Does conservation require quantization?

[md65536]

 

http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

 

The BanachTarski paradox as I've heard it basically says that you can "disassemble an orange and then reassemble it into 2 oranges identical to the first." You can do this mathematically on a set of points but you can't do it with a real orange because it requires taking it apart into uncountably infinite points, I think, and you can't take apart quanta of matter like that.

 

I'm wondering if reasoning in the opposite direction is useful.

 

Outline:

1. Assume that matter isn't quantized, and any piece can be split into several smaller pieces.

2. Find a way to partition a sphere according to the B-T paradox without changing the density of the matter.

3. Create 2 spheres, thus mass is not conserved.

 

Step 2 is the speculative part, I'm not sure it could be meaningfully done since we start by assuming something unreal. But perhaps there might be no meaningful mathematical way to have matter that isn't quantized yet can't be partitioned like that??? (Whether that's true or not is beyond my ability to figure out.) One problem is that the result of splitting up a piece of matter might always result in a countable number of pieces.

 

 

 

Anyway I wondered further if it might be possible to apply the idea to anything that follows a conservation law, requiring that thing to be quantized.

[timo]

Iirc, the Banach-Tarski paradox is an effect of the axion of choice which is required/introduced in the context of real numbers. Rational numbers would not fall under this paradox, I think. Rational numbers are not quantized.