Quantum Milestones, 1928: The Dirac Equation Unifies Quantum Mechanics and Special Relativity

Ijaz Durrani

Zhu Hengjiang

Quantum mechanics and classical mechanics are closely related to each other.The Hamilton-Jacobi Equation of classical mechanics plays an important role in the extension of Schrdinger equation,and it is also a way to extend Schrdinger equation in the teaching of quantum mechanics.Besides,the Hamilton-Jacobi Equation under the theory of relativity can extend Klein-Gordon equation of quantum mechanics of special relativity.Considering and analyzing the differences betweeen Klein-Gordon equation and Dirac Equation further,this paper linearized the Hamilton-Jacobi Equation under the theory of relativity,and extended Dirac Equation which was more important than quantum mechanics of special relativity.It made the route by which Hamilton-Jacobi Equation,as the classical mechanics,goes to quantum mechanics more extensive,and this study further explored the corresponding relationship between classical mechanics and quantum mechanics.

Maria Cristina De Sanctis

By using the general concepts of special relativity and the requirements of quantum mechanics, Dirac equation is derived and studied. Only elementary knowledge of spin and rotations in quantum mechanics and standard handlings of linear algebra are employed for the development of the present work.

shifa liu

Starting from three fundamental duality relations (time-energy, space-momentum, angle-angular momentum), we establish an integral equation of quantum action and its dual counterpart. This integral equation directly links the evolution of the wave function to the accumulated action of spacetime and rotational degrees of freedom, forming a unified framework for quantum theory. Through semiclassical expansion, local approximation, and relativistic covariance, we rigorously derive the Schr\"odinger equation, Klein-Gordon equation, Dirac equation, Heisenberg equation, Schwinger-Dyson equation, Feynman path integral, Yang-Mills equations, Einstein field equations, and the Friedmann and Schwarzschild solutions. The framework also yields the kinematic effects of special relativity (time dilation, length contraction, mass-energy equivalence) as consequences of the duality structure. Moreover, the nonlocal kernel in the integral equation gives an exponential damping factor in momentum space, providing an intrinsic regularization mechanism for ultraviolet divergences. We extend the framework to quantum electrodynamics (QED), quantum chromodynamics (QCD), the electroweak theory, the full Standard Model, and quantum gravity, obtaining dual formulations for each. From the integral equation and its dual, we automatically derive quantization conditions for energy, momentum, angular momentum, and their duals, and propose experimental schemes to observe these discrete effects. All derivations are complete and rigorous, based on the same integral equation, providing a possible answer to Dirac's criticism that "quantum electrodynamics lacks a fundamental equation" and opening a new path toward a unified theory.