Imaginary numbers can be essential in describing reality

The mathematicians were disturbed, centuries ago, to find that calculating the properties of certain curves required what seemed impossible: numbers which, multiplied by themselves, become negative.

All the numbers on the number line, when squared, give a positive number; 22 = 4 and (-2)2 = 4. Mathematicians began to call these familiar numbers “real” and the seemingly impossible breed of “imaginary” numbers.

Imaginary numbers, labeled with units of I (where, for example, (2I)2 = -4), have gradually become essential in the abstract field of mathematics. For physicists, however, real numbers were enough to quantify reality. Sometimes what are called complex numbers, with both real and imaginary parts, such as 2 + 3I, have simplified the calculations, but in an apparently optional manner. No instrument has ever returned a reading with a I.

Yet physicists may have just shown for the first time that imaginary numbers are, in a sense, real.

A group of quantum theorists have devised an experiment whose outcome depends on whether nature has an imaginary side. Provided quantum mechanics are correct – a hypothesis few could complain about – the team’s argument essentially ensures that complex numbers are an inevitable part of our description of the physical universe.

“These complex numbers are usually just a practical tool, but here it turns out that they really have a physical meaning,” said Tamás Vértesi, a physicist at the Institute for Nuclear Research of the Hungarian Academy of Sciences. who, years ago, argued otherwise. “The world is such that it really needs these complex numbers,” he said.

In quantum mechanics, the behavior of a particle or group of particles is encapsulated by a wave entity known as a wave function, or ψ. The wave function predicts possible results of measurements, such as the position or possible momentum of an electron. The so-called Schrödinger equation describes how the wave function changes over time – and this equation has a I.

Physicists have never been quite sure what to think about it. When Erwin Schrödinger derived the equation which now bears his name, he hoped to rub the I outside. “What is unpleasant here, and in fact directly questionable, is the use of complex numbers,” he wrote to Hendrik Lorentz in 1926. “ψ is surely a fundamentally real function. “

Schrödinger’s desire was certainly mathematically plausible: any property of complex numbers can be captured by combinations of real numbers plus new rules to keep them in line, opening up the mathematical possibility of an entirely real version of quantum mechanics.

Indeed, the translation turned out to be simple enough that Schrödinger almost immediately discovered what he believed to be the “true equation of the wave”, the one that avoided I. “Another heavy stone has been rolled away from my heart,” he wrote to Max Planck less than a week after his letter to Lorentz. “Everything came out exactly as we would have liked.”

But using real numbers to simulate complex quantum mechanics is a clunky and abstract exercise, and Schrödinger admitted that his fully real equation was too heavy for everyday use. Within a year, he was describing wave functions as complex, just as physicists think today.

“Anyone who wants to work uses the complex description,” said Matthew McKague, quantum computer scientist at Queensland University of Technology in Australia.

Yet the true formulation of quantum mechanics has remained proof that the complex version is simply optional. Teams such as Vértesi and McKague, for example, have shown 2008 and 2009 that – without a I in sight – they could perfectly predict the outcome of a famous quantum physics experiment known as the Bell test.

The new research, which was posted on the Arxiv.org scientific preprint server in January, finds that these early Bell test proposals simply did not go far enough to shatter the real-number version of quantum physics. It offers a more complex Bell experiment which seems to require complex numbers.

Previous research has led people to conclude that “in quantum theory, complex numbers are only practical, but not necessary,” wrote the authors, including Marc-Olivier Renou from the Institute of Photonic Sciences in Spain and Nicolas Gisin from the University of Geneva. “Here we prove that this conclusion is wrong.”