Ut times is not custom, we will provide following-sales service from January 12. This indeed attracted an unusual number of reactions on the site of the World, who invite you to respond. We reported on Schrödinger’s astonishment when he realized that the equation which today bears his name might only be written using the imaginary number i (defined by the equation i² = –1 and “increasing” the ordinary numbers whose squares are all positive to constitute the set of so-called complex numbers), which showed the irruption of the most abstract mathematics at the heart of modern physics.
Some readers have rightly pointed out that Schrödinger’s equation can actually do without the number i because it is equivalent to two real equations (without i). The “complex” formulation of quantum mechanics would therefore only be an ease of writing, which was believed by Schrödinger and many other physicists following him, who have sought until today an axiomatic reformulation which allows to do without i. The task was not so simple, because quantum theory is not limited to Schrödinger’s equation. Everything related to the process of physical measurement (i.e. the interaction of a quantum particle with a macroscopic measuring device) brings into play other axioms, which also naturally use i (via complex Hilbert spaces).
A swapping experience
By the greatest chance, this question very recently received a definitive answer which allows us to give this to our demanding readers: taken as a whole, quantum theory cannot do without complex numbers! In a very brilliant theoretical paper from 2021, published in Nature, an international team has shown that it is possible to design a so-called “swapping” experiment that would give a result that is absolutely impossible to obtain with a theory that dispenses with numbers i. In this experiment, three physicists A, B, C observe two pairs of particles P1 and P2 (photons or electrons) emitted at the same time from two separate sources. The particles constituting a pair are quantumly correlated, which means that, despite the distance that will increase between them, they will continue to constitute a single quantum entity. The experiment is so organized that A and C receive a single particle coming from P1 and P2 respectively, B receiving two particles, one coming from P1 the other of P2. When A and C compare their results once B has learned regarding the quantum state of “its” pair halves, their statistical properties not only are not independent (as one might expect), but they have plus an “essentially complex” property: no quantum theory dispensing with i (assuming there is one) will not be able to account for it! All that was left was to experience…
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