The quantum theory of electromagnetic field, developed by Fermi and Dirac, revealed some difficulties in quantizing this field. Although these difficulties have been considered in numerous publications, the situation was far from clear. Currently, this problem is even more urgent, because quantum properties of electromagnetic field are proposed to use in a number of devices. In this paper we consider the quantization problems anew. Some features of scalar potential quantization are revealed. A basis system of stationary states, which is conventionally normalized, is constructed. This construction made it possible to reject an unnecessary design: the so-called indefinite metric. The requirements to the field state with respect to the Lorentz condition are formulated. A state satisfying the Lorentz condition and describing the state of the Coulomb field is constructed. It is shown that the component of the four-potential of electromagnetic field (the difference component), which describes the Coulomb field, has zero quantum-mechanical uncertainty. The other (sum) component of the four-potential of electromagnetic field has an infinite quantum dispersion or, in other words, infinite quantum-mechanical uncertainty. Apparently, specifically this circumstance served a basis for numerous statements about unobservability of these potentials. However, it is only the aforementioned sum component of the four-potential of electromagnetic field that is in fact unobservable.
The quantum theory of electromagnetic field, developed by Fermi and Dirac, revealed some difficulties in quantizing this field. Although these difficulties have been considered in numerous publications, the situation was far from clear. Currently, this problem is even more urgent, because quantum properties of electromagnetic field are proposed to use in a number of devices. In this paper we consider the quantization problems anew. Some features of scalar potential quantization are revealed. A basis system of stationary states, which is conventionally normalized, is constructed. This construction made it possible to reject an unnecessary design: the so-called indefinite metric. The requirements to the field state with respect to the Lorentz condition are formulated. A state satisfying the Lorentz condition and describing the state of the Coulomb field is constructed. It is shown that the component of the four-potential of electromagnetic field (the difference component), which describes the Coulomb field, has zero quantum-mechanical uncertainty. The other (sum) component of the four-potential of electromagnetic field has an infinite quantum dispersion or, in other words, infinite quantum-mechanical uncertainty. Apparently, specifically this circumstance served a basis for numerous statements about unobservability of these potentials. However, it is only the aforementioned sum component of the four-potential of electromagnetic field that is in fact unobservable.
We examine a covariant quantization of electromagnetic fields by using an operator derived from a constant scalar that can be called extended Lorentz gauge. The quantization can avoid an inconsistency between Lorentz gauge and a commutation relation, which can eliminate the need for introduction of physical state defined by a subsidiary condition and auxiliary field in Lagrangian density in Lorentz gauge. By using this quantization and indefinite metric straightforwardly, all quantum phenomena can be provided without enigmatic and paradoxical "probability interpretation".
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. 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. 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 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 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.
We extend classical Maxwell field theory to a first quantized theory of the photon by deriving a conserved Lorentz four-current whose zero component is a positive-definite number density. Fields are real and their positive (negative) frequency parts are interpreted as absorption (emission) of a positive-energy photon. With invariant plane-wave normalization, the photon position operator is Hermitian with instantaneously localized eigenvectors that transform as Lorentz four-vectors. Reality of the fields and wave function ensure causal propagation and zero net absorption of energy in the absence of charged matter. The photon probability amplitude is the real part of the projection of the photon's state vector onto a basis of position eigenvectors and its square implements the Born rule. Manifest covariance and consistency with quantum field theory is maintained through use of the electromagnetic four-potential and the Lorenz gauge.
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.
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 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. 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 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.