A century ago, the strange behavior of atoms and elementary particles led physicists to formulate a new theory of nature. That theory, quantum mechanics, found immediate success, proving its worth with accurate calculations of hydrogen’s emission and absorption of light.
There was, however, a snag. The central equation of quantum mechanics featured the imaginary number *i*, the square root of −1. Physicists knew *i* was a mathematical fiction. Real physical quantities like mass and momentum never yield a negative amount when squared. Yet this unreal number that behaves as *i² = −1* seemed to sit at the heart of the quantum world.
After deriving the *i*-riddled equation—essentially the law of motion for quantum entities—Erwin Schrödinger expressed the hope that it would be replaced by an entirely real version. “There is undoubtedly a certain crudeness at the moment” in the equation’s form, he wrote in 1926. Schrödinger’s distaste notwithstanding, *i* stuck around, and new generations of physicists took up his equation without much concern.
Then, in 2021, the role of imaginary numbers in quantum theory attracted newfound interest. A team of researchers proposed a way to empirically determine whether *i* is essential to quantum theory or merely a mathematical convenience. Two teams quickly followed up to perform the intricate experiments and found supposedly unequivocal evidence that quantum theory needs *i*.
This year, however, a series of papers has overturned that conclusion. In March, a group of theorists based in Germany rebutted the 2021 studies, putting forward a real-valued version of quantum theory that’s exactly equivalent to the standard version. Two theorists in France followed up with their own formulation of a real-valued quantum theory. And in September, another researcher approached the question from the perspective of quantum computing and arrived at the same answer: *i* isn’t necessary for describing quantum reality after all.
Although the real-valued theories avoid explicit use of *i*, they do retain hallmarks of its distinct arithmetic. This leads some to wonder whether the imaginary aspect of quantum mechanics—or even reality itself—is truly vanquished. “The mathematical formulation does guide what we infer about the nature of the physical world,” said Jill North, a philosopher of physics at Rutgers University.
### Impossible Values
Living in Amsterdam in 1637 at the peak of tulip mania—the Dutch frenzy for flowers which led to impossibly valued tulip bulbs—René Descartes grappled with equations whose solutions also seemed to have impossible values. Using *x³ − 6x² + 13x − 10 = 0* as an example, Descartes wrote that its solutions “are not always real; but sometimes only imaginary. There is sometimes no quantity that corresponds to what one imagines.”
The three numbers you can plug in for *x* are 2, 2 − *i*, and 2 + *i*. The latter two numbers, each of which has both a real part and an imaginary part in the form *a + ib*, came to be called complex numbers. Descartes viewed them with derision, but complex numbers were later adopted for their utility in fields as diverse as geometry, optics, and signal analysis.
Schrödinger grudgingly acknowledged their ease of use in quantum theory. His equation governs the evolution of the wave function, an entity representing the possible quantum states of an object. These states can interfere destructively and constructively like waves. Schrödinger’s wave function was complex-valued, even though actual measurements of quantum systems always return real values.
“Quantum theory really is the first physical theory where the complex numbers seem to be right smack in the middle of the theory,” said Bill Wootters, a quantum information theorist at Williams College.
One way to represent a complex number like *a + ib* is as a point on a plane, where *a* is the position on the x-axis (which can be thought of as the real number line) and *b* is the position on an imaginary y-axis. Each complex number is an arrow, called a vector, pointing from the origin to the complex coordinate (*a, b*).
These complex vectors obey the unusual math of complex numbers: Multiplying by *i*, for example, rotates the vector 90 degrees. These properties made them a natural fit for the quantum states of the wave function—also vectors obeying odd combination rules.
https://www.quantamagazine.org/physicists-take-the-imaginary-numbers-out-of-quantum-mechanics-20251107/
