The QED-Physical (QED-P) theory described in four previous papers [J. H. Wilson, Phys. Essays 28, 1 (2015); 29, 402 (2016); 31, 59 (2018); 34, 17 (2021)] is combined with QED into a single theory in this paper, since both are based on the same Dirac Equation four current c(α,I). QED couples this four-vector with an external electromagnetic (EM) field and uses covariant perturbation theory to produce results that are very accurate computationally [M. L. Eides et al., Phys. Rep. 342, 63 (2001)], except for the electron self-energy, which is infinite. The reason for QED’s accuracy is its Dirac equation velocity operator, cα, with highly unusual properties compared to classical velocity vectors. QED could not produce highly accurate answers without the 4 × 4 complex matrix cα as the electron “velocity operator.” QED-P starts with the same Dirac Equation four current used so successfully in QED, and the discrete internal spatial and time coordinate operators (ISaTCOs) are derived in QED-P to give the electron internal structure a very specific, but highly, nonclassical geometric description with no ad hoc assumptions. The QED/QED-physical (QED-P) unification theorem is stated and proved in Sec. VII. However, one should ask what new physical measurements are predicted by QED-P that are not determined by the very accurate QED? The answer is none yet, due to the very rapid fluctuation of the ISaTCOs with a period of ∼6.4 × 10−22 s. This induced electron field fluctuation is in addition to any vacuum fluctuation that exists without the electron field present. This paper presents a highly speculative, but testable, experiment, in which the ISaTCO rapid fluctuations may be confirmed indirectly as the physical basis of field quantum entanglement (QE). The electron’s ISaTCO produce a digital internal clock that’s locked out of phase with its entangled positron, and they communicate spin states at the phase speed of 2c2/vCoM. The communication of spin state information is instantaneous for vCoM = 0 but “slows down” to 4c at vCoM = 0.5c, still very fast.
The QED-Physical (QED-P) theory described in four previous papers [J. H. Wilson, Phys. Essays 28, 1 (2015); 29, 402 (2016); 31, 59 (2018); 34, 17 (2021)] is combined with QED into a single theory in this paper, since both are based on the same Dirac Equation four current c(α,I). QED couples this four-vector with an external electromagnetic (EM) field and uses covariant perturbation theory to produce results that are very accurate computationally [M. L. Eides et al., Phys. Rep. 342, 63 (2001)], except for the electron self-energy, which is infinite. The reason for QED’s accuracy is its Dirac equation velocity operator, cα, with highly unusual properties compared to classical velocity vectors. QED could not produce highly accurate answers without the 4 × 4 complex matrix cα as the electron “velocity operator.” QED-P starts with the same Dirac Equation four current used so successfully in QED, and the discrete internal spatial and time coordinate operators (ISaTCOs) are derived in QED-P to give the electron internal structure a very specific, but highly, nonclassical geometric description with no ad hoc assumptions. The QED/QED-physical (QED-P) unification theorem is stated and proved in Sec. VII. However, one should ask what new physical measurements are predicted by QED-P that are not determined by the very accurate QED? The answer is none yet, due to the very rapid fluctuation of the ISaTCOs with a period of ∼6.4 × 10−22 s. This induced electron field fluctuation is in addition to any vacuum fluctuation that exists without the electron field present. This paper presents a highly speculative, but testable, experiment, in which the ISaTCO rapid fluctuations may be confirmed indirectly as the physical basis of field quantum entanglement (QE). The electron’s ISaTCO produce a digital internal clock that’s locked out of phase with its entangled positron, and they communicate spin states at the phase speed of 2c2/vCoM. The communication of spin state information is instantaneous for vCoM = 0 but “slows down” to 4c at vCoM = 0.5c, still very fast.
The electron model developed by the author is unique and is derived directly from the Dirac equation velocity operator, c α , whose four current c ( α, I) in quantum electrodynamics (QED) is coupled with the external electromagnetic field to produce highly accurate results. The issue of whether “zitterbewegung (zbw)” exists, or is an artifact, is widely misunderstood. There are two connected zbw components. First, the velocity operator, c α , contains the electron’s center of charge (CoC) “velocity zitter.” If the electron’s time averaged center of mass (CoM) velocity, v CoM , were used, QED would predict highly inaccurate estimates. Second, integrating the electron velocity operator, c α , produces the unique electron CoC internal position and time coordinate operators that describe the CoC “position zitter.” We call this QED companion theory QED-position (QED-p) and the very successful theory QED-velocity (QED-v) to emphasize their direct connection. The physical description of the free electron requires both QED-p and its companion QED-v theory. QED-v is based on the “strange” Dirac equation velocity operator c α that has eigenvalues ± c . The c α eigenvalues of ± c are certainly not the instantaneous electron CoM velocity but the velocity zitter of the electron CoC “point” vibrating rapidly over time. QED has always incorporated CoC velocity zitter through its velocity operator, c α since its inception in the 1940s.
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.
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.
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.
Abstract This book is unique in covering phenomena in photon- matter interactions in a unified way over a range of many orders in energy. The quantum field theoretic approach to the fully relativistic theory of quantum electrodynamics (QED) is presented together with the non- relativistic theory in both confined and unconfined geometries. The predictions of QED have been verified to a greater accuracy than any other physical theory. Moreover QED is a paradigm for other gauge theories and is presented in such a way that the generalisation to other gauge theories is natural. Gauge and Poincare symmetry properties and the non-existence of a photon wave function are thoroughly discussed. Starting from the Dirac equation the non-relativistic interaction of the electron with the electromagnetic field is derived as an effective Hamiltonian of multipole expansions. Much of quantum optics is based on the lowest order dipole approximation. From this point on the treatment of fully relativistic QED and quantum optics is done in parallel. Applications of perturbation theory such as Compton and Moller scattering and the theory photdetection are given. After the impressive successes of QED, the limitation of the theory and the necessity of electroweak theory and quantumchromodynamics are discussed. The remaining chapters are devoted to quantum optics inside cavaties. Various approaches to open systems such as master equations are discussed within the context of active systems (e.g. the laser) and passive systems. Semi- classical approximarions are shown to imply a rich non- linear dynamics including chaos for certain parameter regimes. The effect of fluctuations on such non-linear dynamics is also studied. The final chapter is devoted to highly non- classical states of the light field such as photon number, squeezed and two photon entangled states. The latter are studied for the important system of parametric down conversion and the localisation properties of photons are characterised in terms of asympotic tails in photodetection probabilities as a function of time delay. The range of the book has wider benefits. Workers in quantum optics will gain a deeper understanding of the foundations of their subject and field theorists will see concrete examples of open systems, which are beginning to impinge on fundamental theories.