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From UD Theory to Quantum Mechanics: The Spinor Projection and the Rigorous Emergence of Quantum Postulates

Dan Zhu

Published Jun 7, 2026• 0 citations
Publisher

Abstract

We derive the foundational postulates of quantum mechanics from the UD axioms. Starting from the four-aspect field structure, we construct the spinor projection $\psi = (U_u + iU_d,\; D_d + iD_u)^{\mathsf{T}}$. The $U(1)$ Noether current from the Dirac action gives the conserved probability density, and the global normalization axiom yields the Born rule. The $D_u$ field (U in matter) is continuous and linear, explaining wave-like phenomena such as interference. The $D_d$ 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 $D_u$ field with a macroscopic $D_d$ apparatus, leading to dissipation of $D_u$ coherence and the Born rule without collapse. The $U\leftrightarrow D$ duality implies the existence of antiparticles with opposite phase. All derivations are self-contained and logically inevitable.

DOI
10.5281/zenodo.20582561

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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\psi = (U_u + iU_d,\; D_d + iD_u)^{\mathsf{T}}ψ=(Uu​+iUd​,Dd​+iDu​)T. The U(1)U(1)U(1) Noether current from the Dirac action gives the conserved probability density, and the global normalization axiom yields the Born rule. The DuD_uDu​ field (U in matter) is continuous and linear, explaining wave-like phenomena such as interference. The DdD_dDd​ 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 DuD_uDu​ field with a macroscopic DdD_dDd​ apparatus, leading to dissipation of DuD_uDu​ coherence and the Born rule without collapse. The U↔DU\leftrightarrow DU↔D duality implies the existence of antiparticles with opposite phase. All derivations are self-contained and logically inevitable.
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.
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.
We derive universal quantum entanglement from the unity of the continuous field in UD theory. Matter formation converts the UUU attribute in space (UuU_uUu​) into the UUU attribute in matter (DuD_uDu​) via the vertex λUuDd\lambda U_u D_dλUu​Dd​; all DuD_uDu​ excitations are perturbations of the same underlying UuU_uUu​ field. The correlation function of these perturbations inherits the connectedness of space. Because the physical observable ⟨S^AS^B⟩∝e−meffr\langle\hat S_A\hat S_B\rangle\propto e^{-m_{\rm eff}r}⟨S^A​S^B​⟩∝e−meff​r, this background correlation does not yield observable Bell violation; a conditionally normalized E=cos⁡ΔθE=\cos\Delta\thetaE=cosΔθ is a formal construct. When two (or more) particles originate from the same matter formation event, their DuD_uDu​ excitations share the same UuU_uUu​ source, and momentum conservation ∑ka=0\sum\mathbf{k}_a=0∑ka​=0 elevates the weak background correlation to strong, distance-independent multi-particle entanglement, with ⟨B⟩=22\langle\mathcal{B}\rangle=2\sqrt{2}⟨B⟩=22​. The measurement process is the unitary coupling of DuD_uDu​ to DdD_dDd​ via H^meas=(m02/8)∫d3x φ^Duφ^Dd\hat H_{\rm meas}=(m_0^2/8)\int d^3x\,\hat\varphi_{D_u}\hat\varphi_{D_d}H^meas​=(m02​/8)∫d3xφ^​Du​​φ^​Dd​​; the Born rule follows from the reduced density matrix. All results are derived from first principles without collapse postulates or adjustable parameters.
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 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.
We derive universal quantum entanglement from the unity of the continuous field in UD theory. Matter formation converts the U attribute in space (UU) into the U attribute in matter (DU) via the vertex λUUDD; all DU excitations are perturbations of the same underlying UU field. The correlation function of these perturbations inherits the connectedness of space. Because the physical observable ⟨ŜAŜB⟩ ∝ e^(-meff r), this background correlation does not yield observable Bell violation; a conditionally normalized E = cos Δθ is a formal construct. When two (or more) particles originate from the same matter formation event, their DU excitations share the same UU source, and momentum conservation ∑ka = 0 elevates the weak background correlation to strong, distance-independent multi-particle entanglement, with ⟨B⟩ = 2√2. The measurement process is the unitary coupling of DU to DD via Ĥmeas = (m0²/4) ∫d³x φ̂DU φ̂DD; the Born rule follows from the reduced density matrix. All results are derived from first principles without collapse postulates or adjustable parameters.