The Formula Remains the Same, but the Mathematics Changes: Physicists Show Quantum World Can Be Described Without Imaginary Numbers

Quantum mechanics has relied for more than a century on numbers that cannot be placed on the ordinary number line. Now physicists have shown that it may be possible to dispense with these so-called imaginary numbers without altering any of the theory’s predictions.

Specialists at Heinrich Heine University Düsseldorf and the German Aerospace Center have re-examined one of the central mathematical pillars of quantum mechanics. Quantum mechanics describes the behavior of matter and energy at the scale of atoms and subatomic particles. The theory explains why particles can exhibit wave-like properties, tunnel through barriers, and maintain non-local correlations with one another—phenomena that underpin emerging technologies such as quantum computing and quantum communication.

Conventionally, quantum states are represented using complex numbers, each composed of a real part and an imaginary part. The real component encodes the magnitude of the state, while the imaginary component captures its phase. For decades, physicists regarded this mathematical structure as indispensable.

In 2021, another research group concluded that imaginary numbers cannot be eliminated from quantum mechanics, a claim supported by experimental results at the time. However, the Düsseldorf team led by Professor Dagmar Bruß and doctoral student Pedro Barrios Hita decided to scrutinize the assumptions underlying that earlier work.

The researchers discovered that one of the conditions imposed in the 2021 study was stricter than required by physical reality. By replacing it with a different mathematical requirement that still correctly describes composite quantum systems, they obtained a broad family of theories formulated exclusively with real numbers.

When expressed in this revised form, the new framework yields exactly the same observable results as standard quantum mechanics for every possible experiment. Consequently, the two mathematical models cannot be distinguished from each other through measurement.

The authors emphasize that they are not proposing to rewrite conventional quantum mechanics. Complex numbers remain a highly convenient and efficient calculational tool. Their study simply shows that the imaginary component does not necessarily reflect an intrinsic feature of nature; it is merely one of several mathematically valid ways to describe the quantum world.