Gauge Theories

A gauge theory explains a force by imposing symmetries on the equations of motion.

The phase of a wavefunction is the current position of its peaks and troughs. If you don't know, peaks and troughs are the maximums and minimums you see in a wave.

The phase should be a degree of freedom, as we should be able to change where it is in space and the observables (we cannot directly observe the wavefunction as it would collapse) should stay the same. In order to do this, we need to impose some symmetries onto the Schrodinger equation, making it so that the equation must be invariant to local changes in the wave function. Invariant means that it stays the same under the transformation, which in this case is the local changes in the wave function.

It turns out that what we added turns out to describe the electromagnetic field, and if we quantize that field, we get the photon, the gauge boson of the electromagnetic field.

This tells us what gauge bosons are. Gauge bosons carry and transmit the four fundamental forces as well as make them up. Bosons are also unconstrained by the Pauli Exclusion principle, so there can be more than one in the same place at the same time.

Image from: https://en.wikipedia.org/wiki/Standard_Model

By imposing a set of symmetries on the equations of motion, we get a gauge field and a gauge boson. The set of symmetries we imposed falls under a category called U(1) symmetry, which has to do with group theory if you are interested.

Two examples of gauge theories are Quantum Electrodynamics (QED) describing electrically charged particles and the electromagnetic force and Quantum Chromodynamics (QCD), describing quarks and the strong force.