A Study on the Mathematical Unification of Classical Field Theory and Quantum Mechanics

Classical physics, relativity, quantum mechanics and quantum field theory have long existed as separate theoretical frameworks in modern physics. Quantum field theory is plagued by the problem of ultraviolet divergence, while quantum phenomena such as wave-particle duality and the uncertainty principle mostly rely on artificial postulates. In addition, a unified description of the four fundamental interactions has not yet been achieved. Based on the existing classical field system, conjugate golden decay law and discrete spatial lattice model of Real-Virtual Dual Field Theory (RVDT), this paper expands the theory via canonical quantization. Throughout the research, the underlying axioms and core equations are not reconstructed; only formal upgrades of physical quantities are implemented. Ultimately, the substantive unification of classical physics, relativity, quantum mechanics and quantum fields is realized within a single mathematical framework. This paper constructs the universal Lagrangian density and the master equation of classical universal fields, and completes the canonical quantization of classical fields. It is verified that classical field equations and quantum field equations share identical mathematical forms. Relying on four core characteristics including discrete lattice, topological vortex structure, conjugate golden decay and intrinsic field resistance, this paper interprets typical quantum effects such as wave-particle duality, uncertainty principle, quantum entanglement and quantum tunneling from first principles. The universal master equation can naturally degenerate into the standard equations for gravitation, electromagnetism, strong interaction, weak interaction and non-relativistic quantum mechanics with varying spatial scales and field resistance parameters, and it is fully compatible with special and general relativity. Furthermore, the discrete spatial lattice inherently limits the range of field quantities, so the ultraviolet divergence in quantum field theory can be eliminated without adopting traditional renormalization methods. The intrinsic field resistance coefficient acts as a continuous parameter to realize the smooth transition between classical and quantum systems. The whole theory is logically self-consistent and systematically complete, providing new ideas and a mathematical framework for the research on the universal framework of physics.