Does the Quantization of Space-Time Imply a Finite Number of Spatial Locations?

[nomadreid]

If space-time itself is quantified, and the spatial universe has, at anyone time, a finite volume, would this not imply that at anyone moment there are a finite number of spatial locations? (If so, then integrating over an infinite number of points would only give an approximation.)

 

[xts]

Of course, you are absolutely right.

Integration is just much easier to perform than summing up discrete finite series.

Just keep in mind that the same argument may be used regarding any quantizied value. Try to make electronic design, counting individual electrons, rather than using 'approximation' of continuous current. Or try to construct a bridge, counting every atom, rather that taking steel as a continuous medium.

But, of course, in reality (whatever you mean by 'reality') steel is not continous, and consists of individual atoms, occupying their individual locations.

 

[nomadreid]

Thanks, xts. But in that case arguments with infinity no longer should work. For example, the cardinality argument for the Casimir effect, in which it is stated that a countably infinite number of virtual particles can appear between the plates, whereas a continuum-number of virtual particles can appear outside, hence accounting for the greater energy density outside. But both inside and outside the number of possible virtual particles should be finite, trashing that argument. No?

 

[xts]

It would just refolmulate argument in terms of finite numbers: between plates you have some finite number of virtual particles, leading to some energy density, outside you have also finite number, but leading to different density, the pressure still is generated.

It is like with simple cylinder/piston/gas examples: you may analyse it in terms of continuous gas of different pressures or in terms of different (finite) numbers of gas atoms bumping from each side. Both views lead to the same results, at least until you don't go to low with the scale, making statistical fluctuation of particle number visible.

 

[nomadreid]

That sounds reasonable. Would this apply to all cardinality arguments in Physics?

 

[xts]

That sounds reasonable. Would this apply to all cardinality arguments in Physics?

I guess so - at least to those arguments, which are used to explain experimentally verifable phenomena.

Such arguments should always be checked against common sense thought experiments. If the Casimir's effect would really require infinite, continuous space outside, it would give different results if performed in the lab room or at the open area. Actually it was confirmed within the lab (and even within a pretty small apparatus).