Editor's Note: Excerpted with permission from “Love and Math: The Heart of Hidden Reality” by Edward Frenkel. Available from Basic Books, a member of the Perseus Books Group. Copyright © 2013.
We are all familiar with the electric and magnetic forces. Electric force is what makes electrically charged objects attract or repel each other depending on whether their charges are of the same or opposite signs. For example, an electron has negative electric charge, and a proton has a positive charge (of opposite value). The attractive force between them is what makes the electron spin around the nucleus of the atom. Electric forces create what is called an electric field. We have all seen it in action during a lightning strike, which is caused by the movement of warm wet air through an electric field.
Photo by Shane Lear. NOAA photo library.
Magnetic force has a different origin. It is the force that is created by magnets or by moving electrically charged particles. A magnet has two poles: north and south. When we place two magnets with opposite poles facing each other, they attract, whereas the same poles repel each other. The Earth is a giant magnet, and we take advantage of the magnetic force it exerts when we use a compass. Any magnet creates a magnetic field, as we can see clearly on the picture.
Photo by Dayna Mason.
In the 1860s, British physicist James Clerk Maxwell developed an exquisite mathematical theory of electric and magnetic fields. He described them by a system of differential equations that now carry his name. You might expect these equations to be long and complex, but in fact they are quite simple: there are only four of them, and they look surprisingly symmetrical. It turns out that if we consider the theory in the vacuum (that is, without any matter present), and exchange the electric field and magnetic fields, the system of equations will not change. In other words, the switching of the two fields is a symmetry of the equations. It is called the electromagnetic duality. This means the relationship between the electric and magnetic fields is symmetrical: each of them affects the other in exactly the same way.
Now, Maxwell’s beautiful equations describe classical electromagnetism, in the sense that this theory works well at large distances and low energies. But at small distances and high energies, the behavior of the two fields is described by the quantum theory of electromagnetism. In the quantum theory, these fields are carried by elementary particles, photons, which interact with other particles. This theory goes under the name of quantum field theory.
To avoid confusion, I want to stress that the term “quantum field theory” has two different connotations: in a broad sense, it means the general mathematical language that is used to describe the behavior and interaction of elementary particles; but it may also refer to a particular model of such behavior – for example, quantum electromagnetism is a quantum field theory in this sense. I will mostly use the term in the latter sense.
In any such theory (or model), some particles (like electrons and quarks) are the building blocks of matter, and some (like photons) are the conduits of forces. Each particle has various characteristics: some familiar ones, like mass and electric charge, and some less familiar, like “spin.” A particular quantum field theory is then a recipe to combine them together.