system
Given the following post, respond as if you're commenting on Reddit:
human
What is the relationship of quantum field theory to quantum mechanics?

The way I see QM is as a sort of general "protocol" for making predictions at the quantum scale. First, choose a Hilbert space representing all the possible states of your system; then choose a self-adjoint operator A representing the variable you want to measure; find the eigenvalues and eigenvectors of A; use the Born rule to calculate the probability distribution; exponentiate iA to find the time evolution.

My understanding of QFT is that it uses this protocol to model fields like the EM field, so that the basic procedure is the same; the difference is that you use classical fields as a guide for choosing Hilbert spaces and operators, and this leads to things like Feynman diagrams and whatnot. How correct is this?
gpt
Correct. QM is a framework in which to describe physical systems. QFT is basically a certain subset of QM where you

1. Can create and destroy particles, and

2. Treat things relativistically (as in *special* relativity).

To make a classical theory quantum, you just apply the postulates of QM. You describe your system as a state vector in a Hilbert space, you promote dynamical quantities to operators, and "have at it".

To turn a classical theory into a QFT following canonical quantization, you promote your fields to operators which create and destroy particles, and impose the canonical commutation relations.

To get from there to Feynman diagrams, you just consider time evolution and expand the time evolution operator (or the S-matrix for scattering problems) in a perturbation series. There are some cool tricks due to Feynman, which allows you to do everything in a relativistically invariant way, and Wick, which allows you to easily work out the kinds of diagrams you need to draw.