Natural numbers are needed to count days, fractional numbers to distribute a pizza, and real numbers to measure time. Beyond these types of numbers, in high school, those who obtain a technological diploma are also told about imaginary (or complex) numbers, strange devices apparently without daily use.
Complex numbers arose from the impossibility of solving the equation x² = -1 using real numbers, since these, multiplied by themselves, always give another positive number. Therefore, mathematicians introduced the imaginary unit, i, an abstract entity satisfying i² = -1. In general, complex numbers are those that can be written as TO + biWhere TO AND B They are real numbers. For example, 2/7 + 8i.
Initially they did not model any physical reality, but over time they found applications. They began to be used in fields of physics such as optics and electromagnetism, if only to simplify calculations; all derivations of these theories can be stated without using complex numbers. In the development of quantum mechanics, the physical theory that describes the behavior of the smallest particles, imaginary numbers appeared in a more fundamental way: it is the only physical theory in whose postulates this type of number appears.
However, its ideologists were not satisfied with the use of complex numbers in the foundations of quantum physics, since it contradicts the fact that everything measurable in the laboratory is quantifiable with real numbers. In one of his letters to Hendrik Lorentz, Erwin Schrödinger stated: “The use of complex numbers is troublesome. We should be able to formulate quantum theory without them.”
For years, some scientists have tried, unsuccessfully, to develop a quantum doctrine that could circumvent complex numbers, while others argued that they were inevitability. Providing arguments for this second position, in 2021, an international group of scientists, with extensive Spanish participation, devised an experiment that, if carried out, would demonstrate that imaginary numbers are essential in quantum theory.
From a theoretical point of view it is nothing more than a game, similar to the one that John Bell designed in 1964 to demonstrate that quantum mechanics is outside the scope of classical physics. Three players participate in the game, Alicia, Belén and Carla, and a referee, who in each round asks each of them a question at the same time. The players know in advance which trio of answers they need to use to correctly answer each trio of questions and win the round of the game, but they are in separate rooms, so they do not know the questions asked of their two partners, on which the correct answer depends. Before starting you can meet and agree on a strategy. Furthermore, they can access an additional resource to coordinate their responses: entangled quantum particles.
Winning percentages in the game are given by values (called norms) associated with certain functions (so-called multilinear forms), which are studied in functional analysis. The key to the proposed game lies in the theoretical predictions, since they allow us to estimate the percentage of victories that players would obtain if they had access to entangled particles described by a quantum theory that uses only real numbers, and also in the case of having particles coming from a theory based on imaginary numbers. They also ensure that the first percentage is lower than the second. Accordingly, to verify the need for complex numbers, it is enough to play many rounds of this game and get a higher winning percentage than predicted by the theory based on real numbers.
Despite the simplicity of the proposal, playing the game requires the manipulation of quantum particles, which represents a challenge from a practical point of view: the systems must be handled at -270ºC, close to the temperature of absolute zero; or control light particles, photons, with high precision. There have already been several experiments that have achieved this goal, but not entirely satisfactorily. The quantum community is confident that the experiment will be fully completed, establishing that imaginary numbers are inevitable to describe the physical world.
Francisco Escudero Gutierrezpre-doctoral researcher at Dutch Center for Mathematics and Computer Science (Wiskunde & Informatics Center, CWI)
Editing and coordination: Agata Timone (Institute of Mathematical Sciences)
Coffee and theorems is a section dedicated to mathematics and the environment in which it is born, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest progress of this discipline, share meeting points between mathematics and other social and cultural expressions and remember those who marked its development and were able to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems”.
