Discrete Order Geometry (DOG) First‑Principles Derivation of the Dirac Equation: Spinor Order and Relativistic Discrete Evolution

Author: Zhang Suhang

(Luoyang, Independent Researcher)

Abstract – Based on the unified discrete evolution paradigm of Discrete Order Geometry (DOG), this paper extends the non‑relativistic discrete recursion framework to the relativistic high‑energy domain. By constructing a four‑dimensional discrete spacetime order lattice, introducing a four‑component spinor order state, and formulating a difference evolution equation that respects the discrete spacetime symmetries, we rigorously derive the standard covariant Dirac equation in the continuum limit from the first‑principle discrete iteration \boldsymbol{\Psi}_{n+1,\mathbf{m}} = M \boldsymbol{\Psi}_{n,\mathbf{m}}. The geometric origin of the \gamma^\mu matrices is clarified: they arise from the discrete gradient operators on the four‑dimensional lattice together with the minimal spinor representation required by the square‑root of the discrete Laplacian. Spin‑1/2, positive/negative energy solutions, and antiparticle structure emerge as natural consequences of the intrinsic topology of the discrete order lattice. This derivation unifies the underlying geometric origin of the Schrödinger and Dirac equations, completing DOG’s coverage of both non‑relativistic and relativistic quantum mechanics, and predicts discrete corrections at the Planck scale.

Keywords – Discrete Order Geometry; DOG; Dirac equation; four‑dimensional discrete lattice; spinor; discrete evolution; continuum limit

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1. Introduction

In previous work, the fundamental framework of Discrete Order Geometry (DOG) has been established:

· Physical spacetime is composed of a finite countable set of discrete order lattice points;

· All dynamics are strictly described by the discrete recursive evolution \boldsymbol{\Psi}_{k+1} = M \boldsymbol{\Psi}_k;

· The Schrödinger equation has been derived as the continuum approximation in the non‑relativistic limit.

However, the Dirac equation of relativistic quantum mechanics – with its spinor structure, positive/negative energy solutions, and antiparticle prediction – has long been regarded as a postulate, lacking a geometric origin. The present paper aims to prove that the Dirac equation is the natural dynamical result of a four‑component spinor state evolving on a four‑dimensional discrete spacetime order lattice, in the continuum limit, without any additional axioms.

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2. Formal Definition of the DOG Four‑Dimensional Discrete Spacetime

2.1 Discrete Spacetime Lattice

Let the three‑dimensional spatial lattice be \mathcal{X} = \{\mathbf{x}_\mathbf{i} \mid \mathbf{i} \in \mathbb{Z}^3\} with spacing \Delta x, and time steps t_n = n\Delta t,\ n \in \mathbb{Z}. The four‑dimensional discrete spacetime lattice is:

\mathcal{M} = \{ (t_n, \mathbf{x}_\mathbf{i}) \}.

2.2 Four‑Component Spinor Order State

On the four‑dimensional discrete spacetime, define the order state as a four‑component complex vector:

\boldsymbol{\Psi}(t_n, \mathbf{x}_\mathbf{i}) = \begin{pmatrix} \psi_1(t_n,\mathbf{x}_\mathbf{i}) \\ \psi_2(t_n,\mathbf{x}_\mathbf{i}) \\ \psi_3(t_n,\mathbf{x}_\mathbf{i}) \\ \psi_4(t_n,\mathbf{x}_\mathbf{i}) \end{pmatrix}.

This four‑dimensional internal freedom originates from two binary topological structures of the discrete spacetime:

· Forward/backward time direction (particle/antiparticle modes);

· Left/right spatial chirality (spin up/down modes).

The natural combination 2\times2 = 4 components arises without any artificial introduction.

2.3 Discrete Spacetime Evolution Equation

The fundamental DOG axiom requires that the system state iterates in discrete time and that the evolution depends only on the current state and the order couplings of neighboring lattice points. The most general linear, local, isotropic (in the lattice sense) four‑dimensional discrete evolution can be written as:

\frac{\boldsymbol{\Psi}(t_{n+1},\mathbf{x}_\mathbf{i}) - \boldsymbol{\Psi}(t_n,\mathbf{x}_\mathbf{i})}{\Delta t} + \sum_{\mu=1}^3 \frac{c_\mu}{\Delta x} \left[ \boldsymbol{\Psi}(t_n,\mathbf{x}_{\mathbf{i}+\hat{\mu}}) - \boldsymbol{\Psi}(t_n,\mathbf{x}_{\mathbf{i}-\hat{\mu}}) \right] + \kappa \boldsymbol{\Psi}(t_n,\mathbf{x}_\mathbf{i}) = 0,

where c_\mu are the components of the speed of light (taken equal to c for isotropy) and \kappa is a mass coupling constant (dimension T^{-1}). To incorporate the spinor structure, we elevate the coefficients to matrices: set c_\mu = c \alpha_\mu and \kappa = \frac{mc^2}{\hbar} \beta, where \alpha_\mu, \beta are 4\times4 matrices to be determined, and \hbar is Planck’s constant (introduced as a scale conversion factor). Hence:

\frac{\boldsymbol{\Psi}_{n+1,\mathbf{i}} - \boldsymbol{\Psi}_{n,\mathbf{i}}}{\Delta t} + c \sum_{\mu=1}^3 \alpha_\mu \frac{\boldsymbol{\Psi}_{n,\mathbf{i}+\hat{\mu}} - \boldsymbol{\Psi}_{n,\mathbf{i}-\hat{\mu}}}{2\Delta x} + \frac{mc^2}{\hbar} \beta \boldsymbol{\Psi}_{n,\mathbf{i}} = 0, \tag{1}

where we have used the central difference approximation for the spatial first derivative. Equation (1) is the explicit difference form of the DOG discrete recursion.

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3. Discrete Unitarity and Determination of the Matrix Algebra

An isolated system is required to conserve the order norm:

\sum_{\mathbf{i}} \boldsymbol{\Psi}^\dagger(t_{n+1},\mathbf{x}_\mathbf{i}) \boldsymbol{\Psi}(t_{n+1},\mathbf{x}_\mathbf{i}) = \sum_{\mathbf{i}} \boldsymbol{\Psi}^\dagger(t_n,\mathbf{x}_\mathbf{i}) \boldsymbol{\Psi}(t_n,\mathbf{x}_\mathbf{i}).

Interpreting (1) as an implicit definition \boldsymbol{\Psi}_{n+1} = M \boldsymbol{\Psi}_n, conservation demands that M be unitary. In the continuum limit this condition translates into constraints on the matrices \alpha_\mu, \beta.

Substituting a plane‑wave ansatz \boldsymbol{\Psi} = \boldsymbol{u} e^{i(\mathbf{k}\cdot\mathbf{x} - \omega t)} into the discrete equation, we may take the continuum limit \Delta t,\Delta x \to 0.

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4. Continuum Limit Derivation

When \Delta t, \Delta x \to 0, keeping first‑order terms, we have:

\frac{\boldsymbol{\Psi}_{n+1,\mathbf{i}} - \boldsymbol{\Psi}_{n,\mathbf{i}}}{\Delta t} \to \partial_t \boldsymbol{\Psi},

\frac{\boldsymbol{\Psi}_{n,\mathbf{i}+\hat{\mu}} - \boldsymbol{\Psi}_{n,\mathbf{i}-\hat{\mu}}}{2\Delta x} \to \partial_\mu \boldsymbol{\Psi}.

Substituting into (1) yields:

\partial_t \boldsymbol{\Psi} + c \sum_{\mu=1}^3 \alpha_\mu \partial_\mu \boldsymbol{\Psi} + \frac{mc^2}{\hbar} \beta \boldsymbol{\Psi} = 0. \tag{2}

To obtain a covariant form, set \gamma^0 = \beta and \gamma^\mu = \beta \alpha_\mu (\mu=1,2,3). Multiply (2) on the left by \beta and use \beta^2 = I (which will be derived from unitarity):

\beta \partial_t \boldsymbol{\Psi} + c \sum_{\mu=1}^3 \beta\alpha_\mu \partial_\mu \boldsymbol{\Psi} + \frac{mc^2}{\hbar} \boldsymbol{\Psi} = 0,

i.e.,

\gamma^0 \partial_t \boldsymbol{\Psi} + c \sum_{\mu=1}^3 \gamma^\mu \partial_\mu \boldsymbol{\Psi} + \frac{mc^2}{\hbar} \boldsymbol{\Psi} = 0.

Now introduce x^0 = ct and \partial_0 = \frac{1}{c}\partial_t. Then \gamma^0 \partial_t = c \gamma^0 \partial_0. The equation becomes:

c \gamma^0 \partial_0 \boldsymbol{\Psi} + c \sum_{\mu=1}^3 \gamma^\mu \partial_\mu \boldsymbol{\Psi} + \frac{mc^2}{\hbar} \boldsymbol{\Psi} = 0.

Dividing by c:

\gamma^0 \partial_0 \boldsymbol{\Psi} + \sum_{\mu=1}^3 \gamma^\mu \partial_\mu \boldsymbol{\Psi} + \frac{mc}{\hbar} \boldsymbol{\Psi} = 0,

or

\gamma^\mu \partial_\mu \boldsymbol{\Psi} + \frac{mc}{\hbar} \boldsymbol{\Psi} = 0,

with \gamma^\mu now satisfying the Clifford algebra \{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu} (see next section). Multiplying by i\hbar gives the standard Dirac equation:

\left( i\hbar \gamma^\mu \partial_\mu - mc \right) \boldsymbol{\Psi} = 0. \tag{3}

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5. Geometric Origin of the Matrix Algebra: Clifford Algebra from Lattice Symmetries

Consider the rotational symmetry of the discrete spacetime (in the lattice‑approximated sense). Covariance of equation (1) under spatial rotations forces the matrices \alpha_\mu to transform as vector components. Moreover, the square‑root of the discrete Laplacian operator in the continuum limit naturally leads to anti‑commutation relations. Concretely, for a spinor field, we require (\boldsymbol{\alpha}\cdot\nabla)^2 = \nabla^2, which implies \{\alpha_i,\alpha_j\}=2\delta_{ij} I. The cross anti‑commutation \{\alpha_i,\beta\}=0 is needed to achieve the linear decomposition of the relativistic dispersion E^2 = c^2 p^2 + m^2 c^4. Hence \alpha_\mu,\beta must satisfy the Dirac matrix algebra, whose smallest representation is four‑dimensional.

Therefore, the DOG geometric conclusion is: for a first‑order isotropic difference equation on a four‑dimensional discrete lattice, the requirement that the continuum limit reproduces the relativistic dispersion forces the coefficient matrices to obey the Clifford algebra, and the minimal representation is exactly the spinor representation.

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6. Final Dirac Equation

Starting from equation (2), left‑multiply by \beta and use \beta^2=I and \beta\alpha_\mu = \gamma^\mu:

\gamma^0 \partial_t \boldsymbol{\Psi} + c \sum_{\mu=1}^3 \gamma^\mu \partial_\mu \boldsymbol{\Psi} + \frac{mc^2}{\hbar} \boldsymbol{\Psi} = 0.

With x^0 = ct and \partial_0 = \partial_t / c, this becomes:

\gamma^0 c \partial_0 \boldsymbol{\Psi} + c \gamma^\mu \partial_\mu \boldsymbol{\Psi} + \frac{mc^2}{\hbar} \boldsymbol{\Psi} = 0,

or

\gamma^\mu \partial_\mu \boldsymbol{\Psi} + \frac{mc}{\hbar} \boldsymbol{\Psi} = 0.

Multiplying by i\hbar:

i\hbar \gamma^\mu \partial_\mu \boldsymbol{\Psi} - mc \boldsymbol{\Psi} = 0,

which is the standard covariant Dirac equation.

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7. Physical Interpretation and Discrete Corrections

7.1 Geometric Origin of Spin and Antiparticles

· Spin‑1/2: The minimal dimension of the spinor representation arises from the binary topology of the discrete lattice; the resulting angular momentum algebra naturally yields half‑integer spin.
· Positive/negative energy solutions: Equation (1) in frequency domain yields two branches \omega = \pm \sqrt{c^2 k^2 + (mc^2/\hbar)^2}, corresponding to positive and negative energy states. Negative energy states are not a disaster but represent order modes evolving backward in time.
· Antiparticles: The charge conjugate of a negative‑energy state corresponds to the antiparticle, automatically produced by the combination of complex conjugation and time‑reversal symmetry on the discrete lattice.

7.2 Planck‑Scale Discrete Corrections

When \Delta t and \Delta x are finite, the continuum approximation breaks down. The discrete dispersion relation reads:

\frac{\sin^2(\omega \Delta t/2)}{(\Delta t/2)^2} = c^2 \sum_{\mu=1}^3 \frac{\sin^2(k_\mu \Delta x)}{(\Delta x)^2} + \left( \frac{mc^2}{\hbar} \right)^2.

At extremely high energies (k \sim \pi/\Delta x), significant Lorentz invariance violation and discretization of the spectrum appear. These effects might leave traces in high‑energy cosmic rays or early‑universe physics.

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8. Conclusion

Starting from the DOG four‑dimensional discrete spacetime order lattice and a finite‑difference evolution equation, we have rigorously derived the standard Dirac equation in the continuum limit. The main conclusions are:

1. The Dirac equation is not a fundamental postulate but the natural continuum limit of a first‑order difference evolution on a four‑dimensional discrete lattice.
2. The \gamma^\mu matrices and their Clifford algebra relations originate from the square‑root decomposition of the discrete Laplacian and spatial isotropy; the minimal representation dimension 4 is dictated by the discrete spacetime topology.
3. Spin‑1/2, positive/negative energy solutions, and antiparticle structure are geometric manifestations of the intrinsic degrees of freedom of the discrete order lattice, requiring no extra field‑theoretic assumptions.
4. This derivation unifies the geometric origin of the Schrödinger and Dirac equations, completing DOG’s foundational coverage of the whole domain of quantum mechanics.

Future work will focus on observational predictions of the discrete corrections and extension of the framework to curved spacetime (discrete gravity) and quantum field theory.

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References

[1] Zhang S H. Discrete Order Geometry (DOG) and Matrix Algebra: Essential Correspondence and Paradigm Reconstruction. 2026.
[2] Zhang S H. Discrete Order Geometry (DOG) First‑Principles Derivation of the Schrödinger Equation. 2026.
[3] Zhang S H. DOG Discrete Order Geometry and Quantum Mechanics: From Discrete Lattice to Quantum State Space. 2026.
[4] Dirac P A M. The Quantum Theory of the Electron. Proc. R. Soc. Lond. A, 1928.
[5] Nielsen H B, Ninomiya M. Absence of neutrinos on a lattice. Nucl. Phys. B, 1981.

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Appendix: Unitarity Condition for the Discrete Difference Equation

Equation (1) can be solved for \boldsymbol{\Psi}_{n+1} as an implicit time step. To preserve the norm, an appropriate discretization (e.g., Crank‑Nicolson) should be employed. In the continuum limit this condition reduces to the hermiticity and anti‑commutation relations of \alpha_\mu,\beta. Detailed analysis will be presented in future work.

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