3.3 Universes don’t split
In the previous Step 3.2, I referred to Everett’s universal wave function branching and how DeWitt extended and popularized the notion to branching – or splitting – universes, rather than the wave function itself. Everett, nevertheless, did seem to be amenable to the idea of splitting universes, since he not only wrote in the footnote to his paper that no observer would ever be aware of any “splitting” process, implying that the observer splits with the universe, but he also went along with Podolsky’s phrase, “a non-denumerable infinity of worlds” (see page 95 of “Transcript of Conference on the Foundations of Quantum Mechanics”).
The question is, exactly how is this splitting supposed to happen? At this point, I’m going to announce a spoiler alert, because I don’t want you to get too invested in the detailed mechanism of splitting, which would be messy: we are going to find that it seems there is no splitting after all! In order to see that, we need first of all to examine the detailed anatomy of the putative split.
Suppose that Alice and Bob, who are situated far apart from each other, are each independently conducting a Stern-Gerlach filter experiment, Alice performing her usual 60º experiment and Bob doing something similar, except that, in his case, the particle enters his filter with a spin prepared in the horizontal direction and his spin filter is vertically inclined, so that the (Born) probability of him finding either spin-up or spin-down is ½ in either case.
We can see the four possible outcomes of Alice’s and Bob’s experiment in the figure. Note that the Alice’s and Bob’s electrons are not entangled, because their particles have never been close to each other.
In the upper half of the figure, sections (a) and (b) show what happens when Alice measures spin-up. At the moment she makes her measurement, according to MWI, her universe has split into two, and she finds herself in the universe where her outcome was spin-up, labelled ¾ Au. The fraction in front of “Au” is just the probability of Alice finding spin-up. Of course, Bob doesn’t know Alice’s outcome immediately – the earliest this can happen is the time it would take for Alice’s “news” to reach him at the limiting speed in the universe, the speed of light. We can think of the split in the universe as expanding outwards in a spherical shell at the speed of light, centred on the measurement (or the interaction) that caused the split. This is illustrated by the pink triangle within the light cone with its apex on Alice’s measurement.
This illustrates how, in MWI, the universes of Alice and Bob are each split by the outcomes of their experiment. Only when Alice’s and Bob’s light cones intersect (shown as the red triangles), can they both be said to inhabit the same split universe (see text). “Au” means “Alice detects spin-up”, “Ad” means “Alice detects spin-down” and “Bu” and “Bd”; mean the equivalent for Bob. A question mark “B?” means that the outcome is not yet known for Alice and “A?” means the equivalent for Bob.
Like Alice, Bob will split his universe into two, and you can see the two alternative splits in (a) and (b), each expanding alongside Alice’s split, with its apex on Bob’s measurement. The ½ in front of “Bu” and “Bd” is again just the probability for each of those outcomes. Until Bob’s light cone reaches Alice, her new, split universe contains only her spin-up outcome: she has to put a question mark for the outcome of Bob’s experiment: ¾ AuB?. Once Bob’s split overlaps with Alice’s world line, then spin-up Alice’s universe is split again – one in which Bob has also measured spin-up and one where he has measured spin-down. These two universes are labelled ⅜ AuBu and ⅜ AuBd, respectively, shown as the red overlapping triangles. The fraction, ⅜, is the product of the two probabilities for the two independent outcomes, namely ¾ and ½, and is the overall probability of getting either AuBu or AuBd.
In the lower half of the figure, sections (c) and (d) show what happens in the universe where Alice has measured spin-down: again, this splits into two when Bob’s split reaches her world line. These two universes are labelled ⅛ AdBu and ⅛ AdBd. Notice that we have used the Born probability of Alice finding spin-down, namely ¼ (because the probability of her finding spin-up is ¾, and the total probability of either spin-up or spin-down is 1). The fraction, ⅛ is the product of the two probabilities for the two independent outcomes, namely ¼ and ½, and is the overall probability of getting either AdBu or AdBd.
The four perspectives, (a), (b), (c) and (d), are captured in these two tree diagrams. The tree in the upper part of the diagram shows the branching from Alice’s perspective. Before either Alice or Bob has made a measurement, they share the same universe, indicated by the two question marks against their initials in the tree trunk. When Alice has made her measurement and the outcome is spin-up, then she is in the branch labelled ¾ AuB?, because, at this stage, “news” of Bob’s outcome has not yet reached her. This corresponds to her light cone in (a) and (b) in the previous figure, before Bob’s light cone intersects her own. Once it does, if Bob’s result was spin-up, then she is in the left-most branch ⅜ AuBu and, if Bob’s result was spin-down, then she is in the adjacent branch ⅜ AuBd.
If Alice’s outcome was spin-down, then she temporarily occupies branch ¼ AdB? until Bob’s outcome has reached her. This corresponds to her light cone in (c) and (d) in the previous figure before it is intersected by Bob’s light cone. Finally, she finds herself either in branch ⅛ AdBu or ⅛ AdBd, depending upon the outcome of Bob’s experiment.
This presents the information in the previous figure in the form of an MWI tree. The upper tree is from Alice’s point of view and the lower tree is from Bob’s point of view. Notice that the two trees are fundamentally different. This is ultimately due to MWI being what Everett called a “relative state formulation”.
The lower tree in the figure shows the branching from Bob’s perspective, where ½ BuA? corresponds to (a) and (c) in the previous figure and ½ BdA? corresponds to (b) and (d) in the same figure.
You will see that I have drawn the branches in the figure with thicknesses that are proportional to their probabilities. So, for example, in the tree in the upper part of the figure, which is drawn from Alice’s perspective, the branch that splits to the left of the trunk, labelled ¾ AuB?, is three times the thickness of the branch that splits to the right, labelled ¼ AdB?, and so on.
Notice that the branch labelled ⅜ AuBu in Alice’s tree is the very same branch shown as ⅜ AuBu in Bob’s tree, and corresponds to the overlapping triangles in (a) in the previous figure. Similarly, branches ⅜ AuBd, ⅛ AdBu and ⅛ AdBd in either Alice’s or Bob’s tree correspond respectively to the overlapping triangles in (b), (c) and (d) in the previous figure.
Notice that the branch ⅛ AdBu is a darker shade in both Alice’s tree and in Bob’s tree: they are the same universe, shared by both Alice and Bob. In this branch Alice can say to Bob:
“Our branch universe split from one in which I had observed Ad (but I didn’t yet know your outcome). So, having now seen that your outcome is Bu, so that our two outcomes are AdBu, I know that our other branch universe must have been AdBd.”
Now here’s the thing – Bob can reply to Alice, saying:
“No, I disagree. Our branch universe split from one in which I had observed Bu (but I didn’t yet know your outcome). So, having now seen that your outcome is Ad, so that we agree that our two outcomes are indeed AdBu, I know that our other branch universe must have been AuBu rather than AdBd as you claim.”
It is clear why Alice and Bob disagree: if you look at the figure, you can see clearly that their trees have different shapes (or, to use the mathematical jargon, different topologies). This has happened because the tree structure is always relative – it depends upon the perspective of the observer (Alice or Bob in this case) from which the tree is drawn.
But, if the tree structure of the multiverse depends on who is looking at it, then it isn’t an independent structure. Actually, died-in-the-wool Everettians (yes, there really is such a term for Everett’s followers) don’t appear to be particularly concerned about this seemingly existential threat to a philosophy that has been their touchstone for decades, because, they point out, the branching refers to the wave function, and not the universe. Nevertheless, the consequence of a branch of the wave function splitting is that the universe represented by that branch must also undergo splitting. As Wallace puts it:
“If we apply to quantum mechanics the same principles we apply right across science, we find that a multiplicity of quasi-classical worlds are emergent from the underlying quantum physics. These worlds are structures instantiated within the quantum state, but they are no less real for all that.” (Wallace D. “The emergent multiverse: quantum theory according to the Everett interpretation”. Oxford University Press, Oxford (2012) page 63.)
In other words, if the wave function branches, or splits, then so do its associated universes, which highlights the observer-dependence of the Everettian tree structure, ruling out its independent existence.
If we regard the multiverse as a mathematical structure, then the structure can’t depend upon observations of its inhabitants, which is what an observer-dependent splitting would entail. So, we need to find a structure that retains the essence of the Many Worlds Interpretation but without the splitting.
