This work reformulates quantum mechanics on the real plane \((\mathbb{R}^2)\), establishing an algebraic isomorphism to the complex field \((\mathbb{C})\) and demonstrating that the complex structure inherent to the Schrödinger equation can be fully captured by coupled real-valued dynamics. By constructing a field \((\mathbb{R}^2, +, \cdot)\) isomorphic to \(\mathbb{C}\), 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 \(\mathcal{J}\) 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 \(\mathbb{R}^2\) 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.
This work reformulates quantum mechanics on the real plane ((\mathbb{R}^2)), establishing an algebraic isomorphism to the complex field ((\mathbb{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 ((\mathbb{R}^2, +, \cdot)) isomorphic to (\mathbb{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 (\mathcal{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 (\mathbb{R}^2) 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.
In this paper, we demonstrate the equivalence between the complex Hilbert space and real Kahler space formulations of quantum mechanics. Complex numbers play an important role in the traditional formulation of quantum mechanics in complex Hilbert spaces. However, the necessity of complex numbers--as opposed to their mere convenience--remains a subject of debate. Several alternative formulations of quantum mechanics using real numbers have been proposed. In this paper, we demonstrate that standard quantum mechanics, formulated in a complex Hilbert space, admits an equivalent reformulation in a real Kahler space. By establishing a natural isomorphism between the operator theories of the complex Hilbert space and the real Kahler space, we prove the equivalence of the two formulations including composite system. This Kahler-space framework preserves all essential features of quantum mechanics while offering a key advantage: it inherently incorporates a Hamiltonian symplectic structure analogous to classical mechanics. This structural alignment provides a unified geometric perspective for both classical and quantum dynamics. Additionally, we show that the ergodicity of finite-dimensional quantum systems becomes manifest in this framework, resolving interpretational ambiguities present in conventional complex formulations.
While the complex formalism of quantum mechanics has been extraordinarily successful, the fundamental geometric origin of its core elements—the imaginary unit, operator non-commutativity, quantum phase, and the intrinsic geometry with quantum spin—has remained unresolved for more than a century. Here we resolve these foundational questions by deriving an exact and physically equivalent formulation of the Schrödinger equation as a Real Geometric Flow—a moving-frame evolution on a 2𝑁-dimensional real state space. In this restored framework, the global imaginary unit 𝑖 fully decomposes into a set of local skew-symmetric generators that drive planar rotations in real space, revealing that the traditional complex formalism is a compressed representation of an intrinsic rotational geometry.Within the real geometric flow, energy eigenvalues appear as the Curvatures of the state trajectory, energy gaps correspond to Torsions that couple internal planes—naturally generating the geometry of quantum spin—and quantum phase emerges naturally as the Holonomy of the real moving frame. This formulation preserves the full empirical content of quantum mechanics while exposing the geometric mechanisms hidden beneath its notation, providing the first coherent explanation of curvature, torsion, non-commutativity, spin behavior, and quantum phase within a single framework.By demonstrating that the algebraic postulates of quantum mechanics arise from a deeper geometric order—one that stands alongside curvature-based theories such as General Relativity—this work establishes a unified differential-geometric foundation for quantum dynamics.It opens new directions for quantum control, quantum information, and fundamental physics.
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.
The necessity of complex numbers in quantum mechanics has long been debated. This paper develops a real Kahler space formulation of quantum mechanics [19], asserting equivalence to the standard complex Hilbert space framework. By mapping the complex Hilbert space Cn to a real Kahler space K2n, i.e. R2n equipped with a metric, a symplectic structure and an automorphism, we establish a correspondence between Hermitian operators in Cn and real operators in K2n. While the isomorphism appears straightforward some subtleties emerge: (i) the overcounting of composite system states under tensor products in R2n, and (ii) the double degeneracy of operator spectra in the real formulation. Through a systematic investigation of these challenges, we clarify the structural relationship between real and complex formulations, resolve ambiguities in composite system representations, and analyze spectral consequences. Our results reaffirm the equivalence of the two frameworks while highlighting nuanced distinctions with implications for foundational debates on locality, phase invariance, and the role of complex numbers in quantum theory.
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.
Standard quantum mechanics employs complex Hilbert spaces, but whether complex numbers are fundamental or merely convenient has long been debated. For decades, real-valued equivalents were considered mathematically possible but cumbersome. However, a highly cited 2021 result claimed that any quantum theory based on real numbers is experimentally falsifiable via network Bell experiments. Yet, it remains an open question whether this falsification applies to all real-valued theories. Here we show that this conclusion rests on an incomplete real formulation, and we present a rigorous real-valued framework that perfectly reproduces all predictions of standard quantum mechanics. We demonstrate that the standard real tensor product (⊗R) used in previous no-go theorems is algebraically incompatible with the rich structure of conventional quantum mechanics. We present a real framework based on Kähler space and prove that it is exactly isomorphic to established quantum mechanics via an explicit bijection γ. The isomorphism extends to composite systems through a symplectic composition rule $\otimes^{\ks}$ that replaces the Kronecker product. Consequently, our formulation achieves the maximal CHSH3 violation of 62 using purely real variables, demonstrating that the no-go theorem is specific to a particular real representation of states and operators and to the composition rule ⊗R built upon it, neither of which extends to the present Kähler framework. These results demonstrate that complex numbers are not fundamentally required by nature; rather, they encode a deeper real geometric structure that governs quantum interference and entanglement, settling this long debate.