"Field" is the abstraction we use in physics to say "there is a value associated with each point, and we're just not going to say whether it's representing the displacement of some invisible jelly, or the rotational velocity of some invisible tiny gears, or if it doesn't even make sense to ask the question". When we say the electromagnetic field wobbles, we just mean some abstract value is going from positive to negative and back.

I know that sounds unsatisfying, but there's good historical reason for doing this. Maxwell, for example, had a rich intuitive understanding of the electromagnetic field as a set of many invisible jellies and gears pushing on each other. His original equations were also 5x longer than the modern form and impossible for anybody else to understand, plus all the extra ingredients made tons of predictions that played poorly with relativity. The sheer butchery one needs to make intuitive pictures like this compatible with relativity is so high that we've basically collectively decided to give up. We now say fields make up real jellies and gears, they're not made of jellies and gears. They're just fields.

I gotcha. Not the right way to think about it. To be clear, you're saying fields are essentially virtual mathematical notions from which emerges actual stuff?

Given the premise that fields are explicitly not consisted of an "invisible jelly", I'll ask what is definitely a "half-assed question" -

How and where _do_ fields "exist"? Where are they in the standard model? Intuition says maybe they reside in spacetime near particles. I have a hunch from QFT that the fields are technically "everywhere" and are merely highly excited into particles when energy transforms the field in certain spots.

I doubt either of my guesses is the right answer but I'm mostly just curious how literally you mean to take the viewpoint that fields are "virtual" in nature.

Since this is a half-assed question, feel free to give a half-assed answer.

> To be clear, you're saying fields are essentially virtual mathematical notions from which emerges actual stuff?

Perhaps, but I think that's a bit unfair. There's no dividing line between "mathematical notions" and "actual stuff". For example, when you poke a bowl of jello, it springs back, so it feels like "actual stuff". But in terms of the mathematics, it's because the configuration of fields that corresponds to compressed jello has a higher energy value than when uncompressed. In other words, while we don't assign intuitive properties like "solidity" and "elasticity" to fields, the relation between the fields and those properties in macroscopic objects is quite direct!

> How and where _do_ fields "exist"? Where are they in the standard model? Intuition says maybe they reside in spacetime near particles. I have a hunch from QFT that the fields are technically "everywhere" and are merely highly excited into particles when energy transforms the field in certain spots.

Indeed, they're technically "everywhere". You commonly hear that for the Higgs field, but it's true for all others as well, such as the electron field, whose excitations are electrons.