4.1 A toy multiverse
While the mathematics is flawless, it may nevertheless strike you as dubious, not to say reckless, for Everett to have extended an equation describing tiny quantum effects to implying the existence of a whole multiverse of parallel universes, and indeed I promised to give you a more convincing argument for such parallel universes.
This is a template for producing “toy” universes containing some, or all, of the three events shown.
Just to consolidate the Many Discrete Worlds picture, consider a “toy multiverse”, comprising block universes each containing no more than three events/interactions labelled A, B and C as in the diagram. Events A and B are Alice’s and Bob’s experiments as shown in the earlier tree diagrams, with outcomes A1 and A2 occurring with probabilities of ¼ and ¾, and outcomes B1 and B2 each occurring with a probability of ½. Finally, event C has three possible outcomes, C1, C2 and C3, with outcome probabilities of ⅕, ⅗ and ⅕ respectively. I have also made the event C dependent upon the outcome of A: if the outcome is A2, then the event takes place, but not if the outcome is A1. However, there is no entanglement between A, B and C, which means that their Born probabilities can be simply multiplied together. To address entanglement, we could consider any one of these three events to be a composite entangled event involving two or more particles, but we won’t do that here, to avoid complicating the picture further.
Remember that these are block universes, and so, in each universe, every outcome within it is fixed. The outcomes are not determined by the probability of their occurring; rather, the probabilities observed in any given universe are determined by the likelihood of being in a universe with those particular outcomes, as we saw in the diagrams of Step 3.4 (Discrete universes).
This shows the Toy Multiverse with its eight different types of block universe. Each universe is represented by a long, light-green filament. The tree structure corresponds to the MWI structure as viewed from the perspective of event A. Remember, from the description of the toy block universe, event C only occurs if the outcome of event A is A2.The numbers of each type of universe, represented by the eight blocks, are given by the product written below each one. You are more likely to be in a universe with given outcomes where there are more copies of such a universe, and that is what determines the probabilities that you calculate if you are in any one of these universes.
How an observer within such a block universe sees these events depends on their reference frame. In this diagram, you see the range of possible outcomes from the perspective of an observer at event A in the block universe of the previous diagram. We assume that event B is sufficiently far from A that its future light cone doesn’t reach A before C has occurred in A’s world line (that is, in those universes in which the outcome of event A is A2).
There are 40 universes in the trunk, and, to make things general, I have labelled the number as N, so N could be 80 or 80 billion. I have also indicated the Born probabilities for each outcome as we rise through the branches of the tree. So, for example, starting from the trunk, we encounter event A and we can follow along the A2 branch. The Born probability for that outcome is ¾, and so the number of universes in this branch is ¾ ⨯ N, or 30 in the diagram. Further on, we come to event C, and the Born probabilities for the three branch outcomes are indicated as ⅕ ⨯ ¾ ⨯ N; ⅗ ⨯ ¾ ⨯ N and ⅕ ⨯ ¾ ⨯ N for the branches C1, C2 and C3 respectively. Further along the C2 branch, we find event B, and, taking the B2 branch, we come to the final “twig”. There are nine identical universes in this twig, given by ½ ⨯ ⅗ ⨯ ¾ ⨯ N, where N is 40. The ½, of course, is just the probability of going into the B2 branch from the C2 branch. You will see that each of the eight block universes pointing to a twig contains the outcomes encountered as you travel from the trunk to that twig.
Since, as we have seen, the same set of block universes represents the MWI tree seen from any perspective, then we should be able to use the set of eight parallel universes, shown from A’s perspective in the figure, to represent the tree from B’s perspective instead. Our next figure does just that – each of the eight sets of universes in the previous figure is reproduced in this one. It is from B’s perspective because A’s light cone reaches B’s world line before C’s light cone, and this is reflected in the order in which the events appear as you go up the trunk and along the branches. While the tree in this figure looks quite different from the tree in the previous figure, you can see that the sequence of outcomes as you travel from the trunk to the twigs nevertheless leads to exactly the same set of eight parallel universes, with the same number in each twig.
This shows the Toy Multiverse from an alternative perspective.
Just to make this point clear with an example, in the previous figure, starting from the rightmost twig, take the second twig and note that its block universe contains outcomes A2, B1 and C3 and that there are three such parallel universes in the twig. The derivation of this quantity, three, is written under the picture of the block universe as ½ ⨯ ⅕ ⨯ ¾ ⨯ N, which is three if N is 40. (Remember, we can multiply these probabilities together because A, B and C are not entangled with each other.) Now we can find the same set of three identical parallel universes in the current figure: starting from the leftmost twig, it is the fourth twig, where the block universe again contains outcomes A2, B1 and C3, and again the number of such block universes is ½ ⨯ ⅕ ⨯ ¾ ⨯ N = 3.
