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.