Curled up dimensions and Lorentz invariance

[JustinLevy]

If we start with minkowski spacetime in 4 dimensions and then add several curled up spatial dimensions attached at every spacetime point, then:

I'll label a spacetime point as:
(ct,x,y,z)[a1,a2,a3,..,an]
where the bracketted coordinates are the 'curled' coordinates.

- If we label the coordinates of the curled dimensions as [0,0,..,0] where they attach to the minkowski spacetime point, does that mean you can't move from point (ct,x,y,z)[anything] to (ct',x',y',z')[something] without going through (ct,x,y,z)[0,0,..,0] and (ct',x',y',z')[0,0,..,0] ? Or am I misunderstanding what is meant by "attach at every spacetime point"?

- While spacetime is still continuous instead of discrete, it seems like there is still a preferred frame now: The frame in which the 'density' of the curled up dimensions in each direction is equal. After doing a boost, the 'density' of the curled up dimensions is greater in one direction. This, in my mind, is analogous to the famous example of the twin's paradox in a closed universe ... while there is of course still no local preferred frame, there is a global preferred frame now. Since strings are "global" in the sense that they can go all the way around a curled dimension, wouldn't they make such a global preferred frame acutely apparent?What I'm trying to think out comes down to something like this (poorly worded, I know, but hopefully gets the point across):
How can local poincare symmetry in the full dimension spacetime still yield "apparent local" poincare symmetry after "coarse graining" over the global structure of the curled dimensions?

Can anyone help me understand better? (even just correcting my terminology here, or giving me terminology to use, to make the discussion more precise would be very helpful)
Thank you.

 

[tom.stoer]

You should try to do the following: construct a simple theory on a 3-torus, or if you like in 1+1 dimensions = on S¹ with periodic boundary conditions. You will see that the Poincare algebra can be constructed w/o problems.

Of course string theory is not as simple as e.g. QED in 1+1 dimensions, but it may give you a hint that compactification does not necessarily break such an invariance. Of course the representation will differ.

 

[arivero]

- If we label the coordinates of the curled dimensions as [0,0,..,0] where they attach to the minkowski spacetime point, does that mean you can't move from point (ct,x,y,z)[anything] to (ct',x',y',z')[something] without going through (ct,x,y,z)[0,0,..,0] and (ct',x',y',z')[0,0,..,0] ? Or am I misunderstanding what is meant by "attach at every spacetime point"?

They are not "attached", it is the usual cartesian product, in he same way that the z dimension is attached to the xy plane.

Now, instead attaching z (a real line) to the xy plane, try to add a short segment, say [-0.0001, +0.0001] plus some condition to preserve trajectories, say bouncing. Any particle moving in this space (bouncing at the z=+0.0001 walls) will still look as it were moving in a plane xy.

You can even remove the walls and define that a particle reaching +0.0001 will appear at -0.0001. This is to put a S^1 manifold attached to the xy plane.

More fun. Now you can define the *mass* of the particle to be the speed (ok, the momentum) of this particle in the z direction. This comes from, ahem, E=mc2: you compare

$$E^2-p_x^2-p_y^2=m_0^2$$
with
$$E^2-p_x^2-p_y^2-p_z^2=0$$

So really you don't need, if you don't want it, massive particles to exist. It was klein's idea.