Many science students may imagine a ball rolling down a hill or a car skidding because of friction as prototypical examples of the systems physicists care about. But much of modern physics consists of searching for objects and phenomena that are virtually invisible: the tiny electrons of quantum physics and the particles hidden within strange metals of materials science along with their highly energetic counterparts that only exist briefly within giant particle colliders.
In their quest to grasp these hidden building blocks of reality scientists have looked to mathematical theories and formalism. Ideally, an unexpected experimental observation leads a physicist to a new mathematical theory, and then mathematical work on said theory leads them to new experiments and new observations. Some part of this process inevitably happens in the physicist’s mind, where symbols and numbers help make invisible theoretical ideas visible in the tangible, measurable physical world.
Sometimes, however, as in the case of imaginary numbers – that is, numbers with negative square values – mathematics manages to stay ahead of experiments for a long time. Though imaginary numbers have been integral to quantum theory since its very beginnings in the 1920s, scientists have only recently been able to find their physical signatures in experiments and empirically prove their necessity.
In December of 2021 and January of 2022, two teams of physicists, one an international collaboration including researchers from the Institute for Quantum Optics and Quantum Information in Vienna and the Southern University of Science and Technology in China, and the other led by scientists at the University of Science and Technology of China (USTC), showed that a version of quantum mechanics devoid of imaginary numbers leads to a faulty description of nature. A month earlier, researchers at the University of California, Santa Barbara reconstructed a quantum wave function, another quantity that cannot be fully described by real numbers, from experimental data. In either case, physicists cajoled the very real world they study to reveal properties once so invisible as to be dubbed imaginary.
For most people the idea of a number has an association with counting. The number five may remind someone of fingers on their hand, which children often use as a counting aid, while 12 may make you think of buying eggs. For decades, scientists have held that some animals use numbers as well, exactly because many species, such as chimpanzees or dolphins, perform well in experiments that require them to count.
Counting has its limits: it only allows us to formulate so-called natural numbers. But, since ancient times, mathematicians have known that other types of numbers also exist. Rational numbers, for instance, are equivalent to fractions, familiar to us from cutting cakes at birthday parties or divvying up the cheque after dinner at a fancy restaurant. Irrational numbers are equivalent to decimal numbers with no periodically repeating digits. They are often obtained by taking the square root of some natural numbers. While writing down infinitely many digits of a decimal number or taking a square root of a natural number, such as five, seems less real than cutting a pizza pie into eighths or 12ths, some irrational numbers, such as pi, can still be matched to a concrete visual. Pi is equal to the ratio of a circle’s circumference and the diameter of the same circle. In other words, if you counted how many steps it takes you to walk in a circle and come back to where you started, then divided that by the number of steps you’d have to take to make it from one point on the circle to the opposite point in a straight line passing through the centre, you’d come up with the value of pi. This example may seem contrived, but measuring lengths or volumes of common objects also typically produces irrational numbers; nature rarely serves us up with perfect integers or exact fractions. Consequently, rational and irrational numbers are collectively referred to as ‘real numbers’.
Negative numbers can also seem tricky: for instance, there is no such thing as ‘negative three eggs’. At the same time, if we think of them as capturing the opposite or inverse of some quantity, the physical world once again offers up examples. Negative and positive electric charges correspond to unambiguous, measurable behaviour. In the centigrade scale, we can see the difference between negative and positive temperature since the former corresponds to ice rather than liquid water. Across the board then, with positive and negative real numbers, we are able to claim that numbers are symbols that simply help us keep track of well-defined, visible physical properties of nature. For hundreds of years, it was essentially impossible to make the same claim about imaginary numbers.
In their simplest mathematical formulation, imaginary numbers are square roots of negative numbers. This definition immediately leads to questioning their physical relevance: if it takes us an extra step to work out what negative numbers mean in the real world, how could we possibly visualise something that stays negative when multiplied by itself?
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