Complex numbers, quantum mechanics and reality

Kwon, Se Kyun

Mathematically, the imaginary unit 𝑖 has long been regarded as an algebraic generator extending the real numbers to the complex field. Yet a century after the birth of quantum mechanics (1925–2025), the answer to its oldest question is now clear: quantum mechanics is not merely a mathematical structure—it is a physical reality, and all quantities appearing in its non-commutative relations are physical. Therefore, 𝑖 is not a mere algebraic extension but the universal operator demanded by the geometric completeness of non-commutative reality between position and momentum. The imaginary unit is a physical reality. The canonical relation [𝑥, 𝑝] = 𝑖ℏ is not merely algebraic—it is a geometric law, showing that the complex structure of quantum mechanics arises inevitably from physical reality itself. It stands without contradiction, without complexity. This is nature. In the (𝑥, 𝑝) plane, 𝑖 represents a 90° rotation and ℏ quantizes the curvature of that rotation. Complex numbers thus arise as the natural geometric language of non-commutative physics, providing a geometric resolution to the century-old question of physical completeness first raised by Einstein, Podolsky and Rosen (1935).

changzheng zhou, ZIQING zhou

The indispensability of the complex structure in quantum mechanics has beenexperimentally confirmed, yet its ontological status—whether complex numbers aremerely a mathematical tool or a constitutive element of physical reality—has longremained undecided. Starting from mathematical necessity, this paper argues thatthe complex structure is not only the descriptive language of quantum mechanicsbut also an ontological constraint condition of physical reality. Based on Frobenius’ theorem on the classification of division algebras over the real numbers, wereveal: if the descriptive framework is required to be both commutative and algebraically closed, then the complex structure is the unique logically necessary choice.This mathematical constraint, when mapped onto physical reality, implies that the”imaginary part” corresponds to a modal dimension that is not directly observablebut constrains observable phenomena. The measurement process is thus reconceptualized as a projection from the complex structure onto a real subspace, whereinformation loss is an inherent feature of the mathematical structure rather thanan additional dynamical assumption. We demonstrate that this framework is compatible with research on imaginarity as a resource in quantum resource theory andwith analyses of Fisher information limits in quantum metrology, and we proposethree testable predictions with quantum metrological precision. This interpretationadvances the quantum measurement problem from a matter of interpretational debate to a level of structural analysis, providing a rigorous mathematical foundationfor understanding the quantum-classical transition.

Bryce Seligman DeWitt

Ijaz Durrani

Eizo Nakaza

Quantum mechanics came to employ complex probability amplitudes and the abstract state space known as Hilbert space because the behavior of electrons and light could not be described by the intuition of classical three-dimensional space alone. Complex numbers retain the magnitude and phase of an amplitude within a single quantity, while Hilbert space unifies linear superposition, inner products, observables, and time evolution in one mathematical framework. This made it possible to describe point-like particle detections and wave-like interference within the same theory. Yet this mathematical success also raised an ontological question: should the state structure in Hilbert space be read as physical reality itself? That question developed into long-standing debates over Schrödinger’s cat, the measurement problem, quantum entanglement, and nonlocality. This paper traces that history for a general readership and applies to quantum theory the distinction, emphasized by NNRT in relativity, between “successful mathematics” and “the physical reality of the mathematical object.” It then argues that physics remains incomplete at that stage: the next central task is to identify the physical mechanism—what acts on what, and how—that gives rise to the observed phenomena. Central thesis: Three questions must be distinguished: whether mathematics correctly describes a phenomenon, whether the mathematical structure itself is physical reality, and whether the physical mechanism producing the phenomenon is actually understood.