Physicists say quantum mechanics may not need imaginary numbers after all

- Date:
- July 13, 2026
- Source:
- Heinrich-Heine University Duesseldorf
- Summary:
- Physicists from Heinrich Heine University Düsseldorf (HHU) have examined a fundamental property of quantum mechanics in collaboration with the German Aerospace Center (DLR). In the scientific journal Physical Review Letters, they show that this theory does not necessarily need to be formulated with imaginary numbers – real numbers can in fact also be used. The American Physical Society has also dedicated a “Highlight” to these findings in its Physics Magazine.
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Quantum mechanics is the branch of physics that explains how matter and energy behave at the atomic and sub atomic scale. Developed in the early 1900s by pioneers including Max Planck, Niels Bohr, Werner Heisenberg, and Erwin Schrödinger, it has become one of the most successful scientific theories ever created.
The theory accurately describes a wide range of microscopic phenomena. These include the famous double slit experiment, in which particles also display wave like behavior, and quantum tunneling, where particles have a probability of passing through a barrier even when they do not have enough energy to overcome it in the classical sense. Other key quantum effects, such as entanglement and coherence, now form the foundation of emerging technologies including quantum computing and quantum communication.
Are Complex Numbers Really Essential?
For decades, quantum mechanics has relied on complex numbers, which combine a real component with an imaginary component. In the mathematical description of a quantum state, the real part represents the amplitude, while the imaginary part represents the phase. This framework has long been considered essential for describing many quantum processes.
Even so, physicists have continued to debate whether complex numbers are truly a fundamental part of nature or simply a convenient mathematical tool. That question naturally leads to another: Could quantum mechanics be formulated using only real numbers?
Revisiting a Key Quantum Assumption
A 2021 study concluded that complex numbers are indispensable under the standard postulates of quantum mechanics (Renou et al., Nature 600, 625 (2021)). Experimental results also supported that conclusion.
Researchers from Heinrich Heine University Düsseldorf (HHU) and the German Aerospace Center (DLR), led by Professor Dr Dagmar Bruß and doctoral researcher Pedro Barrios Hita, decided to take another look at the assumptions behind that earlier work.
In a new study published in Physical Review Letters, they found that one of the postulates used in the 2021 analysis was more restrictive than necessary. By replacing it with a different, physically motivated approach for describing how quantum systems combine, they identified a family of theories that can be expressed entirely with real numbers while remaining experimentally indistinguishable from conventional quantum mechanics.
Professor Bruß said: "This means that both frameworks yield identical predictions for any conceivable experiment. Within this framework, imaginary numbers are thus not fundamentally necessary in quantum mechanics and can in principle be replaced by alternative formulations using real numbers."
Story Source:
Materials provided by Heinrich-Heine University Duesseldorf. Note: Content may be edited for style and length.
Journal Reference:
- Pedro Barrios Hita, Anton Trushechkin, Hermann Kampermann, Michael Epping, Dagmar Bruß. Quantum Mechanics Based on Real Numbers: A Consistent Description. Physical Review Letters, 2026; 136 (24) DOI: 10.1103/4k13-sdjh
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Facts Only

* Physicists from Heinrich Heine University Düsseldorf and the German Aerospace Center collaborated on an examination of quantum mechanics properties.
* The research was published in Physical Review Letters.
* The study found that quantum mechanics does not necessarily need to be formulated with imaginary numbers; real numbers can be used.
* Complex numbers were previously used, where the real part represented amplitude and the imaginary part represented phase.
* A previous 2021 study concluded that complex numbers are indispensable under standard postulates of quantum mechanics.
* The new research identified a family of theories expressible entirely with real numbers that remain experimentally indistinguishable from conventional quantum mechanics.

Executive Summary

Physicists from the Heinrich Heine University Düsseldorf and the German Aerospace Center examined a fundamental property of quantum mechanics concerning the necessity of imaginary numbers. The study, published in Physical Review Letters, demonstrated that the theory of quantum mechanics does not necessarily require the use of imaginary numbers; real numbers can be used instead. This finding stems from re-evaluating a postulate from a previous 2021 analysis. Researchers found that by adopting a different, physically motivated approach for describing how quantum systems combine, they could formulate theories entirely in terms of real numbers that yield results indistinguishable from conventional quantum mechanics experimentally.

Full Take

The core tension explored here is the distinction between mathematical convenience and fundamental physical necessity in theoretical frameworks. The finding suggests that the choice of mathematical structure for describing quantum reality might be a matter of convention or a specific mathematical mapping rather than an ontological requirement. The fact that alternative formulations using only real numbers produce identical experimental predictions challenges the long-held assumption that complex numbers are intrinsically woven into the fabric of nature at the quantum level. This opens up a critical avenue for questioning the foundational axioms upon which modern physics is built. The implication moves beyond mere mathematical preference to question what 'reality' fundamentally means when describing microscopic behavior, suggesting that established theoretical structures might be highly contingent on initial assumptions rather than being necessary truths. What physical intuition or empirical constraint dictates the necessity of phase representation over amplitude representation? What are the long-term consequences if a mathematically simpler description replaces the complex formalism currently used in quantum information theory?

Sentinel — Human

Confidence

The text appears to be a sound journalistic report synthesizing a specific, peer-reviewed physics finding, exhibiting the structure and citation habits of expert reporting rather than pure generative output.

Signals Detected

Natural variation in sentence length and intellectual flow.

Logical progression from established theory to specific research findings without overt, forced balancing of opposing viewpoints.

Proper citation structure and integration of primary source material (journal reference, named researchers).

Claims are directly tied to specific academic findings and formal citations, indicating grounding in research.

Human Indicators

The inclusion of a highly specific, cited academic paper (Physical Review Letters reference) as the core finding strongly suggests human editorial oversight or direct reporting of scientific results.

The narrative flow successfully transitions from a general overview of quantum mechanics to a nuanced theoretical debate.