For the first time in the history of physics, researchers have developed a version of quantum mechanics that works without the need for complex numbers. These numbers, which combine a real part with an imaginary part (the square root of -1), were considered crucial to the theory nearly a century ago, resolving a question that has persisted since the field’s inception.
Complex numbers are made up of an ordinary “real” number and an “imaginary” number, which is a multiple of “i”, the symbol for the square root of -1. An example is 3 + 4i. The root of -1 does not represent a quantity that can be measured directly, which is why it is called imaginary in mathematics.
Despite their abstract nature, complex numbers have several practical applications. Engineers use them to describe alternating electrical current, while physicists use them to explain waves. Since quantum mechanics was formalized in the 1920s, these numbers have been incorporated directly into its equations, describing particles through a wave function that depends on their complexity.
In 2021, a team of physicists predicted that a formulation of quantum mechanics based exclusively on real numbers would generate incorrect predictions in experiments involving multiple particles. The following year, other researchers ran these experiments, and the results confirmed standard quantum mechanics, not the real-number version, suggesting that complexes were, in fact, indispensable.
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However, the 2021 conclusion rested on one specific assumption: a mathematical rule for combining particles. This has raised the question of whether complex numbers are intrinsically necessary to describe reality at the quantum level, or whether they merely represent a tool of convenience. The debate had been going on for decades in the scientific community.
Now, in a new study published June 18 in the journal Physical Review Letters, scientists have found an alternative way to get around the 2021 result. This breakthrough, led by Pedro Barrios Hita, a theoretical physicist and doctoral candidate at the German Aerospace Center and Heinrich Heine University Düsseldorf, suggests a fundamental shift in understanding the mathematical basis of the theory.
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“Complex numbers are not necessary for quantum mechanics,” said Pedro Barrios Hita, lead author of the study, in an interview with Live Science, indicating a reevaluation of the need for these elements.
A new mathematical approach to combining particles
The 2021 study was anchored in a specific mathematical rule known as tensor product, used to integrate two separate quantum systems into a single one. This rule, fundamental in all quantum mechanics textbooks, allows two particles to be combined into a single mathematical description.
The tensor product works effectively in conventional quantum mechanics that employ complex numbers, but previous attempts to construct a real-number version based on this same rule have faced difficulties. They were unable to reproduce the correlations observed in experiments involving three or more entangled particles.
In the most recent work, Barrios Hita and his team demonstrated that the tensor product is not the only alternative. They formulated quantum mechanics based on a different rule, based on the principle that an action carried out in one part of a system should not affect a separate part of it.
In traditional quantum mechanics, multiplying a particle’s state by ‘i’ is undetectable in isolation. However, when two particles combine, this ‘i’ can shift and effectively bind to the other particle. This phenomenon is known as phase regression and is automatically present in the tensor product.
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Barrios Hita’s team managed to recreate this setback using only real numbers. They added a little “flag” to each particle to record what the imaginary part would normally store. Then certain combinations of these flags were treated as physically identical, even though they looked different on paper. This grouping step allowed the real-number version to match all the predictions of standard quantum mechanics, including the multiparticle cases that had defied previous attempts.
At its core, the strategy is simple: a complex number, like 3 + 4i, is actually a pair of real numbers (3 and 4), with the ‘i’ just afor the imaginary part. “A complex number is nothing more than two real numbers,” explained Barrios Hita. The team developed an accounting system that tracks these two real numbers separately, rather than grouping them into a single complex number. Although it took time to perfect this approach consistently for multiple combined particles, the underlying structure turned out to be elegant.
The results obtained place quantum mechanics in the same category as other physical theories that often use complex numbers for mere convenience, as highlighted by Barrios Hita.
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“There are many other theories, for example electromagnetism,” added Barrios Hita, “which have complex numbers at their core. So these theories are formulated using complex numbers, but [they] are not fundamental. They are just useful tools to help express equations.”
It is important to emphasize that the work does not change any experimental predictions or point to new quantum technologies. It is currently limited to systems with a finite number of quantum states. Expanding to infinite-dimensional systems, common in many real physics problems, is the natural next step, and other researchers are already investigating this possibility. Barrios Hita, in turn, will dedicate himself to other research, focused on how quantum properties such as entanglement can be explored as a resource.
Still, the study ends a debate that has lasted decades. Although complex numbers make writing quantum mechanics easier, they are not a requirement for its operation, marking a significant advance in understanding the theoretical foundations of quantum physics.