For nearly a century, quantum mechanics has been written with complex numbers at its core, even though those “imaginary” ingredients do not correspond to any directly countable or measurable quantity. Now that assumption just got punched in the face. In a new study published June 18 in Physical Review Letters, physicists including Pedro Barrios Hita show they can build a quantum model that uses only “real” numbers, while still producing all the same predictions as standard quantum mechanics.

The practical payoff for decision-makers is that the model reproduces the tricky multi-particle results that previously seemed to prove complex numbers were unavoidable. The 2021 “real-number quantum mechanics” prediction had argued that a real-number version would fail in experiments involving multiple particles, and the next year, other researchers ran those experiments and found results matching standard quantum mechanics, not the real-number approach. This new work breaks the stalemate by changing the rule used to combine quantum systems, not the experimental outcomes they match.

To understand why this matters, it helps to unpack what complex numbers actually do in quantum theory. Complex numbers combine a regular real number with an imaginary component, represented by the symbol i, like 3 + 4i. Physicists use them because quantum states are described by a wave function, and that description has long been built into the equations using complex values. A single factor of i multiplied into a particle’s state is undetectable on its own. But once you combine systems, that hidden phase can “show up” through math that effectively transfers the influence between particles. In standard treatments, the specific mechanism that makes this happen is tied to how quantum systems combine.

The earlier 2021 result turned on a textbook rule called the tensor product, which combines two separate quantum systems into one combined mathematical description. The tensor product is foundational, and it works smoothly in the standard complex-number framework. But when researchers attempted to build a real-number alternative while using that same tensor product logic, they couldn’t reproduce correlations seen in experiments involving three or more entangled particles. Those multi-particle correlation failures were the “proof by embarrassment” moment for the real-number idea.

Barrios Hita’s team approached the problem by asking a sharper question than “can we hack the equations,” and more like “what exactly makes complex numbers fundamental versus convenient.” Their solution is to stop assuming the tensor product is the only way to combine quantum systems. Instead, they built quantum mechanics around a different rule based on an idea that an action taken on one part of a system should not affect a separate part of it. Then they recreated the phase kickback effect, but using only real numbers.

In plain terms, phase kickback is what allows an otherwise undetectable “i” factor to become relevant when particles combine. In standard quantum mechanics, that behavior is baked into the tensor product. To replicate it with real numbers, Barrios Hita’s team attached a small “flag” to each particle to keep track of which real bookkeeping corresponds to what the imaginary component would have stored. They then treated certain flag combinations as physically identical, even if they look different in the raw mathematics. That extra step is what lets the real-number model match standard quantum predictions, including the multiparticle cases that had tripped earlier attempts.

Under the hood, the core move is almost deceptively simple: a complex number like 3 + 4i is equivalent to a pair of ordinary real numbers, 3 and 4, with the i serving as a label for which component is the “imaginary part.” The team essentially built a consistent tracking system so the two real components can stay separate until the right moment. Barrios Hita said it took a long time to make the bookkeeping work consistently across multiple combined particles. But once it did, he described the underlying structure as elegant.

This also matters because it reframes how boards and research leaders should think about foundational “rules.” The study does not change experimental predictions, does not point to new quantum technology, and currently is limited to systems with a finite number of quantum states. Extending the approach to infinite-dimensional systems, which show up in many real physics problems, is identified as a natural next step, and other researchers are already looking into it.

Still, the result “settles a decades-long debate,” according to Barrios Hita: complex numbers make quantum mechanics easier to write down, but they are not required for it to work. That is an unusual kind of win. It is not a new device, not a faster algorithm, not a regulatory-approved breakthrough. It is a conceptual closure that can clean up how researchers teach, reason, and audit the foundations of quantum theory. In an industry where grants, model choices, and theoretical assumptions can cascade into years of work, clearing up what is fundamental versus convenience is not trivia. It is strategic infrastructure.