Discrete Order Geometry (DOG) and Quantum Mechanics: Ontological Unification from Discrete Lattices to Quantum Representation

Author: Zhang Suhang

Affiliation: Luoyang

Abstract

Since its inception, quantum mechanics has been regarded as harboring inherent "peculiarities" in the microscopic world, including nonlocality, probabilistic essence, wave-particle duality and the measurement problem. The root cause lies in the adoption of continuous spacetime mathematical frameworks (Hilbert space, wave function) to describe intrinsically discrete quantum processes. Based on the axiomatic system of Discrete Order Geometry (DOG), this paper proposes that quantum mechanics serves as an effective continuous approximation and probabilistic representation of DOG at microscopic scales, while DOG constitutes the fundamental geometric ontology underlying quantum mechanics. By establishing four-fold correspondence relations between DOG discrete lattices and quantum states, order hierarchies and quantum energy levels, remote coupling and quantum entanglement, as well as order evolution equations and the Schrödinger equation, this study verifies that quantization phenomena, the uncertainty principle, entanglement correlation and wave function collapse can all be naturally derived from finite discrete order lattices, continued-fraction hierarchical convergence and order resonance transitions within DOG. DOG eliminates the ontological dependence of quantum mechanics on probability and randomness, reduces quantum behaviors to inevitable consequences of discrete geometry, and realizes the geometric unification of the microscopic quantum realm and the macroscopic classical world.

Keywords: Discrete Order Geometry; Quantum Mechanics; Discrete Lattice; Quantum State; Entanglement; Uncertainty Principle; Ontological Unification

1. Introduction

The formal mathematical system of quantum mechanics, covering Hilbert space, operator algebra and the Schrödinger equation, has achieved unprecedented experimental validation and predictive success, yet its physical interpretation remains controversial for a long time. The probabilistic interpretation of wave functions, wave function collapse during measurement, and non-local entanglement phenomena still lack a self-consistent ontological foundation. We hold that the core dilemma originates from describing inherently discrete quantum events via continuous geometric language.

Discrete Order Geometry (DOG) replaces continuous spacetime points with finitely countable order lattices, substitutes mechanical interactions with order coupling, and adopts discrete recursion in place of differential equations. This paper aims to demonstrate that quantum mechanics is the effective representation of DOG on microscopic scales, and DOG acts as the underlying geometric ontology of quantum mechanics. This viewpoint not only dispels the perceived peculiarities of quantum theory, but also lays a natural foundation for quantum gravity research.

2. Fundamental Predicaments of Quantum Mechanics: Mismatch with Continuous Geometry

Traditional quantum mechanics is constructed upon the following continuous structural frameworks:

- Hilbert space: a continuous, infinitely divisible complex vector space

- Wave function: a continuous field defined over spacetime

- Schrödinger equation: a differential equation governing continuous temporal evolution

- Measurement theory: reliant on the external continuous macroscopic classical world

Nevertheless, actual microscopic phenomena manifest prominent discreteness, including discrete atomic energy levels, quantized angular momentum, photon counting and binary spin states. Describing discrete physical realities with continuous frameworks inevitably introduces probability and randomness as remedial supplements. From the perspective of DOG, such probabilities are not intrinsic properties of nature, but coarse-grained descriptions of discrete geometry generated by continuous approximation.

3. Review of DOG Axioms (Microscopic Formulation)

For logical consistency, the microscopic corresponding forms of the three core DOG axioms are summarized as follows:

- Axiom 1 (Spacetime Discretization): Microscopic spacetime is composed of finitely countable discrete order lattices. Particles can only occupy or transit between these lattices, with no continuous motion paths existing in nature.

- Axiom 2 (Order Priority): Physical configurations are determined by the arrangement of order lattices; fundamental interactions essentially take the form of remote coupling between homologous order lattices, rather than continuous transmission of force fields.

- Axiom 3 (Closed Recursion): The evolutionary process of system states follows the discrete recursive equation:

\boldsymbol{\Omega}_{k+1} = \mathcal{D}(\boldsymbol{\Omega}_k, \mathbb{O})

independent of continuous integral operations.

The Planck scale l_p is defined as the minimum spacing between adjacent lattices, and continuous spacetime emerges as the asymptotic limit when lattice spacing approaches zero.

4. Four-Fold Correspondence between DOG and Quantum Mechanics

4.1 Discrete Lattices ↔ Quantum States

- DOG: Physical space is a set of order lattices \{\mathcal{L}_i\}, where each lattice represents an indivisible fundamental geometric unit.

- QM: Each distinguishable quantum eigenstate |i\rangle corresponds to a specific lattice \mathcal{L}_i, and the topological arrangement of lattices defines adjacency relations within state space.

- Conclusion: Hilbert space is the linearized approximation of DOG lattice space under the continuous limit. The amplitude coefficient c_i in the state vector \psi = \sum c_i |i\rangle denotes the coarse-grained weight of macroscopic observation acting on lattice clusters.

4.2 Order Hierarchies ↔ Quantum Energy Levels

- DOG: Physical systems possess inherent order hierarchies n = 0,1,2,\dots determined by the convergence order of continued fractions, and each hierarchy corresponds to specific lattice density and resonant frequency.

- QM: Energy levels of atomic and bound systems are discrete, and energy level intervals are governed by resonance conditions between different order hierarchies.

- Conclusion: Quantization is not an artificial theoretical assumption, but a natural manifestation of DOG hierarchies in the microscopic domain. The energy difference E_{n+1}-E_n can be directly calculated via order coupling constants and fixed continued-fraction convergence values.

4.3 Remote Coupling ↔ Quantum Entanglement and Nonlocality

- DOG: Distant lattices belonging to the same nested cluster with homologous order can generate "order resonance", achieving synchronous evolution without intermediate signal transmission.

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- QM: Measurement results of entangled particle pairs exhibit instantaneous non-classical strong correlations that violate Bell inequalities.

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- DOG Interpretation: Entanglement is not superluminal action at a distance, but topological correlation derived from two particles sharing an identical order lattice cluster. Measuring one particle selects a definite lattice within the cluster, and the state of the paired particle is uniquely constrained by the conservation law of cluster order, which is an inevitable outcome of discrete geometry and requires no non-local hidden variables.

4.4 Discrete Evolution Equation ↔ Schrödinger Equation

- DOG: System evolution complies with the deterministic discrete recurrence formula:

\boldsymbol{\Omega}_{k+1} = \mathcal{D}(\boldsymbol{\Omega}_k, \mathbb{O})

where the evolution operator \mathcal{D} derives subsequent system states based on current states and fundamental order benchmarks \mathbb{O}.

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- QM: Continuous temporal evolution of wave functions is governed by the Schrödinger equation:

i\hbar \frac{\partial}{\partial t}\psi = \hat{H}\psi

which describes unitary evolution of quantum states.

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- Correlation relation: In the continuous limit where lattice spacing tends to zero, discrete recursion formulas can deduce the Schrödinger equation. In this scenario, the wave function \psi(x,t) is the projection of DOG order state vector \boldsymbol{\Omega}_k onto continuous coordinate bases, and the Hamiltonian operator \hat{H} serves as the differential approximation of the evolution operator \mathcal{D}. Probability amplitudes in quantum mechanics originate from macroscopic averaging effects on discrete lattice clusters, rather than inherent microscopic randomness.

5. Resolution of Century-Old Quantum Mechanics Paradoxes via DOG

5.1 Wave-Particle Duality

- Traditional interpretation: Electrons simultaneously possess particle and wave properties, interpreted through the complementarity principle.

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- DOG interpretation: Particle characteristics correspond to localized lattice occupation, while wave characteristics correspond to propagation modes of order resonance within lattice clusters. A single physical entity presents two different observational perspectives, rendering the complementarity principle unnecessary.

5.2 Uncertainty Principle

- Traditional interpretation: Derived from non-commutative operators, reflecting intrinsic randomness of the microscopic world.

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- DOG interpretation: Lattices have an irreducible minimum spatial scale \Delta x_{\min} \sim l_p, and physical momentum is defined by lattice transition frequencies. The product of position and momentum uncertainties is constrained by the adjacency topology of discrete lattices. The uncertainty relation is a geometric necessity instead of mysterious intrinsic randomness.

5.3 Measurement Problem and Wave Function Collapse

- Traditional interpretation: Spontaneous random wave function collapse lacks effective dynamic mechanism explanations.

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- DOG interpretation: Wave function collapse does not exist in essence. Measurement behavior refers to coupling resonance between macroscopic detection devices and microscopic order lattice clusters, and measurement outcomes correspond to selected specific lattices. The statistical probability of acquiring definite results stems from coarse-grained sampling effects of macroscopic instruments on lattice clusters, while physical systems always evolve strictly following deterministic discrete recursion rules.

5.4 Quantum Gravity Research

- Traditional dilemma: Continuous spacetime defined by general relativity is inherently incompatible with the discrete nature of quantum mechanics.

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- DOG solution: Gravitational effects correspond to collaborative bending of large-scale order lattices determined by the spatial distribution of order benchmarks \mathbb{O}, while quantum effects describe microscopic lattice behaviors. Both physical laws are unified within the same discrete geometric system with no fundamental logical conflicts.

6. Comparison with Existing Quantum Interpretations

- Copenhagen Interpretation: Centered on inherent probabilistic nature, dependent on external classical worlds, lacking solid ontological foundations.

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- Many-Worlds Interpretation: Involves continuous universe splitting and excessive entropy increment assumptions.

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- Hidden Variable Theory: Constrained by nonlocality and overly complex potential field construction.

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- DOG Interpretation: Adopts deterministic evolution rules for discrete lattices; statistical probability originates from macroscopic coarse graining; measurement processes are defined as order resonance behaviors. Featuring concise logical construction and fewer hypothetical premises, it establishes direct inherent connections with fundamental geometric laws.

7. Verifiable Predictions

The DOG theoretical framework predicts observable discrete physical characteristics near the Planck scale:

- High-energy photon propagation velocities may present minor discrete modified deviations, accompanied by slight violations of Lorentz invariance.

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- Delayed-choice experiments on quantum entanglement may display systematic deviations from traditional continuous theoretical predictions, such as stepwise discrete variations of Bell parameters varying with spatial distance.

Specific physical magnitude calculations require further derivation combining DOG order coupling constants and cosmic hierarchical order parameters, which will be completed in follow-up research works.

8. Conclusion

This paper demonstrates that Discrete Order Geometry (DOG) acts as the fundamental geometric ontology of quantum mechanics. The probabilistic characteristics, nonlocality, energy level discretization and measurement peculiarities of quantum mechanics can all be logically deduced from DOG’s core properties including finite discrete lattices, hierarchical order structures, remote order coupling and deterministic recursive evolution. Quantum mechanics is essentially a continuous approximation and macroscopic probabilistic representation of DOG applied to microscopic physical domains. DOG eliminates long-standing ontological ambiguities within quantum theory, and provides an innovative geometric theoretical foundation for the research of quantum gravity and universal unified physical theories.

References

[1] Zhang Suhang. Discrete Order Geometry (DOG): Foundation of a New Geometric Paradigm Based on Fractal Nesting and Continued-Fraction Scales. 2026.

[2] Zhang Suhang. Comparative Paradigm Analysis and Defect Elimination between DOG Discrete Order Geometry and Traditional Fractal Geometry. 2026.

[3] Zhang Suhang. Subversive Underlying Influence of the DOG Paradigm on Global Modular Systems. 2026.
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[5] Bell J S. On the Einstein Podolsky Rosen Paradox. Physics, 1964.