Werner Heisenberg

The famous German physicist Werner Heisenberg was born on this day in 1901. He was awarded the Nobel Prize in 1932 for the creation of quantum mechanics. One of his works include uncertainty principle, one of the few terms that managed to get out of physics domain and used (abeit superficially) in other literatures and some movies.

The quantum world is described by beautiful mathematics, and they show the magic of mirror symmetries. They are equivalence of spaces that has revolutionized geometry. They are originally from enumerative geometry, a well-established, but not very exciting branch of algebraic geometry. The mathematicians who work on enumerative geometry solve problems like counting the number of curves on Calabi-Yau spaces — six-dimensional solutions of Einstein’s equations of gravity that are of particular interest in string theory, where they are used to curl up extra space dimensions.

Just as you can wrap a rubber band around a cylinder multiple times, the curves on a Calabi-Yau space are classified by an integer, called the degree, that measures how often they wrap around. Finding the numbers of curves of a given degree is a famously hard problem, even for the simplest Calabi-Yau space, the so-called quintic. A classical result from the 19th century states that the number of lines — degree-one curves — is equal to 2,875. The number of degree-two curves was only computed around 1980 and turns out to be much larger: 609,250. But the number of curves of degree three required the help of string theorists.

Around 1990, a group of string theorists asked geometers to calculate this number. The geometers devised a complicated computer program and came back with an answer. But the string theorists suspected it was erroneous, which suggested a mistake in the code. Upon checking, the geometers confirmed there was, but how did the physicists know?

String theorists had already been working to translate this geometric problem into a physical one. In doing so, they had developed a way to calculate the number of curves of any degree all at once. It’s hard to overestimate the shock of this result in mathematical circles. It was a bit like devising a way to climb each and every mountain, no matter how high!

Within quantum theory it makes perfect sense to combine the numbers of curves of all degrees into a single elegant function. Assembled in this way, it has a straightforward physical interpretation. It can be seen as a probability amplitude for a string propagating in the Calabi–Yau space, where the sum-over-histories principle has been applied. A string can be thought to probe all possible curves of every possible degree at the same time and is thus a super-efficient “quantum calculator.”

But a second ingredient was necessary to find the actual solution: an equivalent formulation of the physics using a so-called “mirror” Calabi–Yau space. The term “mirror” is deceptively simple. In contrast to the way an ordinary mirror reflects an image, here the original space and its mirror are of very different shapes; they do not even have the same topology. But in the realm of quantum theory, they share many properties. In particular, the string propagation in both spaces turns out to be identical. The difficult computation on the original manifold translates into a much simpler expression on the mirror manifold, where it can be computed by a single integral.

Schrödinger equation $$ \hat{H}|\psi(t)\rangle = i\hbar \frac{\partial}{\partial t} | \psi(t)\rangle $$