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3.4 Discrete universes
For convenience, this figure is reproduced from the previous Step 3.3 Universes don’t split.
Fortunately, it turns out that we can have the best of both worlds (or, indeed, many worlds!). We can keep Everett’s central idea of the branching universal wave function and yet not have any universes splitting. Look again at the Figure from Step 3.3, reproduced here for convenience. Each of the four scenarios may be seen in the following way (taking (c) for illustration). The universe in (c) doesn’t split: the outcomes of the two independent experiments are unequivocally Ad and Bu. The pink light cone centred on Ad merely represents Alice’s knowledge of the two possible outcomes of Bob’s experiment. When “news” of Bob’s result eventually reaches her (once the red light cone reaches her world line), nothing has changed: that universe is a block universe in which the outcomes Ad and Bu are embedded and in which Alice and Bob were always going to find outcomes of spin-down and spin-up respectively. The same goes for the block universes in (a), (b) and (d).
Figure reproduced from Step 3.3 (Universes don’t split) for convenience
The MWI tree from Alice’s point of view in the previous figure can be reconstructed using eight discrete parallel universes. Each universe is represented in the tree by a filament bent in the manner shown, so that, together, they reproduce the shape of the MWI tree. Remember – each universe is a block universe: the outcomes are fixed and never change.
To see how this works, look at the figure reproduced from Step 3.3 (Universes don’t split). The tree representing the multiverse seen from Alice’s perspective is redrawn in shades of green in the following figure. I have added a semi-circular array of eight blue block universes under the tree, with an arrow pointing from each block universe to a green zig-zag strip, or filament, in Alice’s tree. So each filament represents a single, discrete block universe. Each block universe contains the outcomes of Alice’s and Bob’s experiments, shown as starbursts. Notice that, with all the filaments being the same width, we can fit six universes into the left ¾ AuB? branch and two into the right ¼ AdB? branch, so maintaining the probability ratio of 3:1 when Alice’s universe splits (in the MWI model). Following the left branch, we see that three of the six universes fork into the ⅜ AuBu branch and three into the ⅜ AuBd branch, reflecting the probability of ½ for each of Bob’s two possible outcomes. The remaining two universes end up in the ⅛ AdBu branch and the ⅛ AdBd branch.
So, instead of each universe continually branching, we can represent Alice’s tree with eight discrete (blue) universes that never branch. Of course, that number of eight universes is only for convenience: we used one filament to represent one discrete universe, but each filament could equally have stood for 10 identical, discrete universes, or, for that matter, any finite number you care to name.
Equally, the same block universes in the previous figure can also be used to reconstruct the tree from Bob’s point of view, simply by swapping the assignments of some of the universes, leaving the group of eight discrete parallel block universes unchanged.
Now here’s the thing: those very same eight discrete universes in the figure can represent Bob’s tree just as well as Alice’s tree – see the next figure. All I’ve done is to move a couple of the arrows around to fit the branches as Bob sees them. The point is that you can do this for any tree seen from any perspective: the complete range of perspectives of any given part of the branching multiverse tree can be represented by one single, unique set of discrete, non-branching parallel universes.
So, the branching universes represent Everett’s branching universal wave function, but we have seen that the shape – the topology – of the branching universes depends upon who is observing it, calling into question the very existence (or ontology, to use the fancy, philosophical word) of branching universes.
Now we see why we can have the best of both worlds. In the previous figure, the tree from Alice’s perspective represents the branching of Everett’s universal wave function. Equally, the tree from Bob’s perspective in this figure represents the very same branching of Everett’s universal wave function, although Bob’s tree has a different shape from Alice’s tree. So, while the trees in the two figures are observer dependent, the set of discrete universes representing them is unique and, moreover, since it represents either tree, then it also represents the very same branching of Everett’s universal wave function. So, while the branching multiverse may be regarded as notional, observer-dependent, the unique set of non-branching, observer-independent, discrete universes is as real as the universal wave function.
