An Introduction to String Theory

Slide 14 of 37
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To work out what diagrams describe a particular quantum field theory, we look at the "Lagrangian" function that defines the theory. The Lagrangian has a lot of deep physical meaning, but for our purposes we can think of it as a simple catalog of the particle "fields" in the theory and their possible interactions.

In this simple example, the Lagrangian for Quantum Electrodynamics (QED), we see two types of particle fields. Psi represents the electrons, and A_µ is the vector potential of electromagnetism. The vector potential represents photons in the theory: photons are the carriers of electromagnetism in addition to being particles of light. The fact that A_µ is a vector (which has both a length and a direction) corresponds to the fact that photons are polarized in a specific direction.

So what do the terms represent? They're a bit complicated, but their broad meaning is easy to understand. The first term is the "electron kinetic term": it has two electron (psi) pieces together with a derivative (and some other things). Since derivatives involve changes in functions from point to point, it makes some sense that this term describes an electron starting at one point and moving to another. So Feynman diagrams for QED can include segments like the one at left.

The second term is the "photon kinetic term": when you multiply it out, you can see that every term has two photon (A_µ) pieces, with derivatives. This describes a photon starting at one point and moving to another, with a specified polarization at each end. That means that Feynman diagrams in QED can include segments like the one in the middle.

Finally, the last term is the interaction term, the one that actually describes the electromagnetic force. It includes the interaction strength (or "electron charge") e, two electron pieces, and a photon piece. That means that the line segments we saw earlier can join up in a specific way: two electron legs come together with a photon (for example, an electron can come in, emit a photon, and go out).