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.
Actually, the word “recipe” points us toward a useful analogy: think of a quantum field theory as a culinary recipe. Then the ingredients of the dish we are cooking are the analogues of particles, and the way we mix them together is like the interaction between the particles.
For example, let’s look at this recipe of the Russian soup borscht, a perennial favorite in my home country. My mom makes the best one (of course!). Here’s what it looks like (the picture was taken by my dad):
Obviously, I have to keep my mom’s recipe secret. But here’s a recipe I found online:
8 cups of broth (beef or vegetable)
1 pound slice of bone-in beef shank
1 large onion
4 large beets, peeled
4 carrots, peeled
1 large russet potato, peeled
2 cups of sliced cabbage
3/4 cup of chopped fresh dill
3 table spoon of red wine vinegar
1 cup sour cream
salt & pepper
Think of this as the “particle content” of our quantum field theory. What would the duality mean in this context? It would simply mean exchanging some of the ingredients (“particles”) with others, so that the total content stays the same.
Here is how such a duality could work:
beet → carrot
carrot → beet
onion → potato
potato → onion
salt → pepper
pepper → salt
All other ingredients stay put under the duality; that is,
broth → broth
beef shank → beef shank
and so on.
Since the amounts of the ingredients we exchange are the same, the result will be the same recipe! This is the meaning of duality.
If, on the other hand, we exchanged beets for potatoes, we would get a different recipe: one that would have four potatoes and only one beet. I haven’t tried it, but I am guessing it would taste awful.
It should be clear from this example that a symmetry of a recipe is a rare property, from which we can learn something about the dish. The fact that we can switch beets with carrots without affecting the outcome means that our borscht is well-balanced between them.
Let’s go back to quantum electromagnetism. Saying that there is a duality in this theory means that there is a way to exchange the particles so that we end up with the same theory. Under the electromagnetic duality we want all “things electric” to become “things magnetic,” and vice versa. So, for instance, an electron (an analogue of a beet in our soup) carries an electric charge, so it should be exchanged with a particle that carries a magnetic charge (an analogue of a carrot).
The existence of such a particle contradicts our everyday experience: a magnet always has two poles, and they cannot be separated! If we break a magnet in two pieces, each of them will also have two poles.
Nonetheless, the existence of a magnetically charged elementary particle, called magnetic monopole, has been theorized by physicists; the first was one of the founders of quantum physics, Paul Dirac, in 1931. He showed that if we allow something funny to happen to the magnetic field at the position of the monopole (this is what a mathematician would call a “singularity” of the magnetic field), then it will carry magnetic charge.
Alas, magnetic monopoles have not been discovered experimentally, so we don’t know yet whether they exist in nature. If they don’t exist, then an exact electromagnetic duality does not exist in nature at the quantum level.
The jury is still out on whether this is the case or not. Regardless, we can try to build a quantum field theory that is close enough to nature and exhibits the electromagnetic duality. Going back to our kitchen analogy, we can try to “cook up” new theories that possess dualities. We can change the ingredients and their quantities in recipes we know, get rid of some of them, throw in something extra, and so on. This kind of “experimental cuisine” may lead to variable results. We may not necessarily want to “eat” these imagined dishes. But edible or not, it may be worthwhile to study their properties in our dreamed-up kitchen – they may give us some clues about the dishes that are edible (that is to say, the models that could describe our universe).
This trial-and-error “model building” is a path along which progress has been made in quantum physics for decades (just as it was in the culinary art). And symmetry is a powerful guiding principle that has been used in creating these models. The more symmetrical a model is, the easier it is to analyze.
At this point, it is important to note that there are two kinds of elementary particles: fermions and bosons. The former are the building blocks of matter (electrons, quarks, etc.), and the latter are the particles that carry forces (such as photons). The elusive Higgs particle, discovered recently at the Large Hadron Collider under Geneva, is also a boson.
There is a fundamental difference between the two types of particles: two fermions cannot be in the same “state” simultaneously, whereas any number of bosons can. Because their behavior is so radically different, for a long time physicists assumed that any symmetry of a quantum field theory had to preserve a distinction between the fermionic and bosonic sectors – that nature forbids them to be mixed together. But in the mid-1970s several physicists suggested what looked like a crazy idea: that a new type of symmetry was possible that would exchange bosons with fermions. It was christened supersymmetry.
As Niels Bohr, one of the creators of quantum mechanics, famously said to Wolfgang Pauli, “We are all agreed that your theory is crazy. The question that divides us is whether it is crazy enough to have a chance of being correct.”
In the case of supersymmetry, we still don’t know whether it is realized in nature, but the idea has become popular. The reason is that many of the issues that plague conventional quantum field theories are eliminated when supersymmetry is introduced. Supersymmetric theories are generally more elegant and easier to analyze.
Quantum electromagnetism is not supersymmetric, but it has supersymmetric extensions. We throw in more particles, both bosons and fermions, so that the resulting theory exhibits supersymmetry.
In particular, physicists have studied the extension of the electromagnetism with the maximal possible amount of supersymmetry. And they showed that in this extended theory the electromagnetic duality is indeed realized.
To summarize, we don’t know whether a form of quantum electromagnetic duality exists in the real world. But we do know that in an idealized, supersymmetric, extension of the theory, the electromagnetic duality is manifest.
There is one important aspect of this duality that we haven’t yet discussed. The quantum field theory of electromagnetism has a parameter: the electric charge of the electron. It is negative, so we write it as -e, where e = 1.602 · 10-19 Coulombs. It is very small. The maximal supersymmetric extension of electromagnetism has a similar parameter, which we will also denote by e. If we perform the electromagnetic duality and exchange all things electric by all things magnetic, we will get a theory in which the charge of the electron will be not e, but its inverse, 1/e.
If e is small, then 1/e is large. So if we start with the theory with a small charge of the electron (as is the case in our world), then the dual theory will have a large charge of the electron.
This is hugely surprising! In terms of our soup analogy, imagine that e is the soup temperature. Then the duality would mean that switching the ingredients such as carrots and beets would suddenly convert a cold borscht into a hot one.
This inversion of e is in fact a key aspect of the electromagnetic duality, which has far-reaching consequences. The way quantum field theory is set up, we have a good handle on the theory only for small values of the parameter such as e. We don’t even know a priori that the theory makes sense at large values of the parameter. Electromagnetic duality tells us not only that it makes sense, but that it is in fact equivalent to the theory with small values of the parameter. This means that we have a chance to describe the theory for all values of the parameter. That’s why this kind of duality is considered as a Holy Grail of quantum physics.
