Can quantum mechanics do without imaginary numbers? For decades, physicists assumed the answer was no, that the complex numbers at the heart of the theory were not merely convenient but fundamentally necessary. A new analysis published in Physical Review Letters suggests otherwise.
Pedro Barrios Hita, a doctoral researcher at Heinrich Heine University (HHU) Dusseldorf working with Professor Dagmar Bruss and colleagues at the German Aerospace Center (DLR), has shown that quantum mechanics can be formulated entirely with real numbers while reproducing every experimentally testable prediction of the standard theory.
The paper, published June 18 as an Editors’ Suggestion in PRL 136, 240202 (DOI: 10.1103/4k13-sdjh), does not rewrite quantum mechanics from scratch. It identifies and relaxes one specific mathematical assumption about how quantum systems are combined, and in doing so, opens a door to a class of theories that need no imaginary numbers at all.
A Decade-Long Debate Over the Letter i
The question of whether complex numbers are essential to quantum theory dates back to the 1960s, when Swiss physicist Ernst Stueckelberg first sketched a real-Hilbert-space formulation. But the debate gained urgency in 2021, when a team led by researchers at the Austrian Academy of Sciences published a paper in Nature claiming that any real-number formulation preserving the standard tensor product, the mathematical rule used to describe composite systems of multiple particles, would produce experimentally different predictions than standard quantum mechanics. Experiments in 2022 confirmed those predictions, seemingly ruling out any real-number alternative.
“The 2021 result appeared to settle the matter,” Barrios Hita said in a press release. “But we noticed that one of the postulates they used, the way composite systems are formed, could be relaxed without violating any physical principle.”
Relaxing the Tensor Product
The tensor product is a purely mathematical construction: it specifies how the Hilbert space of a combined system is built from the Hilbert spaces of its parts. Barrios Hita and colleagues replaced it with a physically motivated locality principle: a local operation on one subsystem should not affect another independently prepared subsystem.
Under this relaxed rule, each quantum system carries a small auxiliary element, a “flag”, that tracks the information the imaginary unit would otherwise encode. When particles interact, flag configurations that differ on paper but produce identical physical outcomes are treated as equivalent. This step recreates the phase relationships that the standard tensor product handles automatically through complex arithmetic.
The result is a fully consistent real-number quantum theory that reproduces every known multipartite quantum experiment. “Both frameworks yield identical predictions for any conceivable experiment,” Bruss said.
What the Formulation Does and Does Not Say
It is important to be precise about what this result means, and what it does not mean.
The formulation does not eliminate complex numbers from the working mathematics of quantum mechanics. Physicists will continue to use complex numbers in calculations because they are far more convenient. What it shows is that these numbers are a matter of convenience, not necessity. Complex numbers encode phase and amplitude simultaneously in a single object, but the same physical content can, in principle, be expressed in a real-number framework.
The formulation also does not produce any new, testable predictions that differ from standard quantum mechanics. This is by design: the whole point is to show that real-number quantum mechanics cannot be experimentally distinguished from the complex version. “Real-valued quantum mechanics cannot be falsified,” the authors write, meaning that complex numbers are an optional mathematical framework, not a compulsory feature of nature.
Prior Work and the Question of Novelty
Headlines calling this the “first ever” real-number quantum model must be read with caution. Real-Hilbert-space formulations have existed for decades. What is genuinely new about the Barrios Hita result is that it is the first such formulation to pass all multipartite experimental consistency checks, directly countering the specific falsification claim made by the 2021 Nature paper. Where earlier attempts either could not reproduce certain entangled-state predictions or relied on ad hoc mathematical constructions, this one provides a physically motivated justification for modifying the composition rule.
Limitations
The current formulation works only for finite-dimensional quantum systems, those with a finite number of quantum states. Extending to infinite-dimensional systems (common in continuous-variable settings such as quantum optics) remains an open problem, and one that other groups are already investigating. The construction also relies on additional assumptions about single-system representation and the preservation of independent preparation; a full derivation from purely physical first principles is not yet at hand. Whether the locality principle can be consistently applied to indistinguishable particles, where composition is handled via second quantization rather than the tensor product, also remains unclear.
Barrios Hita told reporters he is moving on to explore how quantum properties such as entanglement can be used as a resource, leaving extension of the real-number framework to others.
A Shift in Perspective, Not a Revolution
The paper is best read not as a challenge to quantum mechanics but as a clarification of its logical structure. Complex numbers remain the most efficient tool for quantum calculations, no one is suggesting they be abandoned. What the result clarifies is that they are a convenience, not a metaphysical necessity. Quantum theory, it turns out, can rest on real numbers alone, provided the right physical principle governs how its parts fit together.
Source: Barrios Hita, P., Trushechkin, A., Kampermann, H., Epping, M., & Bruss, D. “Quantum Mechanics Based on Real Numbers: A Consistent Description.” Physical Review Letters 136, 240202 (2026). DOI: 10.1103/4k13-sdjh