Physicists from Heinrich Heine University Düsseldorf and the German Aerospace Center have studied whether quantum mechanics requires complex numbers. Complex numbers contain both a real part and an imaginary part, where the imaginary part involves the square root of negative one. In standard quantum mechanics these numbers help describe a quantum state through its amplitude and its phase.
A study published in 2021 concluded that complex numbers are essential under the usual basic assumptions of the theory. That conclusion was supported by experiments. The new research re-examined those basic assumptions. The physicists found that one of the assumptions used in the earlier work was too restrictive.
Alternative way to combine quantum systems
The researchers identified a different assumption: when you have two separate quantum systems (or subsystems), any operation or measurement performed on one system has no measurable effect on the other system. In mathematical language, the operators describing local actions on each subsystem commute with each other.
This is a physically reasonable assumption - it reflects the idea that you cannot send information or influence one part of a system instantly by acting only on another distant part (a form of no-signaling or locality principle).
This alternative approach produces a version of quantum mechanics that can be written entirely with real numbers. The new framework makes exactly the same predictions as standard quantum mechanics for every possible experiment. As a result imaginary numbers are not required as a fundamental part of the theory.
The new findings are published in Physical Review Letters and were highlighted by the American Physical Society. The work shows that quantum mechanics remains consistent when formulated with only real numbers provided the rules for combining systems are adjusted in a suitable way. "Our construction is based on the notion of the composite Hilbert space as a real quotient space," note the physicists. This opens the possibility of describing the same physical phenomena through alternative mathematical structures without changing any observable outcomes.