For the first time, physicists have built a working version of quantum mechanics without complex numbers — numbers that have been considered essential to the theory for nearly a century.
Complex numbers combine a regular “real” number with an “imaginary” one — a multiple of the square root of -1, represented by the symbol i — into a single value, like 3 + 4i. The square root of -1 doesn’t correspond to any quantity you could count or measure directly (you can’t have negative one apple, for instance), which is why mathematicians call it imaginary.
Still, complex numbers have many useful applications. Engineers use them to describe alternating electrical current. Physicists use them to describe waves. And ever since quantum mechanics was first documented in the 1920s, complex numbers have been built directly into its equations. Quantum mechanics describes particles using something called a wave function, and that description relies on complex numbers.
In 2021, a team of physicists predicted that a version of quantum mechanics built with only real numbers would make incorrect predictions in certain experiments involving multiple particles. The following year, other researchers ran those experiments, and the results matched standard quantum mechanics, not the real-number version. Complex numbers seemed unavoidable.
But that 2021 result rested on one specific assumption: a particular mathematical rule for combining particles. That led physicists to ask a question: Are complex numbers actually necessary to describe reality at the quantum level, or are they just a convenience?
Now, in a new study published June 18 in the journal Physical Review Letters, researchers have found a way around the 2021 result.
“Complex numbers are not needed for quantum mechanics,” study first author Pedro Barrios Hita, a theoretical physicist and doctoral student at the German Aerospace Center and Heinrich Heine University Düsseldorf, told Live Science.
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A different rule
The 2021 result relied on a specific mathematical rule called the tensor product, which combines two separate quantum systems into one. If you have two particles and you want to combine them into a single mathematical description, you can use the tensor product. It’s a rule taught in every quantum mechanics textbook.
It works well for ordinary complex-number quantum mechanics, but past attempts to build a real-number version around that same rule ran into trouble. They couldn’t reproduce the correlations seen in experiments involving three or more entangled particles.
In their new study, Barrios Hita and his colleagues found that the tensor product isn’t the only option. They built quantum mechanics around a different rule based on an idea: An action taken on one part of a system shouldn’t have any effect on a separate part of it.
Entanglement is just one aspect of quantum mechanics that seems to defy reality. Now, the math behind such phenomena can be expressed with only “real” numbers for the first time.
(Image credit: koto_feja/Getty Images)
In ordinary quantum mechanics, multiplying a particle’s state by i is undetectable on its own. But when two particles combine, that i can shuffle over and effectively attach itself to the other particle instead. Physicists call this phase kickback, and it’s built automatically into the tensor product.
Barrios Hita’s team had to recreate that shuffling using only real numbers. They attached a small “flag” to each particle to keep track of what the imaginary part used to store. Then, they treated certain flag combinations as physically identical, even though they looked different on paper. That grouping step allowed their real-number version to match every prediction of standard quantum mechanics, including the multiparticle cases that had tripped up earlier attempts.
At its core, the trick is simple. A complex number, like 3 + 4i, is really just a pair of ordinary real numbers (3 and 4) — the i is only a label marking which one is the imaginary part. “A complex number is nothing but two real numbers,” Barrios Hita said. His team essentially built a bookkeeping system that tracks those two real numbers separately, instead of combining them into one complex number. It took a long time to figure out how to make that work consistently across multiple combined particles. But once they did, Barrios Hita said, the underlying structure turned out to be elegant.
The result puts quantum mechanics in the same boat as other physics theories that are often written using complex numbers purely for convenience, Barrios Hita said.
“There are many other theories, like, for example, electromagnetism,” Barrios Hita added, “which has complex numbers at its core. So, these theories are formulated using complex numbers, but [they] are not fundamental. They’re just helpful tools to help express equations.”
The work doesn’t change any experimental predictions or point to new quantum technology. It’s also currently limited to systems with a finite number of quantum states. Extending it to infinite-dimensional systems, which show up in many real physics problems, is a natural next step, and other researchers are already looking into it. Barrios Hita is moving on to different research, on how quantum properties like entanglement can be used as a resource.
Still, the study settles a decades-long debate. Complex numbers make quantum mechanics easier to write down, but they aren’t required to make it work.
Hita, P. B., Trushechkin, A., Kampermann, H., Epping, M., & Bruß, D. (2026). Quantum Mechanics based on real Numbers: A consistent description. Physical Review Letters, 136(24). https://doi.org/10.1103/4k13-sdjh
