This paper derives quantum mechanics as an effective theory in the microscopic limit from a single fundamental axiom. The axiom is based on the universe originating from the critical state U = D, and through the logical necessity of “mutual containment”, the four-aspect structure UU , UD, DD, DU is derived. In the microscopic limit, the quantum fluctuation component DU dominates, while other components can be neglected. Starting from the UD dynamical equations, through small parameter expansion and non-relativistic approximation, the Schro¨dinger equation is rigorously derived. Deffning a conserved current naturally yields the probabilistic interpretation of the wave function. This demonstrates that quantum mechanics is not an independent fundamental theory, but a natural manifestation of the primordial theory under microscopic conditions.
This paper derives quantum mechanics as an effective theory in the microscopic limit from a single fundamental axiom. The axiom is based on the universe originating from the critical state U = D, and through the logical necessity of “mutual containment”, the four-aspect structure UU , UD, DD, DU is derived. In the microscopic limit, the quantum fluctuation component DU dominates, while other components can be neglected. Starting from the UD dynamical equations, through small parameter expansion and non-relativistic approximation, the Schro¨dinger equation is rigorously derived. Deffning a conserved current naturally yields the probabilistic interpretation of the wave function. This demonstrates that quantum mechanics is not an independent fundamental theory, but a natural manifestation of the primordial theory under microscopic conditions.
This paper derives quantum mechanics as an effective theory in the microscopic limit from a single fundamental axiom. The axiom is based on the universe originating from the critical state U = D, and through the logical necessity of “mutual containment”, the four-aspect structure UU , UD, DD, DU is derived. In the microscopic limit, the quantum fluctuation component DU dominates, while other components can be neglected. Starting from the UD dynamical equations, through small parameter expansion and non-relativistic approximation, the Schro¨dinger equation is rigorously derived. Defining a conserved current naturally yields the probabilistic interpretation of the wave function. This demonstrates that quantum mechanics is not an independent fundamental theory, but a natural manifestation of the primordial theory under microscopic conditions.
This paper is Part II of the UD theory foundational series. Part I derives general relativity from the scalar projection (doi: 10.5281/zenodo.19494052). Abstract: This paper derives quantum mechanics from the spinor projection of UD theory. The four-aspect structure (U_U, U_D, D_D, D_U) naturally forms an SU(2) doublet ψ = (U_U + iU_D, D_D + iD_U)^T. Using the equilibrium background values and the mass scale from Part I, we derive the coupled field equations and prove their equivalence to the Dirac equation. In the non-relativistic limit, the Pauli equation follows, reducing to the Schrödinger equation for spinless particles. The SU(2) symmetry is intrinsic, and the U(1) subgroup gives the conserved probability current. Quantum phenomena such as interference are explained directly in UD terms: the D_U field is continuous and interferes; "particles" are discrete coupling events between D_U and D_D. The standard quantum mechanical notions of superposition and collapse are not fundamental—they are artifacts of describing a continuous field in particle language. The UD theory is a metatheoretical framework. It does not replace existing theories but reveals their origins. This paper demonstrates that quantum mechanics is the spinor projection of UD theory—the natural mathematical consequence of extracting a two-component spinor from the four-aspect structure.
This work derives the fundamental framework of quantum mechanics intrinsically from the first-principle axioms of UD duality theory. Starting from the inherent binary expansive–condensative attributes of the UD ontology, four fundamental physical fields are naturally decomposed by spontaneous symmetry breaking across spatial and material existential domains. A two-component complex spinor structure is uniquely constructed, and the imaginary unit i is proven to be not merely a mathematical convention but a structurally necessary ingredient required by both the compact SU(2) symmetry and the orthogonal inter-containment of U and D attributes. Within the flat spacetime limit of the UD gravitational action, a system of coupled Klein–Gordon field equations is established, governed by a single self-consistent fundamental mass scale m₀=1/(2πe^π) with no adjustable free parameters. The coupling constant is determined analytically from the UD vacuum equilibrium background values and the U↔D duality symmetry. Through explicit construction of four-component Dirac spinors and auxiliary field decomposition, the on-shell equivalence between the UD coupled field equations and the free Dirac equation is rigorously demonstrated. The non-relativistic limit further recovers the Pauli equation and the Schrödinger equation sequentially, reconstructing the core dynamical equations of quantum mechanics from a unified foundational origin. Global U(1) phase invariance of the UD action is verified via total-derivative decomposition, yielding the standard Noether conserved current. The Born probability interpretation is deduced directly from the global normalization axiom by isolating localized laboratory-scale excitations from the cosmic background, eliminating the need for independent quantum postulates. Based on the intrinsic properties of the UD field components, a self-consistent ontological interpretation is provided for wave–particle duality, double-slit interference, and the quantum measurement problem. In this framework, the measurement problem is dissolved rather than solved: no collapse postulate is required, and determinism is restored at the ontological level. The theory naturally implies that the continuous D_U field permits quantum superposition, while condensed D_D intrinsic entities possess definite classicality without physical particle-state superposition. Three testable theoretical predictions are presented, establishing falsifiable physical consequences of the UD quantum formulation. This paper completes the intrinsic derivation of the foundational structure of quantum mechanics from UD theory. Together with the previously derived general relativity framework, it realizes the homologous unification of gravity and quantum mechanics within a single self-contained axiomatic system.
We derive the foundational postulates of quantum mechanics from the UD axioms. Starting from the four-aspect field structure, we construct the spinor projection ψ=(Uu+iUd,Dd+iDu)T. The U(1) Noether current from the Dirac action gives the conserved probability density, and the global normalization axiom yields the Born rule. The Du field (U in matter) is continuous and linear, explaining wave-like phenomena such as interference. The Dd field (D in matter) is localized and determinate, explaining particle-like behavior and the absence of macroscopic superpositions. The measurement process is described by a coupling Hamiltonian derived from the UD interaction, which entangles the Du field with a macroscopic Dd apparatus, leading to dissipation of Du coherence and the Born rule without collapse. The U↔D duality implies the existence of antiparticles with opposite phase. All derivations are self-contained and logically inevitable.
This article is the second part of a three-paper sequence developing Ontological Field Theory (OFT) as a unified framework grounding quantum mechanics in foundational ontological principles. The trilogy is structured as follows: Part I — Conceptual Necessity:Why Quantum Mechanics Is Inevitable: Ontological Continuity and the Necessity of Quantum Dynamics establishes that quantum mechanics follows necessarily from minimal requirements on coherent being, independently of specific physical postulates. Part II — Technical Realization:Derivation of the Schrödinger Equation from Ontological Field Theory provides the explicit field-theoretic derivation showing how the Schrödinger equation emerges as the non-relativistic limit of a covariant ontological field. Part III — Critical Context:Why Classical Physics Is an Approximation: Decoherence, Coarse-Graining, and the Limits of Determinism explains why classical determinism and trajectories arise only as emergent, approximate descriptions of fundamentally quantum and relational dynamics. Each article is logically self-contained, but the full argument is distributed across the three contributions. The conceptual analysis motivates the mathematics; the mathematics grounds the physics; and the critical examination situates classical intuition as a limiting case rather than a foundation. The papers are intended to be read as a coherent whole. We argue that quantum mechanics is not an arbitrary empirical framework but an inevitable consequence of minimal ontological requirements. Assuming only that being must exist necessarily, remain consistent, be relational, and admit self-reference, we show that classical deterministic physics is structurally insufficient. Quantum dynamics emerges as the unique low-energy realization of coherent ontological continuity. This work complements explicit field-theoretic derivations by clarifying why quantum mechanics could not have been otherwise.
Phase Differential Theory (PDT) proposes that quantum mechanics emerges as an effective description of a deeper dynamical phase manifold. In this framework, four scalar phase fields define a locked background whose long‑wavelength envelope dynamics reproduce the Schrödinger equation. The Born rule arises as the natural conserved probability density compatible with locality and continuity of phase flow, and measurement is interpreted as phase synchronisation between subsystems rather than wavefunction collapse. This paper develops a controlled derivation of the Schrödinger equation as an envelope approximation, identifies the conserved probability current associated with phase flow, and shows that |ψ|² is the natural density selected by locality, positivity, and conservation. Measurement is modelled as a dynamical locking of relative phase between system and apparatus, yielding discrete outcomes without introducing non‑unitary dynamics at the fundamental level. All results are obtained within a low‑energy, long‑wavelength effective regime. The microscopic equations of motion for the phase manifold are intentionally deferred, as the effective envelope theory presented here is self‑contained and does not require explicit microscopic specification.