Parallel world: a world that is qualitatively identical or very similar to ours.
Parallel worlds do not just exist in the worlds posited by ►many worlds theories, in ►multiverses, or in other ►dimensions. They are present right here in our standard ►universe. This follows from the assumption of an infinite universe, or even just a sufficiently large one, together with the following reasoning:*
The observable part of the universe — the ►Hubble volume — has a radius of approximately 46 giga-lightyears. Quantum theory stipulates that at a temperature below 108 degrees, this volume can hold at most 10115 elementary particles, which establish at most 2 to the power of 10115 distinguishable quantum states.** Thus, the maximum number of existing different Hubble volumes at this maximum temperature cannot exceed 2 to the power of 10115.
2 to the power of 10115 is an enormously large — albeit finite — number representing an upper limit for the plurality of worlds. Now, imagine we pack one of each theoretically possible version of the world — that is to say, one of each of its possible Hubble volumes — together into one box. Then any remaining Hubble volumes outside the box could only be repetitions of those inside the box, being identical to — and consequently parallel worlds of — worlds in the box. The size of the box, and thus the average distance of a Hubble volume to its identical parallel world, amounts to 2 to the power of 10115 multiplied by 46 giga-lightyears.*** We here presuppose that the parallel worlds are more or less evenly distributed; we assume, that is, that none of them is duplicated considerably more or less often than another. If Hubble volumes that are similar to ours occur considerably more frequently than others then their distance from ours will be shorter.
If the size of the universe exceeds this critical distance, there may be parallel words of our own Hubble volume — and if the universe is infinitely large then the existence of parallel words becomes inevitable. This is why it is possible that at this moment one of your doppelgangers is reading a copy of this article. Or an infinite number of doppelgangers are reading infinitely many copies. Additional conditions such as restriction to the average density and temperature currently prevailing in the universe will reduce the number of distinguishable Hubble volumes to as few as 10 to the power of 1080. Thus, your doppelganger may be closer than you think.
Now, does the existence of infinitely many worlds necessarily mean that somewhere in the universe there is a second planet Earth where ripened fruit flies up into the sky instead of falling to earth, or where there is someone who is both your doppelganger and your grandfather's? Not at all. The existence of infinitely many worlds does not imply the existence of all the worlds that can be imagined; the ►laws of nature and of logic must hold in all worlds that exist in this universe.
* From Max Tegmark, "Science and Ultimate Reality", in Parallel Universes, Cambridge: Cambridge University Press, 2003.
** The number 2 here derives from the fact that there are exactly two possibilities for each particle location: Either it is taken up by a particle or it isn't.
*** This is equivalent to 10 to the power of 10115 meters, for 2 to the power of 10115 is such an enormous number that its multiplication by 46 giga-lightyears does not alter the way it is written — not even if you replace the 2 by a 10 and lightyears by meters. Numbers of this order gradually approach the calculating rules for infinite ►cardinals.
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