How Feynman Diagrams Revolutionized Physics

Feynman discovered another very important item, the path integral formulation of physics. This was important for his derivation of Feynman diagrams and also it is a good conceptual tool.

Think of basic quantum mechanics and firing an electron through a double slit at a screen. In quantum mechanics the eletron does not have a single trajectory from the gun to the screen. Rather, it takes all the trajectories in parallel, like a wave. We can add the contribution as if the particle went over each possible trajectory and this is the same as treating the electron as if its position was given by a propogating wave. This sum over possible histories is the interpretation of Feynmans path integrals. And it is a nice way to think about quantum mechanics - multiple things are happening in parallel.

Taking the example in the video of two electrons scattering off each other, each Feynman diagram represents a possible history for the two particles, including their trajectory and any interactions between them. These interactdions are drawn as a connecting line, which is a "virtual particle" being exchanged.

More specifically, the diagram doesn't represent a single history, rather it represents all histories that have a ceratain topology, meaning here for example one photon is exchanged between the electrons. (There is an integral done to add up all the different ways this can happen.) To do the full calculation, there are many diagrams that must be included. As it is a perterbation theory, you can choose to get more accurate by include more diagrams. The expansion parameter is basically a vertex on the diagram. The more vertices you include, the more accurate you will be (assuming you include all diagrams with that number of vertices).

So basically the feynman diagram is a bookkeeping mechanism to account for all possible histories of the particles in the interaction (electon scattering here). We sum up the contribution from all these histories to find out the quantum amplitude for this scattering scenario. This is exactly analogous to adding the contribution from different paths to fine the amplitude (~probability) for our electron in the double slit experiment hitting a particular location on the screen.

To get to this intuitive result mathematically, the perturbation expansion and renormalization mentioned above are both involved.

If you look at how most physicists use the gamma matrices, they are very reluctant to treat them as algebraic objects and rely heavily on their matrix representation. A proponent of GA would say this is like using the matrix

    [ 0 1
    [-1 0]

For what it's worth, I have a soft spot in my heart from Geometric Algebra, but I think it still needs a lot of notational improvement before it'll ever see any real adoption.

If you're curious about GA as it currently stands, I'd check out Geometric Algebra for Physicists by Doran and Lasenby.