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# MATH AND LOVE: AN UNCOMMON COMBINATION, ALTHOUGH........

by Edward Frenkel, Ph.D

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.

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.

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.

book hint:

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contact: www.edwardfrenkel.com

#mathematikliebe

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