Quantum mechanics in focus: Do we need hypercomplex numbers?

Quantum mechanics in focus: Do we need hypercomplex numbers?

On March 8, 2025, the Friedrich-Alexander University Erlangen-Nuremberg (FAU) published news about current research on quantum mechanics carried out by a team led by Ece Ipek Saruhan, Prof. Dr. Joachim von Zanthier and Dr. Marc Oliver Pleinert will be directed. The scientists are investigating the question of whether hypercomplex numbers, such as quaternions, are necessary to accurately describe quantum mechanics. Traditionally, quantum mechanics is depicted with complex numbers that consist of a real and an imaginary part.

The fundamentals of quantum mechanics were formulated about 100 years ago by prominent physicists such as Werner Heisenberg, Max Born and Pascual Jordan. At the same time, Erwin Schrödinger presented an alternative wave mechanics. Despite the different mathematical approaches, both theories are physically identical. Schrödinger even speculated that quantum mechanics could possibly be formulated using real numbers, but this was refuted. The debate about whether hypercomplex numbers are needed remains current.

Historical development and basics

Quantum mechanics was introduced to overcome the inadequate explanations of classical physics for certain physical phenomena. It developed between 1925 and 1926 through the work of Schrödinger, Heisenberg, Born and Dirac. The goal was to develop a theory that adequately describes the wave properties of particles. By the 1930s, quantum mechanics proved successful in explaining numerous observations in physics and chemistry. Remarkably, to date there have been no experiments that have contradicted the predictions of quantum mechanics.

A central postulate of this theory is that the state of a particle is described by a wave function that contains all information about its quantum mechanical properties. Measurements of physical quantities such as position and momentum are based on the expected values ​​of these states, while the Schrödinger equation defines the time evolution of the state vector.

The Peres test and the current investigations

In the 1970s, physicist Asher Peres proposed a test to verify the need for hypercomplex numbers in quantum mechanics. This test involves comparing interference patterns of light waves produced by different interferometers. Some previous experiments performed simplified versions of this test, but without clear evidence as to whether hypercomplex numbers are needed.

The researchers at FAU have further developed the Peres test theoretically. Their new approach allows the test results to be interpreted as volumes in a three-dimensional space. An important criterion here is that if the volume is zero, complex numbers are sufficient; however, if it has a positive value, hypercomplex numbers would be necessary. Currently, measurements show that the result is always zero, suggesting that complex numbers may be sufficient.

The researchers' goal is to carry out more precise tests in order to finally clarify the crucial question of the necessity of hypercomplex numbers in quantum mechanics. The original publication on these developments is entitled “Multipath and Multiparticle Tests of Complex versus Hypercomplex Quantum Theory” and will soon be published in the renowned journal “Physical Review Letters”.

The deeper mathematical formulation of quantum mechanics, as developed by John von Neumann in 1932, describes a physical system in terms of states, observables and dynamics. In the Copenhagen interpretation, the physically measurable quantities are defined by Hermitian operators in the state space. These sound mathematical concepts form the basis for understanding the complex phenomena that quantum mechanics describes.

The ongoing studies at FAU illustrate the dynamics of scientific progress in quantum mechanics and open up new perspectives for investigating fundamental questions that could shape physics in the coming years.