Physics Question #5297

Peter Schwarz, a 67 year old male from Toronto asks on June 1, 2011,

I understand that even in a finite volume, there are an infinite number of points. Why does the Planck area not apply here?

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The answer

William George Unruh answered on June 1, 2011

Yes, any volume has an ifinite number of points. That is an aspect of mathematics of the real numbers. If you represent the volume as a part of R3 (the 3-dimensional space of reals) it is trivial to show that it has an infinite number of points.

Now, the question can be asked as to whether or not R3 is a good model of a volume, whether a manifold captures the essense of the physics of a spatial volume. That is where the Planck scale comes in. Does the best model of space have something like quantization of spatial volumes in terms of planck scale volumes, or is the Manifold structure a better model? No one knows. There is a suspicion that the former is somwhow true, but we have no adequate mathematical model which would tell us. The loop quantum gravity people would argue that their results suggest that there is a minimal volume and that any volume, if you measure it somehow, would give a finite number of something like planck volumes. String theory suggests there is a complementarity between volumes (or any dimensions) and energy, such that if you try to measure to smaller and smaller scales, you will need more and more energy in the string, which wil then correspond to more and more curved spacetime which makes the concept of the volume become more and more problematic. But the "right answer" is liable to be even weirder.

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