[albatross79]

"And, as for what precisely a quantum field is, this is somewhat mysterious, but generally it is a quantum version of a classical field, where there is a value (e.g. “value of the electromagnetic field”) at each point in spacetime."

But what does this mean concretely? Do you believe there is a real field out there with a value at each point in space time? What's it made of, what is the value a value of? If there no real field where is the accounting done and by what? I understand that when we run it through our models that assume a field like thing we get the right predictions, but what's the mechanism out there?

[drdeca]

Something which I found surprising is that it appears that a Gaussian random field in more than one dimension apparently has to be distribution valued, such that with probability 1 one can’t really evaluate it a particular point.

Even setting that aside, I wouldn’t expect the state to be an eigenstate for that even if the “value of the field at this location” was an actual observable rather than a like, operator valued measure, so, even then I wouldn’t expect the value to be determinate, no.

If spacetime turns out to be discrete, that would resolve the “the distribution over the values for the field are distribution valued, not valued in genuine functions” issue, (and the other reason for it not having a determinate value is actually normal) but it is hard to see how this would fit with our non-observation of violations of Lorentz invariance.

I don’t know what you are asking for when you ask about a mechanism. Do you mean a classical mechanism? Nature isn’t classical.

[albatross79]

Sounds like you might have gotten lost in abstractions. It's a simple question. There is a box. I cannot see inside. I can model the output based on my input to it. Is that enough to tell me everything I want to know about the box? If that is all we can know about it, if we can never see inside, or there is no inside, then what do we know? Is that enough to satisfy everything you want to know about the nature of the universe?