> Even so I don't get how electrons a meter apart interact through that stuff
Very roughly: It's possible for point-like (or tiny looped) particles to interact as long as they take every possible path instead of just the one path that would cause a collision. How you interpret this is... up for debate. I prefer the many-world interpretation (MWI), but not everyone agrees.
> "if you cut down the spatial dimensions to just 1D" doesn't sound very physical to me
That's just a simplification to aid understanding, it's now how the theory actually works.
Yeah maybe. My flatmate of some years was doing a PhD in string theory at UT Austin along the lines of "if you cut down the spatial dimensions to just 1D" but he was a mathematician, not really a physicist and was ok with that if it produced interesting mathematics. For real physical things like wiring the lighting system I'd do it because he wasn't so good with that.
I think he went into string research because he was good at maths and there was grant money available for that rather than a deep belief that that was the nature of reality, which is kind of what I mean by sociological factors.
I think much string theory may be like that. Interesting maths but not good at figuring where electrons go.
Except this is how electrons actually go, and it has real testable consequences. The question I'm aware of (because it related to my degree in nanotechnology) was: are metals conductive at different dimensionalities?
Because at the nanoscale, you in fact can have 1D, 2D and 3D metals. 3D metals are bulk solids - like we're familiar with. 2D metals are planes of single (or very few) atoms. 1D metals are lines - think placing individual metal atoms down in a row - nanotubes are a practical example.
All real, possible structures to build.
When you do measurements on all these structures you get...weird answers. Like is a nanotube a superconductor? And the answer is...yes, but also no. Yes because you'll in fact view superconductivity like behavior, but no because actually it's a ballistic electron conductor - at the right energy level an electron bounces through the thing without hitting it, but not all electrons can do that at all energy levels, so you still measure a voltage across a nanotube between two conductors.
But a nanotube is 1D - we only have 1 dimension things move in (from one end to the other). So - conductive, not a superconductor, but you can kind of use it like one sometimes. And we know 3D metals are conductors - that's obvious right...so what are 2D metals? Presumably conductors right...?
And the answer is...nope, insulators - at least sometimes. And the reason is because the sum of all possible electron paths in a 2D metal is the electron always returns back to where it started - and those grow much faster mathematically then paths where the electron ends up somewhere else.
But only in 2D: in the 1D case most paths take you out of the conductor. And in the 3D case, the number of paths which land you somewhere else grows much faster then those which loop back, due to the extra dimension of freedom. But 2D metals are constrained - for any given path elsewhere, there are mathematically far more that land back where you started. This is observable, measurable behavior which is a topic of research for future semiconductors. Yet it's almost entirely quantum probability based behavior.
I think the 1D mentioned in jiggawatts comment is a different thing to a confining electrons to a nanotube type of setup.