# Question about spacetime quantization

I have a question about spacetime...if spacetime was quantized, would we still be considered to have 3 spatial dimensions?

As far as I understand, 3 numbers are the minimum that we currently need to specify a location somewhere in space after selecting an arbitrary origin (the numbers are usually presented as (x,y,z) coordinate triplets). I'm dropping the 4th time coordinate because i'm not concerned with a specific event, but only a point at which events keep occuring as time flows.

If spacetime is quantized, then wouldn't we be able to describe points in space with fewer numbers, and thus we wouldn't actually have 3 spacial dimensions? If i simplify to a "2 dimensional" plane, we would currently need to pick an origin and then specify each point with an x and y coordinate, and there would be an infinite number of such points for any specific area. But, if there was a discrete number of points (quantized spacetime) in that same area, then wouldn't we be able to pick an origin, and go around the origin in a spiral, numbering every discrete point we encounter? And we'd be able to assign a single number to every point on the plane, instead of needing 2 numbers per point?

## Answers and Replies

You have to take time into consideration if you aim to use it in any type of application.

If three-dimensional space were quantized then you would still need three numbers to describe a location in it. Essentially you would be changing your space from $$\mathbb{R}^3$$ (triplets of real numbers) to $$\mathbb{Z}^3$$ (triplets of integers). In some sense you are right that there are fewer numbers, since the cardinality of $$\mathbb{R}$$ is larger than the cardinality of $$\mathbb{Z}$$. Even though there are infinitely many integers, there is a "larger" infinity of real numbers (you might want to read about the http://en.wikipedia.org/wiki/Cardinality_of_the_continuum" [Broken]).

You could imagine quantized space as being more like how we locate points on a street grid -- you might say that something was on the 11th floor of the building at 42nd Street and 3rd Avenue, or (42, 3, 11), but you wouldn't say that it was on the 11.325th floor at 42.08 St and 2.71 Ave, or (42.08, 2.71, 11.325). If the grid were actually quantized, then you could only ever be exactly at an intersection, and exactly on a given floor, never part-way between them.

It is true as Division said that in any application to physics you would have to include the time dimension, although this is not necessarily a requirement when trying to form a picture in your mind of what quantized space might look like.

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But, if there was a discrete number of points (quantized spacetime) in that same area, then wouldn't we be able to pick an origin, and go around the origin in a spiral, numbering every discrete point we encounter? And we'd be able to assign a single number to every point on the plane, instead of needing 2 numbers per point?

The process you described is sometimes considered in mathematics, for example the points of the plane with integer coordinates could be put into one-to-one correspondence with the integers on a line.

There is, however, no possibility for spacetime to be discrete in this simplistic sense. It would violate all kinds of observations that we can already make. Even fringe theories like loop quantum gravity are not proposing a discrete spacetime: what they mean by 'quantization' is not the same as 'turn into a discrete set of points.'

Thanks for the help everyone :) I think I understand this a bit better now