On Charge Conservation and the Continuity Equation for Real Wavefunctions in Unified Quantum Mechanics

This paper presents a rigorous mathematical derivation of the continuity equation within a reformulated quantum mechanical framework-Unified Quantum Mechanics (UQM)-where the fundamental imaginary unit \(i\) is systematically replaced by the temporal Hilbert transform operator \(\mathcal{H}_t\). This substitution enables a complete description of quantum dynamics using exclusively real-valued wavefunctions while preserving the essential conservation laws of quantum theory. We demonstrate that this real-valued formalism naturally gives rise to a conserved quantity analogous to the probability density in conventional quantum mechanics. By constructing a coupled system of evolution equations for the primary wavefunction \(\psi\) and its Hilbert transform \(\phi = \mathcal{H}_t[\psi]\), we identify an underlying global SO(2) symmetry that is mathematically isomorphic to the U(1) gauge symmetry responsible for charge conservation in standard complex-valued quantum mechanics. Through direct computation, we derive the continuity equation \(\frac{\partial\rho}{\partial t} + \nabla \cdot \vec{j} = 0\) and obtain explicit expressions for the conserved density \(\rho\) and current density \(\vec{j}\). This result establishes that the fundamental principle of local charge conservation emerges naturally within the UQM dynamics, thereby validating its mathematical consistency and physical completeness. Moreover, we extend the result to demonstrate the local conservation of mass, deriving the corresponding continuity equation for mass density.