Quantum mechanics: Hypercomplex, or ‘just’ complex?

Today, physicists are still asking themselves whether quantum mechanics needs hypercomplex numbers. FAU researchers Ece Ipek Saruhan, Prof. Dr. Joachim von Zanthier and Dr. Marc Oliver Pleinert have been investigating this question in their research.

Exactly 100 years ago, Werner Heisenberg, Max Born and Pascual Jordan formulated quantum mechanics and revolutionized our understanding of physics. Only a few months later, Erwin Schrödinger presented an alternative but mathematically equivalent formulation: Wave mechanics based on Schrödinger’s equation.

Even though these approaches appeared to be competing at first, they turned out to be physically identical—they describe the same reality, but use different mathematical approaches.

To describe quantum mechanics in mathematical terms, both Heisenberg and Schrödinger used the system of complex numbers. In contrast to real numbers, those we use every day to describe distances or temperatures, complex numbers comprise a real part, like a number on an x axis, and an imaginary part, located on a y axis. This enables them to be shown as points on a two-dimensional plane.

While Schrödinger later speculated that quantum mechanics could be formulated only using real numbers, this theory has since been disproved by a series of experiments.

An additional dimension?

However, another question that remains unanswered to this day goes in the other direction: Could it be necessary to go above and beyond complex numbers in quantum mechanics and to describe it using hypercomplex numbers, for example by using something known as quaternions?

Hypercomplex numbers extend the concept of complex numbers by adding additional dimensions to the imaginary part. Instead of points on a two-dimensional plane, they therefore represent points in multi-dimensional spaces. To this day, there is no conclusive evidence that such a description is necessary, but it has not been ruled out. However, for our understanding of nature, such an extension would be revolutionary.

The founding fathers of quantum mechanics examined this question and investigated hypercomplex formulations of quantum physics. To gain some clarity, physicists also later looked for experiments to prove this hypothesis.

One of the first people to examine this question was Asher Peres. In the 1970s, he proposed an experiment to demonstrate whether quantum mechanics can actually be fully described using complex numbers or whether hypercomplex numbers are required. His basic idea: If quantum mechanics is hypercomplex, certain outcomes of an experiment he proposed should behave differently than if standard quantum mechanics applied.

Specifically, he proposed sending lightwaves through different interferometers with two or three slits and to compare the resulting interference patterns. In standard quantum mechanics, certain combinations of the resulting interference patterns cancel each other out. When this is not the case, this could be an indication for hypercomplex quantum mechanics.

Since then, several researchers have carried out the Peres test in experiments. Early experiments initially used simplified versions of the test with neutrons. Only recently have scientists conducted measurements in the optical and microwave range in accordance with Peres’ original concept. Due to the limited measurement accuracy, however, no clear evidence for or against hypercomplex quantum mechanics has been found to date in an experiment.

Saruhan, von Zanthier and Pleinert at the Department of Physics at FAU recently carried out a thorough theoretical investigation of the Peres test and extended it. The paper is published in the journal Physical Review Letters.

Here, they discuss their findings:

You developed the Peres test. What exactly did you change?

Saruhan: We put the test on a mathematical foundation, making it generally more applicable. In addition, our approach means that we can interpret the results of the test as volumes in a three-dimensional space. If a description with complex numbers is sufficient, all measurements are located on a plane in this space and the test results in a volume of zero. In contrast, if the volume were different to zero…

…a hypercomplex description would be necessary?

Pleinert: Exactly. Then physical dimensions could only be fully described with hypercomplex numbers. Our extended test method also makes it possible to expand the test to any number of dimensions: You can think of it like this—each additional slit in the experiment adds a further dimension, allowing us to systematically investigate all hypercomplex number systems.

Why is it important?

von Zanthier: The Schrödinger equation is the established standard in quantum mechanics, but its validity has never been formally proven—it just worked. This means that complex numbers seem to have always been enough to describe the observed phenomena. Our test extends the experimental verification of whether complex numbers are actually sufficient to describe quantum mechanics to higher dimensions and can help to clarify this fundamental question of physics.

You have improved the Peres test even further?

Saruhan: Yes, we have extended the procedure so that not just a single light particle, but several light particles are sent through an interferometer with any number of slits at the same time. This allows us to further increase the informative value of the test.

Has the extended test provided new results?

Pleinert: All measurements up to now show that the result is always “zero,” which means complex numbers are sufficient for the achieved measurement accuracy. The tests require extremely precise measurements, however. Above all, our aim is to encourage physicists all over the world to use our extended version in order to carry out even more precise tests.

Maybe that will enable us one day to provide a definitive answer as to whether complex numbers are sufficient for quantum mechanics, or whether hypercomplex numbers are required. This would be a significant step forward for quantum research.