A Question of Reality
Quantum mechanics, the theory describing the universe at the smallest scales, is famously weird. But perhaps its strangest feature is the maths it relies on. Since its inception in the 1920s, the theory's core equations have used 'complex numbers' — numbers which
include the 'imaginary' square root of -1. For decades, physicists have debated a fundamental question: are these imaginary numbers a true feature of reality, or just a convenient mathematical shortcut? Until recently, it was a philosophical argument. But as of June 2026, new research published in the journal Physical Review Letters has provided a startling answer, developing a working model of quantum mechanics without using complex numbers at all.
The Imaginary Rulebook
When physicists like Erwin Schrödinger first wrote down the equations for quantum mechanics, they found that particles like electrons couldn't be described as just tiny balls. Instead, they were better represented as 'wave functions,' which describe the probability of finding a particle in a certain place. To make the maths work, these wave functions needed complex numbers. A complex number has two parts: a 'real' part, like the numbers we use every day, and an 'imaginary' part. The real part could represent the wave's size (amplitude), while the imaginary part represents its position in its cycle (phase). This framework has been incredibly successful, forming the basis for everything from lasers to computer chips. Yet, the reliance on an 'imaginary' concept to describe the real world has always been a point of discomfort for some scientists.
Putting Math to the Test
The debate took a fascinating turn a few years ago. In 2021, a study proposed an experiment that could, in theory, prove whether complex numbers were truly necessary. When the experiment was carried out, the results seemed to confirm that standard, complex-number-based quantum mechanics made the correct predictions, while a simplified real-number version would fail. It appeared the case was closed: the imaginary numbers were here to stay. However, that conclusion rested on a key assumption about how to mathematically combine two separate quantum systems, using a standard textbook rule called the tensor product. This is where the new breakthrough comes in.
A New Way to Combine Worlds
The researchers behind the new study, led by physicists from Heinrich Heine University Düsseldorf and the German Aerospace Center, took a closer look at that assumption. They argued that the standard tensor product rule was too restrictive. They developed an alternative way to combine quantum systems, based on the simple physical idea that an action on one part of a system shouldn't affect a separate part. Using this new approach, they successfully built a version of quantum mechanics using only real numbers that makes all the same predictions as the standard theory. As study author Pedro Barrios Hita explained, this means complex numbers are not fundamentally needed; they are a convenience, not a necessity.
A Simpler, Stranger Reality
So, what does this mean for our understanding of the universe? For one, it doesn't mean quantum mechanics is wrong. On the contrary, it shows that the theory is even more robust than we thought, capable of being built on different mathematical foundations while giving the same results. It resolves a century-old question that bothered even the theory's founders. While the new 'real' formulation is more complicated to write down, its existence is a major philosophical and theoretical breakthrough. It forces physicists to re-evaluate which parts of their mathematical toolkit are essential descriptions of nature and which are just helpful habits. This deeper understanding could subtly influence future research, especially in the burgeoning fields of quantum computing and the search for a unified theory of physics.
