Explanatory diagram for the research question – is quantum mechanics possible with only real numbers? – and results of the study.
Image Credit: © HHU / Pedro Barrios Hita
Scientific Frontline: Extended "At a Glance" Summary: Real-Number Quantum Mechanics
The Core Concept: Quantum mechanics, the physical theory describing the behavior of atomic and subatomic particles, can be successfully formulated using solely real numbers. This mathematically rigorous alternative challenges the traditional reliance on complex numbers, which incorporate both real and imaginary components, to describe quantum states.
Key Distinction/Mechanism: Standard quantum mechanics uses complex numbers where a state's amplitude is represented by the real part and its phase by the imaginary part. By utilizing a physically motivated, less restrictive postulate for system composition, researchers have developed an alternative framework that strictly uses real numbers while remaining experimentally indistinguishable from standard quantum mechanics.
Origin/History: The development of quantum mechanics began in the 1900s through the foundational work of physicists such as Max Planck, Niels Bohr, Werner Heisenberg, and Erwin Schrödinger. The modern debate over the mathematical necessity of imaginary numbers was highlighted by a 2021 study declaring them essential, which was subsequently overturned in 2026 by physicists from Heinrich Heine University Düsseldorf and the German Aerospace Center.
Major Frameworks/Components:
- Complex number integration (the standard use of real and imaginary coordinates for calculation).
- Quantum state representations (the mathematical modeling of amplitude and phase).
- Core quantum phenomena (wave-particle duality, the quantum tunneling effect, entanglement, and coherence).
- System composition postulates (the fundamental theoretical rules dictating how interacting quantum systems are formalized).
Branch of Science: Quantum Mechanics, Theoretical Physics, and Quantum Information Science.
Future Application: Refining the theoretical foundations and mathematical models used to develop quantum computers, advanced quantum communication networks, and cryptography.
Why It Matters: Proving that both frameworks yield identical predictions for any conceivable experiment demonstrates that imaginary numbers are a practical calculation tool rather than a fundamental physical necessity, opening new pathways for understanding and formalizing quantum theory.
Physicists from Heinrich Heine University Düsseldorf (HHU), in collaboration with the German Aerospace Center (DLR), have examined a fundamental property of quantum mechanics. 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 physical theory of quantum mechanics describes the world of atomic and subatomic particles. Its development began in the 1900s with physicists such as Max Planck, Niels Bohr, Werner Heisenberg, and Erwin Schrödinger. Quantum mechanics can effectively describe phenomena at microscopic scales, including, for example, the diffraction of particles at a double slit—which shows that particles also exhibit wavelike behavior—and the quantum tunneling effect, in which a certain probability exists that particles can penetrate a barrier even if they lack sufficient energy to do so. Particularly important phenomena today include entanglement and coherence, which are key for applications such as quantum computers and communication.
Complex numbers are an important tool in quantum mechanics. A number is represented by two coordinates—a real and an imaginary part; a quantum state has an amplitude represented by the real part and a phase represented by the imaginary part. Without this construct, many processes could not previously be described using quantum mechanics. However, it remains disputed whether complex numbers are fundamentally necessary in quantum mechanics or whether they are simply a practical calculation tool. This raises the question: Is quantum mechanics also possible using only real numbers?
In a study published in 2021, researchers concluded that complex numbers are essential for quantum mechanics under the standard postulates. This finding was also corroborated experimentally.
Now, a team of physicists from HHU and the DLR, led by Professor Dagmar Bruß and her doctoral researcher Pedro Barrios Hita, has examined the postulates used in the earlier study. In a paper published in Physical Review Letters, they show that one of these postulates is too restrictive. Instead, the authors identified a physically motivated alternative for formalizing system composition, which gives rise to a class of theories that can be formulated entirely with real numbers and are experimentally indistinguishable from standard quantum mechanics.
As Professor Bruß stated, “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.”
Research material: Quantum theory based on real numbers can be experimentally falsified (Nature)
Published in journal: Physical Review Letters
Title: Quantum Mechanics Based on Real Numbers: A Consistent Description
Authors: Pedro Barrios Hita, Anton Trushechkin, Hermann Kampermann, Michael Epping, and Dagmar Bruß
Source/Credit: Heinrich-Heine-Universität Düsseldorf | Arne Claussen
Edited by: Scientific Frontline
Reference Number: qs062226_01
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