[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?

[drdeca]

I believe I answered the question? You asked whether these quantum fields have values at points. I believe there is a field-of-sorts, but that unless spacetime is discrete, the value of it at an individual point isn’t really a meaningful question, and even if spacetime is discrete, while the question becomes meaningful (as in, it is an observable), typically it will not have a determinate answer.

If there is no inside to a box, then knowing everything about how the box interacts with things outside the box, is pretty much everything there is to know about the box, yeah.

The study of physics concerns only that which we can observe/measure. Now, like I implied before, I’m not a scientific materialist, and I don’t claim that all-that-there-is is amenable to understanding through the lens of physics. So, like, I guess the answer is “No, I don’t expect physics to tell us everything I want to know about the nature of the universe, just all of it that is accessible to experiment.”.