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."