In quantum mechanics (QM), the electron of spin-charge in probabilistic distribution about a nucleus of a hydrogen atom is described by non-relativistic Schrödinger’s wave equation. Its transformation to Dirac fermion of a complex four-component spinor is incorporated into relativistic quantum field theory (QFT). The quantum system in both cases is represented by a wave function but how it collapses to a point at observation omits out both the hidden variables of the spinor and their evolutionary paths into space-time. This presents an ontological dilemma that cannot be satisfactorily explained by conventional theories either driven by experiments or vice versa. In this study, the ontology of Dirac fermion within a proposed MP model of 4D quantum space-time of hydrogen atom type is examined. The electron of a point-particle and its transformation to Dirac fermion appears consistent with Dirac belt trick for the generation of superposition states of spin-charge and their antimatter. Center of mass reference frame relevant to Newtonian gravity is assigned to the spherical point-boundary of the model. This offers an intricate dynamic tool compatible with basic aspects of both QM and QFT. Some of these are explored for non-relativistic wave function and its collapse, quantized Hamiltonian, Dirac spinors, Weyl spinors, Majorana fermions and Lorentz transformation. How the model could further relate to space-time geometry for a body-mass in an elliptical orbit is plotted from an alternative interpretation of general relativity and a multiverse of the MP models at a hierarchy of scales is proposed for further considerations.
In quantum mechanics (QM), the electron of spin-charge in probabilistic distribution about a nucleus of a hydrogen atom is described by non-relativistic Schrödinger’s wave equation. Its transformation to Dirac fermion of a complex four-component spinor is incorporated into relativistic quantum field theory (QFT). The quantum system in both cases is represented by a wave function but how it collapses to a point at observation omits out both the hidden variables of the spinor and their evolutionary paths into space-time. This presents an ontological dilemma that cannot be satisfactorily explained by conventional theories either driven by experiments or vice versa. In this study, the ontology of Dirac fermion within a proposed MP model of 4D quantum space-time of hydrogen atom type is examined. The electron of a point-particle and its transformation to Dirac fermion appears consistent with Dirac belt trick for the generation of superposition states of spin-charge and their antimatter. Center of mass reference frame relevant to Newtonian gravity is assigned to the spherical point-boundary of the model. This offers an intricate dynamic tool compatible with basic aspects of both QM and QFT. Some of these are explored for non-relativistic wave function and its collapse, quantized Hamiltonian, Dirac spinors, Weyl spinors, Majorana fermions and Lorentz transformation. How the model could further relate to space-time geometry for a body-mass in an elliptical orbit is plotted from an alternative interpretation of general relativity and a multiverse of the MP models at a hierarchy of scales is proposed for further considerations.
In quantum mechanics (QM), the electron of spin-charge in probabilistic distribution about a nucleus of a hydrogen atom is described by non-relativistic Schrödinger’s wave equation. Its transformation to Dirac fermion of a complex four-component spinor is incorporated into relativistic quantum field theory (QFT). The quantum system in both cases is represented by a wave function but how it collapses to a point at observation omits out both the hidden variables of the spinor and their evolutionary paths into space-time. This presents an ontological dilemma that cannot be satisfactorily explained by conventional theories either driven by experiments or vice versa. In this study, the ontology of Dirac fermion within a proposed MP model of 4D quantum space-time of hydrogen atom type is examined. The electron of a point-particle and its transformation to Dirac fermion appears consistent with Dirac belt trick (DBT) for the generation of superposition states of spin-charge and their antimatter. Center of mass reference frame owed to DBT offers an alternative version of Newtonian gravity and this is assigned to a specific spherical point-boundary of the model as an emergent geometry. This offers an intricate dynamic tool compatible with basic aspects of both QM and QFT. Some of these are explored for non-relativistic wave function and its collapse, quantized Hamiltonian, Dirac spinors, Standard Model and Lorentz transformation. The space-time geometry of the MP model is presented in the last section and a multiverse of the models at a hierarchy of scales is suggested for a planet in orbit of the sun. These outcomes offer a dynamic intuitive tool that can be modelled and tested for the hydrogen atom in order to probe the fundamentals of physics.
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.
The Dirac equation stands as the cornerstone of relativistic quantum mechanics, successfully predicting spin and antimatter. However, its derivation is typically presented as a formal algebraic requirement—the linearization of the Hamiltonian—without a clear physical ontology explaining why nature demands such a structure. In this work, building upon a previously established model of particles as topologically anchored wave-defects in a continuous medium, we demonstrate that the Dirac equation is not merely an algebraic curiosity but a structural necessity. We show that the requirement to describe the causal, first-order evolution of a relativistic standing wave compels the state variable to possess a multi-component spinor structure. We explicitly derive the Clifford algebra of the dynamical operators and identify the emergent 4-component state space with the physical degrees of freedom of the defect: a local vibrational duality (Spin) coupled to a global topological orientability (Charge). Thus, the ”quantum strangeness” of the fermion—its spin-1/2 nature and antiparticle partner—are demystified, emerging as the natural description of a relativistic object with non-trivial topology.
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.
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.
The axiomatic foundations of quantum mechanics, while operationally successful, obscure the underlying physical reality. This work introduces a mechanical topological ontology intended to provide a causal basis for the theory. We propose a single postulate: that a fundamental fermion is the 0-dimensional boundary of a 1-dimensional, massive, string-like object topologically anchored to a dynamic vacuum field. From this postulate, we argue that the core structures of single-particle relativistic quantum mechanics can be understood as derivable theorems. First, we construct the object’s state space by identifying two independent, binary properties—a local, vibrational/topological duality (spin) and a global, orientational duality (matter/antimatter)—whose tensor product naturally yields the 4-component Dirac spinor. Second, we show that the requirement of relativistic consistency uniquely determines the algebraic operators (gamma matrices and their Clifford algebra) and the lowest-order dynamical law, leading to the Dirac equation as a theorem of consistency. Third, we demonstrate that the non-relativistic Schrödinger and Pauli equations emerge as its correct low-energy limits, where the quantum potential V (x) is identified with a local variation in the object’s rest mass. Finally, we argue that the foundational tenets of quantum measurement—the spin-1/2 property, the probabilistic Born rule, and the correlations of entanglement—can be reframed as necessary consequences of the object’s intrinsic topology and interaction mechanics. This framework thus offers a path toward demoting the axioms of quantum theory to derivable theorems, aiming to resolve its long-standing paradoxes by providing a clear and coherent physical picture.