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.
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.
We extend the framework of Unified Quantum Mechanics (UQM) to the relativistic domain by formulating the Dirac equation entirely in terms of real-valued fields. In this approach, the complex imaginary unit i is systematically replaced by the temporal Hilbert transform operator Ht, which satisfies the anti-involution property Ht2=−I and acts as a dimensionless isometric operator. The resulting theory is expressed as a coupled system of real-valued equations for a primary field ψ and its Hilbert-transformed partner ϕ=Ht[ψ], whose evolution is governed exclusively by local differential operators. We demonstrate that all essential algebraic and symmetry structures of relativistic quantum mechanics are preserved within this real-valued formulation. In particular, the full Clifford algebra of the Dirac matrices, Lorentz covariance, and the relativistic energy-momentum relation are recovered exactly. A rigorous isomorphism with the conventional complex Dirac equation is established via the analytic signal representation ψa=ψ−iϕ, under which the standard U(1) gauge symmetry emerges naturally as a global SO(2) rotation in the real field space. The resulting formulation is therefore operationally and predictively equivalent to the standard complex theory. The conserved scalar density ρ=ψ†ψ+ϕ†ϕ=∣ψa∣2 is shown to arise as the squared envelope (intensity) of a real physical field and admits a direct interpretation as a local field intensity with well-defined physical dimensions. Probability conservation is recovered through the continuity equation, while normalization in standard quantum mechanics is reinterpreted as a rescaling by the conserved total field. Stability of matter, expressed as energy positivity, is shown to restrict the physical solution space to analytic signals, enforcing a Hilbert-pair structure and eliminating independent complex degrees of freedom. This work demonstrates that the complex structure of relativistic quantum mechanics is not fundamental but emerges from the analytically constrained dynamics of real-valued fields, providing a unified and physically transparent foundation for the Dirac theory.
We present Unified Quantum Mechanics (UQM) by unifying real and standard complex quantum theories via real energy and momentum operators derived from the Hilbert Transform. This paper especially presents a fully real-valued formulation of quantum mechanics, showing that while complex numbers are commonly used in the standard theory, they are not fundamentally necessary for describing quantum phenomena. We develop Real Quantum Mechanics (RQM) by defining the real energy and momentum operators using the temporal Hilbert transform which shifts the phases of wave functions while preserving all essential quantum behavior. These real operators unify the proposed RQM and standard Complex Quantum Mechanics (CQM), as these are valid for both real and complex wavefunctions. We introduce a real-valued algebraic structure constructed from the identity operator and the Hilbert transform operator. This framework is mathematically equivalent to the system of complex numbers, preserving all fundamental algebraic properties while operating entirely within the realm of real operators. The isomorphism provides a novel representation of complex quantum mechanics using only real-valued components, offering potential advantages for both theoretical analysis and physical implementations. Importantly, we establish that the Hilbert transform operator is a generalization of the imaginary unit. The key insight is that the Hilbert transform generalizes the role of imaginary numbers in standard quantum mechanics, allowing us to construct real operators for both real and complex wave functions that produce the same physical predictions. We derive real versions of energy and momentum operators, leading to a modified wave equation that replaces the Schrödinger equation. Despite its non-local time evolution, this framework retains local spatial behavior and reproduces the usual energy-momentum relations. We rigorously test this approach on fundamental quantum systems, including the hydrogen atom and relativistic particles, confirming that it yields identical energy levels and wave functions as conventional quantum mechanics. Notably, we establish a consistent probabilistic framework by introducing real analogues of quantum probabilities via the analytic Born rule. We derive a unified uncertainty principle that coincides with the standard Heisenberg uncertainty bound for complex-valued wave functions, while admitting a vanishing lower bound in the case of real-valued wave functions. The implications of this work are significant: (1) it demonstrates that complex numbers are a mathematical convenience rather than a fundamental requirement in quantum theory, (2) it introduces new tools for analyzing quantum dynamics using signal-processing techniques, (3) it provides insights into non-local quantum evolution, (4) it connects quantum mechanics to real-field theories. Ultimately, RQM reproduces all known quantum predictions while offering a fresh perspective on operator structure, time evolution, and the mathematical foundations of the theory, and (5) it provides a real qubit representation on the modified Bloch sphere for quantum computing.
We establish a state-dependent quantum uncertainty principle within Unified Quantum Mechanics (UQM), demonstrating vanishing bounds for real-valued wavefunctions via the temporal Hilbert transform operator Ht. UQM generalizes the imaginary unit i with the Ht to obtain real energy and momentum operators. This leads to non-canonical, operator-valued commutation relations [x^,p^]=ℏHt and [t^,E^]=−ℏHt. These relations generate the generalized uncertainty principles, ΔxΔp≥2ℏ∣⟨Ht⟩∣ and ΔtΔE≥2ℏ∣⟨Ht⟩∣, whose lower bound is state-dependent. For conventional complex-valued states, the framework recovers the standard Heisenberg limit as ⟨Ht⟩=i. However, for any real-valued wave function, the expectation value can vanish (⟨Ht⟩=0), predicting the zero-uncertainty bounds, ΔxΔp≥0 and ΔtΔE≥0. These results facilitate a seamless classical-quantum transition, showing how macroscopic determinism (ΔxΔp=0) can emerge from the theory without wavefunction collapse. Furthermore, the formalism resolves foundational paradoxes by circumventing Pauli's theorem against a self-adjoint time operator while maintaining a bounded-below energy spectrum. UQM thus reinterprets quantum uncertainty not as a universal constant, but as a property contingent on the mathematical representation of the state.
This work introduces a complete and self-consistent framework for quantum computation within the domain of real-valued quantum mechanics. Standard quantum computation is predicated on the manipulation of states in a complex Hilbert space, where a complex phase is a critical resource. We demonstrate that this complex structure is not fundamental but can be fully reproduced by a real-operator algebra. Our formalism is built upon a rigorous algebraic isomorphism between the imaginary unit i and the temporal Hilbert transform operator Ht, which acts as a physical generator of phase. Leveraging this isomorphism, we systematically redefine the fundamental quantum logic gates-including the Pauli, Hadamard, phase rotation, and CNOT gates-entirely within a real operator algebra. We prove that each gate's action on a real-valued qubit state representation is mathematically and operationally equivalent to its conventional complex counterpart. The Pauli-Y, phase, and controlled-phase gates, which are intrinsically complex in the standard model, are shown to correspond to real-operator matrices involving the Hilbert transform. This formulation provides a physically grounded interpretation of quantum logic, where abstract phase manipulation corresponds to the application of a concrete, albeit non-local, integral transform. We demonstrate how this framework correctly reproduces the generation of quantum entanglement and establishes a consistent measurement theory via an Analytic Born Rule in Unified Quantum Mechanics. This work solidifies the viability of a complete real-valued model for quantum information processing and suggests new paradigms for the physical implementation of quantum hardware.
The historical debate surrounding the necessity of complex numbers in quantum mechanics has been revitalized by recent experimental falsifications of standard real-amplitude models. However, Barrios Hita et al. (2026) demonstrated that a physically consistent real-number quantum theory can be constructed by mapping complex Hilbert spaces onto real quotient Hilbert spaces with local flags. In this work, we extensively expand this framework to full quantum field theory and numerous applications. We present explicit, index-level, and operator-level derivations of non-relativistic dynamics (the Schr¨odinger equation, Ehrenfest’s theorem, uncertainty principle), relativistic quantum mechanics (Klein-Gordon and Dirac equations), and quantum field theory (canonical quantization, Fock space, propagators, path integrals). We further apply the real formulation to harmonic oscillators, spin systems, perturbation theory, scattering theory, quantum entanglement, gauge theories and Quantum computing. We show that U(1) gauge transformations are rigorously isomorphic to SO(2) rotations in the real flag space, and that all physical predictions—including Bell violations and scattering amplitudes—are exactly reproduced. All derivations are presented at the explicit index level with full algebraic details, using a symplectic structure J that replaces the complex unit i.
The persistent challenge of unifying quantum mechanics with general relativity is often traced to a series of apparent structural tensions: complex versus real ontology, probabilistic versus deterministic dynamics, linear versus nonlinear evolution, and spectral discreteness versus geometric continuity. We argue that these tensions do not signal a fundamental incompatibility between the theories, but rather originate from surplus structure in the conventional formulation of quantum mechanics. Beginning from the physically indispensable requirement of matter stability, we prove that the Hamiltonian spectrum can always be shifted to become non-negative. The consequent positivity of energy enforces one-sided temporal frequency support for all physical states, selecting the Hardy space of analytic signals as the admissible solution space. In this space, the real and imaginary parts of the wave function are not independent; they are linked uniquely via the temporal Hilbert transform. Hence, the complex structure of quantum mechanics emerges as a representational artifact of an underlying real operator algebra. Building on our earlier work on Unified Quantum Mechanics and the Unified Real Wave Equation, we establish a rigorous proof of exact equivalence between the real-field evolution and the Schrödinger equation on the physical subspace. Probabilistic quantities arise naturally as quadratic invariants of analytic signals, rather than as independent postulates. The resulting framework provides a realfield formulation of quantum mechanics and gravitation, with stability and causality implemented as foundational principles, and demonstrates that the Einstein field equations are compatible with a unified field-theoretic description of spacetime and matter. Extending the same analyticity principle to spacetime geometry, we propose an analytic formulation of the Einstein field equations, from which classical general relativity is recovered exactly as the real projection.