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I didn’t find math particularly exciting when I was in high school. To be honest, I only studied it when I went to university because it initially seemed quite easy to me. But in my very first math lecture as an undergraduate, I realized that everything I thought I knew about math was wrong. It was anything but easy. Mathematics, I soon discovered, can be really exciting—especially if you go beyond the realm of pure arithmetic.

In physics, the truly surprising content—concepts that go against your intuition about the universe—emerges around high school, when students can glimpse the strange quantum world and encounter Einstein’s general and special theories of relativity. School mathematics cannot keep up with these wonders. You learn elementary arithmetic operations, integration, and derivation, the basic handling of probabilities and vectors. If you’re lucky, ambitious teachers might show you a simple proof. And that’s it. So it’s no wonder that many pupils fail to develop a real passion for the subject.

Yet mathematics offers all sorts of surprises, such as the Banach-Tarski paradox, which states that you can double a sphere almost magically, or the fact that there are infinitely many different infinities. What really blew me away was discovering how deeply mathematics is interwoven with the strangest physical phenomena. It’s not necessarily quantum physics itself that gives rise to the incredible effects; no, the systems always follow the strict rules of mathematics. As chemist Peter Atkins put it in his 2003 book Galileo’s Finger, “Determining where mathematics ends and science begins is as difficult, and as pointless, as mapping the edge of a morning mist.”

Few examples illustrate the mixing of math and physics better than a discovery made by physicist Michael Berry. In 1984 Berry revealed a profound and largely unexpected geometric side to quantum mechanics. This geometry, Berry realized, gives quantum particles a kind of memory.

Nothing Should Actually Happen

At the time, Berry was investigating a very simple system: the quantum state of a particle, such as a neutron, in a changing environment. Neutrons have a quantum property called spin, which acts like a tiny magnet that the particles carry with them. This spin can either be oriented with the north pole facing upward or downward—so physicists speak of neutrons having “spin up” or “spin down.” The spin of a neutron is influenced by external magnetic fields.

Berry used mathematical means to investigate what would happen to the neutron if the direction of the magnetic field changed slowly. According to the so-called adiabatic theorem, which was introduced in the early 20th century, the quantum properties of the particle should not change as a result: its energy, momentum, mass, and spin remain the same.

If you slowly turn the direction of the magnetic field and then move it back in the original direction, this action should, in principle, not actually change anything. “That, at any rate, was the prevalent opinion among physicists for many years,” wrote Berry in an article in Scientific American in December 1988. But a “change on the phase of a wave function was overlooked.”

One of the strangest phenomena of quantum mechanics is wave-particle duality: quantum objects can be imagined as pointlike shapes, but they also exhibit wave behavior like water. A phase describes a displacement of the wave by a certain angle—for example, the cosine function is nothing other than a phase-shifted sine function.

As Berry recognized in his calculations, a slow change in the magnetic field causes the wave function of the neutron to rotate by a certain phase. This means that the wave function of the particle shows what happened in the past (in this case, the change in the magnetic field). Further, Berry recognized that this phase does not only occur in the special case of a particle in a magnetic field. Various situations in which a quantum system is slowly changed and then returned to its original conditions leave traces in the wave function.

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