system
Given the following post, respond as if you're commenting on Reddit:
human
Are quantum fields in any way similar to classical fields?

I understand classical fields as being functions of space with spatial(-temporal) coordinate inputs and scalar, vector or tensor outputs (like temperature being a function T(x,y,z,t) with a single scalar output). But when people talk of quantum fields, they talk of "exciting the field to produce particles" and other things that refer to quantum fields more as physical media that can be disturbed and picked apart. Am I missing something big here? Or is QFT just so complicated that it's not worth comparing quantum fields to classical ones?
gpt
In the canonical picture, quantum fields are operator-valued fields (distributions, to be more precise). The Hilbert space on which they act is spanned by number states. Each number state consists of a given number of particles with given momenta and other quantum numbers (spin, color etc.). In fact the \*free\* fields behave like a set of harmonic oscillators for each momentum/spin/color/etc. Of course not every state is a number state, but they provide a basis by which you can express any state by linear combination.

In this picture, the fields are just operators that create and destroy particles, but the particle content is in the abstract quantum state.

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The path integral picture is closer to the classical interpretation, but the relationship to particles is a bit more difficult to see. Basically, there is the path integral/partition functional, which consists of a sum over all possible classical field configurations weighted by the value of the action on the given field configuration. Thus all possible field configurations contribute to the amplitudes and expectation values. In the classical limit, only those configurations contribute for which the action is stationary, eg. those that satisfy the classical equations of motion.

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TL;DR: In the canonical approach, fields are just tools. In the path integral approach, physical processes happen by summing over classical fields, even those that do not solve the (classical) equations of motion.