The Schrödinger equation can be rewritten as a second-order equation for a single real-valued scalar field. Taken at face value, however, this one-field reduction obscures several local structures of quantum mechanics: the Born density and current, the role of momentum as a generator, and the usual derivation of the uncertainty relation. We argue that these difficulties do not show that real variables fail. They show that the reduced equation is not, by itself, a complete local representation of the theory. Its coherent real form is obtained by restoring the underlying Hamiltonian phase-space structure, where the missing component reappears as a canonical partner. In this real phase-space formulation, the standard structures reappear locally, and minimal magnetic coupling reveals an internal \(SO(2)\) gauge structure. Thus, the symbol \(i\) can be removed, but the symplectic and complex structure it encodes cannot.
The Schrödinger equation can be rewritten as a second-order equation for a single real-valued scalar field. Taken at face value, however, this one-field reduction obscures several local structures of quantum mechanics: the Born density and current, the role of momentum as a generator, and the usual derivation of the uncertainty relation. We argue that these difficulties do not show that real variables fail. They show that the reduced equation is not, by itself, a complete local representation of the theory. Its coherent real form is obtained by restoring the underlying Hamiltonian phase-space structure, where the missing component reappears as a canonical partner. In this real phase-space formulation, the standard structures reappear locally, and minimal magnetic coupling reveals an internal (SO(2)) gauge structure. Thus, the symbol (i) can be removed, but the symplectic and complex structure it encodes cannot.
Classical mechanics admits multiple equivalent formulations, from Newton's equations to the variational Lagrange-Hamilton framework and the scalar Hamilton-Jacobi (HJ) theory. In the HJ formulation, classical ensembles evolve through the continuity equation for a real density ρ=R2 coupled to Hamilton's principal function S. Here we develop a complementary formulation, the Hamilton-Jacobi-Schrödinger (HJS) theory, by embedding the pair (R,S) into a single complex field. Starting from a completely general complex ansatz ψ=f(R,S)eig(R,S), and imposing two minimal structural requirements, we obtain a unique map $ψ= R\, e^{iS/κ}\, $ together with a linear HJS equation whose ∣κ∣→0 limit reproduces the HJ formulation exactly. Remarkably, when Re(κ)=0, essential features of quantum mechanics, superposition, operator algebra, commutators, the Heisenberg uncertainty principle, Born's rule and unitary evolution, follow naturally as structural consistency conditions. HJS thus provides a unified mathematical viewpoint in which classical and quantum dynamics appear as different limits of a single underlying structure.
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.
This work reformulates quantum mechanics on the real plane (R2), establishing an algebraic isomorphism to the complex field (C) and demonstrating that the complex structure inherent to the Schrödinger equation can be fully captured by coupled real-valued dynamics. By constructing a field (R2,+,⋅) isomorphic to C, the real counterpart of the Schrödinger equation is derived, yielding a system of coupled real partial differential equations. This system reveals a symplectic structure where the symplectic matrix J geometrically represents the imaginary unit i, acting as a phase-space rotation operator. Time-independent solutions decouple into identical real Helmholtz equations, preserving spectral equivalence. The reformulation provides geometric insights into foundational aspects: the geometric representation of i, the emergence of probability conservation from real-component interactions, and the encoding of quantum phase relationships through the multiplicative structure of the R2 field. This perspective reinforces connections to classical Hamiltonian mechanics and geometric quantization, suggesting that complex numbers are structural, emerging from field operations on the real plane, rather than fundamental. Furthermore, quantum computation elements like qubits and gates are shown to have equivalent representations as real symplectic and special orthogonal matrices, maintaining unitary equivalence. We introduce a framework for quantum computation within a real vector space and provide a generalized protocol for generating entangled Bell states. These results offer a new perspective on the necessity of complex numbers in quantum theory, providing a framework with potential computational advantages for hybrid quantum-classical systems and enhanced geometric interpretability.
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 analyze the Schrödinger dynamics and the Schrödinger function (or the so-called wavefunction) in the following four aspects. (1) The Schrödinger equation is reconstructed from scratch in the real field only, without referring to Newtonian mechanics nor optics. Only the very simple conditions such as the space-time translational symmetry and the conservation of flux and energy are imposed on the factorization of the density distribution in configuration space, giving rise to a two-dimensional real vector. On returning to the original Schrödinger equation, the imaginary number arises naturally. (2) Like the Langevin equation in a Brownian dynamics, we formulate a single-event path dynamics in quantum mechanics, contrasting with the Schrödinger distribution function. The path thus attained is referred to as one-world path, which represents, for instance, a path of a singly launched electron in the double-slit experiment that leaves a spot at the measurement board, while many of accumulated spots give rise to the fringe pattern. We start from the Feynman-Kac formula to draw a relation between a stochastic dynamics and the parabolic differential equations, to one of which the Schrödinger equation is transformed. (3) To highlight the roles of the flux and energy conservation in the Schrödinger dynamics, we build that the quantum Maupertuis-Hamilton principle, which reveals the symplectic structure in the parameter space. (4) We track how the inherent quantum nature like the Huygens-principle-like properties is built. We show that classical trajectory components in semiclassics are demanded to branch into many coherent pieces beyond the semiclassical regime and dissolve into the deep dynamics of genuine full quantum dynamics.