Physics scientists developed, in a study released on June 18, in an unprecedented way, a version of quantum mechanics that does not use complex numbers, mathematical substances previously considered indispensable for understanding the theory for almost a century.
Complex numbers are formed by the union of a conventional ‘real’ number with an ‘imaginary’ number, which is a multiple of the square root of -1, symbolized by ‘i’, resulting in a single piece of data such as 3 + 4i. The square root of -1 does not have a direct representation in the physical world or in counts, which justifies its designation as ‘imaginary’ by mathematicians.
Despite their abstract nature, these numbers have vast utility. Engineering professionals use them to model alternating electrical currents, while physicists apply them to describe waves. Since the conception of quantum mechanics in the 1920s, complex numbers have been integrated directly into its formulations, being crucial for the description of particles through the wave function.
In 2021, a group of scientists predicted that a formulation of quantum mechanics based exclusively on real numbers would fail in its predictions for experiments involving multiple particles. In 2022, other studies confirmed this hypothesis, with test results aligned with traditional quantum mechanics rather than its actual variation. This scenario reinforced the apparent need for complex numbers.
However, the 2021 conclusion was based on a specific assumption: the application of a particular mathematical rule for combining particles. This point raised a question among researchers: were complex numbers truly essential to describe the universe at the quantum level, or would they just serve as a convenience tool?
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A new investigation, published on June 18 in the publication Physical Review Letters, revealed an innovative strategy that allows us to overcome the obstacle identified in 2021.
Pedro Barrios Hita, theoretical physicist and doctoral candidate at the German Aerospace Center and Heinrich Heine University Düsseldorf, stated that “complex numbers are not necessary for quantum mechanics.”
How a new rule simplifies quantum mechanics
The 2021 discovery was based on a specific mathematical rule known as tensor product, a method that unifies two distinct quantum systems. This product is the standard approach to combining particles into a single mathematical representation and is widely taught in all books on quantum mechanics.
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While it works perfectly for quantum mechanics employing complex numbers, previous attempts to develop a real-number version based on the same rule have faced considerable challenges. These approaches failed to replicate the correlations observed in experiments with three or more entangled particles.
In the new work, Barrios Hita and his team realized that the tensor product does not represent the only viable alternative. They reformulated quantum mechanics using a distinct rule, based on the principle that an action applied to one part of a system should not influence another part of it.
In conventional quantum mechanics, the multiplication of the state of a particle by ‘i’ (imaginary number) is, in itself, imperceptible. However, when combining two particles, this ‘i’ can move and link to the other particle, a phenomenon known as phase regression, which is already intrinsic to the tensor product.
Barrios Hita’s team sought to recreate this rearrangement using exclusively real numbers. To do this, they attached a small ‘flag’ to each particle, managing the information that was previously contained in the imaginary part. Later, certain combinations of these flags were considered physically equivalent, even with different appearances on a theoretical level. This grouping process made it possible for the new version, based on real numbers, to match all the predictions of standard quantum mechanics, including the multiple particle scenarios that were previously a challenge.
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The essence of the solution appears to be simple. A complex number, like 3 + 4i, is nothing more than a set of two real numbers (3 and 4), where the ‘i’ only serves as a marker for the imaginary part. As Barrios Hita explained, “a complex number is nothing more than two real numbers.” His team established a control system that monitors these two real numbers independently, rather than consolidating them into a single complex. Achieving consistency for multiple combined particles took considerable time, but once achieved, Barrios Hita highlighted the elegance of the resulting structure.
According to Barrios Hita, this discovery places quantum mechanics on a similar level to other physics theories that often use complex numbers for mere convenience in their formulation.
He added: “There are several other theories, such as electromagnetism, that incorporate complex numbers into their structure. However, these formulations employ complex numbers as mere useful tools to express equations, without them being fundamental to the theoretical basis.”
It is important to highlight that this research does not modify any experimental predictions nor does it suggest the emergence of new quantum technologies. Furthermore, current application is restricted to systems with a limited number of quantum states. The next natural step is to expand this concept to infinite-dimensional systems, common in several real physical problems, and other scientists are already exploring this possibility. Pedro Barrios Hita, in turn, focuses on a different study, investigating the use of quantum properties, such as entanglement, as a resource.
Still, the work ends a discussion that has gone on for decades, confirming that, although complex numbers can simplify the description of quantum mechanics, they are not absolutely essential for its intrinsic operation.