June 19, 2026 -- The physical theory of quantum mechanics describes the world of atomic and sub-atomic 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 e.g. the diffraction of particles at a double slit – which shows that particles also exhibit wave-like behaviour – and the quantum tunnelling effect in which a certain probability exists that particles can also penetrate a barrier even if they have insufficient energy to do so. Particularly important phenomena today include entanglement and coherence, which are key for applications such as quantum computers and communication.

So-called 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 these numbers are simply a practical calculation tool. This consequently poses the question: Is quantum mechanics also possible with only real numbers?

In a study published in 2021, the authors concluded that complex numbers are essential for quantum mechanics under the standard postulates (Renou et al., Nature 600, 625 (2021).This was also corroborated experimentally.

Now, a team of physicists from HHU and the DLR led by Professor Dr Dagmar Bruß and her doctoral researcher Pedro Barrios Hita have examined the postulates used in the earlier study. In a paper now published in Physical Review Letters, they show that one of these postulates is too restrictive. Instead, the authors identified a physically motivated alternative for formalising 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.

Professor Bruß: “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.”