Physicists Show Quantum Mechanics Can Work Without Imaginary Numbers, Overturning a 2021 Verdict

For nearly a century, imaginary numbers have seemed woven into the very fabric of quantum mechanics. Now physicists in Germany say the theory does not actually need them — a result that reopens one of the deepest questions about what the mathematics of the quantum world is really telling us.

Imaginary numbers are built from the square root of minus one, a quantity that has no place on the ordinary number line but that mathematicians and physicists use constantly. In quantum mechanics, the equations that describe how particles behave are traditionally written with these so-called complex numbers, and many physicists have long assumed they are an unavoidable feature of reality at the smallest scales.

Researchers at Heinrich Heine University Düsseldorf (HHU), working with the German Aerospace Center (DLR), have now shown that the theory can be reformulated using only real numbers without changing any of its testable predictions. Reporting in the journal Physical Review Letters, the team demonstrated that a version of quantum mechanics built entirely from real numbers yields the same results as the standard complex-number version for any experiment one could actually perform.

The finding directly challenges a prominent 2021 study that had concluded imaginary numbers were essential — that no real-number formulation could reproduce all of quantum theory's predictions. The Düsseldorf group, led by Professor Dagmar Bruß and doctoral researcher Pedro Barrios Hita, argued that the earlier claim rested on particular assumptions, and that once those are relaxed, the two mathematical frameworks become physically indistinguishable. "Both descriptions lead to identical predictions," the researchers said, meaning imaginary numbers can, in principle, be replaced.

The result does not mean physicists will abandon complex numbers, which remain by far the most compact and convenient way to write quantum equations. Instead, it sharpens a philosophical point: the imaginary unit may be an elegant bookkeeping tool rather than an irreducible ingredient of nature. The work drew enough attention that the American Physical Society featured it as a "Highlight" in its Physics magazine. For students who have long puzzled over why the universe seems to run on numbers that don't exist, the message is oddly reassuring — the strangeness may lie in the math we choose, not in reality itself.

The debate is more than an academic curiosity. Questions about which mathematical structures are truly fundamental touch on how physicists interpret quantum theory and how they might one day extend or replace it. If imaginary numbers are merely a convenient language rather than a hard constraint of nature, that could subtly change how researchers think about the foundations of quantum information, entanglement and the boundary between the quantum and classical worlds. The Düsseldorf team said it plans to explore whether real-number formulations offer any practical advantages, or whether, as most physicists still expect, the complex-number version will remain the tool of choice simply because it is so much easier to work with.