The authors after showing that a confined helical wave possesses rest mass, electric charge and half spin [1][2] now proves that such a wave is a solution of the Dirac equation. Using the confined helical wave representation of electron, the authors explain in a simple manner why the eigen value for Dirac’s velocity operator is ± c while the average value of the velocity of the wave packet formed by its positive energy solutions is the particle velocity. Further, the confined helical wave representation is seen to be consistent with van der Waerden equation and the zig-zag picture of electron proposed by Pen Rose.
The authors after showing that a confined helical wave possesses rest mass, electric charge and half spin [1][2] now proves that such a wave is a solution of the Dirac equation. Using the confined helical wave representation of electron, the authors explain in a simple manner why the eigen value for Dirac’s velocity operator is ± c while the average value of the velocity of the wave packet formed by its positive energy solutions is the particle velocity. Further, the confined helical wave representation is seen to be consistent with van der Waerden equation and the zig-zag picture of electron proposed by Pen Rose.
Abstract Steven Weinberg, one of the founders of the Standard Model of Physics, asked years ago [1]: is the wave function of quantum mechanics real, or is it just used by scientists to calculate? We try to find the answer. We study the general solution of the Schrödinger equation mathematically. We will prove that for satisfying the Schrödinger equation, the particle should possess a helical structure of mass density. Then we will prove further that (i) charged elementary particles and atomic nucleus fall into two categories: they have charge or, but different directions of charges helices, like right-handed electron and left-handed electron etc. (ii) Charge (or) distributes double helically on the particle side boundary. It generates and carries a double helix external electric field with the same velocity. (iii) This E-wave will be proved is just the de Broglie wave. (iv) Its function is particle’s state function in quantum mechanics. (v) Charged particle itself and the de Broglie E-wave it carries form a particle-wave hybrid structure. The E -wave exhibits all the possible states and probabilities; as for the particle itself it can take only one basis state randomly, such as at an eigen state in the atom and molecular or at a point on the interference pattern. (vi) The orbital motion of an electron in the lower energy layers of atom and molecule does not radiate EM energy to collapse into the nucleus. (vii) Self-interference of the hybrid structure can better explain the single particle double-slit interference
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.
In quantum mechanics, atomic and subatomic particles are treated as waves. The idea of wave function and wave function collapse brought us great success but left us different opinions. Is there a way it can avoid the different opinions and give us more? We try to postulate directly that these particles are size particles and satisfy the Schrodinger equation. Start from the general solution of free particle’s Schrodinger equation, we will prove: (i) To satisfy the Schrodinger equation, the size particle must possess a circular polarized structure of the plane vector（a complex number）like the electric vector E in the circular polarized light; its modular is mass density. (ii) These particles fall into two categories: they have the same sign of negative charge e or positive charge e, but different directions of mass density and charged helices, like left handed electron and right handed electron etc. (iii) The spin of the particle has its mechanism, it is formed by the motion of the helices structure of mass plus intrinsic self rotation. (iv) Negative charge e (or positive charge e) distributes double helically on the particle side boundary. It produces and carries a circular polarized electric field E with the same velocity of the particle. It forms a circular polarized E-wave. This wave will be proved is just the de Broglie wave and the E-wave function is just the particle’s state function in quantum mechanics. Such charged particle is consisted of the particle itself and the de Broglie wave; they form a particle-wave hybrid structure. The E- wave exhibits all possible states with possibility in the atom, molecule or interference pattern; as for the particle itself, it cannot split into two or move with two velocity at the same time. It can only locate at one basic state any moment, such as in an eigen state of the atom or at a point in the interference pattern. It is the self-interference of the hybrid structure of the particle makes the one after one single particle form double-slits interference.
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, impulse 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. At the end of the paper the question on a possibility of second quantization of the electron field in the spinor coordinate space is touch upon. The statement that the electron and positron have the same, and exactly positive energy, and have opposite signs of charge and opposite signs of mass is also justified. Accordingly, in the process of annihilation their total energy, momentum, charge and mass do not change. It is shown that if we take as an axiom the conservation at interaction of the total mass of the system of particles taking into account its sign, then it is possible to explain the difference between bosons and fermions in the statistics to which they obey.
Dirac's equation with static external fields is considered as an equation of relativistic quantum theory, which describes the quantum dynamics of an electron in four-dimensional space-time. It is shown that the Dirac electron has only one intrinsic degree of freedom -- spin. The quantum ensemble of an electron with a given energy consists of two inseparable from each other subensembles of heavy and light spin-1/2 quasiparticles to have opposite (relative) intrinsic parities. In the case of electrical scalar potential nonuniform in the direction of a particle motion, these subensembles behave as analogs of the electrical and magnetic components of the plane monochromatic electromagnetic wave propagating in a dispersive medium, nonuniform in the direction of a wave motion. The role of states with negative energies and the properties of the velocity operator are explained.
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, impulse 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. At the end of the paper the question on a possibility of second quantization of the electron field in the spinor coordinate space is touch upon. The statement that the electron and positron have the same, and exactly positive energy, and have opposite signs of charge and opposite signs of mass is also justified. Accordingly, in the process of annihilation their total energy, momentum, charge and mass do not change. It is shown that if we take as an axiom the conservation at interaction of the total mass of the system of particles taking into account its sign, then it is possible to explain the difference between bosons and fermions in the statistics to which they obey.