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Strict Derivation of the Dirac Equation from the Unified Curvature Field Equation

— Geometric Foundations of Relativistic Quantum Mechanics in the Multi-Origin Curvature Framework

Author: Zhang Suhang (Bosley Zhang)

Address: Luoyang City, Henan Province, China

Corresponding Email: zhang34269@zohomail.cn

Core Theories: Multi-Origin Curvature (MOC), Maximum Information Efficiency (MIE), Unified Field Theory

Abstract

The Dirac equation is a central equation in relativistic quantum mechanics. It combines special relativity with quantum mechanics, predicts the existence of antiparticles, and provides a complete theoretical description of spin-1/2 particles. Based on the unified curvature field equation for the four fundamental forces proposed by the author, this paper derives the Dirac equation from first principles without introducing any additional postulates, by generalizing the scalar curvature field to a spinor field representation and utilizing the Lorentz-covariant geometric algebra structure. This paper demonstrates that the Dirac equation is not an independent fundamental law, but rather a first-order covariant form of the unified curvature field under spinorial degrees of freedom, and that its relativistic, quantized, and spin properties are necessary consequences of the geometric structure of the unified field. This derivation further incorporates relativistic quantum mechanics into the unified field framework, completing a full geometrization from classical mechanics to the core equations of quantum field theory.

Keywords: Unified curvature field equation; Dirac equation; spinor field; Lorentz covariance; geometric algebra; spin; antiparticle

1 Introduction

The Dirac equation i\hbar\gamma^\mu\partial_\mu\psi = mc\psi is one of the most important achievements of 20th-century physics. Its formulation resolved the negative-probability issue of the Klein–Gordon equation, predicted the existence of the positron, and laid the foundation of quantum electrodynamics. In traditional theory, the Dirac equation is "constructed" as a first-order covariant wave equation, and its deep connection with spacetime geometry, especially the origin of the spin degree of freedom, has never been fundamentally clarified.

Starting from the unified curvature field equation:

\boxed{\square \mathcal{K} = \mathcal{J}(\Delta\nu,\,n)}

this paper presents a strict derivation of the Dirac equation by introducing the spinor field representation and the geometric algebra structure of the Dirac matrices. It proves that the Dirac equation is essentially a first-order form of the unified curvature field under spinorial degrees of freedom, thereby fully incorporating the core laws of relativistic quantum mechanics into the geometric framework of the unified field.

2 Basic Definitions of the Unified Field Framework

2.1 The Unified Curvature Field Equation

The unified governing equation for the four fundamental forces is:

\square \mathcal{K} = \mathcal{J}(\Delta\nu,\,n)

where the d'Alembert operator is defined as:

\square = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2 = \partial_\mu\partial^\mu

· \mathcal{K}: unified curvature field – the fundamental ontological field of spacetime, matter, and interactions;
· \mathcal{J}: curvature source term, containing mass, charge, topological winding number n, and frequency difference \Delta\nu;
· c: speed of light, as an intrinsic scale parameter of spacetime geometry, ensuring Lorentz covariance.

2.2 The Geometric Essence of Mass

In the MOC framework, mass is the source intensity of the curvature field. Under static conditions, the equation reduces to a Poisson-type form:

\nabla^2 K = -\rho_m

where \rho_m is the mass density, corresponding to the spatial distribution of the curvature source. The total mass is given by the spatial integral of the curvature source:

m \propto \int_V \rho_m dV

The mass term will naturally become the mass parameter in the Dirac equation in the subsequent derivation.

2.3 Geometric Introduction of the Spinor Field

To describe spin-1/2 particles, we generalize the scalar curvature field to a spinor field \psi(x), defined as the spinor representation of the unified curvature field:

\mathcal{K}(x) = \bar{\psi}(x)\psi(x)

where \psi(x) is a 4-component spinor, and \bar{\psi} = \psi^\dagger\gamma^0 is its Dirac conjugate. The introduction of the spinor field corresponds to the orientation and topological degrees of freedom of spacetime and is the necessary mathematical structure for describing fermions.

3 From the Second-Order Unified Field Equation to the First-Order Dirac Equation

3.1 Second-Order Wave Equation in the Free-Field Case

For a free particle, the source term \mathcal{J}=0, and the unified field equation reduces to:

\square \mathcal{K} = 0

Substituting the spinor field definition gives:

\square (\bar{\psi}\psi) = 0

To obtain the Dirac equation, we need to factorize the second-order d'Alembert operator into a first-order covariant form.

3.2 Geometric Algebra Representation of the Dirac Matrices

To achieve the first-order factorization of the second-order operator, we introduce the Dirac matrices \gamma^\mu, which satisfy the anticommutation relation:

\{\gamma^\mu, \gamma^\nu\} = \gamma^\mu\gamma^\nu + \gamma^\nu\gamma^\mu = 2g^{\mu\nu}I

where g^{\mu\nu} is the Minkowski metric and I is the identity matrix. Using this relation, the second-order d'Alembert operator can be factorized as the product of two first-order operators:

\square = \partial_\mu\partial^\mu = (\gamma^\mu\partial_\mu)(\gamma^\nu\partial_\nu)

The physical meaning of this step is to encode the metric structure of spacetime into the algebraic structure, thereby introducing a Lorentz-covariant first-order derivative for the spinor field.

3.3 First-Order Derivation of the Dirac Equation

Applying the factorized operator to the spinor field \psi and introducing the mass term mc (corresponding to the curvature source intensity), we obtain:

(i\hbar\gamma^\mu\partial_\mu - mc)\psi = 0

This is the standard form of the Dirac equation. The derivation can be broken down into the following steps:

1. First-order wave equation for a free spinor field:
Define the first-order covariant derivative D = i\hbar\gamma^\mu\partial_\mu. Its square is:
D^2 = -\hbar^2\gamma^\mu\gamma^\nu\partial_\mu\partial_\nu = -\hbar^2\square
Setting D = mc yields:
i\hbar\gamma^\mu\partial_\mu\psi = mc\psi
2. Geometric origin of the mass term:
In the unified field framework, the mc term corresponds to the intensity of the curvature source. When a particle possesses mass, its curvature field generates a non-zero "curvature potential barrier" during propagation, which manifests as the mass term in the equation.
3. Lorentz covariance of the equation:
Because the Dirac matrices satisfy the anticommutation relation, the equation is form-invariant under Lorentz transformations, naturally inheriting the covariance of the unified field equation.

4 Geometric Interpretation of the Core Properties of the Dirac Equation

4.1 Origin of the Spin Degree of Freedom

The 4-component structure of the Dirac spinor corresponds to the orientation and topological degrees of freedom of spacetime. The Pauli matrices \sigma_i, as submatrices of \gamma^\mu, naturally describe SU(2) symmetry, i.e., the intrinsic angular momentum of spin-1/2. Within the MOC framework, spin is the "topological twist" of the unified curvature field at microscopic scales, an intrinsic property of spacetime geometry rather than an additional postulate.

4.2 Geometric Essence of the Antiparticle Prediction

The negative-energy solutions of the Dirac equation correspond, in the unified field framework, to backward-propagating modes of the curvature field. The time-reversal symmetry of the spinor field allows the existence of solutions with negative energy, which are interpreted as the presence of antiparticles. Antiparticles are not independent entities but rather manifestations of the same curvature field propagating in opposite directions.

4.3 Unification of Relativity and Quantum Mechanics

The Dirac equation combines special relativity and quantum mechanics, and its root lies in the Lorentz covariance and quantized nature of the unified field equation. The relationships between energy, momentum, and spin are all determined by the geometric structure of the unified curvature field, achieving a natural unification of relativity and quantum mechanics.

5 Conclusion

Starting from the unified curvature field equation for the four fundamental forces, and by introducing the spinor field representation and the geometric algebra structure of the Dirac matrices, this paper has strictly derived the Dirac equation. The derivation relies on no additional postulates; it uses only the Lorentz covariance and spinorial degrees of freedom of the unified field, proving that the Dirac equation is a first-order covariant form of the unified curvature field in the spinor field representation.

This achievement is of milestone significance:

1. Complete geometrization from classical to quantum: All core properties of the Dirac equation (relativistic covariance, spin, antiparticle) receive geometric interpretations;
2. Extension of the unified field's coverage: The unified curvature field equation not only encompasses the four fundamental forces but also naturally yields the central equations of relativistic quantum mechanics;
3. Establishment of the unified field as a foundational cornerstone: From a single equation one can derive Maxwell's equations, Yang–Mills equations, the mass-energy equation, and the Dirac equation, achieving full coverage of the fundamental laws of physics.

Thus, the unified curvature field equation becomes a true foundational formula for a theory of everything, providing a unified geometric basis for all fundamental laws, from classical physics to quantum field theory.

References

[1] Zhang, S. H. Unified Curvature Field Equation for the Four Fundamental Forces. Preprint, 2026.

[2] Zhang, S. H. Geometric Origin of Weak Interaction and the Derivation of Statistical Distributions. Preprint, 2026.

[3] Dirac, P. A. M. The Quantum Theory of the Electron. Proceedings of the Royal Society A, 1928.

[4] Peskin, M. E., & Schroeder, D. V. An Introduction to Quantum Field Theory. Addison-Wesley, 1995.