paper, we analyze the quadratic form for an arbitrary four-component complex vector based on Pauli matrices. The form is invariant with respect to Lorentz transformations including any rotations and boosts. The invariance of the form allows us to construct on its basis an equation for a free particle combining the properties of the relativistic wave equation and the Dirac equation. For an electron in the presence of an electromagnetic potential it is shown that taking into account the commutation relations between the momentum and coordinate components allows us to obtain from this equation the known results describing the interactions of the electron spin with the electric and magnetic field. In the presence of a potential the momentum components cease to commute with each other. To neutralize this effect, the Schrödinger equation is supplemented by several equations with mixed partial derivatives on coordinates. Section 3 presents a version of quantum theory in spinor coordinate space. Section 4 presents a modification of the theory of the path integral, which consists in considering the path integral in the spinor coordinate space. An equation of motion for the electron is proposed that is relativistically invariant. This novel equation permitted the construction of an actually invariant procedure for the second quantization of the fermion field in spinor coordinate space. Furthermore, it is demonstrated that the field operators are a combination of plane waves in spinor or vector space, with the coefficients of which being pseudospinors or pseudovectors. Furthermore, each plane wave possesses an additional coefficient in the form of a creation or annihilation operator. In vector space, these operators commute, whereas in spinor space they anticommutate. The paper presents the spinor and vector representations of the field operators in explicit form, comprising sets of 16 pseudospinors or 4 pseudovectors. An explicit form of the symmetric traceless tensor with spin two, zero mass and two polarizations are presented, which can serve as a model of the graviton. Section 5 presents Einstein's inhomogeneous equation, which features the Riemann tensor rather than the Ricci tensor, with the energy-momentum tensor of a charged particle having four indices rather than two. In the case where a particle of matter has a charge, another Riemann tensor is added to Einstein’s equation, in addition to the usual Riemann curvature tensor, which describes the additional curvature created for a charged particle by an electromagnetic field.
Physical processes are usually described using four-dimensional vector quantities - coordinate vector, momentum vector, current vector. But at the fundamental level they are characterized by spinors - coordinate spinors, momentum spinors, spinor wave functions. The propagation of fields and their interaction takes place at the spinor level, and since each spinor uniquely corresponds to a certain vector, the results of physical processes appear before us in vector form. For example, the relativistic Schrödinger equation and the Dirac equation are formulated by means of coordinate vectors, momentum vectors and quantum operators corresponding to them. In the Schrödinger equation the wave function is represented by a single complex quantity, in the Dirac equation a step forward is taken and the wave function is a spinor with complex components, but still coordinates and momentum are vectors. For a closed description of nature using only spinor quantities, it is necessary to have an equation similar to the Dirac equation in which momentum, coordinates and operators are spinors. It is such an equation that is presented in this paper. Using the example of the interaction between an electron and an electromagnetic field, we can see that the spinor equation contains more detailed information about the interaction than the vector equations. This is not new for quantum mechanics, since it describes interactions using complex wave functions, which cannot be observed directly, and only when measured goes to probabilities in the form of squares of the moduli of the wave functions. In the same way spinor quantities are not observable, but they completely determine observable vectors. In Section 2 of the paper, we analyze the quadratic form for an arbitrary four-component complex vector based on Pauli matrices. The form is invariant with respect to Lorentz transformations including any rotations and boosts. The invariance of the form allows us to construct on its basis an equation for a free particle combining the properties of the relativistic wave equation and the Dirac equation. For an electron in the presence of an electromagnetic potential it is shown that taking into account the commutation relations between the momentum and coordinate components allows us to obtain from this equation the known results describing the interactions of the electron spin with the electric and magnetic field. In section 3 of the paper this quadratic form is expressed through momentum spinors, which makes it possible to obtain an equation for the spinor wave function in spinor coordinate space by replacing the momentum spinor components by partial derivative operators on the corresponding coordinate spinor component. Section 4 presents a modification of the theory of the path integral, which consists in considering the path integral in the spinor coordinate space. The Lagrangian densities for the scalar field and for the electron field, along with their corresponding propagators, are presented. An equation of motion for the electron is proposed that is relativistically invariant, in contrast to the Dirac equation, which lacks this invariance. This novel equation permitted the construction of an actually invariant procedure for the second quantization of the fermion field in spinor coordinate space. Furthermore, it is demonstrated that the field operators are a combination of plane waves in spinor or vector space, with the coefficients of which being pseudospinors or pseudovectors. Each of these pseudovectors or pseudospinors corresponds to one of the particles presented in the theory of electrodynamics. Furthermore, each plane wave possesses an additional coefficient in the form of a birth or annihilation operator. In vector space, these operators commute, whereas in spinor space they anticommutate. The paper presents the spinor and vector representations of the field operators in explicit form, comprising sets of 16 pseudospinors or 4 pseudovectors corresponding to particles represented in electrodynamics.