Three years ago one of us, Toni, asked another of us, Marco, to come to his office at the Institute of Photonic Sciences, a large research center in Castelldefels near Barcelona. “There is a problem that I wanted to discuss with you,” Toni began. “It is a problem that Miguel and I have been trying to solve for years.” Marco made a curious face, so Toni posed the question: “Can standard quantum theory work without imaginary numbers?”

Imaginary numbers, when multiplied by themselves, produce a negative number. They were first named “imaginary” by philosopher René Descartes, to distinguish them from the numbers he knew and accepted (now called the real numbers), which did not have this property. Later, complex numbers, which are the sum of a real and an imaginary number, gained wide acceptance by mathematicians because of their usefulness for solving complicated mathematical problems. They aren't part of the equations of any fundamental theory of physics, however—except for quantum mechanics.

The most common version of quantum theory relies on complex numbers. When we restrict the numbers appearing in the theory to the real numbers, we arrive at a new physical theory: real quantum theory. In the first decade of the 21st century, several teams showed that this “real” version of quantum theory could be used to correctly model the outcomes of a large class of quantum experiments. These findings led many scientists to believe that real quantum theory could model any quantum experiment. Choosing to work with complex instead of real numbers didn't represent a physical stance, scientists thought; it was just a matter of mathematical convenience.

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