The Story of Anti-Matter and Quaternions

Transcript

It's the 16th of October 1843 in Dublin, Ireland. The legendary mathematician Sir William Rowan Hamilton is walking to the Royal Irish Academy when a flash of genius enters his mind. Inspired by his epiphany, he carves a simple formula onto a stone of the Broom Bridge. Little did Hamilton realize at the time that what he had just discovered would become the key to discovering the first antiarticle some 85 years later. In doing so, a man named Paul Durac would bring together two of the greatest theories of his time, quantum mechanics and special relativity to launch us into a new and vastly strange realm of reality.

Our story begins with a mathematician who was for years stuck on a single problem. This was Sir Hamilton who for years was looking to generalize the success that the complex numbers had in two dimensions to the case of three dimensions. As many of you might know, the complex numbers which are what you get when you start considering a number I which is the square root of -1 are incredibly successful at describing rotations in two dimensions. Several beautiful and elegant formulas using I allow you to easily do geometry and rotations in a two-dimensional plane. Now, Hamilton wanted to have something like the complex numbers, but this time for rotations in three dimensions.

For years, he searched for a number system that would allow him to concisely work with three-dimensional rotations. But for the longest time, he could not find a solution. But on Broombridge, a flash of brilliance brought the solution to his mind. It was the quatians. This is a number system that is similar to the complex numbers.

But instead of having just i as the square root of -1, you also have two new variables j and k which are also square roots of -1. The rules for multiplying between i, j and k can be summarized by these formulas. And so this number system looks a lot like the complex numbers but with extra square roots of -1. The quturnians turned out to elegantly describe rotations in three dimensions solving Hamilton's original problem. But Hamilton wanted to go further.

He wanted to apply his new quturnionic numbers to physics. There's a famous differential equation in physics called the wave equation that pops up all over the natural world. It looks something like this. Now, what Hamilton realized was that his quatanionic numbers gave him a way to take the square root of part of the wave equation just like this. Hamilton himself wrote down this observation and even said he believed that it would be important for physics one day in the future, but he couldn't tell for what.

Unfortunately, Hamilton would never live to see the applications of his marvelous discovery. For years, his quernionic numbers were put on the side and not given much importance by the mathematical community. Hamilton knew that his new number system would one day be important, but the strange discovery that they would lead to, I doubt even Hamilton would have imagined. Our story resumes in 1928, a time when physics was rapidly changing. Einstein had just released his theory of special relativity 23 years earlier which described strange phenomena like time slowing down or speeding up when traveling near the speed of light.

At the same time, perhaps an even stranger theory emerged in quantum mechanics that said that the universe at small scales was much more random and unpredictable than we could have imagined. The great physicist Paul Durk was working on a way to combine these two theories. He wanted to know how tiny particles described by quantum mechanics would behave when their speed approached the speed of light where the rules of special relativity would start to take effect. Durk was attempting to take the famous Schrodinger equation from quantum mechanics and modify it in a way that reflected the laws of special relativity. But how does one do this? The story goes that the flash of insight came to Durk when he thought about taking the square root of a part of the related Klein cordon equation.

In particular, the part of the equation called the wave operator. By taking the square root of the wave operator, Durac obtained an equation of a form like this. This was precisely what Hamilton had done 85 years before. Although to the best of my knowledge, Durac was not aware of Hamilton's quartians when doing this. By taking the square root of the wave operator, Durac wrote the following equation down.

Notice how similar this looks to the quturnionic equation produced by Hamilton. But here, instead of using quturnians, Derak used matrices which are denoted as the gamma variables here. This is how the famous DRA equation is fabled to have been born. The equation describes the behavior of certain particles, namely the spin 1/2 particles like the electron. But the strange thing about this equation is that when you solve it, you get two possible solutions for energy.

One that is positive and the other negative. This negative solution is what corresponds to antimatter. A particle that in our case has all the same properties as the electron but with opposite charge. Such a particle is named a posetron. Durac realized that his equation predicted a particle that no one had ever seen before.

A strange anti-electron that almost seemed like science fiction. He was troubled by his discovery and didn't fully embrace the prediction of antiparticles at first. It wasn't until 1932 that a clever experiment confirmed the prediction of the anti-electron for the first time. This was the work of the great physicist Carl Anderson. Anderson created an experiment that measured particles that fall from space onto Earth's surface, which we call cosmic rays.

These cosmic rays would go through a highly pressurized cloud chamber on Earth where electrons would leave a cloud trail that you could see and measure. By looking at the trajectory of the trail, you could tell that the particle that left the trail was an electron by measuring the curvature of its path and comparing this to theoretical predictions. Anderson applied a magnetic field to his pressurized cloud chamber. And what he discovered was stunning. When cosmic rays came down, there were tracks in the cloud chamber that were identical to the electrons tracks but moved in the opposite direction.

This meant that a particle was falling through the clouds that was identical to the electron in every way except with opposite charge. In other words, a posetron had just fallen through. This was the beginning of our understanding of the strange world of antimatter, a realm of physics that almost sounds like science fiction. Although Hamilton's original idea on the broom bridge wasn't directly used by Durak or by Anderson, the spirit of his quitterians were directly seen in Durak's derivation of his famous equation. The strange connection between Hamilton's prediction that the square root of the wave operator would one day be important in physics and then decades later Derak using this square root to predict antimatter is a story that speaks to the bizarre nature of discovery in physics.

It makes you wonder how much maths we have invented today that is perhaps laying dormant which may be used centuries from now to discover mindbendingly strange secrets in our universe.