Physicists prove quantum mechanics functions using only real numbers

Physicists have established that the mathematical framework of quantum mechanics can be fully formulated using only real numbers, settling a debate that has persisted since the 1920s. The findings, published in the journal Physical Review Letters on 18 June 2026, demonstrate that while complex numbers — which incorporate the imaginary unit i — have long served as a standard tool for modeling quantum systems, they are a choice of convenience rather than a fundamental requirement of nature.

Historical Context and the Rise of i

The reliance on complex numbers began early in the development of quantum theory. Erwin Schrödinger, whose wave equation forms the foundation of the field, expressed discomfort with the use of the complex wave function, once describing its use as carrying a certain crudeness. Despite these concerns, physicists adopted complex numbers because they provided a compact and elegant way to model wave-like behavior, phase rotations, and quantum state evolution. A complex number, such as 3 + 4i, combines a real and an imaginary component; in quantum mechanics, these components traditionally map to a state's amplitude and phase. Because the imaginary unit i does not correspond to a directly measurable quantity in the physical world, its central role in the equations of motion has remained a point of conceptual inquiry for decades.

Media additions

Challenging the 2021 Consensus

The necessity of these numbers appeared to be confirmed by research conducted in 2021, when a study proposed an experiment involving entangled particles to test whether a real-number version of quantum mechanics could reproduce standard experimental results. When researchers subsequently performed these tests, the data aligned with standard complex-number theory, leading many to conclude that complex numbers were essential. However, recent work by teams in Germany and France has overturned that conclusion. Researchers, including Pedro Barrios Hita, Anton Trushechkin, Hermann Kampermann, Michael Epping, and Professor Dr. Dagmar Bruß of Heinrich Heine University Düsseldorf and the German Aerospace Center, identified that the 2021 experiments relied on a restrictive assumption regarding the tensor product, the rule used to combine separate quantum systems into one. By identifying a physically motivated alternative for formalizing system composition, these researchers constructed a model that is experimentally indistinguishable from standard theory.

Mechanisms of the Real-Valued Model

The team achieved this by implementing a bookkeeping system that tracks the real and imaginary parts of a number separately, rather than combining them into a single complex value. By attaching flags to particles to monitor their states and employing specific grouping steps, the researchers successfully recreated phenomena like phase kickback, where a state change on one particle impacts another, using only real numbers.

Implications and Future Directions

This re-evaluation does not alter existing experimental predictions or current technologies, but it fundamentally shifts the theoretical understanding of quantum mechanics. By stripping the theory down to real-valued components, physicists hope to better reconcile quantum mechanics with gravity, a long-standing challenge in the field. Further investigation is required to address several areas:

• Infinite-Dimensional Systems: Current real-number models are restricted to finite quantum states; expanding this framework to infinite systems remains a primary goal.

While the mathematical reliance on complex numbers persists due to its ease of use and computational elegance, the emerging consensus is that the imaginary nature of quantum mechanics is a tool of language rather than a reflection of reality. As researchers continue to explore these foundations, the focus remains on whether this shift toward real-valued formulations will eventually provide new pathways for understanding the quantum world.