Research Description This research develops an explicit 3+1-dimensional candidate field-theoretic framework integrating Yang–Mills gauge fields, Maxwell electromagnetism, Dirac fermions, and a complex scalar matter order parameter expressed through the Madelung representation, also identified as the Scretching–Planck (SP) formulation. The construction investigates how gauge interactions, relativistic matter dynamics, probability-density evolution, and quantum-potential effects can be represented within a common mathematical framework. The study incorporates the author-identified Scretching–Maxwell–Dirac (SMD) formulation by preserving the supplied Maxwell–Dirac action in its original SI conventions. Additional Yang–Mills and scalar-field interaction terms are explicitly distinguished as proposed extensions rather than components of the original SMD action. This separation preserves the distinction between the author's source formulation and the newly constructed theoretical model. Using the variational principle, the research derives the associated Euler–Lagrange field equations, including the equations governing the gauge connections, electromagnetic field, Dirac spinor, and scalar matter sector. The complex scalar field is decomposed into probability-density and phase variables, yielding a probability conservation equation, an SP phase-evolution equation, and the corresponding quantum-potential contribution. These results clarify the mathematical relationship between conventional field-theoretic dynamics and the density-based SP representation. The investigation also connects the proposed framework to spectral-gap analysis through the finite-lattice weighted Poincaré identity, providing a precise mathematical relationship between ground-state probability distributions and excitation energies under the stated operator assumptions. Controlled algebraic examples and an independently generated Gaussian probability-density dataset supplement the derivations, illustrating selected properties of the formulation and supporting reproducible computational examination. The principal contribution is the explicit organization of Maxwell–Dirac dynamics, Yang–Mills gauge structure, and SP density-phase variables within a single proposed 3+1-dimensional action-based construction. The work establishes a mathematically specified starting point for investigating interactions among gauge fields, fermionic matter, scalar density evolution, and spectral properties. Importantly, the research distinguishes derived mathematical relations from unproven physical claims. The proposed construction does not establish perturbative or nonperturbative renormalizability, demonstrate a previously unknown fundamental interaction, prove confinement, or resolve the existence and positive mass-gap problem for four-dimensional continuum Yang–Mills theory. Rather, it provides a candidate mathematical framework whose consistency, quantum realization, continuum behavior, and physical predictions require further independent investigation. Keywords: Scretching–Planck formulation; Scretching–Maxwell–Dirac theory; Yang–Mills gauge fields; Maxwell–Dirac equations; Madelung transformation; quantum potential; scalar field theory; probability conservation; Euler–Lagrange equations; weighted Poincaré inequality; spectral gap; Gaussian probability density; quantum field theory; 3+1-dimensional spacetime.
Research Description This research develops an explicit 3+1-dimensional candidate field-theoretic framework integrating Yang–Mills gauge fields, Maxwell electromagnetism, Dirac fermions, and a complex scalar matter order parameter expressed through the Madelung representation, also identified as the Scretching–Planck (SP) formulation. The construction investigates how gauge interactions, relativistic matter dynamics, probability-density evolution, and quantum-potential effects can be represented within a common mathematical framework. The study incorporates the author-identified Scretching–Maxwell–Dirac (SMD) formulation by preserving the supplied Maxwell–Dirac action in its original SI conventions. Additional Yang–Mills and scalar-field interaction terms are explicitly distinguished as proposed extensions rather than components of the original SMD action. This separation preserves the distinction between the author's source formulation and the newly constructed theoretical model. Using the variational principle, the research derives the associated Euler–Lagrange field equations, including the equations governing the gauge connections, electromagnetic field, Dirac spinor, and scalar matter sector. The complex scalar field is decomposed into probability-density and phase variables, yielding a probability conservation equation, an SP phase-evolution equation, and the corresponding quantum-potential contribution. These results clarify the mathematical relationship between conventional field-theoretic dynamics and the density-based SP representation. The investigation also connects the proposed framework to spectral-gap analysis through the finite-lattice weighted Poincaré identity, providing a precise mathematical relationship between ground-state probability distributions and excitation energies under the stated operator assumptions. Controlled algebraic examples and an independently generated Gaussian probability-density dataset supplement the derivations, illustrating selected properties of the formulation and supporting reproducible computational examination. The principal contribution is the explicit organization of Maxwell–Dirac dynamics, Yang–Mills gauge structure, and SP density-phase variables within a single proposed 3+1-dimensional action-based construction. The work establishes a mathematically specified starting point for investigating interactions among gauge fields, fermionic matter, scalar density evolution, and spectral properties. Importantly, the research distinguishes derived mathematical relations from unproven physical claims. The proposed construction does not establish perturbative or nonperturbative renormalizability, demonstrate a previously unknown fundamental interaction, prove confinement, or resolve the existence and positive mass-gap problem for four-dimensional continuum Yang–Mills theory. Rather, it provides a candidate mathematical framework whose consistency, quantum realization, continuum behavior, and physical predictions require further independent investigation. Keywords: Scretching–Planck formulation; Scretching–Maxwell–Dirac theory; Yang–Mills gauge fields; Maxwell–Dirac equations; Madelung transformation; quantum potential; scalar field theory; probability conservation; Euler–Lagrange equations; weighted Poincaré inequality; spectral gap; Gaussian probability density; quantum field theory; 3+1-dimensional spacetime.
We develop a Maxwell–Schwartz field-theoretic formulation of spin-1/2 quantum systems within the framework of tempered distribution spaces and Schwartz linear algebra. In this approach, the fundamental objects of the theory are Maxwellian fields, namely elements of S′(M4,C3), while scalar quantum states arise as derived projections carrying the spectral structure of the theory. Relativistic dynamics is formulated through square-root Hamiltonian operators defined via spectral calculus. On a distinguished subspace of Maxwellian fields, the curl operator realizes the momentum magnitude operator, allowing a direct correspondence between scalar and vectorial relativistic dynamics. This leads to a Schrödinger-type formulation of relativistic evolution without requiring Dirac’s matrix factorization as a primitive assumption. Spin-1/2 structures emerge naturally from two-component Cartesian configurations of Maxwellian fields. The Pauli algebra is realized as an internal symmetry acting on such pairs, and spin observables correspond to explicit transformations of field components. A first-order Dirac-type dynamics is obtained through operator factorization, yielding a Maxwellian realization of relativistic two-component equations, including the massless Weyl regime. The tensorial structure of the theory provides a natural setting for entanglement. Bell-type correlations are derived intrinsically from Maxwellian field configurations, reproducing the standard quantum predictions and their violation of Bell inequalities. A central result of the present work is the energetic interpretation of quantum probability. The spectral coefficients of a state determine the amplitudes of its Maxwellian realization, whose energy density is quadratic in these amplitudes. The Born rule is thus recovered as a normalized energy distribution law. Moreover, the electromagnetic continuity equation induces a probability continuity equation, with the probability current identified as the normalized Poynting flux. This transport interpretation extends to entangled states, where Bell correlations are interpreted as correlations in the transport of energy across field channels. Finally, the framework is extended to massive Maxwellian fields, where probability currents are expressed through nonlocal spectral kernels associated with square-root Hamiltonians, yielding a unified continuity structure for both massless and massive regimes. These results suggest that spinorial quantum structures, probabilistic interpretation, and quantum correlations may be understood as emerging from the internal organization, spectral structure, and energy transport of Maxwellian fields in the Schwartz distributional setting.
We develop a constructive, entanglement-first derivation of quantum field theory in which all physical structure arises from a single operator relation, ρ = e −S(H) . The fundamental object is the correlation matrix Cij = ⟨λi |ρ|λj ⟩, from which locality, spectral geometry, renormalization flow, and interaction structure are derived without prior assumptions of spacetime, fields, or gauge symmetry. We show that the conventional kernel formulation is a projection of the full entanglement structure and is therefore insufficient to reconstruct physical dynamics, establishing operator entanglement as the unique fundamental layer. Within this framework, non-Abelian gauge structure emerges from symmetry-induced degeneracies of the entanglement operator, yielding a natural construction of gauge connections and curvature. The Yang–Mills action is derived as the leading gauge-invariant functional consistent with entanglement-induced locality and spectral scaling, rather than postulated. In the strong-coupling regime, exponential decay of correlations implies clustering, a nonzero mass gap, and an area law for Wilson loops, providing a nonperturbative mechanism for confinement. We further construct the full quantum theory via an entanglement-induced measure, verifying all Osterwalder–Schrader axioms and establishing Hilbert space reconstruction, uniqueness of the continuum limit, universality, vacuum uniqueness, and operator-domain control. The resulting framework provides a unified and constructive foundation for quantum field theory in which Yang–Mills dynamics and nonperturbative phenomena arise directly from operator entanglement.
The article contains a review and new results of some mathematical models relevant to the interpretation of quantum mechanics and emulating well-known quantum gauge theories, such as scalar electrodynamics (Klein–Gordon–Maxwell electrodynamics), spinor electrodynamics (Dirac–Maxwell electrodynamics), etc. In these models, evolution is typically described by modified Maxwell equations. In the case of scalar electrodynamics, the scalar complex wave function can be made real by a gauge transformation, the wave function can be algebraically eliminated from the equations of scalar electrodynamics, and the resulting modified Maxwell equations describe the independent evolution of the electromagnetic field. Similar results were obtained for spinor electrodynamics. Three out of four components of the Dirac spinor can be algebraically eliminated from the Dirac equation, and the remaining component can be made real by a gauge transformation. A similar result was obtained for the Dirac equation in the Yang–Mills field. As quantum gauge theories play a central role in modern physics, the approach of this article may be sufficiently general. One-particle wave functions can be modeled as plasma-like collections of a large number of particles and antiparticles. This seems to enable the simulation of quantum phase-space distribution functions, such as the Wigner distribution function, which are not necessarily non-negative.
This work develops a Lorentz-invariant variational framework in which Fisher-information geometry appears as an intrinsic structural contribution to quantum dynamics. Motivated by longstanding attempts to connect quantum mechanics with information-theoretic principles, we introduce an action functional depending on the density and phase fields in the Madelung representation. Variation of this action yields a modified Klein–Gordon equation containing a single nonlinear term proportional to the four-dimensional Fisher-information curvature of the probability density. The standard Klein–Gordon equation is recovered when the structural parameter vanishes, ensuring full compatibility with established relativistic dynamics. Taking the nonrelativistic limit, we obtain a uniquely determined nonlinear Schrödinger equation in which the correction term is the functional derivative of the Fisher information. The resulting dynamics preserve probability, maintain the Hamilton–Jacobi correspondence, and contain the linear Schrödinger equation as a special case. Analytical expressions for Gaussian and superposed states demonstrate how the structural modification scales with spatial localization and interference structure, providing clear qualitative signatures that distinguish the model from previous nonlinear extensions and offer a theoretical basis for future experimental verification. The results establish a mathematically transparent link between information geometry and quantum dynamics and provide a foundation for future extensions to fermionic, gauge, and many-body systems.
We extend the framework of Unified Quantum Mechanics (UQM) to the relativistic domain by formulating the Dirac equation entirely in terms of real-valued fields. In this approach, the complex imaginary unit i is systematically replaced by the temporal Hilbert transform operator Ht, which satisfies the anti-involution property Ht2=−I and acts as a dimensionless isometric operator. The resulting theory is expressed as a coupled system of real-valued equations for a primary field ψ and its Hilbert-transformed partner ϕ=Ht[ψ], whose evolution is governed exclusively by local differential operators. We demonstrate that all essential algebraic and symmetry structures of relativistic quantum mechanics are preserved within this real-valued formulation. In particular, the full Clifford algebra of the Dirac matrices, Lorentz covariance, and the relativistic energy-momentum relation are recovered exactly. A rigorous isomorphism with the conventional complex Dirac equation is established via the analytic signal representation ψa=ψ−iϕ, under which the standard U(1) gauge symmetry emerges naturally as a global SO(2) rotation in the real field space. The resulting formulation is therefore operationally and predictively equivalent to the standard complex theory. The conserved scalar density ρ=ψ†ψ+ϕ†ϕ=∣ψa∣2 is shown to arise as the squared envelope (intensity) of a real physical field and admits a direct interpretation as a local field intensity with well-defined physical dimensions. Probability conservation is recovered through the continuity equation, while normalization in standard quantum mechanics is reinterpreted as a rescaling by the conserved total field. Stability of matter, expressed as energy positivity, is shown to restrict the physical solution space to analytic signals, enforcing a Hilbert-pair structure and eliminating independent complex degrees of freedom. This work demonstrates that the complex structure of relativistic quantum mechanics is not fundamental but emerges from the analytically constrained dynamics of real-valued fields, providing a unified and physically transparent foundation for the Dirac theory.
This paper extends the preceding Schrödinger-based analysis to the relativistic Dirac framework and examines how gauge covariance, spin magnetism, and quantum-vacuum effects deepen the connection between quantum theory and Maxwell electrodynamics. Local phase invariance of the Dirac equation yields the electromagnetic field tensor, while the conserved Dirac four-current provides a covariant source for Maxwell’s equations. The Pauli limit recovers g=2g=2 and the Bohr magneton, and the Gordon decomposition separates convection and spin-magnetization currents. The Foldy–Wouthuysen expansion further connects relativistic quantum dynamics to spin–orbit and Darwin terms involving the electromagnetic fields. Hydrogen fine-structure and Darwin-shift benchmarks are evaluated quantitatively, while the Dirac velocity operator and Zitterbewegung scale are compared with the magnetic dipole length appearing in the SQC/MSC M1 formulation. The analysis is then extended to the quantum vacuum through the Schwinger critical fields and Euler–Heisenberg effective theory, including vacuum birefringence. The paper distinguishes structural consequences of relativistic quantum mechanics from electromagnetic field dynamics that must still be specified independently.