# Imaginary numbers could be needed to describe reality, new studies find

*The researchers used an upgraded version of the Bell test, an experiment that relies on quantum entanglement, to assess how important imaginary numbers were in describing reality. (Image credit: Jurik Peter via Shutterstock)*

If standard quantum theory holds up, imaginary numbers are critical. Imaginary numbers are necessary to accurately describe reality, two new studies have suggested.

When you take the square root of a negative number, you get imaginary numbers, which have long been used in the most essential equations of quantum mechanics, the branch of physics that describes the world of the very small. When you add imaginary numbers and real numbers, the two form complex numbers, which enables physicists to write out quantum equations in simple terms. However, whether quantum theory requires these mathematical chimeras or just uses them as useful shortcuts has long been controversial.

Even the founders of quantum mechanics were concerned about the implications of using complex numbers in their equations. In a letter to his friend Hendrik Lorentz, physicist Erwin Schrödinger — the first person to introduce complex numbers into quantum theory, with his quantum wave function (ψ) — wrote, “What is unpleasant here, and indeed directly to be objected to, is the use of complex numbers. ψ is unquestionably a real function.”

Schrödinger did discover ways to describe his equation using only real numbers, as well as an additional set of rules for how to use the equation, and subsequent physicists have done the same with other parts of quantum theory. But in the absence of hard experimental evidence to rule upon the predictions of these “all real” equations, a question has lingered: Are imaginary numbers an optional simplification, or does trying to work without them rob quantum theory of its ability to describe reality?

Two new studies, published in the journals Nature and Physical Review Letters, have shown Schrödinger wrong. They show, via a relatively simple experiment, that if quantum mechanics is right, imaginary numbers are a necessary part of our universe’s mathematics.

“The early founders of quantum mechanics could not find any way to interpret the complex numbers appearing in the theory,” lead author Marc-Olivier Renou, a theoretical physicist at the Institute of Photonic Sciences in Spain, told Live Science in an email. “Having them [complex numbers] worked very well, but there is no clear way to identify the complex numbers with an element of reality.”

The authors of the first study designed a variant on a classic quantum experiment known as the Bell test to assess if complex numbers were truly vital. The test was first proposed in 1964 by physicist John Bell as a technique to show that quantum entanglement — the strange connection between two far-apart particles that Albert Einstein objected as “spooky action at a distance” — was required by quantum theory.

The physicists created an experiment in which two independent sources (called S and R) would be put between three detectors (A, B, and C) in an elementary quantum network in an updated version of the classic Bell test. The source S would then emit two entangled light particles, or photons, one to A and the other to B. The source R would also emit two entangled photons, which would be sent to nodes B and C. The photons that arrived at detectors A and C would not need to be entangled if the cosmos were characterized by standard quantum mechanics based on complex numbers, but they would in a quantum theory based on real numbers.

To test this setup, the researchers of the second study performed an experiment in which they shone laser beams onto a crystal. The energy given to some of the crystals’ atoms by the laser was later released as entangled photons. By examining the states of photons arriving at their three detectors, the researchers discovered that the states of photons arriving at detectors A and C were not entangled, implying that their data could only be described by a quantum theory based on complex numbers.

The conclusion makes logical sense; photons must physically interact to become entangled, so photons arriving at detectors A and C should not be entangled if they are produced by a different physical source. However, the researchers highlighted that their experiment only rules out theories that do not use imaginary numbers if the current quantum mechanics conventions are correct. Most scientists are certain that this is the case, but there is one important caveat.

The result suggests that the possible ways we can describe the universe with math are actually much more constrained than we might have thought, Renou said.

“Just by observing what’s coming out of some experiments, we can rule out many potential descriptions without making any assumptions [on the] reliability of the physical devices used in the experiment,” Renou said. In the future, this could mean that it might just take a small number of experiments, building from first principles, for physicists to arrive at a complete quantum theory.

Aside from that, the researchers stated that their experimental setup, a rudimentary quantum network, could be beneficial for describing the piled up a future quantum internet.