Geometric Law of Quantum Mechanics and the Imaginary Unit

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).