Introduction

Congratulations on making it this far! We have covered all of the necessary background material to understand quantum field theory. We extensively studied quantum mechanics, special relativity, and classical field theory. Now we are ready to dive into quantum field theory.

The basis of quantum field theory lies in quantization; First quantization is the process of taking a classical system and turning it into a quantum system. Then, field quantization (or historically known as second quantization) is the process of taking a quantum system and turning it into a field theory. For example, take Einstein's energy-momentum relation:

First quantization involves replacing the classical variables , , and with their quantum mechanical counterparts, which are operators:

This is the Klein-Gordon equation. Finally, we can take the Klein-Gordon equation and promote it to a field theory by replacing the wavefunction with a field .

This is the Klein-Gordon field equation.

Natural Units

In theoretical physics, we often use natural units, where . There are many advantages to this. For one, it simplifies equations and makes them easier to read. For example, the Klein-Gordon equation in natural units is:

Second, in computation, it reduces floating point errors. Constants like are very small (), and when we multiply them by other small numbers, we can get floating point errors. This is especially true in quantum field theory, where we often deal with very small numbers.