The proofs in mathematics that I like best are those which are simple but unexpected. I’m thinking of proofs like the one about the infinity of primes, or the proof that the square root of 2 cannot be expressed as a ratio of two whole numbers. These particularly appeal to me because you start out with what sounds like a reasonable premise, you follow the logic, and you end up with a contradiction to the original premise. So the premise was wrong all the time.
Well, that happens in The Finite Multiverse. I just followed the logic of a multiverse of parallel block universes, using a Many-Worlds Interpretation tree to represent all of the possible outcomes of all of the different events. (Using a MWI tree doesn’t violate the block-universe requirement of no branching, because the tree comprises many filaments, extending from trunk to the outermost twigs, each filament representing a non-branching block universe.)
To my surprise, it turned out that the thickness of any branch in the tree – in other words, the number of block universes for a given quantum outcome – depends upon the thickness of every branch and twig in the whole multiverse, even those entirely unconnected apart from at the trunk itself.
Another consequence of the parallel block universe hypothesis is that no branch can contain an infinite number of universes because, if it did, then the relative probabilities of the outcome of a quantum event would depend upon ratios of infinite quantities, which are meaningless.
Now, although the last paragraph argues that no individual branch can contain an infinite number of parallel block universes, that, in itself, doesn’t rule out an infinite number of branches – and therefore, an infinite number of parallel universes – in the multiverse.
This is where the contradiction comes in. If you assume that there is indeed an infinite number of branches in the multiverse, even while the branches themselves each contain a finite number of universes, then, because the number of universes in any branch depends upon the number in every branch (see above), then the number of universes in every branch would be infinite.
Because of this contradiction, the assumption that there is an infinite number of branches in the multiverse must be wrong.
The details are in The Finite Multiverse, but there’s more.
If you assume that any individual universe is infinite either in space or in time, or both, then there will be an infinite number of quantum events in that universe. So, for each quantum event to be represented by a branch, there would have to be an infinite number of branches in the multiverse. But we have just seen that this is impossible – another contradiction. So all universes in the multiverse are finite both in space and in time. (Notice, by the way, that when I talk about a “universe” here, I mean a block universe. That is different, and much greater, than the observable universe that people generally mean when they refer to the universe.)
Since no individual branch can contain an infinite number of parallel universes (as we saw above) and since there is not an infinite number of branches in the multiverse (again, as we saw above), then there cannot be an infinite number of parallel universes in the multiverse.
A consequence of this is that the multiverse does not contain every conceivable universe! This will surely disappoint aficionados of alternate-history novels (like Philip K Dick’s The Man in the High Castle).
Another consequence of the parallel block universe hypothesis is that quantum-mechanical probabilities are never irrational, which means that quantum theory – and probably general relativity – must be re-cast in terms of a discrete theory.
So the parallel block universe hypothesis is rich in predictions:
· The number of parallel universes in the multiverse is finite;
· No universe within the multiverse is infinite;
· Not every possible universe exists;
· Quantum probabilities are never irrational;
· Quantum mechanics is only an approximation to an equivalent theory which must be discrete (this probably applies to general relativity as well).
A theoretical physicist, whose work I admire, suggested to me that I should submit The Finite Multiverse, duly spruced up, to Cornell University’s arXiv, which I have now done, and which prompted me to write this post.