system
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human
What does it mean when physicists refer to several additional "small"  dimensions?

I've just reread a couple of Hawking's books, and I can't figure out how to conceptualize additional dimensions.  Three spacial dimensions plus time as a fourth seem straightforward enough (though obviously not always easy to visualize), but what does it mean to speak of additional dimensions that are "small" or "curled up"?  How exactly are they different from the 4 space-time dimensions?

You can measure a particle's relative position or velocity in space-time.  Does that particle have a coordinate in each of the other existing dimensions as well?  

If dimensions are "small", does that mean they are local, or do they exist throughout spacetime?
gpt
>but what does it mean to speak of additional dimensions that are "small" or "curled up"? How exactly are they different from the 4 space-time dimensions?

What we mean by "small" is that the symmetry groups describing translations in those directions are "compact."  What compactness means is that if you go far enough in a compact direction, you get back to where you started.  Rotations of solid bodies form a compact group, because after rotating by 2*pi you're back where you started.  Similarly, directions on the surface of a sphere are compact: if you travel for long enough in a given direction, you get back to where you started.

So the thought to stringy or Kaluza-Klein models is to consider (an) extra dimension(s) which fields or strings are allowed to propagate in, but their "size" (how far you can move in those extra directions) is smaller than around 2*10^-19 meters (1 TeV^-1 ).

>You can measure a particle's relative position or velocity in space-time. Does that particle have a coordinate in each of the other existing dimensions as well?

Classically yes, but from a quantum mechanical standpoint, these other dimensions are so small that particles are kind of smeared through this entire other dimension.  In KK models, for example, typically there's an infinite number of ways they can be smeared out, each having a different amount of energy, which would look to us like, for example, an infinite number of excited electrons each with higher mass than the last.

>If dimensions are "small", does that mean they are local, or do they exist throughout spacetime?

They would exist throughout spacetime, in the same way that the three spatial dimensions we know of exist throughout spacetime.

EDIT: unrelated, but can anyone tell me how to cancel superscripts without putting a space in?