In this work we discuss the nature of Maxwell and Dirac field by examining the mathematical structures of the subfields that are coupled to form these two physical fields. We show that while Maxwell and Dirac field are hyperbolic fields which are described by wave equations the subfields are elliptic fields which are described by elliptic equations. Therefore, it is reasonable to suggest that the subfields are more fundamental and should be used to represent quantum particles in stable states with invariant physical properties. Furthermore, since the subfields are described by elliptic equations therefore they comply with the Euclidean relativity rather than the pseudo-Euclidean relativity as Maxwell and Dirac fields do, and this results in profound implications such as if quantum particles possess physical properties that are represented by subfields which are described by elliptic equations, hence acting in accordance with the Euclidean relativity, then they can be used to explain physical phenomena that require physical transmissions with speeds greater than the speed of light in vacuum, such as the Einstein-Podosky-Rosen paradox in quantum entanglement. As a further discussion, we also show that it is possible to formulate a Dirac-like elliptic field that complies with the Euclidean relativity.
In this work we discuss the nature of Maxwell and Dirac field by examining the mathematical structures of the subfields that are coupled to form these two physical fields. We show that while Maxwell and Dirac field are hyperbolic fields which are described by wave equations the subfields are elliptic fields which are described by elliptic equations. Therefore, it is reasonable to suggest that the subfields are more fundamental and should be used to represent quantum particles in stable states with invariant physical properties. Furthermore, since the subfields are described by elliptic equations therefore they comply with the Euclidean relativity rather than the pseudo-Euclidean relativity as Maxwell and Dirac fields do, and this results in profound implications such as if quantum particles possess physical properties that are represented by subfields which are described by elliptic equations, hence acting in accordance with the Euclidean relativity, then they can be used to explain physical phenomena that require physical transmissions with speeds greater than the speed of light in vacuum, such as the Einstein-Podosky-Rosen paradox in quantum entanglement. As a further discussion, we also show that it is possible to formulate a Dirac-like elliptic field that complies with the Euclidean relativity.
In this work we discuss the nature of matter wave of quantum particles whose dynamics are described by Dirac equation. Since we have shown that both Dirac equation and Maxwell field equations can be derived from a general system of linear first order partial differential equations, it is reasonable to suggest that matter wave may have similar physical formation to that of the electromagnetic field in the sense that matter wave is also the result of a coupling of two physical fields, such as the electric field and the magnetic field in electromagnetism. In particular, we show that when Dirac equation is reformulated as a system of real equations, like Maxwell field equations, then Dirac equation describes a quantum particle as a string-like object whose cross-section vibrates as a membrane.
: In our previous works, we showed that the Einstein-Podolsky-Rosen (EPR) paradox could be resolved by constructing a Euclidean relativity that not only leads to the same results obtained from Einstein general relativity but also permits an instantaneous transmission of interaction. However, there still remains the question about the nature of these physical fields and their mathematical formulations if they exist. In this work we show that it is possible to formulate Euclidean relativistic field equations similar Dirac equation from a general system of linear first order partial differential equation. Since the speeds of the Euclidean relativistic fields have no upper values, they can be used to rectify the quantum entanglement in quantum mechanics.
Maxwell's equations describe the interactions of the electromagnetic field at a macroscopic level. In the 1920s, Louis DeBroglie demonstrated that every moving particle (including an electron) has a wave nature, and we know from Einstein that every wave has a particle nature, which we call the photon. Later in the 1930s, Paul Dirac's development of the famous Dirac equation showed the quantum nature of the electron at relativistic speeds. Then in 1948 Richard Feynman and Julian Schwinger extended these concepts in the development of quantum electrodynamics which gives a full accounting (although a very strange one) of how an electron can borrow energy from the vacuum of space and return it legally as long it does so within limits of the uncertainty principle.
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 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 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. 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 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 corresponding to particles represented in electrodynamics. An explicit form of the symmetric traceless tensor with spin two, zero mass and two polarizations is presented, which can serve as a model of the graviton. The results obtained may prompt changes in some aspects of the construction of Feynman diagrams. Among other things, it presents a purely mathematical derivation of Maxwell's inhomogeneous equations without reference to empirical data on the action of electric current, which is usually referred to when deriving equations. Section 4 presents Einstein's inhomogeneous equation, which features the Riemann tensor rather than the Ricci tensor, with the energy-momentum tensor having four indices rather than two.
Recent experimental results have shown a violation of Bell’s inequalities, which are a mathematical formulation of Einstein-Podolsky-Rosen (EPR) paradox. The violation leads to the conclusion that there are no local hidden variable theories that underlie quantum mechanics. However, the Bell’s inequalities do not rule out the possibility to construct non-local hidden variable theories that comply with quantum mechanics, in particular, a theory of special and general relativity that permits an instantaneous transmission of interaction. In this work we show that a special relativity with a Euclidean metric that allows not only local interactions but also interactions that can be transmitted instantaneously can be constructed and, furthermore, such special relativity can also be generalised to formulate a general theory of relativity that leads to the same experimental results as Einstein theory of general relativity. We also show that it is possible to formulate Dirac-like relativistic wave equations in this Euclidean relativity with either real mass or imaginary mass, which suggests that the proper mass of a quantum particle may be defined in terms of a differential operator that is associated with a spacetime substructure of the particle.
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 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 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 derivatives on coordinates. 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 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 corresponding to particles represented in electrodynamics. An explicit form of the symmetric traceless tensor with spin two, zero mass and two polarizations is presented, which can serve as a model of the graviton. The results obtained may prompt changes in some aspects of the construction of Feynman diagrams. Among other things, it presents a purely mathematical derivation of Maxwell's inhomogeneous equations without reference to empirical data on the action of electric current, which is usually referred to when deriving equations.