Having said that, I do believe the correct answer is some form of "we don't know". But that would be considered a half baked answer according to the author. It's also deeply unsatisfying on an emotional level.
So all the answers are basically modifying the intent of the asker in order to answer related questions which _are_ satisfying. This is compounded by the StackExchange community upvoting highly knowledgeable answers. Fortunately you can see the disputes on these answers also get upvotes in the answer's comments.
It's completely fair if you don't think any of the answers were satisfying, but it's not like there's another option on the table.
A lot of laypeople think that a truly satisfying answer must be mechanistic -- something like "there's an invisible jelly everywhere and it wobbles at a finite speed, and that is the speed of light" -- but the 20th century taught us that this completely natural desire is actually counterproductive, because you have to add piles of epicycles to hide the jelly's other effects.
>"there's an invisible jelly everywhere and it wobbles at a finite speed, and that is the speed of light"
As a layperson, is this analogous to "the electromagnetic field is everywhere and wobbles at a finite speed"?
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.
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.
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.